$\partial_{t_{1}}A_{j,1}=4\partial_{x_{1}}^{2}A_{j,1}+\left(-1+\frac{\varepsilon_{1}(t_{1})}{2}\right)A_{j,1}-a_{j,1}-4i\partial_{x_{1}}^{3}A_{j-1,1}-\partial_{x_{1}}^{4}A_{j-2,1},\quad
A_{j,1}(x_{1},0)=A_{j,1}^{\ast}(x_{1}),$

where $x_{1}\in\mathbb{R}$, $t_{1}\in[0,T_{1}]$ and $a_{j,1}$ is given by
$a_{j,1}=a_{j,1}^{L}+a_{j,1}^{N}$ with $a_{j,1}^{L}$ as in (81). This equation
also decouples from system (63), and many of the estimates in
$H_{ul}^{\theta}$ can be obtained via arguments similar to those used in Steps
II-III. We therefore present fewer details. Both
$-4i\partial_{x_{1}}^{3}A_{j-1,1}$ and $\partial_{x_{1}}^{4}A_{j-2,1}$ may be
considered as source terms, since we given solutions $A_{j-1,1}(x_{1},t_{1})$
and $A_{j-2,1}(x_{1},t_{1})$ for all $x_{1}\in\mathbb{R}$ and
$t_{1}\in[0,T_{1}]$ either via either Lemma 5.4, Lemma 5.6, or the induction
hypothesis (depending on $j$).

Using the same integrating factor as in Step III leads to

$\partial_{t_{1}}\tilde{A}_{j,1}=4\partial_{x_{1}}^{2}\tilde{A}_{j,1}-\tilde{A}_{j,1}-a_{j,1}^{L}(\tilde{A}_{j,1},\overline{\tilde{A}_{j,1}})-\mathrm{e}^{-I_{1}(t_{1})}\left(a^{N}_{j,1}+4i\partial_{x_{1}}^{3}A_{j-1,1}+\partial_{x_{1}}^{4}A_{j-2,1}\right),$

together with $\tilde{A}_{j,1}(x_{1},0)=A_{j,1}^{\ast}(x_{1})$, after the
transformation
$A_{j,1}(x_{1},t_{1})=:\mathrm{e}^{I_{1}(t_{1})}\tilde{A}_{j,1}(x_{1},t_{1})$.
As in Step III, we apply the variation of constants formula and split the
solution into two parts $\tilde{A}_{j,1}:=\tilde{A}_{j,1,h}+\tilde{A}_{j,1,s}$
given by

$\begin{split}\tilde{A}_{j,1,h}(\cdot,t_{1})&=\mathrm{e}^{t_{1}\Lambda_{1}}A_{j,1}^{\ast}-\int_{0}^{t_{1}}\mathrm{e}^{(t_{1}-s_{1})\Lambda_{1}}a_{j,1}^{L}(\tilde{A}_{j,1}(\cdot,s_{1}),\overline{\tilde{A}_{j,1}}(\cdot,s_{1}))ds_{1},\\\
\tilde{A}_{j,1,s}(\cdot,t_{1})&=-\int_{0}^{t_{1}}\mathrm{e}^{(t_{1}-s_{1})\Lambda_{1}}\mathrm{e}^{-I_{1}(s_{1})}\left(a^{N}_{j,1}(\cdot,s_{1})+4i\partial_{x_{1}}^{3}A_{j-1,1}(\cdot,s_{1})+\partial_{x_{1}}^{4}A_{j-2,1}(\cdot,s_{1})\right)ds_{1},\end{split}$
(85)

for $t_{1}\in[0,T_{1}]$. Note that the $\partial_{x_{1}}^{3}A_{j-1,1}$
contribution has been included in the source terms, in contrast to Step III.
Arguments analogous to those which led to the bound for
$\|A_{2,1,h}(\cdot,T_{1})\|_{H_{ul}^{\theta}}$ in Lemma 5.6 lead to a similar
bound for
$A_{j,1,h}(\cdot,T_{1})=:\mathrm{e}^{I_{1}(T_{1})}\tilde{A}_{j,1,h}(\cdot,T_{1})$.
Specifically, we obtain

$\begin{split}\|\tilde{A}_{j,1,h}(\cdot,t_{1})\|_{H_{ul}^{\theta}}&\leq
C\mathrm{e}^{-ct_{1}}\|A_{j,1}^{\ast}\|_{H_{ul}^{\theta}}+Ci\mathrm{e}^{-ct_{1}}\mathrm{e}^{-(2\tilde{c}-c)/2\varepsilon_{1}^{\ast}}\frac{\|A_{1,1}^{\ast}\|_{H_{ul}^{\theta}}^{2}}{\sqrt{\varepsilon_{1}^{\ast}}}\Gamma\left(\frac{1}{2},-(2\tilde{c}-c)\left(\frac{1}{2\varepsilon_{1}^{\ast}}-s_{1}\right)\right)\bigg{|}_{0}^{t_{1}}\\\
&\leq
C\mathrm{e}^{-ct_{1}}\max_{k=1,\ldots,j}\|A_{k,1}^{\ast}\|_{H_{ul}^{\theta}},\end{split}$
(86)

for all $t_{1}\in[0,\tilde{T}_{j,1}]$, such that

$\|A_{j,1,h}(\cdot,T_{1})\|=\mathrm{e}^{I_{1}(T_{1})}\|\tilde{A}_{j,1,h}(\cdot,T_{1})\|\leq
C\frac{\mathrm{e}^{-c/2\varepsilon_{1}^{\ast}}}{{\varepsilon_{1}^{\ast}}^{3/4}}\max_{k=1,\ldots,j}\|A_{k,1}^{\ast}\|_{H_{ul}^{\theta}}.$

We now consider the second term $\tilde{A}_{j,1,s}$. Using the same bound for
the semigroup as in Step III we obtain the estimate

$\begin{split}\|\tilde{A}_{j,1,s}(\cdot,t_{1})\|_{H_{ul}^{\theta}}&\leq\\\
C\mathrm{e}^{-ct_{1}}&\int_{0}^{t_{1}}\mathrm{e}^{cs_{1}}\sqrt[4]{1-2\varepsilon_{1}^{\ast}s_{1}}\left\|a^{N}_{j,1}(\cdot,s_{1})+4i\partial_{x_{1}}^{3}A_{j-1,1}(\cdot,s_{1})+\partial_{x_{1}}^{4}A_{j-2,1}(\cdot,s_{1})\right\|_{H_{ul}^{\theta}}ds_{1},\end{split}$
(87)

for all $t_{1}\in[0,\tilde{T}_{j,1}]$. The $a^{N}_{j,1}$ term is controlled by
the assumption that $a^{N}_{j,1}(\cdot,s_{1})\in H_{ul}^{\theta}$ for all
$s_{1}\in[0,T_{1}]$, and the derivative terms
$\partial_{x_{1}}^{3}A_{j-1,1}(\cdot,s_{1})$ and
$\partial_{x_{1}}^{4}A_{j-2,1}(\cdot,s_{1})$ are easily controlled because we
assume sufficient regularity. Specifically,

$\begin{split}\|\partial_{x_{1}}^{3}A_{j-1,1}(\cdot,s_{1})\|_{H_{ul}^{\theta}}&\leq
C\|A_{j-1,1}(\cdot,s_{1})\|_{H_{ul}^{\theta+3}}=C\|A_{j-1,1}(\cdot,s_{1})\|_{H_{ul}^{\theta_{j-1,j}}},\\\
\|\partial_{x_{1}}^{4}A_{j-2,1}(\cdot,s_{1})\|_{H_{ul}^{\theta}}&\leq
C\|A_{j-2,1}(\cdot,s_{1})\|_{H_{ul}^{\theta+4}}\leq
C\|A_{j-2,1}(\cdot,s_{1})\|_{H_{ul}^{\theta_{j-2,j}}},\end{split}$

for all $s_{1}\in[0,\tilde{T}_{j,1}]$. Since $A_{j-1,1}(\cdot,t_{1})\in
H_{ul}^{\theta_{j-1,j}}$ and $A_{j-2,1}(\cdot,t_{1})\in
H_{ul}^{\theta_{j-2,j}}$ for all $t_{1}\in[0,T_{1}]$ by the induction
hypothesis, it follows that the $H_{ul}^{\theta}$ norm in the integrand of
(87) is bounded above by a constant for all $s_{1}\in[0,T_{1}]$. It follows
that

$\|\tilde{A}_{j,1,s}(\cdot,t_{1})\|_{H_{ul}^{\theta}}\leq
C\mathrm{e}^{-ct_{1}}\int_{0}^{t_{1}}\mathrm{e}^{cs_{1}}\sqrt[4]{1-2\varepsilon_{1}^{\ast}s_{1}}ds_{1}=C\mathrm{e}^{-ct_{1}}\mathrm{e}^{c/2\varepsilon_{1}^{\ast}}\sqrt[4]{\varepsilon_{1}^{\ast}}\Gamma\left(\frac{5}{4},c\left(\frac{1}{2\varepsilon_{1}^{\ast}}-s_{1}\right)\right)\bigg{|}_{0}^{t_{1}},$

for all $t_{1}\in[0,\tilde{T}_{j,1}]$, where we used Lemma A.5 again. After
multiplying through by the integrating factor we obtain

$\|A_{j,1}(\cdot,t_{1})\|_{H_{ul}^{\theta}}\leq
C\frac{\mathrm{e}^{-ct_{1}}}{\sqrt[4]{1-2\varepsilon_{1}^{\ast}t_{1}}}\max_{k=1,\ldots,j}\|A_{k,1}^{\ast}\|_{H_{ul}^{\theta}}+C\frac{\mathrm{e}^{-c(t_{1}-1/2\varepsilon_{1}^{\ast})}}{\sqrt[4]{1-2\varepsilon_{1}^{\ast}t_{1}}}\sqrt[4]{\varepsilon_{1}^{\ast}}\Gamma\left(\frac{5}{4},c\left(\frac{1}{2\varepsilon_{1}^{\ast}}-s_{1}\right)\right)\bigg{|}_{0}^{t_{1}},$
(88)

for $t_{1}\in[0,\tilde{T}_{j,1}]$. Since (88) is bounded for all
$t_{1}\in[0,T_{1}]$ (this follows directly from the asymptotics of the
incomplete gamma function and the monotonicity properties of the integral
described in Appendix A), it follows that $\tilde{T}_{j,1}=T_{1}$, such that
$A_{j,1}(\cdot,t_{1})\in H_{ul}^{\theta}$ for all $t_{1}\in[0,T_{1}]$.

We now set $\mu(x)\equiv 0$ and assume that the condition (82) holds. In order
to verify (83) we need to improve the bound on the right-most term in (88),
which is currently $O(1)$ at $t_{1}=T_{1}$ as $\varepsilon_{1}^{\ast}\to 0$.
This can be achieved by obtaining better bounds for
$\|a^{N}_{j,1}(\cdot,s_{1})\|_{H_{ul}^{\theta}}$,
$\|\partial_{x_{3}}^{4}A_{j-1,1}(\cdot,s_{1})\|_{H_{ul}^{\theta}}$ and
$\|\partial_{x_{1}}^{4}A_{j-2,1}(\cdot,s_{1})\|_{H_{ul}^{\theta}}$. It
suffices to show that

$\|A_{k,1}(\cdot,t_{1})\|_{H_{ul}^{\theta_{k,j}}}\leq
C\frac{\mathrm{e}^{-\tilde{c}t_{1}}}{\sqrt[4]{1-2\varepsilon_{1}^{\ast}t_{1}}}\max_{i=1,\ldots,k}\|A_{i,1}^{\ast}\|_{H_{ul}^{\theta_{i,j}}},$
(89)

for all $t_{1}\in[0,T_{1}]$ and $k=1,\ldots,j$, where
$\tilde{c}=c-C\tilde{K}^{2}\in(0,1)$ as before. This is because (89) implies

$\|A_{k,1}(\cdot,T_{1})\|_{H_{ul}^{\theta_{k,j}}}\leq
C\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}\max_{i=1,\ldots,k}\|A_{i,1}^{\ast}\|_{H_{ul}^{\theta_{i,j}}}$

for any fixed $\kappa\in(0,\tilde{c})$, for all $k=1,\ldots,j$, assuming that
$\zeta>0$ is fixed sufficiently small. Thus (89) implies (83).

For $k=1$, (89) follows immediately from Lemma 5.4. For $k=2$ we have that
$A_{2,1}=A_{2,1,h}$ when $\mu(x)\equiv 0$. Therefore, equation (78) in the
proof of Lemma 5.6 implies that

$\|A_{2,1}(\cdot,t_{1})\|_{H_{ul}^{\theta_{2,j}}}\leq
C\frac{\mathrm{e}^{-\tilde{c}t_{1}}}{\sqrt[4]{1-2\varepsilon_{1}^{\ast}t_{1}}}\max_{i=1,2}\|A_{i,1}^{\ast}\|_{H_{ul}^{\theta_{i,j}}},$
(90)

for all $t_{1}\in[0,T_{1}]$, as required.

We show (89) for $k=3,\ldots,j$ by induction. For the base case $k=3$,
equation (82) and the bounds in (89) with $k=1,2$ imply that

$\|a^{N}_{3,1}(\cdot,s_{1})\|_{H_{ul}^{\theta_{3,j}}}\leq
C\frac{\mathrm{e}^{-3\tilde{c}s_{1}}}{(1-2\varepsilon_{1}^{\ast}s_{1})^{3/4}}\max_{i=1,2}\|A_{i,1}^{\ast}\|_{H_{ul}^{\theta_{i,j}}}^{3}\leq
C\frac{\mathrm{e}^{-\tilde{c}s_{1}}}{(1-2\varepsilon_{1}^{\ast}s_{1})^{1/4}}\max_{i=1,2}\|A_{i,1}^{\ast}\|_{H_{ul}^{\theta_{i,j}}},$

for $s_{1}\in[0,T_{1}]$ (note that
$\mathrm{e}^{-3cs_{1}}/(1-2\varepsilon_{1}^{\ast}s_{1})^{3/4}\leq\mathrm{e}^{-cs_{1}}/\sqrt[4]{1-2\varepsilon_{1}^{\ast}s_{1}}$
on $s_{1}\in[0,T_{1}]$). Since we assume sufficient regularity, bounds for
$\partial_{x_{1}}^{3}A_{1,2}(\cdot,s_{1})$ and
$\partial_{x_{1}}^{4}A_{1,1}(\cdot,s_{1})$ are derived directly via the
corresponding bounds for $A_{1,2}(\cdot,s_{1})$ and $A_{1,1}(\cdot,s_{1})$
respectively. Specifically, we have

$\begin{split}\|\partial_{x_{1}}^{3}A_{1,2}(\cdot,s_{1})\|_{H_{ul}^{\theta_{3,j}}}&\leq
C\|A_{1,2}(\cdot,s_{1})\|_{H_{ul}^{\theta_{2,j}}}\leq
C\frac{\mathrm{e}^{-\tilde{c}s_{1}}}{\sqrt[4]{1-2\varepsilon_{1}^{\ast}s_{1}}}\max_{i=1,2}\|A_{i,1}^{\ast}\|_{H_{ul}^{\theta_{2,j}}},\\\
\|\partial_{x_{1}}^{4}A_{1,1}(\cdot,s_{1})\|_{H_{ul}^{\theta_{3,j}}}&\leq
C\|A_{1,1}(\cdot,s_{1})\|_{H_{ul}^{\theta_{3,j}+4}}\leq
C\frac{\mathrm{e}^{-\tilde{c}s_{1}}}{\sqrt[4]{1-2\varepsilon_{1}^{\ast}s_{1}}}\|A_{1,1}^{\ast}\|_{H_{ul}^{\theta_{A}(j)}},\end{split}$

for $s_{1}\in[0,T_{1}]$. We therefore obtain

$\big{\|}a^{N}_{3,1}(\cdot,s_{1})+4i\partial_{x_{1}}^{3}A_{1,2}(\cdot,s_{1})+\partial_{x_{1}}^{4}A_{1,1}(\cdot,s_{1})\big{\|}_{H_{ul}^{\theta_{3,j}}}\leq
C\frac{\mathrm{e}^{-\tilde{c}s_{1}}}{\sqrt[4]{1-2\varepsilon_{1}^{\ast}s_{1}}}\max_{i=1,2}\|A_{i,1}^{\ast}\|_{H_{ul}^{\theta_{i,j}}},$

for $s_{1}\in[0,T_{1}]$. Combining this with (87) leads to

$\|\tilde{A}_{3,1,s}(\cdot,t_{1})\|_{H_{ul}^{\theta_{3,j}}}\leq
C\mathrm{e}^{-ct_{1}}\max_{i=1,2}\|A_{i,1}^{\ast}\|_{H_{ul}^{\theta_{i,j}}}\int_{0}^{t_{1}}\mathrm{e}^{(c-\tilde{c})s_{1}}ds_{1}\\\
\leq
C\mathrm{e}^{-\tilde{c}t_{1}}\max_{i=1,2}\|A_{i,1}^{\ast}\|_{H_{ul}^{\theta_{i,j}}},$

for all $t_{1}\in[0,T_{1}]$. After multiplying through by the integrating
factor we therefore obtain the estimate

$\|A_{3,1}(\cdot,t_{1})\|_{H_{ul}^{\theta_{3,j}}}\leq
C\frac{1}{\sqrt[4]{1-2\varepsilon_{1}^{\ast}t_{1}}}\|\tilde{A}_{3,1,h}(\cdot,t_{1})\|_{H_{ul}^{\theta_{3,j}}}+C\frac{\mathrm{e}^{-\tilde{c}t_{1}}}{\sqrt[4]{1-2\varepsilon_{1}^{\ast}t_{1}}}\max_{i=1,2}\|A_{i,1}^{\ast}\|_{H_{ul}^{\theta_{i,j}}},$

for $t_{1}\in[0,T_{1}]$. The desired bound (89) follows after using the bound
for $\|\tilde{A}_{3,1,h}(\cdot,t_{1})\|_{H_{ul}^{\theta_{3,j}}}$ in (86),
thereby proving the desired result for the base case $k=3$.

It remains to carry out the induction step, for which we assume that (89)
holds for $k=3,\ldots,j-1\leq N-3$. We present fewer details since the
structure of the argument is the same. Using (82) it follows that

$\|a^{N}_{j,1}(\cdot,s_{1})\|_{H_{ul}^{\theta}}\leq
C\frac{\mathrm{e}^{-3\tilde{c}s_{1}}}{(1-2\varepsilon_{1}^{\ast}s_{1})^{3/4}}\max_{i=1,\ldots,j-1}\|A_{i,1}^{\ast}\|_{H_{ul}^{\theta_{i,j}}}^{3}\leq
C\frac{\mathrm{e}^{-\tilde{c}s_{1}}}{\sqrt[4]{1-2\varepsilon_{1}^{\ast}s_{1}}}\max_{i=1,\ldots,j-1}\|A_{i,1}^{\ast}\|_{H_{ul}^{\theta_{i,j}}},$

for $s_{1}\in[0,T_{1}]$. Due to our regularity assumptions and induction
hypothesis we have

$\begin{split}\|\partial_{x_{1}}^{3}A_{j-1,1}(\cdot,s_{1})\|_{H_{ul}^{\theta}}&\leq
C\|A_{j-1,1}(\cdot,s_{1})\|_{H_{ul}^{\theta_{j-1,j}}}\leq
C\frac{\mathrm{e}^{-\tilde{c}s_{1}}}{\sqrt[4]{1-2\varepsilon_{1}^{\ast}s_{1}}}\max_{i=1,\ldots,j-1}\|A_{i,1}^{\ast}\|_{H_{ul}^{\theta_{i,j}}},\\\
\|\partial_{x_{1}}^{4}A_{j-2,1}(\cdot,s_{1})\|_{H_{ul}^{\theta}}&\leq
C\|A_{j-2,1}(\cdot,s_{1})\|_{H_{ul}^{\theta_{j-2,j}}}\leq
C\frac{\mathrm{e}^{-\tilde{c}s_{1}}}{\sqrt[4]{1-2\varepsilon_{1}^{\ast}s_{1}}}\max_{i=1,\ldots,j-2}\|A_{i,1}^{\ast}\|_{H_{ul}^{\theta_{i,j}}},\end{split}$

for $s_{1}\in[0,T_{1}]$. The bound for
$\|A_{j,1}(\cdot,t_{1})\|_{H_{ul}^{\theta}}$ on $t_{1}\in[0,T_{1}]$ then
follows analogously to the base case $k=3$. By induction, it follows that (89)
holds for all $k=3,\ldots,j\leq N-2$. ∎

######  Remark 5.8.

In order to verify the assumptions on $a_{j,1}^{N}$ in Lemma 5.7 we need to
control the norm of non-critical modulation functions $A_{mj,1}$ with $|m|\neq
1$. To see that the conditions on $a_{j,1}^{N}$ are ‘natural’, recall from
Remark 4.6 that $a_{j,1}^{N}$ can be written in terms of lower order critical
modulation functions $A_{\pm 1j^{\prime}}$, where $j^{\prime}<j$. It will
follow from Lemma 5.9 and its proof in Section 5.1.1 below that the relevant
conditions are always satisfied.

####  5.1.1 Step V: Estimate $\psi_{1}$

Lemmas 5.2, 5.4, 5.6 and 5.7 provide sufficient information to characterise
the dynamics of the modulation equations (63). In order to properly
characterise the dynamics of the ‘full approximation’ $\psi_{1}$ and the map
$\pi_{1}$ defined in (64), however, we must also control the non-critical
modulation functions $A_{mj,1}$ with $|m|\neq 1$. Moreover, we need to verify
the assumptions on $a_{j,1}^{N}$ (recall Remark 5.8). In the following we
reinstate the full notation $A_{\pm 1j,1}$ instead of $A_{j,1}$ in order to
distinguish $A_{mj,1}$ with $|m|=1$ and $|m|\neq 1$. We also recall that
$\textbf{A}_{mj,1}$ denotes the set of lower order critical modulation
functions $A_{\pm 1j^{\prime},1}$ with
$j^{\prime}\in\\{1,\ldots,\alpha(m)+j-1\\}$.

######  Lemma 5.9.

Let $\theta_{A}=3(N-3)+\theta$ where $\theta>1/2$. We have that $A_{\pm
1j,1}(\cdot,t_{1})\in H_{ul}^{\theta_{j,N-2}}$ for all $j=1,\ldots,N-2$ and
$t_{1}\in[0,T_{1}]$, and

$A_{mj,1}(\cdot,t_{1})=g_{mj,1}^{gl}\left(\textbf{{A}}_{mj,1}(\cdot,t_{1}),\varepsilon_{1}(t_{1})\right)\in
H_{ul}^{\theta_{\alpha(m)+j,N}},$ (91)

for all $t_{1}\in[0,T_{1}]$, $m\in I_{N}\setminus\\{-1,1\\}$ and
$j=1,\ldots,\tilde{\alpha}(m)$. If additionally $\mu(x)\equiv 0$ then for each
$m\in I_{N}\setminus\\{-1,1\\}$ we have

$\big{\|}g_{mj,1}^{gl}\left(\textbf{{A}}_{mj,1}(\cdot,T_{1}),\varepsilon_{1}(T_{1})\right)\big{\|}_{H_{ul}^{\theta_{\alpha(m)+j,N}}}\leq
C\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}\max_{k=1,\ldots,\alpha(m)+j}\|A_{1k,1}^{\ast}\|_{H_{ul}^{\theta_{k,N-2}}},$
(92)

for all $j=1,\ldots,\tilde{\alpha}(m)$.

###### Proof.

Similarly to the established modulation theory for the static SH equation in
e.g. [82, Chapter 10], this follows from the recursive construction of $\Psi$
in Section 4 and the existence of solutions $A_{11,1}\in H_{ul}^{\theta_{A}}$
due to Lemma 5.4. The idea is to proceed iteratively with terms of order
$O(r_{1}^{\alpha(m)+j})$. For the first three orders, we find the following:

  * •

$\alpha(m)+j=1$: In this case $(m,j)\in\\{(\pm 1,1)\\}$, and $A_{\pm
11,1}(\cdot,t_{1})\in H_{ul}^{\theta_{A}}$ by Lemma 5.4.

  * •

$\alpha(m)+j=2$: In this case $(m,j)\in\\{(\pm 1,2),(0,1),(\pm 2,1)\\}$. We
have $A_{\pm 21,1}(\cdot,t_{1})\in H_{ul}^{\theta_{2,N-2}}$ by Lemma 5.6, and
$A_{01,1}\equiv A_{\pm 21,1}\equiv 0$ after restriction to
$\mathcal{M}_{1}^{gl}$; recall (62).

  * •

$\alpha(m)+j=3$: In this case $(m,j)\in\\{(\pm 1,3),(0,2),(\pm 2,2),(\pm
3,1)\\}$. Restriction to $\mathcal{M}_{1}^{gl}$ yields $A_{02,1}=0$, $A_{\pm
22,1}=0$ and $A_{\pm 31,1}=-A_{\pm 11,1}^{3}/64$ (see again (62)). Note that
$A_{\pm 31,1}(\cdot,t_{1})\in H_{ul}^{\theta_{A}}$ since $A_{\pm
11,1}(\cdot,t_{1})\in H_{ul}^{\theta_{A}}$ and $H_{ul}^{\theta_{A}}$ is an
algebra. This implies that $a_{3,1}^{N}(\cdot,t_{1})\in H_{ul}^{\theta_{A}}$
for all $t_{1}\in[0,T_{1}]$. Therefore $A_{\pm 13,1}(\cdot,t_{1})\in
H_{ul}^{\theta_{3,N-2}}$ by Lemma 5.7.

The recursive structure for higher orders is as follows: At the level
$\alpha(m)+j=l\leq N$ one has a finite set of equations for $A_{mj,1}$ with
$(m,j)\in\mathcal{I}^{l}:=\\{(m,j)\in
I_{N}\setminus\\{-1,1\\}\times\\{1,\ldots,\tilde{\alpha}(m)\\}:\alpha(m)+j=l\\}$
determined via restriction to $\mathcal{M}_{1}^{gl}$. The graph equation
defining $\mathcal{M}_{1}^{gl}$ in (28) only depends on
$A_{m^{\prime}j^{\prime},1}$ with $\alpha(m^{\prime})+j^{\prime}<l$, which in
turn depend only on critical modulation functions $A_{\pm 1l^{\prime}}$ of
lower order with $l^{\prime}<l$; recall Remark 4.6. In particular,
$a_{j,1}^{N}$ is a polynomial in modulation functions $A_{\pm 1j^{\prime},1}$
with $j^{\prime}<j$. This allows one to show that $a_{j,1}^{N}(\cdot,t_{1})\in
H_{ul}^{\theta_{j,N-2}}$ for all $t_{1}\in[0,T_{1}]$, since Lemmas 5.4 and 5.6
can be used to initiate an inductive proof which relies on the fact that
$A_{\pm 1j^{\prime},1}(\cdot,t_{1})\in H_{ul}^{j^{\prime},N-2}$ for all lower
orders $j^{\prime}=1,\ldots,j-1$. The details are standard and omitted for
brevity. After showing that $a_{j,1}^{N}(\cdot,t_{1})\in
H_{ul}^{\theta_{j,N-2}}$ for all $t_{1}\in[0,T_{1}]$, Lemma 5.7 can be applied
to establish that $A_{\pm j,1}(\cdot,t_{1})\in H_{ul}^{\theta_{j,N-2}}$ for
all $t_{1}\in[0,T_{1}]$.

Assume now that $\mu(x)\equiv 0$, in which case $\nu_{m}=0$ for all $m\in
I_{N}$ and

$g_{mj,1}^{gl}\left(\textbf{A}_{mj,1},\varepsilon_{1}\right)=-(\mathcal{L}_{m,1}^{(0)})^{-1}\bigg{(}\mathcal{L}_{m,1}^{(1)}A_{mj-1,1}+\tilde{\mathcal{L}_{m,1}^{(2)}}A_{mj-2,1}+\mathcal{L}_{m,1}^{(3)}A_{mj-3,1}+\mathcal{L}_{m,1}^{(4)}A_{mj-4,1}-a_{mj,1}\bigg{)}.$
(93)

In order to prove the bound in (92), we note first that the matching condition
$\alpha(m_{1})+\alpha(m_{2})+\alpha(m_{3})+j_{1}+j_{2}+j_{3}\leq\alpha(m)+j+2$
implies that

$\begin{split}\|a_{mj,1}(\cdot,t_{1})\|_{H_{ul}^{\theta_{\alpha(m)+j,N}}}\leq
C\sup_{\\{(m^{\prime},j^{\prime}):\alpha(m^{\prime})+j^{\prime}\leq\alpha(m)+j-2\\}}\|A_{m^{\prime}j^{\prime},1}(\cdot,t_{1})\|_{H_{ul}^{\theta_{\alpha(m^{\prime})+j^{\prime},N}}},\end{split}$
(94)

for all $t_{1}\in[0,T_{1}]$ and $m\neq 1$. Since assume sufficient regularity
to control the derivatives in the linear operators $\mathcal{L}_{m,1}^{(l)}$
for $l=1,2,3,4$, we obtain

$\begin{split}\|g_{mj,1}^{gl}&\left(\textbf{A}_{mj,1}(\cdot,t_{1}),\varepsilon_{1}\right)\|_{H_{ul}^{\theta_{\alpha(m)+j,N}}}\leq
C\bigg{(}\|A_{mj-1,1}(\cdot,t_{1})\|_{H_{ul}^{\theta_{\alpha(m)+j-1,N}+1}}\\\
&+\|A_{mj-2,1}(\cdot,t_{1})\|_{H_{ul}^{\theta_{\alpha(m)+j-2,N}+2}}+\|A_{mj-3,1}(\cdot,t_{1})\|_{H_{ul}^{\theta_{\alpha(m)+j-3,N}+3}}\\\
&+\|A_{mj-4,1}(\cdot,t_{1})\|_{H_{ul}^{\theta_{\alpha(m)+j-4,N}+4}}+\sup_{\\{(m^{\prime},j^{\prime}):\alpha(m^{\prime})+j^{\prime}\leq\alpha(m)+j-2\\}}\|A_{m^{\prime}j^{\prime},1}(\cdot,t_{1})\|_{H_{ul}^{\theta_{\alpha(m^{\prime})+j^{\prime},N}}}\bigg{)},\end{split}$

for all $t_{1}\in[0,T_{1}]$. If we assume as an induction hypothesis that

$\|A_{mj,1}(\cdot,t_{1})\|_{H_{ul}^{\theta_{\alpha(m)+j,N}}}=\|g_{mj,1}^{gl}\left(\textbf{A}_{mj,1}(\cdot,t_{1}),\varepsilon_{1}\right)\|_{H_{ul}^{\theta_{\alpha(m)+j,N}}}\leq
C\sup_{k=1,\ldots,j-2}\|A_{\pm 1k,1}(\cdot,t_{1})\|_{H_{ul}^{\theta_{k,N-2}}}$

for all $t_{1}\in[0,T_{1}]$, (the base case follows by the arguments above in
dot points), then we obtain

$\|g_{mj,1}^{gl}\left(\textbf{A}_{mj,1}(\cdot,t_{1}),\varepsilon_{1}\right)\|_{H_{ul}^{\theta_{\alpha(m)+j,N}}}\leq
C\sup_{k=1,\ldots,j-2}\|A_{\pm
1k,1}(\cdot,t_{1})\|_{H_{ul}^{\theta_{k,N-2}}},$ (95)

for all $t_{1}\in[0,T_{1}]$. The bound in (93) follows after applying the
bounds for $A_{\pm 1,j}$ when $\mu(x)\equiv 0$ from Lemmas 5.4, 5.6 and (5.7)
when $\mu(x)\equiv 0$ (the condition (82) is satisfied due to (94)). ∎

######  Remark 5.10.

The proof of Lemma 5.9 also shows that the assumptions on $a_{j,1}^{N}$ for
the applicability of Lemma 5.7 are satisfied.

Combining the results of Lemmas 5.2, 5.4, 5.6, 5.7 and 5.9 we obtain the
following result, which summarises the dynamics in chart $\mathcal{K}_{1}$.

######  Proposition 5.11.

Let $\theta_{A}=1+3(N-3)+\theta$ where $\theta>1/2$. Then for $\zeta,K>0$
sufficiently small but fixed we have that $\psi_{1}(\cdot,t_{1})\in
H_{ul}^{\theta}$ for all $t_{1}\in[0,T_{1}]$. In particular, the map
$\pi_{1}:\Sigma_{1}^{in}\to\Sigma_{1}^{out}$ is well-defined and given by

$\pi_{1}:(\psi_{1}^{\ast},\rho_{in}^{1/2},\varepsilon_{1}^{\ast})\mapsto\left(\psi_{1}(\cdot,T_{1}),\rho_{in}^{1/2}\left(\frac{\varepsilon_{1}^{\ast}}{\zeta}\right)^{1/4},\zeta\right),$
(96)

where

$\psi_{1}(x,T_{1})={\varepsilon_{1}^{\ast}}^{1/4}\frac{\rho_{in}^{1/2}}{\zeta^{1/4}}\left(\nu_{1}f_{2,1}(\varepsilon_{1}^{\ast})\mathrm{e}^{ix}+c.c.\right)+{\varepsilon_{1}^{\ast}}^{1/2}h_{rem,1}(x),$
(97)

where $f_{2,1}(\varepsilon_{1}^{\ast})$ is the function described in Lemma 5.6
and $h_{rem,1}\in H_{ul}^{\theta}$. If additionally $\mu(x)\equiv 0$ then we
have

$\|\psi_{1}(\cdot,T_{1})\|_{H_{ul}^{\theta}}\leq
C\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}\max_{j=1,\ldots,N-2}\|A_{j,1}^{\ast}\|_{H_{ul}^{\theta+1}},$

where $\kappa\in(0,\tilde{c})$ can be fixed arbitrarily close to $\tilde{c}$
if $K,\zeta>0$ are sufficiently small.

###### Proof.

The form of the map in (96) follows immediately from Lemma 5.2. The fact that
$\psi_{1}(\cdot,t_{1})\in H_{ul}^{\theta}$ for all $t_{1}\in[0,T_{1}]$ follows
from its definition in (45) and the fact that $A_{mj,1}(\cdot,t_{1})\in
H_{ul}^{\theta+1}$ for all $m\in I_{N}$ with corresponding
$j\in\\{1,\ldots,\tilde{\alpha}(m)\\}$ and for all $t_{1}\in[0,T_{1}]$; recall
Lemma 5.9 and note that we assume one additional degree of regularity. The
extra regularity is needed since we use Lemma A.1 in order to take the
$H_{ul}^{\theta}$ with respect to $x$ instead of $x_{1}$.

In order to obtain the asymptotic expression in (97) we evaluate the series
expansion (45) at $t_{1}=T_{1}$. This yields

$\begin{split}r_{1}(T_{1})\psi_{1}(x,T_{1})&=\sum_{m\in
I_{N}}\sum_{j=1}^{\tilde{\alpha}(m)}r_{1}(T_{1})^{\alpha(m)+j}A_{mj,1}(x_{1},T_{1})\mathrm{e}^{imx}\\\
&=r_{1}(T_{1})\psi_{GL,1}(x,T_{1})+r_{1}(T_{1})^{2}\left(A_{12,1}(x_{1},T_{1})\mathrm{e}^{ix}+c.c.\right)+r_{1}(T_{1})^{3}\tilde{h}_{rem,1}(x,T_{1}),\end{split}$

where $\psi_{GL,1}(x,T_{1})=A_{11,1}(x_{1},T_{1})\mathrm{e}^{ix}+c.c.$ and we
used the fact that $A_{01,1}(\cdot,T_{1})=A_{21,1}(\cdot,T_{1})\equiv 0$;
recall the proof of Lemma 5.9. The remainder function satisfies
$\tilde{h}_{rem,1}(\cdot,T_{1})\in H_{ul}^{\theta}$, since it is a linear
combination of terms of the form
$r_{1}(T_{1})^{\alpha(m)+j}A_{mj,1}(\cdot,T_{1})\mathrm{e}^{imx}$ with
$r_{1}(T_{1})\leq\rho_{in}^{1/2}$ by Lemma 5.2 and $A_{mj,1}(\cdot,T_{1})\in
H_{ul}^{\theta+1}$ by Lemma 5.9. The formal leading order approximation
$\psi_{GL,1}(x,T_{1})$ is in fact higher order, since by Lemmas 5.4 and A.1 we
have

$\|\psi_{GL,1}(x,T_{1})\|_{H_{ul}^{\theta}}\leq
C\|A_{11,1}(\cdot,T_{1})\|_{H_{ul}^{\theta+1}}\leq
C\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}\|A_{1,1}^{\ast}\|_{H_{ul}^{\theta+1}}.$

The expression (97) follows after substituting the asymptotic expression for
$A_{2,1}=A_{12,1}$ in Lemma 5.6, which yields

$\begin{split}\psi_{1}(x,T_{1})=r_{1}(T_{1})\left(\nu_{1}f_{2,1}(\varepsilon_{1}^{\ast})\mathrm{e}^{ix}+c.c.\right)+r_{1}(T_{1})^{2}h_{rem,1}(x),\end{split}$
(98)

where we defined a new remainder term

$h_{rem,1}(x):=r(T_{1})^{-1}(A_{2,1,h}(x_{1},T_{1})\mathrm{e}^{ix}+c.c.)+r_{1}(T_{1})^{-2}\psi_{GL,1}(x,T_{1})+\tilde{h}_{rem,1}(x,T_{1}).$

A direct application of the triangle inequality and Lemma A.1 shows that
$h_{rem,1}\in H_{ul}^{\theta}$ (note that the norm of $A_{2,1,h}$ is
exponentially small due to Lemma 5.6). Substituting the solution
$r_{1}(T_{1})=\rho_{in}^{1/2}\sqrt[4]{\varepsilon_{1}^{\ast}/\zeta}$ into (98)
yields (97).

Now assume that $\mu(x)\equiv 0$. It follows from Lemmas 5.4, 5.6, 5.7 and 5.9
that for each $m\in I_{N}$ and corresponding $j=1,\ldots,N-2$ we have

$\|A_{mj,1}(\cdot,T_{1})\|_{H_{ul}^{\theta+1}}\leq
C\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}\max_{j=1,\ldots,N-2}\|A_{j,1}^{\ast}\|_{H_{ul}^{\theta+1}}.$

Applying this bound to the series expansion for $\psi_{1}(\cdot,T_{1})$
together with a triangle inequality and Lemma A.1 yields the desired result. ∎

######  Remark 5.12.

Recall from Lemma 5.6 that $f_{21}(\varepsilon_{1}^{\ast})\approx\zeta$ for
small $\zeta$. Thus a close numerical approximation is
$\psi_{1}(x,T_{1})\approx
2\nu_{1}\rho_{in}^{1/2}\zeta^{3/4}{\varepsilon_{1}^{\ast}}^{1/4}\cos x$.

###  5.2 Chart $\mathcal{K}_{2}$

We recall the equations in chart $\mathcal{K}_{2}$:

$\begin{split}\partial_{t_{2}}A_{1,2}&=4\partial_{x_{2}}^{2}A_{1,2}+v_{2}A_{1,2}-3A_{1,2}|A_{1,2}|^{2},\\\
\partial_{t_{2}}v_{2}&=1,\\\
\partial_{t_{2}}A_{2,2}&=4\partial_{x_{2}}^{2}A_{2,2}+v_{2}A_{2,2}-a_{2,2}-4i\partial_{x_{2}}^{3}A_{1,2}+\nu_{1},\\\
\partial_{t_{2}}A_{j,2}&=4\partial_{x_{2}}^{2}A_{j,2}+v_{2}A_{j,2}-a_{j,2}-4i\partial_{x_{2}}^{3}A_{j-1,2}-\partial_{x_{2}}^{4}A_{j-2,2},\end{split}$
(99)

for $j=3,\ldots,N-2$. Our aim in chart $\mathcal{K}_{2}$ is to describe the
map

$\pi_{2}:\Sigma^{in}_{2}\ni\left(\psi_{2}^{\ast},-\zeta^{-1/2},r_{2}\right)\mapsto\left(\psi_{2}(\cdot,T_{2}),\zeta^{-1/2},r_{2}\right)\in\Sigma_{2}^{out},$

induced by forward evolution of initial conditions in

$\Sigma_{2}^{in}=\left\\{(\psi_{2}^{\ast},-\zeta^{-1/2},r_{2}):\|\psi_{2}^{\ast}\|_{H_{ul}^{\theta}}\leq\zeta^{1/4}K,r_{2}\in[0,\vartheta]\right\\},$

which corresponds to the exit section $\Sigma_{1}^{out}$ in chart
$\mathcal{K}_{1}$ coordinates ($\vartheta>0$ can be chosen such that
$\Sigma_{2}^{in}=\kappa_{12}(\Sigma^{out}_{1})$), up to the exit section

$\Sigma_{2}^{out}=\left\\{(\psi_{2},\rho_{mid},r_{2}):\|\psi_{2}\|_{H_{ul}^{\theta}}\leq\zeta^{1/4}K,r_{2}\in[0,\vartheta]\right\\},$

which corresponds to the section $\Delta^{mid}$ defined in (36) after blow-
down after fixing $\varepsilon$ and using the natural identification with
points in $\Delta^{mid}_{\varepsilon}=\Delta^{mid}\times[0,\varepsilon_{0}]$.
The analysis proceeds via steps similar to those of the $\mathcal{K}_{1}$
analysis.

#### Step I: Solve for $v_{2}(t_{2})$ and $T_{2}$

We have the following result.

######  Lemma 5.13.

The transition time is $T_{2}=\zeta^{-1/2}+\rho_{mid}$, and on the interval
$t_{2}\in[0,T_{2}]$ we have $v_{2}(t_{2})=-\zeta^{-1/2}+t_{2}$.

###### Proof.

This follows by direct integration and the boundary constraint
$v_{2}(T_{2})=\rho_{mid}$. ∎

#### Step II: Estimate $A_{1,2}(x_{2},t_{2})$

We prove the following result.

######  Lemma 5.14.

Fix $\theta_{A}>1/2$ and assume that $A_{1,2}^{\ast}\in H_{ul}^{\theta_{A}}$.
Then $A_{1,2}(\cdot,t_{2})\in H_{ul}^{\theta_{A}}$ on $t_{2}\in[0,T_{2}]$. In
particular,

$\|A_{1,2}(\cdot,t_{2})\|_{H_{ul}^{\theta_{A}}}\leq
C\|A_{1,2}^{\ast}\|_{H_{ul}^{\theta_{A}}},$ (100)

for all $t_{2}\in[0,T_{2}]$.

###### Proof.

Substituting the expressions for $v_{2}(t_{2})$ into the equation for
$A_{1,2}$ yields a GL-type equations with time-dependent linear damping:

$\partial_{t_{2}}A_{1,2}=4\partial_{x_{2}}^{2}A_{1,2}+v_{2}(t_{2})A_{1,2}-3A_{1,2}|A_{1,2}|^{2},\qquad
A_{1,2}(x_{2},0)=A_{1,2}^{\ast}(x_{2}),$ (101)

where $v_{2}(t_{2})=-\zeta^{-1/2}+t_{2}$ by Lemma 5.13. Equation (101)
decouples from system (99), so we may consider it independently. The effect of
the non-constant coefficient $v_{2}(t_{2})$ can be mitigated via the
introduction of an integrating factor $\mathrm{e}^{I_{2}(t_{2})}$, where

$I_{2}(t_{2})=\int_{0}^{t_{2}}v_{2}(s_{2})ds_{2}=-\zeta^{-1/2}t_{2}+\frac{t_{2}^{2}}{2}.$
(102)

Defining
$A_{1,2}(x_{2},t_{2})=:\mathrm{e}^{I_{2}(t_{2})}\tilde{A}_{1,2}(x_{2},t_{2})$
leads to the equation

$\partial_{t_{2}}\tilde{A}_{1,2}=4\partial_{x_{2}}^{2}\tilde{A}_{1,2}-3\mathrm{e}^{2I_{2}(t_{2})}\tilde{A}_{1,2}|A_{1,2}|^{2},\qquad\tilde{A}_{1,2}(x_{2},0)=A_{1,2}^{\ast}(x_{2}),$

for which the variation of constants formula yields

$\tilde{A}_{1,2}(\cdot,t_{2})=\mathrm{e}^{t_{2}\Lambda_{2}}A_{1,2}^{\ast}-3\int_{0}^{t_{2}}\mathrm{e}^{(t_{2}-s_{2})\Lambda_{2}}\mathrm{e}^{2I_{2}(s_{2})}\tilde{A}_{1,2}(\cdot,s_{2})|\tilde{A}_{1,2}(\cdot,s_{2})|^{2}ds_{2},$
(103)

where $(\mathrm{e}^{t_{2}\Lambda_{2}})_{t_{2}\geq 0}$ is the (heat) semigroup
generated by $\Lambda_{2}:=4\partial_{x_{2}}^{2}$. Since the transition time
$T_{2}$ is finite, there exists a constant $C>0$ such that
$\mathrm{e}^{2I_{2}(t_{2})}\leq C$ for all $t_{2}\in[0,T_{2}]$. Moreover, it
follows from Lemma A.2 that there is a constant $C>0$ such that

$\|\mathrm{e}^{t_{2}\Lambda_{2}}\|_{H_{ul}^{\theta_{A}}\to
H_{ul}^{\theta_{A}}}\leq C,$

for all $t_{2}\in[0,T_{2}]$. Taking the $H^{\theta_{A}}_{ul}$ norm in (103),
we obtain

$\|\tilde{A}_{1,2}(\cdot,t_{2})\|_{H_{ul}^{\theta_{A}}}\leq
C\|A_{1,2}^{\ast}\|_{H_{ul}^{\theta_{A}}}+3C\int_{0}^{t_{2}}\|\tilde{A}_{1,2}(\cdot,s_{2})|\tilde{A}_{1,2}(\cdot,s_{2})|^{2}\|_{H_{ul}^{\theta_{A}}}ds_{2}.$

By Lemma A.3 we have
$\|\tilde{A}_{1,2}(\cdot,s_{2})|\tilde{A}_{1,2}(\cdot,s_{2})|^{2}\|_{H_{ul}^{\theta_{A}}}\leq
C\|\tilde{A}_{1,2}(\cdot,s_{2})\|_{H_{ul}^{\theta_{A}}}^{3}\leq\tilde{C}\tilde{K}^{2}\|\tilde{A}_{1,2}(\cdot,s_{2})\|_{H_{ul}^{\theta_{A}}}$
for all $s_{2}\in[0,\tilde{T}_{2}]$. Therefore

$\|\tilde{A}_{1,2}(\cdot,t_{2})\|_{H_{ul}^{\theta_{A}}}\leq
C\|A_{1,2}^{\ast}\|_{H_{ul}^{\theta_{A}}}+\tilde{C}\tilde{K}^{2}\int_{0}^{t_{2}}\|\tilde{A}_{1,2}(\cdot,s_{2})\|_{H_{ul}^{\theta_{A}}}ds_{2},$

for $t_{2}\in[0,\tilde{T}_{2}]$. Applying Grönwall’s inequality yields the
estimate

$\|\tilde{A}_{1,2}(\cdot,t_{2})\|_{H_{ul}^{\theta_{A}}}\leq
C\|A_{1,2}^{\ast}\|_{{H_{ul}^{\theta_{A}}}}\mathrm{e}^{\tilde{C}\tilde{K}^{2}t_{2}}\leq
C\|A_{1,2}^{\ast}\|_{{H_{ul}^{\theta_{A}}}},$ (104)

for $t_{2}\in[0,\tilde{T}_{2}]$ ($\mathrm{e}^{\tilde{C}\tilde{K}^{2}t_{2}}$ is
bounded since $\tilde{T}_{2}$ is finite). Since (104) is finite for all
$t_{2}\leq T_{2}$, we have that $A_{1,2}(\cdot,t_{2})\in H_{ul}^{\theta_{A}}$
for all $t_{2}\in[0,T_{2}]$. The bound in (100) follows from (104) and the
fact that $\|A_{1,2}(\cdot,t_{2})\|_{H_{ul}^{\theta_{A}}}\leq
C\|\tilde{A}_{1,2}(\cdot,t_{2})\|_{H_{ul}^{\theta_{A}}}$. ∎

#### Step III: Estimate $A_{2,2}(x_{2},t_{2})$

We prove the following result.

######  Lemma 5.15.

Let $\theta_{A}=3+\theta$ where $\theta>1/2$. Then $A_{2,2}(\cdot,t_{2})\in
H_{ul}^{\theta}$ for all $t_{2}\in[0,T_{2}]$. Moreover, we have that

$A_{2,2}(x_{2},T_{2})=\nu_{1}f_{2,2}+A_{2,2,h}(x_{2},T_{2}),$ (105)

where $f_{2,2}$ is a constant

$f_{2,2}:=\sqrt{\frac{\pi}{2}}\mathrm{e}^{\rho_{mid}^{2}/2}\left(\textnormal{erf}\left(\frac{\rho_{mid}}{\sqrt{2}}\right)+\textnormal{erf}\left(\frac{1}{\sqrt{2\zeta}}\right)\right)>0,$

and

$\|A_{2,2,h}(\cdot,T_{2})\|_{H_{ul}^{\theta}}\leq
C\max_{k=1,2}\|A_{k,2}^{\ast}\|_{H_{ul}^{\theta_{k}}},$ (106)

where $\theta_{k}$ is given by (74).

###### Proof.

Given solutions for $v_{2}(t_{2})$ and $A_{1,2}(x_{2},t_{2})$ as described in
Lemmas 5.13 and 5.14 respectively, the equation for $A_{2,2}$ decouples. We
obtain

$\partial_{t_{2}}A_{2,2}=4\partial_{x_{2}}^{2}A_{2,2}+v_{2}(t_{2})A_{2,2}-3A_{1,2}(x_{2},t_{2})^{2}\overline{A_{2,2}}-6|A_{1,2}(x_{2},t_{2})|^{2}A_{2,2}-4i\partial_{x_{2}}^{3}A_{1,2}(x_{2},t_{2})+\nu_{1},$

together with $A_{2,2}(x_{2},0)=A_{2,2}^{\ast}(x_{2})$. After applying the
transformation
$A_{2,2}(x_{2},t_{2})=:\mathrm{e}^{I_{2}(t_{2})}\tilde{A}_{2,2}(x_{2},t_{2})$,
where $I_{2}(t_{2})$ is the same integrating factor (102) as in Step II, we
obtain

$\partial_{t_{2}}\tilde{A}_{2,2}=4\partial_{x_{2}}^{2}\tilde{A}_{2,2}-3A_{1,2}(x_{2},t_{2})^{2}\overline{\tilde{A}_{2,2}}-6|A_{1,2}(x_{2},t_{2})|^{2}\tilde{A}_{2,2}-4i\mathrm{e}^{-I_{2}(t_{2})}\partial_{x_{2}}^{3}A_{1,2}(x_{2},t_{2})+\mathrm{e}^{-I_{2}(t_{2})}\nu_{1},$

together with $A_{2,2}(x_{2},0)=A_{2,2}^{\ast}(x_{2})$. Again we apply the
variation of constants formula and split the solution into two parts via
$\tilde{A}_{2,2}:=\tilde{A}_{2,2,h}+\tilde{A}_{2,2,s}$ defined by

$\begin{split}\tilde{A}_{2,2,h}(\cdot,t_{2})&=\mathrm{e}^{t_{2}\Lambda_{2}}A_{2,2}^{\ast}\
-\\\
\int_{0}^{t_{2}}&\mathrm{e}^{(t_{2}-s_{2})\Lambda_{2}}\left(3A_{1,2}(\cdot,s_{2})^{2}\overline{\tilde{A}_{2,2}}(\cdot,s_{2})+6|A_{1,2}(\cdot,s_{2})|^{2}\tilde{A}_{2,2}(\cdot,s_{2})+\mathrm{e}^{-I_{2}(s_{2})}4i\partial_{x_{2}}^{3}A_{1,2}(\cdot,s_{2})\right)ds_{2},\\\
\tilde{A}_{2,2,s}(\cdot,t_{2})&=\nu_{1}\int_{0}^{t_{2}}\mathrm{e}^{(t_{2}-s_{2})\Lambda_{2}}\mathrm{e}^{-I_{2}(s_{2})}ds_{2},\end{split}$
(107)

where as in Step II, $(\mathrm{e}^{t_{2}\Lambda_{2}})_{t_{2}\geq 0}$ denotes
the heat semigroup generated by $\Lambda_{2}=4\partial_{x_{2}}^{2}$. In order
to estimate the norm of $A_{2,2,h}:=\mathrm{e}^{I_{2}}\tilde{A}_{2,2,h}$,
first recall from Step II above that
$\|\mathrm{e}^{t_{2}\Lambda_{2}}\|_{H_{ul}^{\theta}\to H_{ul}^{\theta}}\leq C$
for all $t_{2}\in[0,T_{2}]$. Then for all $t_{2}\in[0,\tilde{T}_{2,2}]$, where
$\tilde{T}_{2,2}:=\min\\{T_{2},\sup\\{t_{2}>0:\|\tilde{A}_{2,2}(\cdot,t_{2})\|_{H_{ul}^{{\theta}}}\leq\tilde{M}\\}\\}$
for an arbitrarily large but fixed constant $\tilde{M}>0$ (c.f. Remark 5.5),
we obtain

$\|\tilde{A}_{2,2,h}(\cdot,t_{2})\|_{H_{ul}^{\theta}}\leq
C\|A_{2,2}^{\ast}\|_{H_{ul}^{\theta}}+C\tilde{M}^{2}T_{2}\|A_{1,2}^{\ast}\|_{H_{ul}^{\theta_{A}}}\leq
C\max_{k=1,2}\|A_{k,2}^{\ast}\|_{H_{ul}^{\theta_{k}}},$ (108)

where we used Lemma A.3, the fact that
$\|A_{1,2}(\cdot,t_{2})\|_{H_{ul}^{\theta}}\leq
C\tilde{K}\|A_{1,2}^{\ast}\|_{H_{ul}^{\theta}}$ for all $t_{2}\in[0,T_{2}]$
due to Lemma 5.14, and
$\mathrm{e}^{-I_{2}(s_{2})}\|\partial_{x_{2}}^{3}A_{1,2}(\cdot,s_{2})\|_{H_{ul}^{\theta}}\leq
C\|\partial_{x_{2}}^{3}A_{1,2}(\cdot,s_{2})\|_{H_{ul}^{\theta}}\leq
C\|A_{1,2}(\cdot,s_{2})\|_{H_{ul}^{\theta_{A}}}$ in order to derive the the
right-hand side. Since (108) is bounded for all $t_{2}\in[0,T_{2}]$ and the
integrating factor is bounded, $A_{2,2,h}(\cdot,t_{2})\in H_{ul}^{\theta}$ and
satisfies (106) for all $t_{2}\in[0,T_{2}]$.

We can calculate $\tilde{A}_{2,2,s}$ explicitly by viewing it as the solution
to a heat equation with a constant source term $\nu_{1}$ and initial data
$\tilde{A}_{2,2,s}(x_{2},0)\equiv 0$. A direct calculation using the explicit
solution formula gives the spatially homogeneous expression

$\tilde{A}_{2,2,s}(x_{2},t_{2})=\tilde{A}_{2,2,s}(t_{2})=\nu_{1}\sqrt{\frac{\pi}{2}}\mathrm{e}^{1/2\zeta}\left(\textnormal{erf}\left(\frac{\sqrt{\zeta}t_{2}-1}{\sqrt{2\zeta}}\right)+\textnormal{erf}\left(\frac{1}{\sqrt{2\zeta}}\right)\right),$

for all $t_{2}\in[0,T_{2}]$. In particular,

$\tilde{A}_{2,2,s}(T_{2})=\nu_{1}\sqrt{\frac{\pi}{2}}\mathrm{e}^{1/2\zeta}\left(\textnormal{erf}\left(\frac{\rho_{mid}}{\sqrt{2}}\right)+\textnormal{erf}\left(\frac{1}{\sqrt{2\zeta}}\right)\right).$

The expression in (105) follows after multiplying through by the integrating
factor. ∎

#### Step IV: Estimate $A_{j,2}(x_{2},t_{2})$, $j=3,\ldots,N-2$

In order to consider modulation functions $A_{j,2}$ with $j\geq 3$ we use
Lemma 4.3 again in order to separate $a_{j,2}=:a_{j,2}^{L}+a_{j,2}^{N}$ into
two parts. Similarly to the analysis in chart $\mathcal{K}_{1}$ Step IV, we
write

$a_{j,2}^{L}(\tilde{A}_{j,2},\overline{\tilde{A}_{j,2}})=\left(\alpha_{1}A_{1,2}^{2}+\alpha_{2}|A_{1,2}|^{2}+\alpha_{3}\overline{A_{1,2}}^{2}\right)\tilde{A}_{j,2}+\left(\beta_{1}A_{1,2}^{2}+\beta_{2}|A_{1,2}|^{2}+\beta_{3}\overline{A_{1,2}}^{2}\right)\overline{\tilde{A}_{j,2}},$

and note that $a_{j,2}^{N}$ depends only modulation functions
$A_{m^{\prime}j^{\prime},2}$ at lower orders with
$\alpha(m^{\prime})+j^{\prime}<j$.

We prove the following result.

######  Lemma 5.16.

Let $\theta_{A}=\theta_{A}(j)=3(j-1)+\theta$ where $j\in\\{3,\ldots,N-2\\}$
and $\theta>1/2$, and assume that $a_{j,2}^{N}(\cdot,t_{2})\in
H_{ul}^{\theta}$ for all $t_{2}\in[0,T_{2}]$. Then $A_{j,2}(\cdot,t_{2})\in
H_{ul}^{\theta}$ for all $t_{2}\in[0,T_{2}]$. If additionally $\mu(x)\equiv 0$
and

$\|a_{j,2}^{N}(\cdot,t_{2})\|_{H_{ul}^{\theta}}\leq
C\max_{k=1,\ldots,j-1}\|A_{k,2}(\cdot,t_{2})\|_{H_{ul}^{\theta_{k,j}}}^{3},$
(109)

then for $K>0$ sufficiently small it holds that

$\|A_{j,2}(\cdot,T_{2})\|_{H_{ul}^{\theta}}\leq
C\max_{k=1,\ldots,j}\|A_{k,2}^{\ast}\|_{H_{ul}^{\theta_{k,j}}},$ (110)

where $\theta_{k,j}$ is given by (84).

###### Proof.

We proceed similarly to the proof of Lemma 5.7 in Step IV of the chart
$\mathcal{K}_{1}$ analysis, and therefore present fewer details. As an
induction hypothesis, we assume that $A_{k,2}(\cdot,t_{2})\in
H_{ul}^{\theta_{k,j}}$ for all $k=3,\ldots,j-1\leq N-3$. We need to show that
$A_{j,2}(\cdot,t_{2})\in H_{ul}^{\theta}$. We omit the explicit treatment of
the base case again since it is similar to the induction step.

The equation for $A_{j,2}$ is given by

$\partial_{t_{2}}A_{j,2}=4\partial_{x_{2}}^{2}A_{j,2}+v_{2}(t_{2})A_{j,2}-a_{j,2}-4i\partial_{x_{2}}^{3}A_{j-1,2}+\partial_{x_{2}}^{4}A_{j-2,2},\qquad
A_{j,2}(x_{2},0)=A_{j,2}^{\ast}(x_{2}),$

and we obtain a new equation

$\partial_{t_{2}}\tilde{A}_{j,2}=4\partial_{x_{2}}^{2}\tilde{A}_{j,2}-a_{j,2}^{L}\left(\tilde{A}_{j,2},\overline{\tilde{A}_{j,2}}\right)-\mathrm{e}^{-I_{2}(t_{2})}\left(a_{j,2}^{N}+4i\partial_{x_{2}}^{3}A_{j-1,2}+\partial_{x_{2}}^{4}A_{j-2,2}\right),$

with $\tilde{A}_{j,2}(x_{2},0)=A_{j,2}^{\ast}(x_{2})$, after the
transformation
$A_{j,2}(x_{2},t_{2})=:\mathrm{e}^{I_{2}(t_{2})}\tilde{A}_{j,2}(x_{2},t_{2})$.
We now apply the variation of constants formula and split the solution into
two parts $\tilde{A}_{j,2}=\tilde{A}_{j,2,h}+\tilde{A}_{j,2,s}$, where

$\begin{split}\tilde{A}_{j,2,h}(\cdot,t_{2})&=\mathrm{e}^{t_{2}\Lambda_{2}}A_{j,2}^{\ast}-\int_{0}^{t_{2}}\mathrm{e}^{(t_{2}-s_{2})\Lambda_{2}}a_{j,2}^{L}\left(\tilde{A}_{j,2}(\cdot,s_{2}),\overline{\tilde{A}_{j,2}}(\cdot,s_{2})\right)ds_{2},\\\
\tilde{A}_{j,2,s}(\cdot,t_{2})&=-\int_{0}^{t_{2}}\mathrm{e}^{(t_{2}-s_{2})\Lambda_{2}}\mathrm{e}^{-I_{2}(s_{2})}\left(a^{N}_{j,2}(\cdot,s_{2})+4i\partial_{x_{2}}^{3}A_{j-1,2}(\cdot,s_{2})+\partial_{x_{2}}^{4}A_{j-2,2}(\cdot,s_{2})\right)ds_{2},\end{split}$
(111)

for $t_{2}\in[0,T_{2}]$.

Using the same bound for the semigroup as in Steps II-III above, as well as
the bound for $A_{1,2}$ in Lemma 5.14, we obtain

$\|\tilde{A}_{j,2,h}(\cdot,t_{2})\|_{H_{ul}^{\theta}}\leq
C\|A_{j,1}^{\ast}\|_{H_{ul}^{\theta}}+C\tilde{T}_{j,2}\|A_{1,2}^{\ast}\|_{H_{ul}^{\theta}}^{2}\leq
C\max_{k=1,\ldots,j}\|A_{j,1}^{\ast}\|_{H_{ul}^{\theta}},$ (112)

for $t_{2}\in[0,\tilde{T}_{j,2}]$, where
$\tilde{T}_{j,2}:=\min\\{T_{2},\sup\\{t_{2}>0:\|\tilde{A}_{j,2}(\cdot,s_{2})\|_{H_{ul}^{\theta}}\leq\tilde{M}\\}\\}$.

For the source term $\tilde{A}_{j,2,s}$ we have

$\begin{split}\|\tilde{A}_{j,2,s}(\cdot,t_{2})\|_{H_{ul}^{\theta}}&\leq
C\int_{0}^{t_{2}}\big{\|}a^{N}_{j,2}(\cdot,s_{2})+4i\partial_{x_{2}}^{3}A_{j-1,2}(\cdot,s_{2})+\partial_{x_{2}}^{4}A_{j-2,2}(\cdot,s_{2})\big{\|}_{H_{ul}^{\theta}}ds_{2}\\\
&\leq
C\tilde{T}_{j,2}\left(\big{\|}a^{N}_{3,2}(\cdot,t_{2})\big{\|}_{H_{ul}^{\theta}}+\|\partial_{x_{2}}^{3}A_{j-1,2}(\cdot,t_{2})\big{\|}_{H_{ul}^{\theta}}+\|\partial_{x_{2}}^{4}A_{j-2,2}(\cdot,t_{2})\big{\|}_{H_{ul}^{\theta}}\right),\end{split}$
(113)

for $t_{2}\in[0,\tilde{T}_{j,2}]$. Since $a^{N}_{3,2}(\cdot,t_{2})\in
H_{ul}^{\theta}$ by assumption and

$\begin{split}\|\partial_{x_{2}}^{3}A_{j-1,2}(\cdot,t_{2})\big{\|}_{H_{ul}^{\theta}}&\leq
C\|A_{j-1,2}(\cdot,t_{2})\big{\|}_{H_{ul}^{\theta_{j-1,j}}}\leq C,\\\
\|\partial_{x_{2}}^{4}A_{j-2,2}(\cdot,t_{2})\big{\|}_{H_{ul}^{\theta}}&\leq
C\|A_{j-2,2}(\cdot,t_{2})\big{\|}_{H_{ul}^{\theta_{j-2,j}}}\leq C,\end{split}$

by our regularity assumptions and the induction hypothesis,
$\tilde{A}_{j,2,s}(\cdot,t_{2})\in H_{ul}^{\theta}$ for all
$t_{2}\in[0,\tilde{T}_{j,2}]$.

Since the integrating factor $e^{I_{2}(t_{2})}$ is also bounded for all
$t_{2}\in[0,\tilde{T}_{2,j}]$, it follows that
$A_{j,2}(\cdot,t_{2})=\mathrm{e}^{I_{2}(t_{2})}(\tilde{A}_{j,2,h}(\cdot,t_{2})+\tilde{A}_{j,2,s}(\cdot,t_{2}))\in
H_{ul}^{\theta}$ for all $t_{2}\in[0,\tilde{T}_{j,2}]$. We may set
$\tilde{T}_{j,2}=T_{2}$, because the bounds above are finite for all
$t_{2}\in[0,T_{2}]$.

It remains to prove the bound in (110). Under the additional assumptions that
(109) is satisfied and $\mu(x)\equiv 0$ we can obtain a better bound on the
norm of $\tilde{A}_{j,2,s}$. This is achieved by using (109) and

$\begin{split}\|\partial_{x_{2}}^{3}A_{j-1,2}(\cdot,t_{2})\big{\|}_{H_{ul}^{\theta}}&\leq
C\|A_{j-1,2}(\cdot,t_{2})\big{\|}_{H_{ul}^{\theta_{j-1,j}}}\leq
C\max_{k=1,\ldots,j-1}\|A_{k,2}^{\ast}\|_{H_{ul}^{\theta_{k,j}}},\\\
\|\partial_{x_{2}}^{4}A_{j-2,2}(\cdot,t_{2})\big{\|}_{H_{ul}^{\theta}}&\leq
C\|A_{j-2,2}(\cdot,t_{2})\big{\|}_{H_{ul}^{\theta_{j-2,j}}}\leq
C\max_{k=1,\ldots,j-2}\|A_{k,2}^{\ast}\|_{H_{ul}^{\theta_{k,j}}},\end{split}$

where the right-most bounds can be derived using a recursive argument base on
the improved bounds from Lemmas 5.15 and 5.16 in case $\mu(x)\equiv 0$.
Substituting these bounds into the right-hand side of (113), combining the
resulting bound with (112) and multiplying through by the (bounded)
integrating factor yields the result in (110). ∎

####  5.2.1 Step V: Estimate $\psi_{2}$

As for the $\mathcal{K}_{1}$ analysis in Section 5.1, we need to control the
non-critical modulation functions $A_{mj,2}$ with $|m|\neq 1$ in order to
describe the dynamics of $\psi_{2}(\cdot,T_{2})$ and the map
$\pi_{2}:\Sigma^{in}_{2}\to\Sigma^{out}_{2}$. This is also necessary in order
to verify the assumptions on $a_{j,2}^{N}$ in Lemma 5.16. As before, we
reinstate the full notation $A_{\pm 1j,2}$ instead of $A_{j,2}$ in order to
distinguish modulation functions with $|m|=1$ and $|m|\neq 1$, and recall that
$\textbf{A}_{mj,2}$ denotes the set of lower order critical modulation
functions $A_{\pm 1j^{\prime},2}$ with
$j^{\prime}\in\\{1,\ldots,\alpha(m)+j-1\\}$.

######  Lemma 5.17.

Let $\theta_{A}=1+3(N-3)+\theta$ where $\theta>1/2$. Then the conditions on
$a_{j,2}^{N}$ for the applicability of Lemma 5.16 are satisfied. We have that
$A_{\pm 1j,2}(\cdot,t_{2})\in H_{ul}^{\theta_{j,N-2}}$ for all
$j=1,\ldots,N-2$ and $t_{2}\in[0,T_{2}]$, and

$A_{mj,2}(\cdot,t_{1})=g_{mj,2}^{gl}\left(\textbf{{A}}_{mj,2}(\cdot,t_{2}),v_{2}(t_{2})\right)\in
H_{ul}^{\theta_{\alpha(m)+j,N}},$ (114)

for all $t_{2}\in[0,T_{2}]$, $m\in I_{N}\setminus\\{-1,1\\}$ and
$j=1,\ldots,\tilde{\alpha}(m)$. If additionally $\mu(x)\equiv 0$ then for each
$m\in I_{N}\setminus\\{-1,1\\}$ we have

$\big{\|}g_{mj,2}^{gl}\left(\textbf{{A}}_{mj,2}(\cdot,T_{2}),v_{2}(T_{2})\right)\big{\|}_{H_{ul}^{\theta_{j,N-2}}}\leq
C\max_{k=1,\ldots,j-2}\|A_{1k,2}^{\ast}\|_{H_{ul}^{\theta_{k,N-2}}},$ (115)

for all $j=1,\ldots,\tilde{\alpha}(m)$.

###### Proof.

The proof is analogous to the proof of Lemma 5.9, so we omit the details. The
reason that we obtain constant bounds in the case $\mu(x)\equiv 0$ (as opposed
the the exponential bounds in Lemma 5.9) is that we have only constant bounds
when $\nu_{1}=0$ in Lemmas 5.14, 5.15 and 5.16. ∎

Combining the results of Lemmas 5.13, 5.14, 5.15, 5.16 and 5.17 we obtain the
following result, which summarises the dynamics in chart $\mathcal{K}_{2}$.

######  Proposition 5.18.

Let $\theta_{A}=1+3(N-3)+\theta$ where $\theta>1/2$. Then for $\vartheta,K>0$
sufficiently small but fixed we have $\psi_{2}(\cdot,t_{2})\in
H_{ul}^{\theta}$ for all $t_{2}\in[0,T_{2}]$. In particular, the map
$\pi_{2}:\Sigma_{2}^{in}\to\Sigma_{2}^{out}$ is well-defined and given by

$\pi_{2}:(\psi_{2}^{\ast},-\zeta^{-1/2},r_{2})\mapsto\left(\psi_{2}(\cdot,T_{2}),\rho_{max},r_{2}\right),$
(116)

with

$\psi_{2}(x,t_{2})=\psi_{GL,2}(x,t_{2})+r_{2}\left(A_{2,2,h}(x,t_{2})\mathrm{e}^{ix}+c.c.\right)+r_{2}^{2}h_{rem,2}(x,t_{2}),$
(117)

where $\psi_{GL,2}(x,t_{2})=A_{1,2}(x_{2},T_{2})\mathrm{e}^{ix}+c.c.$
satisfies $\|\psi_{GL,2}(\cdot,t_{2})\|_{H_{ul}^{\theta_{A}-1}}\leq
C\|A_{1,2}^{\ast}\|_{H_{ul}^{\theta_{A}}}$, the function $A_{2,2}$ is
described in Lemma 5.15, and the remainder term $h_{rem,2}(\cdot,t_{2})\in
H_{ul}^{\theta}$ for all $t_{2}\in[0,T_{2}]$. If additionally $\mu(x)\equiv 0$
then we have

$\|\psi_{2}(\cdot,t_{2})\|_{H_{ul}^{\theta}}\leq
C\|A_{1,2}^{\ast}\|_{H_{ul}^{\theta+1}}+r_{2}C\max_{k=1,2}\|A_{k,2}^{\ast}\|_{H_{ul}^{\theta+1}}+r_{2}^{2}C\max_{k=1,\ldots,N-2}\|A_{k,2}^{\ast}\|_{H_{ul}^{\theta+1}},$
(118)

for all $t_{2}\in[0,T_{2}]$.

###### Proof.

The form of the map in (116) follows immediately from Lemma 5.13. The fact
that $\psi_{2}(\cdot,t_{2})\in H_{ul}^{\theta}$ for all $t_{2}\in[0,T_{2}]$
follows from its definition in (45) and the fact that
$A_{mj,2}(\cdot,t_{2})\in H_{ul}^{\theta+1}$ for all $m\in I_{N}$,
$j=1,\ldots,\tilde{\alpha}(m)$ and $t_{2}\in[0,T_{2}]$ due to Lemmas 5.14,
5.15, 5.16 and 5.17. Similarly to the proof of Proposition 5.11, we need one
extra degree of regularity wherever we use Lemma A.1 to take the
$H_{ul}^{\theta}$ norm with respect to $x$ instead of $x_{2}$.

The asymptotic expression in (117) is obtained directly by evaluating the
series expansion (45) and evaluating it at $t_{2}=T_{2}$. The details are
similar to the derivation of the asymptotic formula (97) in the proof of
Proposition 5.11, and so omitted here for brevity.

Finally, the bound in (118) follows after applying a triangle inequality to
the series definitions of $\psi_{2}$ and applying the assertions pertaining to
the case $\mu(x)\equiv 0$ in Lemmas 5.14, 5.15, 5.16 and 5.17. ∎

###  5.3 Proof of Lemma 3.14

Lemma 3.14 can be proved with the information obtained in charts
$\mathcal{K}_{1}$ and $\mathcal{K}_{2}$. The idea is to analyse
$\pi^{mid}:\Delta^{in}\to\Delta^{mid}$ via its pre-image
$\pi_{2}\circ\kappa_{12}\circ\pi_{1}:\Sigma_{1}^{in}\to\Sigma_{2}^{out}$ in
the blown-up space, where $\kappa_{12}$ is the change of coordinates map
defined in (46) and $\pi_{1}$, $\pi_{2}$ are the maps described in
Propositions 5.11, 5.18 respectively. The fact that
$\Psi(\cdot,t)=r(\bar{t})\psi(\cdot,\bar{t})\in H_{ul}^{\theta}$ for all
$t\in[0,T_{mid}]$ follows directly from the fact that
$\psi_{l}(\cdot,t_{l})\in H_{ul}^{\theta}$ for all $t_{l}\in[0,T_{l}]$ in
charts $\mathcal{K}_{l}$ and $\mathcal{K}_{2}$; recall Propositions 5.11 and
5.18. It therefore remains to show that the map $\pi^{mid}$ is given by (38).

We consider an initial condition $(\Psi^{\ast},-\rho_{in})\in\Delta^{in}$,
which corresponds to an initial condition of the form
$(\psi_{1}^{\ast},\sqrt{\rho_{in}},\varepsilon_{1}^{\ast})\in\Sigma_{1}^{in}$
defined by

$\psi_{1}^{\ast}=\frac{\Psi^{\ast}}{\sqrt{\rho_{in}}},\qquad
r_{1}^{\ast}=\sqrt{\rho_{in}},\qquad\varepsilon_{1}^{\ast}=\frac{\varepsilon}{\rho_{in}^{2}}.$
(119)

Since $\Sigma_{2}^{in}=\kappa_{12}(\Sigma_{1}^{out})$, we obtain the following
bounds for the initial modulation functions in chart $\mathcal{K}_{2}$ using
the change of coordinates formula
$A_{mj,2}^{\ast}=\zeta^{-(\alpha(m)+j)/4}A_{mj,1}^{\ast}(\cdot,T_{1})$, recall
(47), together with Lemmas 5.4, 5.6, 5.7 and 5.9:

$\begin{split}&\|A_{1,2}^{\ast}\|_{H_{ul}^{\theta_{A}}}=\zeta^{-1/4}\|A_{1,1}(\cdot,T_{1})\|_{H_{ul}^{\theta_{A}}}\leq
C\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}\|A_{1,1}^{\ast}\|_{H_{ul}^{\theta_{A}}},\\\
&\|A_{2,2}^{\ast}\|_{H_{ul}^{\theta_{2,N-2}}}=\zeta^{-1/2}\|A_{2,1}(\cdot,T_{1})\|_{H_{ul}^{\theta_{2,N-2}}}\leq
C\left(|\nu_{1}|f_{2,1}(\varepsilon_{1}^{\ast})+\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}\max_{k=1,2}\|A_{k,1}^{\ast}\|_{H_{ul}^{\theta_{k,N-2}}}\right),\\\
&\|A_{j,2}^{\ast}\|_{H_{ul}^{\theta_{j,N-2}}}=\zeta^{-j/4}\|A_{j,1}(\cdot,T_{1})\|_{H_{ul}^{\theta_{j,N-2}}}\leq
C,\\\
&\|A_{mj,2}^{\ast}\|_{H_{ul}^{\theta_{\alpha(m)+j,N}}}=\zeta^{-(\alpha(m)+j)/4}\|A_{mj,1}(\cdot,T_{1})\|_{H_{ul}^{\theta_{\alpha(m)+j,N}}}\leq
C,\end{split}$ (120)

where $j\geq 3$ in the second-to-last line and $|m|\neq 1$ in the last. This
provides sufficient information on initial conditions

$\left(\psi_{2}^{\ast},-\zeta^{-1/2},r_{2}\right)=\kappa_{12}\left(\psi_{1}(\cdot,T_{1}),\sqrt{\rho_{in}}\left(\frac{\varepsilon_{1}^{\ast}}{\zeta}\right)^{1/4},\varepsilon_{1}^{\ast}\right)=\left(\zeta^{-1/4}\psi_{1}(\cdot,T_{1}),-\zeta^{-1/2},\sqrt{\rho_{in}}{\varepsilon_{1}^{\ast}}^{1/4}\right)$

in $\Sigma_{2}^{in}$, now in chart $\mathcal{K}_{2}$ coordinates. The
evolution from $\Sigma_{2}^{in}$ up to $\Sigma_{2}^{out}$ is described by
Proposition 5.18. Using Proposition 5.18 and the bound for
$\|A_{1,2}^{\ast}\|_{H_{ul}^{\theta_{A}}}$ in (120) we have that

$\|\psi_{GL,2}(\cdot,T_{2})\|_{{H_{ul}^{\theta_{A}-1}}}\leq
C\|A_{1,2}^{\ast}\|_{H_{ul}^{\theta_{A}}}\leq
C\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}\|A_{1,1}^{\ast}\|_{H_{ul}^{\theta_{A}}}.$

Thus $\psi_{GL,2}(\cdot,T_{2})$ can be pushed into the higher order terms in
(117), such that we obtain

$\psi_{2}(x,t_{2})=r_{2}\left(A_{2,2}(x_{2},t_{2})\mathrm{e}^{ix}+c.c.\right)+r_{2}^{2}\tilde{h}_{rem,2}(x,t_{2}),$
(121)

where $\tilde{h}_{rem,2}(\cdot,t_{2})\in H_{ul}^{\theta}$ for all
$t_{2}\in[0,T_{2}]$ and

$\left\|A_{2,2}(r_{2}\cdot,T_{2})\mathrm{e}^{i\cdot}+c.c.\right\|_{H_{ul}^{\theta_{2,N-2}}}\leq
C\left(|\nu_{1}|+\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}\right),$

due to (120) and Lemma 5.15.

In order to express (121) in terms of the original function $\Psi$ we simply
apply the blow-down transformations

$\Psi(x,T_{mid})=r_{2}\psi_{2}(x,T_{2})=\varepsilon^{1/4}\psi_{2}\left(x,\sqrt{\rho_{in}}{\varepsilon_{1}^{\ast}}^{1/4}\right)$

and $\varepsilon_{1}^{\ast}=\varepsilon/\rho_{in}^{2}$, thereby obtaining

$\Psi(x,T_{mid})=\varepsilon^{1/2}\left(A_{2,2}(\varepsilon^{1/4}x,T_{2})\mathrm{e}^{ix}+c.c.\right)+\varepsilon^{3/4}R(x,\varepsilon),$

where

$\left\|A_{2,2}(\varepsilon^{1/4}\cdot,T_{2})\mathrm{e}^{i\cdot}+c.c.\right\|_{H_{ul}^{\theta}}\leq
C\left(|\nu_{1}|+\mathrm{e}^{-\kappa\rho_{in}^{2}/2\varepsilon}\right)$

and $R(\cdot,\varepsilon):=\tilde{h}_{rem,2}(\cdot,T_{2})\in H_{ul}^{\theta}$
as required. This concludes the proof of Lemma 3.14. ∎

###  5.4 Chart $\mathcal{K}_{3}$

In order to prove Lemma 3.16 we also need to understand the dynamics in chart
$\mathcal{K}_{3}$. We recall the equations in chart $\mathcal{K}_{3}$:

$\begin{split}\partial_{t_{3}}A_{1,3}&=4\partial_{x_{3}}^{2}A_{1,3}+\left(1-\frac{\varepsilon_{3}}{2}\right)A_{1,3}-3A_{1,3}|A_{1,3}|^{2},\\\
\partial_{t_{3}}r_{3}&=\frac{1}{2}r_{3}\varepsilon_{3},\\\
\partial_{t_{3}}\varepsilon_{3}&=-2\varepsilon_{3}^{2},\\\
\partial_{t_{3}}A_{2,3}&=4\partial_{x_{3}}^{2}A_{2,3}+\left(1-\frac{\varepsilon_{3}}{2}\right)A_{2,3}-a_{2,3}-4i\partial_{x_{3}}^{3}A_{1,3},\\\
\partial_{t_{3}}A_{j,3}&=4\partial_{x_{3}}^{2}A_{j,3}+\left(1-\frac{\varepsilon_{3}}{2}\right)A_{j,3}-a_{j,3}-4i\partial_{x_{3}}^{3}A_{j-1,3}-\partial_{x_{3}}^{4}A_{j-2,3},\end{split}$
(122)

for $j=3,\ldots,N-2$. Note that we have set $\nu_{1}=0$ in the equation for
$A_{2,3}$ since our goal is to prove Lemma 3.16, which has $\mu(x)\equiv 0$.
Our aim in chart $\mathcal{K}_{3}$ is to describe the map

$\pi_{3}:\Sigma^{in}_{3}\ni\left(\psi_{3}^{\ast},r_{3}^{\ast},\zeta\right)\mapsto\left(\psi_{3}(\cdot,T_{3}),\rho_{out}^{1/2},\varepsilon_{3}^{out}(r_{3}^{\ast})\right)\in\Sigma_{3}^{out},$

induced by forward evolution of initial conditions in

$\Sigma_{3}^{in}=\left\\{(\psi_{3}^{\ast},r_{3},\zeta):\|\psi_{3}^{\ast}\|_{H_{ul}^{\theta}}\leq
K,r_{3}\in[0,\zeta^{-1/4}\vartheta]\right\\},$

which corresponds to the exit section $\Sigma^{out}_{2}$ in chart
$\mathcal{K}_{2}$ coordinates via
$\Sigma_{3}^{in}=\kappa_{23}(\Sigma_{2}^{out})$ if we set
$\rho_{mid}=\zeta^{-1/2}$ (which we shall assume is the case from this point
on), up to the exit section

$\Sigma_{3}^{out}=\left\\{(\psi_{3}^{\ast},\rho_{out}^{1/2},\varepsilon_{3}):\|\psi_{3}^{\ast}\|_{H_{ul}^{\theta}}\leq
K,\varepsilon_{3}\in[0,\zeta]\right\\},$

which corresponds to the original exit section $\Delta^{out}$ defined in (37)
after blowing down, fixing $\varepsilon$, and using the natural identification
with points in
$\Delta^{out}_{\varepsilon}=\Delta^{out}\times[0,\varepsilon_{0}]$. We proceed
again in steps.

#### Step I: Solve for $r_{3}(t_{3})$, $\varepsilon_{3}(t_{3})$ and $T_{3}$

We have the following result.

######  Lemma 5.19.

Let $r_{3}^{\ast}:=r_{3}(0)$. The transition time $T_{3}$ is given by

$T_{3}=\frac{1}{2\zeta{r_{3}^{\ast}}^{4}}\left(\rho_{out}^{2}-{r_{3}^{\ast}}^{4}\right),$

and for $t_{3}\in[0,T_{3}]$ we have

$r_{3}(t_{3})=r_{3}^{\ast}(1+2\zeta
t_{3})^{1/4},\qquad\varepsilon_{3}(t_{3})=\frac{\zeta}{1+2\zeta t_{3}}.$

In particular,
$\varepsilon_{3}^{out}(r_{3}^{\ast})=\zeta{r_{3}^{\ast}}^{4}/\rho_{out}^{2}$.

###### Proof.

The expressions for $r_{3}(t_{3})$ and $\varepsilon_{3}(t_{3})$ follow by
direct integration, and $T_{3}$ is obtained from the requirement that
$r_{3}(T_{3})=\rho_{out}^{1/2}$. The expression for
$\varepsilon_{3}^{out}(r_{3}^{\ast})$ follows after evaluating
$\varepsilon_{3}(T_{3})$. ∎

#### Step II: Estimate $A_{1,3}(x_{3},t_{3})$

We prove the following result.

######  Lemma 5.20.

Fix $\theta_{A}>1/2$ and assume that $A_{1,3}^{\ast}\in H_{ul}^{\theta_{A}}$.
If $K,\zeta>0$ are sufficiently small then for each
$\tilde{\sigma}_{0}\geq\tilde{\sigma}>1$ there is exists a constant $C>0$ such
that if initial conditions satisfy

$\|A_{1,3}^{\ast}\|_{H_{ul}^{\theta_{A}}}\leq
C\mathrm{e}^{-\tilde{\sigma}_{0}\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}},$
(123)

then $A_{1,3}(\cdot,t_{3})\in H_{ul}^{\theta_{A}}$ for all
$t_{3}\in[0,T_{3}]$, with

$\|A_{1,3}(\cdot,T_{3})\|_{H_{ul}^{\theta_{A}}}\leq
C\mathrm{e}^{\tilde{\sigma}\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}}\|A_{1,3}^{\ast}\|_{H_{ul}^{\theta_{A}}}\leq
C\mathrm{e}^{-(\tilde{\sigma}_{0}-\tilde{\sigma})\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}}.$

###### Proof.

The proof proceeds via arguments similar to those used to prove Lemma 5.4.
Substituting the expressions for $\varepsilon_{3}(t_{3})$ into the equation
for $A_{1,3}$ yields

$\partial_{t_{3}}A_{1,3}=4\partial_{x_{3}}^{2}A_{1,3}+\left(1-\frac{\varepsilon_{3}(t_{3})}{2}\right)A_{1,3}-3A_{1,3}|A_{1,3}|^{2},\qquad
A_{1,3}(x_{3},0)=A_{1,3}^{\ast}(x_{3}).$ (124)

Defining an integrating factor $\mathrm{e}^{I_{3}(t_{3})}$ via

$I_{3}(t_{3})=-\int_{0}^{t_{3}}\frac{\varepsilon_{3}(s_{3})}{2}ds_{3}=-\frac{1}{4}\ln(1+2\zeta
t_{3}),\qquad\mathrm{e}^{I_{3}(t_{3})}=\frac{1}{\sqrt[4]{1+2\zeta t_{3}}},$
(125)

and applying the transformation
$A_{1,3}(x_{3},t_{3})=:\mathrm{e}^{I_{3}(t_{3})}\tilde{A}_{1,3}(x_{3},t_{3})$
leads to

$\partial_{t_{3}}\tilde{A}_{1,3}=4\partial_{x_{3}}^{2}\tilde{A}_{1,3}+\tilde{A}_{1,3}-3\mathrm{e}^{2I_{3}(t_{3})}\tilde{A}_{1,3}|\tilde{A}_{1,3}|^{2},\qquad\tilde{A}_{1,3}(x_{3},0)=A_{1,3}^{\ast}(x_{3}).$

Using the variation of constants formula, this equation can be recast in the
integral form

$\tilde{A}_{1,3}(\cdot,t_{3})=\mathrm{e}^{t_{3}\Lambda_{3}}A_{1,3}^{\ast}-3\int_{0}^{t_{3}}\mathrm{e}^{(t_{3}-s_{3})\Lambda_{3}}\mathrm{e}^{2I_{3}(s_{3})}\tilde{A}_{1,3}(\cdot,s_{3})|\tilde{A}_{1,3}(\cdot,s_{3})|^{2}ds_{3},$
(126)

where $(\mathrm{e}^{t_{3}\Lambda_{3}})_{t_{3}\geq 0}$ is the semigroup
generated by $\Lambda_{3}:=4\partial_{x_{3}}^{2}+1$. In order to bound
$\|\tilde{A}_{1,3}(\cdot,t_{3})\|_{H_{ul}^{\theta_{A}}}$ we need to bound
$(\mathrm{e}^{t_{3}\Lambda_{3}})_{t_{3}\geq 0}$ and the cubic nonlinearity
$\tilde{A}_{1,3}(\cdot,s_{3})|\tilde{A}_{1,3}(\cdot,s_{3})|^{2}$ in
$H_{ul}^{\theta_{A}}$. We shall also use the fact that
$\mathrm{e}^{2I_{3}(s_{3})}\leq 1$ for all $s_{3}\geq 0$.

The semigroup can be bounded using Lemma A.2. We find that for each $\sigma>1$
there exists a $C>0$ depending on ${\theta_{A}}$ such that

$\|\mathrm{e}^{t_{3}\Lambda_{3}}\|_{H_{ul}^{\theta_{A}}\to
H_{ul}^{\theta_{A}}}\leq C\mathrm{e}^{\sigma t_{3}}$

for all $t_{3}\geq 0$.

Using the semigroup bound and Lemma A.2 we obtain

$\|\tilde{A}_{1,3}(\cdot,t_{3})\|_{H^{\theta_{A}}_{ul}}\leq
C\mathrm{e}^{\sigma
t_{3}}\|A_{1,3}^{\ast}\|_{H^{\theta_{A}}_{ul}}+C\tilde{K}^{2}\int_{0}^{t_{3}}\mathrm{e}^{\sigma(t_{3}-s_{3})}\|\tilde{A}_{1,3}(\cdot,s_{3})\|_{H^{\theta_{A}}_{ul}}\,ds_{3},$

for $t_{3}\in[0,\tilde{T}_{3}]$, and applying the Grönwall inequality in Lemma
A.4 leads to

$\|\tilde{A}_{1,3}(\cdot,t_{3})\|_{H^{\theta_{A}}_{ul}}\leq
C\mathrm{e}^{\tilde{\sigma}t_{3}}\|A_{1,3}^{\ast}\|_{H^{\theta_{A}}_{ul}},$

for $t_{3}\in[0,\tilde{T}_{3}]$, where
$\tilde{\sigma}:=\sigma+2C\tilde{K}^{2}>1$. Therefore

$\|A_{1,3}(\cdot,t_{3})\|_{H^{\theta_{A}}_{ul}}\leq
C\frac{\mathrm{e}^{\tilde{\sigma}t_{3}}}{\sqrt[4]{1+2\zeta
t_{3}}}\|A_{1,3}^{\ast}\|_{H^{\theta_{A}}_{ul}},$ (127)

for $t_{3}\in[0,\tilde{T}_{3}]$. Since the right-hand side is increasing on
$t_{3}\in[0,T_{3}]$, it follows that $A_{1,3}(\cdot,t_{3})\in
H_{ul}^{\theta_{A}}$ for all $t_{3}\in[0,T_{3}]$ if the initial condition
$A_{1,3}^{\ast}$ is sufficiently small in $H_{ul}^{\theta_{A}}$. A sufficient
condition is that

$\|A_{1,3}(\cdot,T_{3})\|_{H^{\theta_{A}}_{ul}}\leq
C\frac{\mathrm{e}^{\tilde{\sigma}T_{3}}}{\sqrt[4]{1+2\zeta
T_{3}}}\|A_{1,3}^{\ast}\|_{H^{\theta_{A}}_{ul}}<K.$ (128)

This condition is satisfied due to (123), which can be seen by substituting
the expression for $T_{3}$ in Lemma 5.19. In this case we can set
$\tilde{T}_{3}=T_{3}$ in (127), thereby proving the desired result. ∎

######  Remark 5.21.

Since the source term $\varepsilon\mu(x)$ does not influence the lowest order
approximation, Lemma 5.20 applies whether or not we have $\mu(x)\equiv 0$.

#### Step III: Estimate $A_{2,3}(x_{3},t_{3})$

We prove the following result.

######  Lemma 5.22.

Let $\theta_{A}=\theta+3$ where $\theta>1/2$ and assume that

$\|A_{k,3}^{\ast}\|_{H_{ul}^{\theta_{k}}}\leq
C\mathrm{e}^{-\tilde{\sigma}_{0}\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}},$
(129)

for $k=1,2$, where $\tilde{\sigma}_{0}>\tilde{\sigma}>\sigma>1$ is the same
constant as in Lemma 5.20 and $\theta_{k}$ is defined in (74). Then
$A_{2,3}(\cdot,t_{3})\in H_{ul}^{\theta}$ for all $t_{3}\in[0,T_{3}]$ and

$\|A_{2,3}(\cdot,T_{3})\|_{H_{ul}^{\theta}}\leq
C{r_{3}^{\ast}}^{-1}\mathrm{e}^{\tilde{\sigma}\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}}\max_{k=1,2}\|A_{k,3}^{\ast}\|_{H_{ul}^{\theta_{k}}}\leq
C\mathrm{e}^{-(\tilde{\sigma}_{0}-\tilde{\sigma})\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}}.$
(130)

###### Proof.

The proof proceeds via arguments similar to those in the proof of Lemma 5.6.
We have

$\partial_{t_{3}}A_{2,3}=4\partial_{x_{3}}^{2}A_{2,3}+\left(1-\frac{\varepsilon_{3}(t_{3})}{2}\right)A_{2,3}-3A_{1,3}(x_{3},t_{3})^{2}\overline{A_{2,3}}-6|A_{1,3}(x_{3},t_{3})|^{2}A_{2,3}-4i\partial_{x_{3}}^{3}A_{1,3}(x_{3},t_{3}),$

with $A_{2,3}(x_{3},0)=A_{2,3}^{\ast}(x_{3})$ for $t_{3}\in[0,T_{3}]$ and
$x_{3}\in\mathbb{R}$. Following the transformation
$A_{2,3}(x_{3},t_{3})=:\mathrm{e}^{I_{3}(t_{3})}\tilde{A}_{2,3}(x_{3},t_{3})$,
where $\mathrm{e}^{I_{3}(t_{3})}$ is the same integrating factor used in Step
II (recall (125)), we obtain

$\partial_{t_{3}}\tilde{A}_{2,3}=4\partial_{x_{3}}^{2}\tilde{A}_{2,3}+\tilde{A}_{2,3}-3A_{1,3}(x_{3},t_{3})^{2}\overline{\tilde{A}_{2,3}}-6|A_{1,3}(x_{3},t_{3})|^{2}\tilde{A}_{2,3}-4i\mathrm{e}^{-I_{3}(t_{3})}\partial_{x_{3}}^{3}A_{1,3}(x_{3},t_{3}),$

with $A_{2,3}(x_{3},0)=A_{2,3}^{\ast}(x_{3})$. After applying the variation of
constants formula we obtain

$\begin{split}\tilde{A}_{2,3}&(\cdot,t_{3})=\mathrm{e}^{t_{3}\Lambda_{3}}A_{2,3}^{\ast}\
-\\\
\int_{0}^{t_{3}}&\mathrm{e}^{(t_{3}-s_{3})\Lambda_{3}}\left(3A_{1,3}(\cdot,s_{3})^{2}\overline{\tilde{A}_{2,3}}(\cdot,s_{3})+6|A_{1,3}(\cdot,s_{3})|^{2}\tilde{A}_{2,3}(\cdot,s_{3})+4i\mathrm{e}^{-I_{3}(s_{3})}\partial_{x_{3}}^{3}A_{1,3}(\cdot,s_{3})\right)ds_{3},\end{split}$
(131)

where $\Lambda_{3}=4\partial_{x_{3}}^{2}+1$ as in Step II. We have the same
bound for the semigroup as in Step II, i.e. for each $\sigma>1$ there is a
constant $C>0$ depending on $\theta$ such that
$\|\mathrm{e}^{t_{3}\Lambda_{3}}\|_{H_{ul}^{\theta}\to H_{ul}^{\theta}}\leq
C\mathrm{e}^{\sigma t_{3}}$. Given the bounds for $A_{1,3}$ in Lemma 5.20 and
its proof (in particular (127)), it follows via arguments analogous to those
applied in the proof of Lemma 5.6 that for all $t_{3}\in[0,\tilde{T}_{2,3}]$
we have

$\begin{split}\|\tilde{A}_{2,3}(\cdot,t_{3})\|_{H_{ul}^{\theta}}&\leq
C\mathrm{e}^{\sigma
t_{3}}\|A_{2,3}^{\ast}\|_{H_{ul}^{\theta}}+C\mathrm{e}^{\tilde{\sigma}t_{3}}\|A_{1,3}^{\ast}\|_{H_{ul}^{\theta_{A}}}\\\
&+Ci\mathrm{e}^{\sigma
t_{3}}\|A_{1,3}^{\ast}\|_{H_{ul}^{\theta}}^{2}\Gamma\left(\frac{1}{2},-(-\sigma+2\tilde{\sigma})\left(\frac{1}{2\zeta}+s_{3}\right)\right)\bigg{|}_{t_{3}}^{0},\end{split}$
(132)

where $-\sigma+2\tilde{\sigma}=\sigma+4C\tilde{K}^{2}>1$ and
$-\sigma+\tilde{\sigma}=2C\tilde{K}^{2}\in(0,1)$. A direct calculation using
the upper gamma function asymptotics in Appendix A (equation (166)) shows that

$\|\tilde{A}_{2,3}(\cdot,t_{3})\|_{H_{ul}^{\theta}}\leq
C\mathrm{e}^{\tilde{\sigma}t_{3}}\max_{k=1,2}\|A_{k,3}^{\ast}\|_{H_{ul}^{\theta_{k}}}+C\mathrm{e}^{2\tilde{\sigma}t_{3}}\|A_{1,3}^{\ast}\|_{H_{ul}^{\theta}}^{2}\leq
C\mathrm{e}^{\tilde{\sigma}t_{3}}\max_{k=1,2}\|A_{k,3}^{\ast}\|_{H_{ul}^{\theta_{k}}},$
(133)

for $t_{3}\in[0,\tilde{T}_{2,3}]$, where we used the bound on
$\|A_{1,3}^{\ast}\|_{H_{ul}^{\theta}}$ from (129) in the second inequality.
Using the bounds in (129) again leads to

$\|\tilde{A}_{2,3}(\cdot,T_{3})\|_{H_{ul}^{\theta}}\leq
C\mathrm{e}^{-(\tilde{\sigma}_{0}-\tilde{\sigma})\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}}.$
(134)

Since (132) is increasing on $t_{3}\in[0,T_{3}]$ and (134) is bounded because
$\tilde{\sigma}_{0}>\tilde{\sigma}$, we can ensure that
$\tilde{T}_{2,3}=T_{3}$ by choosing $\vartheta>0$ sufficiently small. The
bound in (130) follows after multiplying through by the integrating factor. ∎

#### Step IV: Estimate $A_{j,3}(x_{3},t_{3})$, $j=3,\ldots,N-2$

Similarly to the Step IV analysis in charts $\mathcal{K}_{1}$ and
$\mathcal{K}_{2}$ we write $a_{j,3}=:a_{j,3}^{L}+a_{j,3}^{N}$, where

$a_{j,3}^{L}(\tilde{A}_{j,3},\overline{\tilde{A}_{j,3}})=\left(\alpha_{1}A_{1,3}^{2}+\alpha_{2}|A_{1,3}|^{2}+\alpha_{3}\overline{A_{1,3}}^{2}\right)\tilde{A}_{j,3}+\left(\beta_{1}A_{1,3}^{2}+\beta_{2}|A_{1,3}|^{2}+\beta_{3}\overline{A_{1,3}}^{2}\right)\overline{\tilde{A}_{j,3}},$

and $a_{j,3}^{N}$ depends only modulation functions
$A_{m^{\prime}j^{\prime},3}$ at lower orders with
$\alpha(m^{\prime})+j^{\prime}<j$.

We prove the following result.

######  Lemma 5.23.

Let $\theta_{A}=\theta_{A}(j)=3(j-1)+\theta$ where $j\in\\{3,\ldots,N-2\\}$
and $\theta>1/2$, assume

$\|a_{j,3}^{N}(\cdot,t_{3})\|_{H_{ul}^{\theta}}\leq
C\max_{k=1,\ldots,j-1}\|A_{k,3}(\cdot,t_{3})\|_{H_{ul}^{\theta_{j,k}}}^{3},$
(135)

and that

$\|A_{k,3}^{\ast}\|_{H_{ul}^{\theta_{k,j}}}\leq
C\mathrm{e}^{-\tilde{\sigma}_{0}\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}},$
(136)

for all $k=1,\ldots,j$, where $\tilde{\sigma}_{0}>\tilde{\sigma}>\sigma>1$ is
the same constant as in Lemmas 5.20-5.22 and $\theta_{k,j}$ is given by (84).
Then $A_{j,3}(\cdot,t_{3})\in H_{ul}^{\theta}$ for all $t_{3}\in[0,T_{3}]$ and

$\|A_{j,3}(\cdot,T_{3})\|_{H_{ul}^{\theta}}\leq
C\mathrm{e}^{-(\tilde{\sigma}_{0}-\tilde{\sigma})\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}}.$
(137)

###### Proof.

We proceed similarly to the proof of Lemmas 5.7, 5.16, and show that
$A_{j,3}(\cdot,t_{3})\in H_{ul}^{\theta}$ for all $j=1,\ldots,N-2$ using an
inductive argument over $k=3,\ldots,j-1\leq N-3$. As an induction hypothesis
we assume the existence of solutions $A_{k,3}(\cdot,t_{3})\in
H_{ul}^{\theta_{k,j}}$ for each $k=3,\ldots,j-1\leq N-3$, for all
$t_{3}\in[0,T_{3}]$. As in the proofs of Lemmas 5.7 and 5.16, we skip the base
case for brevity since it is similar to the inductive step.

The equation for $A_{j,3}$ is given by

$\partial_{t_{3}}A_{j,3}=4\partial_{x_{3}}^{2}A_{j,3}+\left(1-\frac{\varepsilon_{3}(t_{3})}{2}\right)A_{j,1}-a_{j,3}-4i\partial_{x_{3}}^{3}A_{j-1,3}-\partial_{x_{3}}^{4}A_{j-2,3},\qquad
A_{j,3}(x_{3},0)=A_{j,3}^{\ast}(x_{3}),$

for all $x_{3}\in\mathbb{R}$ and $t_{3}\in[0,T_{3}]$. Many of the estimates
below are obtained using arguments similar to those used in Steps II-III, so
we present fewer details. Note that $\partial_{x_{3}}^{3}A_{j-1,3}$ and
$\partial_{x_{3}}^{4}A_{j-2,3}$ can be considered as source terms, since we
are given solutions $A_{j-1,3}(\cdot,t_{3})\in H_{ul}^{\theta_{j-1,j}}$ and
$A_{j-2,3}(\cdot,t_{3})\in H_{ul}^{\theta_{j-2,j}}$ for all
$t_{3}\in[0,T_{3}]$ via Lemma 5.20, Lemma 5.22 or induction hypothesis
depending on $j$.

Setting
$A_{j,3}(x_{3},t_{3})=:\mathrm{e}^{I_{3}(t_{3})}\tilde{A}_{j,3}(x_{3},t_{3})$
where $I_{3}(t_{3})$ is given by (125) leads to

$\partial_{t_{3}}\tilde{A}_{j,3}=4\partial_{x_{3}}^{2}\tilde{A}_{j,3}-\tilde{A}_{j,3}-a_{j,3}^{L}(\tilde{A}_{j,3},\overline{\tilde{A}_{j,3}})-\mathrm{e}^{-I_{3}(t_{3})}\left(a^{N}_{j,3}+4i\partial_{x_{3}}^{3}A_{j-1,3}+\partial_{x_{3}}^{4}A_{j-2,3}\right),$

together with $\tilde{A}_{j,3}(x_{3},0)=A_{j,3}^{\ast}(x_{3})$. We apply the
variation of constants formula and split the solution into two parts
$\tilde{A}_{j,3}=:\tilde{A}_{j,3,h}+\tilde{A}_{j,3,s}$, where

$\begin{split}\tilde{A}_{j,3,h}(\cdot,t_{3})&=\mathrm{e}^{t_{3}\Lambda_{3}}A_{j,3}^{\ast}-\int_{0}^{t_{3}}\mathrm{e}^{(t_{3}-s_{3})\Lambda_{3}}a_{j,3}^{L}(\tilde{A}_{j,3}(\cdot,s_{3}),\overline{\tilde{A}_{j,3}}(\cdot,s_{3}))ds_{3},\\\
\tilde{A}_{j,3,s}(\cdot,t_{3})&=-\int_{0}^{t_{3}}\mathrm{e}^{(t_{3}-s_{3})\Lambda_{3}}\mathrm{e}^{-I_{3}(s_{3})}\left(a^{N}_{j,3}(\cdot,s_{3})+4i\partial_{x_{3}}^{3}A_{j-1,3}(\cdot,s_{3})+\partial_{x_{3}}^{4}A_{j-2,3}(\cdot,s_{3})\right)ds_{3},\end{split}$
(138)

for $t_{3}\in[0,T_{3}]$. Arguments analogous to those which led to the bound
for $\|A_{2,3}(\cdot,t_{3})\|_{H_{ul}^{\theta}}$ in Lemma 5.22 show that
$A_{j,3,h}(\cdot,t_{3})=\mathrm{e}^{I_{3}(t_{3})}\tilde{A}_{j,3,h}(\cdot,t_{3})$
has a similar bound. Specifically, we have

$\begin{split}\|\tilde{A}_{j,3,h}(\cdot,t_{3})\|_{H_{ul}^{\theta}}&\leq
C\mathrm{e}^{\sigma
t_{3}}\|A_{j,3}^{\ast}\|_{H_{ul}^{\theta}}+Ci\mathrm{e}^{\sigma
t_{3}}\|A_{1,3}^{\ast}\|_{H_{ul}^{\theta}}^{2}\Gamma\left(\frac{1}{2},-(-\sigma+2\tilde{\sigma})\left(\frac{1}{2\zeta}+s_{3}\right)\right)\bigg{|}_{t_{3}}^{0},\\\
&\leq C\mathrm{e}^{\sigma
t_{3}}\|A_{j,3}^{\ast}\|_{H_{ul}^{\theta}}+C\mathrm{e}^{2\tilde{\sigma}t_{3}}\|A_{1,3}^{\ast}\|_{H_{ul}^{\theta}}^{2}\\\
&\leq C\mathrm{e}^{\sigma
t_{3}}\|A_{j,3}^{\ast}\|_{H_{ul}^{\theta}}+C\mathrm{e}^{\tilde{\sigma}t_{3}}\|A_{1,3}^{\ast}\|_{H_{ul}^{\theta}}\\\
&\leq
C\mathrm{e}^{\tilde{\sigma}t_{3}}\max_{k=1,\ldots,j}\|A_{k,3}^{\ast}\|_{H_{ul}^{\theta}},\end{split}$
(139)

for all $t_{3}\in[0,\tilde{T}_{j,3}]$, where we used the gamma function
asymptotics (166) in Appendix A in the second inequality, and the bound for
$A_{1,3}^{\ast}$ in (136) in the second to last inequality. Using the
expression for $T_{3}$ in Lemma 5.19 and the bounds (136), we obtain

$\|A_{j,3,h}(\cdot,T_{3})\|_{H_{ul}^{\theta}}\leq
C\mathrm{e}^{-(\tilde{\sigma}_{0}-\tilde{\sigma})\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}}.$
(140)

Now consider the source term $\tilde{A}_{j,3,s}$. Using the same bound for the
semigroup as in Steps II-III we obtain the estimate

$\begin{split}\|\tilde{A}_{j,3,s}(\cdot,t_{3})\|_{H_{ul}^{\theta}}&\leq\\\
C\mathrm{e}^{\sigma t_{3}}&\int_{0}^{t_{3}}\mathrm{e}^{-\sigma
s_{3}}\sqrt[4]{1+2\zeta
s_{3}}\left\|a^{N}_{j,3}(\cdot,s_{3})+4i\partial_{x_{3}}^{3}A_{j-1,3}(\cdot,s_{3})+\partial_{x_{3}}^{4}A_{j-2,3}(\cdot,s_{3})\right\|_{H_{ul}^{\theta}}ds_{3},\end{split}$
(141)

for all $t_{3}\in[0,\tilde{T}_{j,3}]$. From this point on the proof resembles
the part of the proof of Lemma 5.7 devoted to the proving the bound (83) when
$\mu(x)\equiv 0$. We show that

$\|\tilde{A}_{j,3,s}(\cdot,t_{3})\|_{H_{ul}^{\theta}}\leq
C\mathrm{e}^{-(\tilde{\sigma}_{0}-\tilde{\sigma})\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}},$
(142)

by an inductive argument over $k=3,\ldots,j$. The idea is to bound each
$\|\tilde{A}_{k,3,s}(\cdot,t_{3})\|_{H_{ul}^{\theta}}$ using bounds
established for lower orders.

In order to prove the base case for $k=3$, we use the bounds

$\|A_{1,3}(\cdot,t_{3})\|_{H_{ul}^{\theta_{A}(j)}}\leq
C\frac{\mathrm{e}^{\tilde{\sigma}t_{3}}}{\sqrt[4]{1+2\zeta
t_{3}}}\|A_{1,3}^{\ast}\|_{H_{ul}^{\theta_{A}(j)}},\qquad\|A_{2,3}(\cdot,t_{3})\|_{H_{ul}^{\theta_{2,j}}}\leq\\\
C\frac{\mathrm{e}^{\tilde{\sigma}t_{3}}}{\sqrt[4]{1+2\zeta
t_{3}}}\max_{i=1,2}\|A_{k,3}^{\ast}\|_{H_{ul}^{\theta_{k,j}}},$

which apply for all $t_{3}\in[0,T_{3}]$ due to the proofs of Lemmas 5.20 and
5.22 (c.f. equations (127) and (133) respectively).

In order to obtain a bound on the integrand in (141) we use (135), which
implies

$\begin{split}\|a^{N}_{3,3}(\cdot,s_{3})\|_{H_{ul}^{\theta_{3,j}}}&\leq
C\max_{i=1,2}\|A_{i,3}(\cdot,t_{3})\|_{H_{ul}^{\theta_{i,j}}}^{3}\\\ &\leq
C\frac{\mathrm{e}^{3\tilde{\sigma}t_{3}}}{(1+2\zeta
t_{3})^{3/4}}\max_{i=1,2}\|A_{i,3}^{\ast}\|_{H_{ul}^{\theta_{i,j}}}^{3}\\\
&\leq C\frac{\mathrm{e}^{\tilde{\sigma}t_{3}}}{\sqrt[4]{1+2\zeta
t_{3}}}\max_{i=1,2}\|A_{i,3}^{\ast}\|_{H_{ul}^{\theta_{i,j}}},\end{split}$

where we used Lemma 5.20 in the second inequality and (136) together with
$(1+2\zeta t_{3})^{-3/4}\leq(1+2\zeta t_{3})^{-1/4}$ in the last inequality.
The derivative terms can be bounded using the bounds for $A_{1,3}$ and
$A_{2,3}$ above. Specifically,

$\begin{split}\|\partial_{x_{3}}^{3}A_{2,3}(\cdot,s_{3})\|_{H_{ul}^{\theta_{3,j}}}&\leq
C\|A_{2,3}(\cdot,s_{3})\|_{H_{ul}^{\theta_{2,j}}}\leq
C\frac{\mathrm{e}^{\tilde{\sigma}t_{3}}}{\sqrt[4]{1+2\zeta
t_{3}}}\max_{i=1,2}\|A_{i,3}^{\ast}\|_{H_{ul}^{\theta_{i,j}}},\\\
\|\partial_{x_{3}}^{4}A_{1,3}(\cdot,s_{3})\|_{H_{ul}^{\theta_{3,j}}}&\leq
C\|A_{1,3}(\cdot,s_{3})\|_{H_{ul}^{\theta_{A}(j)}}\leq
C\frac{\mathrm{e}^{\tilde{\sigma}t_{3}}}{\sqrt[4]{1+2\zeta
t_{3}}}\|A_{1,3}^{\ast}\|_{H_{ul}^{\theta_{A}(j)}},\end{split}$

for $s_{3}\in[0,T_{3}]$. It follows that

$\left\|a^{N}_{3,3}(\cdot,s_{3})+4i\partial_{x_{3}}^{3}A_{2,3}(\cdot,s_{3})+\partial_{x_{3}}^{4}A_{1,3}(\cdot,s_{3})\right\|_{H_{ul}^{\theta_{3,j}}}\leq
C\frac{\mathrm{e}^{\tilde{\sigma}t_{3}}}{\sqrt[4]{1+2\zeta
t_{3}}}\max_{i=1,2}\|A_{i,3}^{\ast}\|_{H_{ul}^{\theta_{i,j}}},$

for all $s_{3}\in[0,\tilde{T}_{3,3}]$. Using this to bound (141) with $j=3$,
we obtain

$\|\tilde{A}_{3,3,s}(\cdot,t_{3})\|_{H_{ul}^{\theta_{3,j}}}\leq
C\mathrm{e}^{\sigma
t_{3}}\max_{i=1,2}\|A_{i,3}^{\ast}\|_{H_{ul}^{\theta_{i,j}}}\int_{0}^{t_{3}}\mathrm{e}^{(\tilde{\sigma}-\sigma)s_{3}}ds_{3}\leq
C\mathrm{e}^{\tilde{\sigma}t_{3}}\max_{i=1,2}\|A_{i,3}^{\ast}\|_{H_{ul}^{\theta_{i,j}}},$

for $t_{3}\in[0,\tilde{T}_{3,3}]$.

Multiplying through by the integrating factor and using the bounds on the
initial conditions in (136) we obtain

$\|A_{3,3,s}(\cdot,T_{3})\|_{H_{ul}^{\theta_{3,j}}}\leq
C\mathrm{e}^{-(\tilde{\sigma}_{0}-\tilde{\sigma})\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}},$
(143)

which is valid for each $\tilde{\sigma}_{0}>\tilde{\sigma}$ if $\vartheta>0$
is sufficiently small. The desired bound for
$\|A_{3,3}(\cdot,t_{3})\|_{H_{ul}^{\theta_{3,j}}}$ is obtained using (140),
(143) and the triangle inequality. Note that $\tilde{T}_{3,3}=T_{3}$ as
required since the right-hand sides in (140) and (143) are bounded below
$\tilde{K}$.

We turn now to the induction step, and assume that (142) holds for
$k=3,\ldots,j-1$. Similarly to the proof of Lemma 5.7, we present fewer
details since the structure of the argument is the same as the base case.
Using (135), the induction hypothesis and the bounds in (136) we obtain

$\|a^{N}_{j,3}(\cdot,s_{3})\|_{H_{ul}^{\theta}}\leq
C\frac{\mathrm{e}^{3\tilde{\sigma}s_{1}}}{(1+2\zeta
s_{3})^{3/4}}\max_{i=1,\ldots,j-1}\|A_{i,3}^{\ast}\|_{H_{ul}^{\theta_{i,j}}}^{3}\leq
C\frac{\mathrm{e}^{\tilde{\sigma}s_{3}}}{\sqrt[4]{1+2\zeta
s_{3}}}\max_{i=1,\ldots,j-1}\|A_{i,3}^{\ast}\|_{H_{ul}^{\theta_{i,j}}},$

for $s_{3}\in[0,T_{3}]$. The derivative terms are controlled because we assume
sufficient regularity, so that the induction hypothesis implies

$\begin{split}\|\partial_{x_{3}}^{3}A_{j-1,3}(\cdot,s_{3})\|_{H_{ul}^{\theta}}&\leq
C\|A_{j-1,3}(\cdot,s_{3})\|_{H_{ul}^{\theta_{j-1,j}}}\leq
C\frac{\mathrm{e}^{\tilde{\sigma}s_{3}}}{\sqrt[4]{1+2\zeta
s_{3}}}\max_{i=1,\ldots,j-1}\|A_{i,3}^{\ast}\|_{H_{ul}^{\theta_{i,j}}},\\\
\|\partial_{x_{3}}^{4}A_{j-2,3}(\cdot,s_{3})\|_{H_{ul}^{\theta}}&\leq
C\|A_{j-2,3}(\cdot,s_{3})\|_{H_{ul}^{\theta_{j-2,j}}}\leq
C\frac{\mathrm{e}^{\tilde{\sigma}s_{3}}}{\sqrt[4]{1+2\zeta
s_{3}}}\max_{i=1,\ldots,j-2}\|A_{i,3}^{\ast}\|_{H_{ul}^{\theta_{i,j}}},\end{split}$

for $s_{3}\in[0,T_{3}]$. From here the bound for
$\|A_{j,3}(\cdot,t_{3})\|_{H_{ul}^{\theta}}$ on $t_{3}\in[0,T_{3}]$ follows
analogously to the base case $k=3$. ∎

####  5.4.1 Step V: Estimate $\psi_{3}$

We need the following result before we can describe the dynamics of
$\psi_{3}$. As in the chart $\mathcal{K}_{2}$ and $\mathcal{K}_{3}$ analyses,
we reinstate the full notation $A_{\pm 1j,3}$ instead of $A_{j,3}$ in order to
distinguish modulation functions with $|m|=1$ and $|m|\neq 1$, and recall that
$\textbf{A}_{mj,3}$ denotes the set of lower order critical modulation
functions $A_{\pm 1j^{\prime},3}$ with
$j^{\prime}\in\\{1,\ldots,\alpha(m)+j-1\\}$.

######  Lemma 5.24.

Let $\theta_{A}=3(N-3)+\theta$ where $\theta>1/2$ and assume that initial
conditions $A_{\pm 1j,3}^{\ast}$ satisfy (136). Then the conditions on
$a_{j,3}^{N}$ for the applicability of Lemma 5.23 are satisfied, and $A_{\pm
1j,3}(\cdot,t_{3})\in H_{ul}^{\theta_{j,N-2}}$ for all $j=1,\ldots,N-2$ and
$t_{3}\in[0,T_{3}]$. Moreover,

$A_{mj,3}(\cdot,t_{3})=g_{mj,3}^{gl}\left(\textbf{{A}}_{mj,3}(\cdot,t_{3}),\varepsilon_{3}(t_{3})\right)\in
H_{ul}^{\theta_{\alpha(m)+j,N}},$ (144)

and

$\big{\|}g_{mj,3}^{gl}\left(\textbf{{A}}_{mj,3}(\cdot,T_{3}),\varepsilon_{3}(T_{3})\right)\big{\|}_{H_{ul}^{\theta_{\alpha(m)+j,N}}}\leq
C\mathrm{e}^{-(\tilde{\sigma}_{0}-\tilde{\sigma})\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}},$
(145)

for all $t_{2}\in[0,T_{2}]$, $m\in I_{N}\setminus\\{-1,1\\}$ and
$j=1,\ldots,\tilde{\alpha}(m)$.

###### Proof.

The proof proceeds similarly to the (second part of the) proof of Lemma 5.9,
so we omit the details for brevity. It is necessary to assume the initial
condition bounds (136) in order to apply Lemmas 5.20, 5.22 and 5.23. ∎

Combining the results of Lemmas 5.19, 5.20, 5.22, 5.23 and 5.24 we obtain the
following result, which summarises the dynamics in chart $\mathcal{K}_{3}$.

######  Proposition 5.25.

Let $\theta_{A}=1+3(N-3)+\theta$ where $\theta>1/2$ and assume that initial
conditions are small according to (136). Then for $\zeta,K>0$ sufficiently
small but fixed we have $\psi_{3}(\cdot,t_{3})\in H_{ul}^{\theta}$ for all
$t_{3}\in[0,T_{3}]$. In particular, the map
$\pi_{3}:\Sigma_{3}^{in}\to\Sigma_{3}^{out}$ is well-defined and given by

$\pi_{3}:\left(\psi_{3}^{\ast},r_{3}^{\ast},\zeta\right)\mapsto\left(\psi_{3}(\cdot,T_{3}),\rho_{out}^{1/2},\zeta\frac{{r_{3}^{\ast}}^{4}}{\rho_{out}^{2}}\right),$
(146)

where $\psi_{3}(x,T_{3})$ satisfies

$\|\psi_{3}(\cdot,T_{3})\|_{H_{ul}^{\theta}}\leq
C\mathrm{e}^{-(\tilde{\sigma}_{0}-\tilde{\sigma})\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}},$
(147)

with $\tilde{\sigma}_{0}>\tilde{\sigma}>\sigma>1$.

###### Proof.

The conditions of Proposition 5.25 imply that Lemmas 5.19, 5.20, 5.22, 5.23
and 5.24 apply. It follows that all modulation functions
$A_{mj,3}(\cdot,t_{3})\in H_{ul}^{\theta+1}$ for all $t_{3}\in[0,T_{3}]$,
implying that $\psi_{3}(\cdot,t_{3})\in H_{ul}^{\theta}$ for all
$t_{3}\in[0,T_{3}]$ (recall the definition of $\psi_{3}$ in (45)).

The form of the map in (146) follows from Lemma 5.19. To verify the bound in
(147), notice that Lemmas 5.20, 5.22, 5.23 and 5.24 imply that for each $m\in
I_{N}$ we have

$\|A_{mj,3}(\cdot,T_{3})\|_{H_{ul}^{\theta+1}}\leq
C\mathrm{e}^{-(\tilde{\sigma}_{0}-\tilde{\sigma})\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}},$

for all $j=1,\ldots,\tilde{\alpha}(m)$. Thus

$\begin{split}\|\psi_{3}(\cdot,T_{3})\|_{H_{ul}^{\theta}}&\leq\frac{1}{r_{3}(T_{3})}\sum_{m\in
I_{N}}\sum_{j=1}^{\tilde{\alpha}(m)}r_{3}(T_{3})^{\alpha(m)+j}\|A_{mj,3}(r_{3}(T_{s})\cdot,T_{3})\mathrm{e}^{imx}\|_{H_{ul}^{\theta}}\\\
&\leq C\max_{m\in
I_{N},j\in\\{1,\ldots,\tilde{\alpha}(m)\\}}\|A_{mj,3}(\cdot,T_{3})\|_{H_{ul}^{\theta+1}}\\\
&\leq
C\mathrm{e}^{-(\tilde{\sigma}_{0}-\tilde{\sigma})\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}},\end{split}$

as required. The bound (147) implies that $\pi_{3}$ is well-defined because
$\|\psi_{3}(\cdot,T_{3})\|_{H_{ul}^{\theta}}\leq K$ for all
$r_{3}^{\ast}\in[0,\zeta^{-1/4}\vartheta]$ with $\vartheta>0$ sufficiently
small. ∎

###  5.5 Proof of Lemma 3.16

We are now in a position to prove Lemma 3.16 by combining results obtained for
the case $\mu(x)\equiv 0$ in each chart $\mathcal{K}_{l}$, $l=1,2,3$. The fact
that $\Psi(\cdot,t)=r(\bar{t})\psi(\cdot,\bar{t})\in H_{ul}^{\theta}$ for all
$t\in[0,T]$ follows directly from the fact that $\psi_{l}(\cdot,t_{l})\in
H_{ul}^{\theta}$ for all $t_{l}\in[0,T_{l}]$ in each chart $\mathcal{K}_{l}$;
recall Propositions 5.11, 5.18 and 5.25. It therefore remains to prove the
bound (41).

Similarly to the proof of Lemma 3.14 in Section 5.3, we consider an initial
condition $(\Psi^{\ast},-\rho_{in})\in\Delta^{in}$, which can be rewritten as
initial condition
$(\psi_{1}^{\ast},\sqrt{\rho_{in}},\varepsilon_{1}^{\ast})\in\Sigma_{1}^{in}$
in chart $\mathcal{K}_{1}$ using (120). Since $\mu(x)\equiv 0$, Proposition
5.11 implies that solutions are exponentially small in $H_{ul}^{\theta}$ at
$\Sigma_{2}^{in}=\kappa_{12}(\Sigma_{1}^{out})$. This time, we obtain the
following bounds for the initial modulation functions
$A_{mj,2}^{\ast}=\zeta^{-(\alpha(m)+j)/4}A_{mj,1}^{\ast}(\cdot,T_{1})$ in
chart $\mathcal{K}_{2}$ using the assertions of Lemmas 5.4, 5.6, 5.7 and 5.9
relevant to the case $\mu(x)\equiv 0$:

$\begin{split}&\|A_{1,2}^{\ast}\|_{H_{ul}^{\theta_{A}}}=\zeta^{-1/4}\|A_{1,1}(\cdot,T_{1})\|_{H_{ul}^{\theta_{A}}}\leq
C\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}\|A_{1,1}^{\ast}\|_{H_{ul}^{\theta_{A}}},\\\
&\|A_{2,2}^{\ast}\|_{H_{ul}^{\theta_{2,N-2}}}=\zeta^{-1/2}\|A_{2,1}(\cdot,T_{1})\|_{H_{ul}^{\theta_{2,N-2}}}\leq
C\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}\max_{k=1,2}\|A_{k,1}^{\ast}\|_{H_{ul}^{\theta_{k,N-2}}},\\\
&\|A_{j,2}^{\ast}\|_{H_{ul}^{\theta_{j,N-2}}}=\zeta^{-j/4}\|A_{j,1}(\cdot,T_{1})\|_{H_{ul}^{\theta_{j,N-2}}}\leq
C\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}\max_{k=1,\ldots,j}\|A_{k,1}^{\ast}\|_{H_{ul}^{\theta_{k,N-2}}},\\\
&\|A_{mj,2}^{\ast}\|_{H_{ul}^{\theta_{\alpha(m)+j,N}}}=\zeta^{-(\alpha(m)+j)/4}\|A_{mj,1}(\cdot,T_{1})\|_{H_{ul}^{\theta_{\alpha(m)+j,N}}}\leq
C\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}\max_{k=1,\ldots,j-2}\|A_{k,1}^{\ast}\|_{H_{ul}^{\theta_{k,N-2}}},\end{split}$
(148)

where $j\geq 3$ in the second-to-last line and $|m|\neq 1$ in the last. The
evolution from $\Sigma_{2}^{in}$ up to $\Sigma_{2}^{out}$ is understood in
chart $\mathcal{K}_{2}$ and summarised in Proposition 5.18. Bounds for the
initial modulation functions
$A_{mj,3}^{\ast}=\zeta^{(\alpha(m)+j)/4}A_{mj,2}^{\ast}(\cdot,T_{2})$ in chart
$\mathcal{K}_{3}$ are obtained via Lemmas 5.14, 5.15, 5.16 and 5.17. These
bounds can be formulated in terms of initial data in $\mathcal{K}_{1}$ using
(148). We obtain

$\begin{split}&\|A_{1,3}^{\ast}\|_{H_{ul}^{\theta_{A}}}=\zeta^{1/4}\|A_{1,2}(\cdot,T_{2})\|_{H_{ul}^{\theta_{A}}}\leq
C\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}\|A_{1,1}^{\ast}\|_{H_{ul}^{\theta_{A}}},\\\
&\|A_{2,3}^{\ast}\|_{H_{ul}^{\theta_{2,N-2}}}=\zeta^{1/2}\|A_{2,2}(\cdot,T_{2})\|_{H_{ul}^{\theta_{2,N-2}}}\leq
C\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}\max_{k=1,2}\|A_{k,1}^{\ast}\|_{H_{ul}^{\theta_{k,N-2}}},\\\
&\|A_{j,3}^{\ast}\|_{H_{ul}^{\theta_{j,N-2}}}=\zeta^{j/4}\|A_{j,2}(\cdot,T_{2})\|_{H_{ul}^{\theta_{j,N-2}}}\leq
C\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}\max_{k=1,\ldots,j}\|A_{k,1}^{\ast}\|_{H_{ul}^{\theta_{k,N-2}}},\\\
&\|A_{mj,3}^{\ast}\|_{H_{ul}^{\theta_{\alpha(m)+j,N}}}=\zeta^{(\alpha(m)+j)/4}\|A_{mj,2}(\cdot,T_{2})\|_{H_{ul}^{\theta_{\alpha(m)+j,N}}}\leq
C\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}\max_{k=1,\ldots,j-2}\|A_{k,1}^{\ast}\|_{H_{ul}^{\theta_{k,N-2}}},\end{split}$
(149)

where again, $j\geq 3$ in the second-to-last line and $m\neq 1$ in the last.
The idea from here is to apply Proposition 5.25. In order to do so, we need to
check the bounds (136).

Since the argument is the same for each $A_{j,1}^{\ast}$, we present the
details for $j=1$ only. A sufficient condition for (136) to be satisfied is

$\|A_{1,3}^{\ast}\|_{H_{ul}^{\theta}}\leq
C\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}\|A_{1,1}^{\ast}\|_{H_{ul}^{\theta}}\leq
C\mathrm{e}^{-\tilde{\sigma}_{0}\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}},$
(150)

which is satisfied for all $\varepsilon_{1}^{\ast}$ and $r_{3}^{\ast}$
sufficiently small if

$\frac{\kappa}{2\varepsilon_{1}^{\ast}}\geq\frac{\tilde{\sigma}_{0}\rho_{out}^{2}}{2\zeta{r_{3}^{\ast}}^{4}}.$
(151)

It follows from the form of the maps $\pi_{1}$ and $\pi_{2}$ in Propositions
5.11 and 5.18 respectively that

$r_{3}^{\ast}=\zeta^{-1/4}r_{2}(T_{2})=\zeta^{-1/4}r_{2}^{\ast}=r_{1}(T_{1})=\rho_{in}^{1/2}\sqrt[4]{\frac{\varepsilon_{1}^{\ast}}{\zeta}}.$
(152)

We also recall that for sufficiently small (but fixed) $K>0$, the constants
$\kappa\in(0,1)$ and $\tilde{\sigma}_{0}>1$ can be chosen arbitrarily close to
$1$. It follows that (151) (and therefore (150)) is satisfied if

$\frac{\rho_{out}}{\rho_{in}}\leq\sqrt{\frac{\kappa}{\tilde{\sigma}_{0}}}=:1-\omega,$

where $\omega\in(0,1)$ can be chosen arbitrarily close to $1$ if $K>0$ is
fixed but sufficiently small, which is precisely the assumption in the
statement of Lemma 3.16.

The preceding arguments show that Proposition 5.25 applies. Setting
$\tilde{\sigma}_{0}=\kappa(\rho_{in}/\rho_{out})^{2}$ in order to maximise the
contraction, we obtain the following via (147), (152) and the blow-down
transformation $\varepsilon_{1}^{\ast}=\varepsilon/\rho_{in}^{2}$:

$\|\psi_{3}(\cdot,T_{3})\|_{H_{ul}^{\theta}}\leq
C\exp\left(-\frac{\kappa}{2\varepsilon}\left(\rho_{in}^{2}-\frac{\tilde{\sigma}}{\kappa}\rho_{out}^{2}\right)\right),$

where $\tilde{\sigma}/\kappa>1$ can be chosen arbitrarily close to $1$ if
$K>0$ is sufficiently small. Setting $\kappa_{-}:=\kappa$ and
$\kappa_{+}:=\tilde{\sigma}/\kappa$ and applying the blow-down transformation

$\psi_{3}(\cdot,T_{3})=\frac{\Psi(\cdot,T)}{\sqrt{\rho_{out}}}$

yields the desired result. ∎

##  6 Proof of Theorem 3.20

In this section we prove Theorem 3.20. In order to do so, we need to bound the
norm of the error $\|u(\cdot,t)-\Psi(\cdot,t)\|_{H_{ul}^{\theta}}$ at
particular times $t=T_{mid}$ and $t=T$. Similarly to the established
approaches for the static SH problem (6) in e.g. [82, Ch. 10], we define a
weighted error function $R$ via

$r(\bar{t})^{\beta}R(x,\bar{t}):=u(x,t)-r(\bar{t})\psi(x,\bar{t}),$ (153)

where the exponent $\beta>0$ is expected to depend upon the order $n$ of the
approximation $\Psi$, but left unspecified for now. The first step is to
obtain an evolution equation for $R$ in terms of $\psi$ and
$\textrm{Res}(r\psi)$. This equation is expected to differ in the dynamic
setting since the variable $r$, which plays the role of the small parameter,
depends on time (recall the discussion following the statement of Theorem
3.20).

Differentiating $u$ with respect to $t$ and applying the definition (153)
leads to

$\begin{split}\partial_{t}u&=\partial_{t}(r^{\beta}R)+\partial_{t}(r\psi)\\\
&=\partial_{t}(r^{\beta}R)-(1+\partial_{x}^{2})^{2}r\psi+r^{3}\bar{v}\psi-r^{2}\psi^{3}+r^{4}\varepsilon\mu(x)-\textrm{Res}(r\psi)\\\
&=-(1+\partial_{x}^{2})^{2}(r^{\beta}R+r\psi)+r^{2}\bar{v}(r^{\beta}R+r\psi)-(r^{\beta}R+r\psi)^{3}+r^{4}\bar{\varepsilon}\mu(x).\end{split}$

Expanding $\partial_{t}(r^{\beta}R)=r^{\beta}\partial_{t}R+\beta
r^{\beta-1}R\partial_{t}r$, changing to the desingularized time $\bar{t}$ and
rearranging a little, we obtain the following equation for the (weighted)
error function $R$:

$\partial_{\bar{t}}R=-r^{-2}(1+\partial_{x}^{2})^{2}R+\left(\bar{v}-\beta
r^{-1}\partial_{\bar{t}}r-3\psi^{2}\right)R-3r^{\beta-1}R^{2}\psi-r^{2\beta-2}R^{3}+r^{-\beta-2}\textrm{Res}(r\psi).$
(154)

Some of the time-dependence in the linear part can be mitigated via the use of
an integrating factor. Defining a new function $\tilde{R}$ via
$R(x,\bar{t})=:\mathrm{e}^{I(\bar{t})}\tilde{R}(x,\bar{t})$, where

$I(\bar{t})=-\beta\int_{0}^{\bar{t}}\frac{\partial_{\bar{s}}r(\bar{s})}{r(\bar{s})}d\bar{s},$

leads to

$\partial_{\bar{t}}\tilde{R}=-r^{-2}(1+\partial_{x}^{2})^{2}\tilde{R}+\bar{v}\tilde{R}-3\psi^{2}\tilde{R}-3r^{\beta-1}\mathrm{e}^{I}\tilde{R}^{2}\psi-r^{2\beta-2}\mathrm{e}^{2I}\tilde{R}^{3}+r^{-\beta-2}\mathrm{e}^{-I}\textrm{Res}(r\psi).$

Using the variation of constants formula we obtain

$\tilde{R}(\cdot,\bar{t})=\mathrm{e}^{\bar{t}\Lambda}R^{\ast}+\int_{0}^{\bar{t}}\mathrm{e}^{(\bar{t}-\bar{s})\Lambda}\left(F(\cdot,\bar{s})+r(\bar{s})^{-\beta-2}\mathrm{e}^{-I(\bar{s})}\textrm{Res}(r(\bar{s})\psi(\cdot,\bar{s}))\right)d\bar{s},$
(155)

where $(\mathrm{e}^{\bar{t}\Lambda})_{\bar{t}\geq 0}$ is the evolution family
generated by
$\Lambda:=-r(\bar{t})^{-2}(1+\partial_{x}^{2})^{2}+\bar{v}(\bar{t})$, the
initial condition satisfies $R^{\ast}:=\tilde{R}(\cdot,0)=\tilde{R}(\cdot,0)$,
and

$F(\cdot,\bar{s}):=-3\psi(\cdot,\bar{s})^{2}\tilde{R}(\cdot,\bar{s})-3r(\bar{s})^{\beta-1}\mathrm{e}^{I(\bar{s})}\tilde{R}(\cdot,\bar{s})^{2}\psi(\cdot,\bar{s})-r(\bar{s})^{2\beta-2}\mathrm{e}^{2I(\bar{s})}\tilde{R}(\cdot,\bar{s})^{3}.$
(156)

In order to estimate the norm of (155) (and thus also the norm of $R$) in
$H_{ul}^{\theta}$, we need to estimate the following:

  * •

The evolution family $(\mathrm{e}^{\bar{t}\Lambda})_{\bar{t}\geq 0}$;

  * •

The residual $\textrm{Res}(r(\bar{s})\psi(\cdot,\bar{s}))$;

  * •

The ‘nonlinear’ terms in $F(\cdot,\bar{s})$.

We derive estimates for each of these terms below. Since the equation for the
error in (154) is posed in the blown-up space, we will often work in
coordinate charts $\mathcal{K}_{l}$.

###  6.1 Evolution family estimates

We prove the following result.

######  Lemma 6.1.

The following estimates for the evolution family are obtained in coordinate
charts $\mathcal{K}_{l}$, and hold for every $\theta\geq 0$:

  1. (i)

Chart $\mathcal{K}_{1}$: For each $c_{1}\in(0,1)$ there exists a constant
$C>0$ depending on $\theta$ such that

$\|\mathrm{e}^{t_{1}\Lambda_{1}}\|_{H_{ul}^{\theta}\to H_{ul}^{\theta}}\leq
C\mathrm{e}^{-c_{1}t_{1}}$

for all $t_{1}\in[0,T_{1}]$, where
$\Lambda_{1}=-r_{1}(t_{1})^{-2}(1+\partial_{x}^{2})^{2}-1$.

  2. (ii)

Chart $\mathcal{K}_{2}$: There exists a constant $C>0$ such that

$\|\mathrm{e}^{t_{2}\Lambda_{2}}\|_{H_{ul}^{\theta}\to H_{ul}^{\theta}}\leq C$

for all $t_{2}\in[0,T_{2}]$, where
$\Lambda_{2}=-r_{2}^{-2}(1+\partial_{x}^{2})^{2}+v_{2}(t_{2})$.

  3. (iii)

Chart $\mathcal{K}_{3}$: For each $c_{3}>1$ there exists a constant $C>0$
depending on $\theta$ such that

$\|\mathrm{e}^{t_{3}\Lambda_{3}}\|_{H_{ul}^{\theta}\to H_{ul}^{\theta}}\leq
C\mathrm{e}^{c_{3}t_{3}}$

for all $t_{3}\in[0,T_{3}]$, where
$\Lambda_{3}=-r_{3}(t_{3})^{-2}(1+\partial_{x}^{2})^{2}+1$.

###### Proof.

We begin in global coordinates. The evolution family
$(\mathrm{e}^{\bar{t}\Lambda})_{\bar{t}\geq 0}$ is induced by the linear
equation

$\partial_{\bar{t}}\tilde{w}(x,\bar{t})=-\frac{(1+\partial_{x}^{2})^{2}}{r_{1}(\bar{t})^{2}}\tilde{w}(x,\bar{t})+\bar{v}(\bar{s})\tilde{w}(x,\bar{t}),\qquad\tilde{w}(x,0)=w^{\ast}(x),\qquad(\bar{t}\geq
0,\ x\in\mathbb{R}).$

Fourier transforming in space leads to a family of linear non-autonomous ODEs

$\partial_{\bar{t}}\hat{\tilde{w}}(\xi,\bar{t})=-\frac{(1-\xi^{2})^{2}}{r(\bar{t})^{2}}\hat{\tilde{w}}(\xi,\bar{t})+\bar{v}(\bar{s})\hat{\tilde{w}}(\xi,\bar{t}),\qquad\hat{\tilde{w}}(\xi,0)=\hat{w}^{\ast}(\xi),\qquad(\bar{t}\geq
0,\ \xi\in\mathbb{R}),$

with solutions

$\hat{\tilde{w}}(\xi,\bar{t})=\exp\left(\int_{0}^{\bar{t}}\left(-\frac{(1-\xi^{2})^{2}}{r(\bar{s})^{2}}+\bar{v}(\bar{s})\right)d\bar{s}\right)\hat{w}^{\ast}(\xi).$

Taking the inverse Fourier transform leads to the solution formula

$\tilde{w}(\cdot,\bar{t})=\mathscr{F}^{-1}\exp\left(\int_{0}^{\bar{t}}\left(-\frac{(1-\xi^{2})^{2}}{r(\bar{s})^{2}}+\bar{v}(\bar{s})\right)d\bar{s}\right)\mathscr{F}w^{\ast}=:P(\bar{t})w^{\ast}.$

Fourier multiplier techniques allow us to study the mapping properties of
$\tilde{P}(\bar{t})\colon H^{\theta}_{ul}\to H^{\theta}_{ul},\ w^{\ast}\mapsto
P(\bar{t})w^{\ast}$ via its symbol

$\tilde{p}(\xi,\bar{t}):=\exp\left(\int_{0}^{\bar{t}}\left(-\frac{(1-\xi^{2})^{2}}{r(\bar{s})^{2}}+\bar{v}(\bar{s})\right)d\bar{s}\right).$

In particular, Lemma A.2 shows that it suffices to derive estimates for the
$C^{2}_{b}$-norm of $\tilde{p}(\cdot,\bar{t})$. These can be estimated
directly using the above and

$\begin{split}\partial_{\xi}\tilde{p}(\xi,\bar{t})&=4\xi(1-\xi^{2})\left(\int_{0}^{\bar{t}}\frac{1}{r(\bar{s})^{2}}d\bar{s}\right)\tilde{p}(\xi,\bar{t}),\\\
\partial_{\xi}^{2}\tilde{p}(\xi,\bar{t})&=\left(4-12\xi^{2}+16\xi^{2}(1-\xi^{2})^{2}\right)\left(\int_{0}^{\bar{t}}\frac{1}{r(\bar{s})^{2}}d\bar{s}\right)^{2}\tilde{p}(\xi,\bar{t}).\end{split}$

In order to estimate $\tilde{p}(\xi,\bar{t})$ and the integral terms, we work
in charts $\mathcal{K}_{l}$.

In $\mathcal{K}_{1}$ we have that
$r(\bar{t})=r_{1}(t_{1})=\rho_{in}^{1/2}(1-2\varepsilon_{1}^{\ast}t_{1})^{1/4}$
and $\bar{v}(\bar{t})=-1$ for all $t_{1}\in[0,T_{1}]$; recall Lemma 5.2. There
are algebraic growth rates in time due to the fact that

$\int_{0}^{t_{1}}\frac{1}{r_{1}(s_{1})^{2}}ds_{1}=\rho_{in}^{-1}{\varepsilon_{1}^{\ast}}^{-1}(1-\sqrt{1-2\varepsilon_{1}^{\ast}t_{1}}),$

but these are compensated by the exponential decay in the expression for
$\tilde{p}(\xi,\bar{t})$, due to the fact that $\bar{v}=-1$ in chart
$\mathcal{K}_{1}$. In particular, letting $\tilde{p}_{1}$ denote the
representation of $\tilde{p}$ in $\mathcal{K}_{1}$, we find that for each
$c_{1}\in(0,1)$ there is a constant $C>0$ depending on $\theta$ such that

$\|\tilde{p}_{1}(\cdot,t_{1})\|_{C_{b}^{2}}\leq C\mathrm{e}^{-c_{1}t_{1}},$

for all $t_{1}\in[0,T_{1}]$. Lemma A.2 implies the corresponding result in
$H_{ul}^{\theta}$, thereby proving Assertion (i).

In $\mathcal{K}_{2}$ we have $r(\bar{t})=r_{2}$ and
$\bar{v}(\bar{t})=v_{2}(t_{2})=-\zeta^{-1/2}+t_{2}$ for all
$t_{2}\in[0,T_{2}]$; recall Lemma 5.13. A direct calculation shows that

$\|\tilde{p}_{2}(\cdot,t_{2})\|_{C_{b}^{2}}\leq
C\frac{t_{2}^{2}}{r_{2}^{4}}\mathrm{e}^{-(1-\xi^{2})^{2}t_{2}/r_{2}^{2}}$

for all $t_{2}\in[0,T_{2}]$. Thus there is exponential decay for all
wavenumbers $\xi$ bounded away from the critical modes $\xi\pm 1$. It turns
out that the unwanted algebraic growth close to $\xi\pm 1$ can be mitigated
using mode filters. Due to the similarity with pre-existing arguments in e.g.
[65, 73, 74, 82], we omit the details, noting simply that close for $\xi=\pm
1\pm r_{2}\bar{\xi}$ close to $\xi=\pm 1$ one has
$-(1-\xi^{2})^{2}=-4(r_{2}\bar{\xi})^{2}+O((r_{2}\bar{\xi})^{4})$ so that

$\exp\left(-\int_{0}^{t_{2}}\left(4\bar{\xi}^{2}+O((r_{2}\bar{\xi})^{2}\right)ds_{2}\right)=\exp\left(-4\bar{\xi}^{2}t_{2}+O((r_{2}\bar{\xi})^{2}t_{2})\right),$

which is bounded for all $t_{2}\in[0,T_{2}]$. This leads to a $C^{2}_{b}$ norm
which is growing at a rate proportional to $t_{2}^{2}$, which is finite on
$[0,T_{2}]$. Applying Lemma A.2 yields the $H_{ul}^{\theta}$ estimate in
Assertion (ii).

In $\mathcal{K}_{3}$ we have $r(\bar{t})=r_{3}(t_{3})=r_{3}^{\ast}(1+2\zeta
t_{3})^{1/4}$ and $\bar{v}(\bar{t})=1$ for all $t_{3}\in[0,T_{3}]$; recall
Lemma 5.19. In this case we also identify algebraic growth rates in time,
however these are dominated by the exponential growth due to the fact that
$\bar{v}=1$ in chart $\mathcal{K}_{3}$. In particular, for each $c_{3}>1$
there is a $c_{3}$ such that

$\|\tilde{p}_{3}(\xi,t_{3})\|_{C_{b}^{2}}\leq C\mathrm{e}^{c_{3}t_{3}}$

for all $t_{3}>0$. The bound in Assertion (iii) follows after applying Lemma
A.2. ∎

###  6.2 Residual estimates

In the following we require extra regularity in order to control higher order
contributions to the residual because of the reasons given in Remark 4.5. In
order to keep track of this extra regularity we use

$\theta_{j,k}:=1+3(j-k+1)+\theta=\begin{cases}\theta_{A}(j)=1+3j+\theta,&k=1,\\\
3(j-k+1)+\theta,&k=2,\ldots,j-1,\\\ 4+\theta,&k=j,\end{cases}$

instead of the earlier definition in (84). We obtain the following result.

######  Lemma 6.2.

Let $\theta_{A}=1+3(n-3)+\theta$ and $\theta\geq 1$. Then

$\|\textup{Res}(r_{l}(t_{l})\psi_{l}(\cdot,t_{l}))\|_{H_{ul}^{\theta}}\leq
Cr_{l}(t_{l})^{n}\left(\delta_{n,4}\max_{m\in I_{N}}|\nu_{m}|+\sup_{m\in
I_{N},j=1,\ldots,\tilde{\alpha}(m)}\|A_{mj,l}(\cdot,t_{l})\|_{H_{ul}^{\alpha(m)+j,N}}\right),$
(157)

for $t_{l}\in[0,T_{l}]$ and $l=1,2$, where

$\delta_{n,4}:=\begin{cases}1,&n=4,\\\ 0,&n\geq 4.\end{cases}$

If additionally $\mu(x)\equiv 0$ and the conditions of Proposition 5.25 are
satisfied, then

$\|\textup{Res}(r_{l}(t_{l})\psi_{l}(\cdot,t_{l}))\|_{H_{ul}^{\theta}}\leq
Cr_{l}(t_{l})^{n}\sup_{k=1,\ldots,N-2}\|A_{1k,l}(\cdot,t_{l})\|_{H_{ul}^{\theta_{k,N-2}}},$
(158)

for all $t_{1}\in[0,T_{l}]$ and $l=1,2,3$.

###### Proof.

We begin in global coordinates. It follows from the formal construction for
$\Psi=r\psi$ in Section 4 that

$\begin{split}\textrm{Res}(r\psi)&=\delta_{n,4}r^{4}\bar{\varepsilon}\sum_{m\in
I_{N}}\nu_{m}\mathrm{e}^{imx}\\\ &+\sum_{m\in
I_{N}}\sum_{j=1}^{\tilde{\alpha}(m)}r^{\alpha(m)+j}\big{(}r\mathcal{L}_{m}^{(1)}+r^{2}\tilde{\mathcal{L}_{m}^{(2)}}+r^{3}\mathcal{L}_{m}^{(3)}+r^{4}\mathcal{L}_{m}^{(4)}\big{)}A_{mj}\mathrm{e}^{imx}-\sum_{m\in
I_{3N}\setminus I_{N}}b_{m}\mathrm{e}^{imx}.\end{split}$ (159)

The first sum is easily controlled. The second sum has terms which vanish up
to and including $O(r^{N})$, however we need to control the higher order terms
at $O(r^{N+1})=O(r^{n})$, $O(r^{n+1})$, $O(r^{n+2})$ and $O(r^{n+3})$. Since
$\tilde{\alpha}(m)\leq N-1$ for all $m\in I_{N}$, terms involving
$r\mathcal{L}_{m}^{(1)}$ must be $O(r^{n})$ with $\alpha(m)+j=N$. It follows
that

$\sum_{m\in
I_{N}}\sum_{j=1}^{\tilde{\alpha}(m)}r^{\alpha(m)+j}r\mathcal{L}_{m}^{(1)}A_{mj}\mathrm{e}^{imx}=r^{n}\mathcal{L}_{m}^{(1)}\left(A_{0N-1}+A_{2N-1}\mathrm{e}^{2ix}+A_{3N-2}\mathrm{e}^{3ix}+\cdots+A_{N1}\mathrm{e}^{Nix}\right)+c.c.,$

the norm of which is bounded by $Cr^{n}$ since the non-critical modulation
functions $A_{mj}$ with $|m|\neq 1$ inside the term in parentheses are bounded
according to Lemmas 5.9, 5.17 and 5.24. More precisely, we have that

$\bigg{\|}x\mapsto\sum_{m\in
I_{N}}\sum_{j=1}^{\tilde{\alpha}(m)}r^{\alpha(m)+j}r\mathcal{L}_{m}^{(1)}A_{mj}\mathrm{e}^{imx}\bigg{\|}_{H_{ul}^{\theta}}\leq
Cr^{n}\sup_{\\{(m,j):\alpha(m)+j=N\\}}\|A_{mj}(\cdot,\bar{t})\|_{H_{ul}^{\theta+2}},$

where we require two extra degrees of regularity (one is needed to apply Lemma
A.1, the other is needed to control the $\partial_{\bar{x}}$ derivative in
$\mathcal{L}_{m}^{(1)}$). Similar arguments relying on the bounds obtained for
the modulation functions $A_{mj}$ in Section 5 show that

$\begin{split}\bigg{\|}x\mapsto\sum_{m\in
I_{N}}\sum_{j=1}^{\tilde{\alpha}(m)}r^{\alpha(m)+j}r^{2}\tilde{\mathcal{L}}_{m}^{(2)}A_{mj}\mathrm{e}^{imx}\bigg{\|}_{H_{ul}^{\theta}}&\leq
Cr^{n}\sup_{\\{(m,j):\alpha(m)+j=N-1\\}}\|A_{mj}(\cdot,\bar{t})\|_{H_{ul}^{\theta+3}}\\\
&+Cr^{n+1}\sup_{\\{(m,j):\alpha(m)+j=N\\}}\|A_{mj}(\cdot,\bar{t})\|_{H_{ul}^{\theta+3}},\\\
\bigg{\|}x\mapsto\sum_{m\in
I_{N}}\sum_{j=1}^{\tilde{\alpha}(m)}r^{\alpha(m)+j}r^{3}\mathcal{L}_{m}^{(3)}A_{mj}\mathrm{e}^{imx}\bigg{\|}_{H_{ul}^{\theta}}&\leq
Cr^{n}\sup_{\\{(m,j):\alpha(m)+j=N-2\\}}\|A_{mj}(\cdot,\bar{t})\|_{H_{ul}^{\theta+4}}\\\
&+Cr^{n+1}\sup_{\\{(m,j):\alpha(m)+j=N-1\\}}\|A_{mj}(\cdot,\bar{t})\|_{H_{ul}^{\theta+4}}\\\
&+Cr^{n+2}\sup_{\\{(m,j):\alpha(m)+j=N\\}}\|A_{mj}(\cdot,\bar{t})\|_{H_{ul}^{\theta+4}},\\\
\bigg{\|}x\mapsto\sum_{m\in
I_{N}}\sum_{j=1}^{\tilde{\alpha}(m)}r^{\alpha(m)+j}r^{4}\mathcal{L}_{m}^{(4)}A_{mj}\mathrm{e}^{imx}\bigg{\|}_{H_{ul}^{\theta}}&\leq
Cr^{n}\sup_{\\{(m,j):\alpha(m)+j=N-3\\}}\|A_{mj}(\cdot,\bar{t})\|_{H_{ul}^{\theta+5}}\\\
&+Cr^{n+1}\sup_{\\{(m,j):\alpha(m)+j=N-2\\}}\|A_{mj}(\cdot,\bar{t})\|_{H_{ul}^{\theta+5}}\\\
&+Cr^{n+2}\sup_{\\{(m,j):\alpha(m)+j=N-1\\}}\|A_{mj}(\cdot,\bar{t})\|_{H_{ul}^{\theta+5}}\\\
&+Cr^{n+3}\sup_{\\{(m,j):\alpha(m)+j=N\\}}\|A_{mj}(\cdot,\bar{t})\|_{H_{ul}^{\theta+5}},\end{split}$
(160)

from which it follows that

$\begin{split}\bigg{\|}x\mapsto\sum_{m\in
I_{N}}\sum_{j=1}^{\tilde{\alpha}(m)}r^{\alpha(m)+j}\big{(}r\mathcal{L}_{m}^{(1)}+r^{2}\tilde{\mathcal{L}_{m}^{(2)}}+r^{3}\mathcal{L}_{m}^{(3)}&+r^{4}\mathcal{L}_{m}^{(4)}\big{)}A_{mj}\mathrm{e}^{imx}\bigg{\|}_{H_{ul}^{\theta}}\leq\\\
&Cr^{n}\sup_{\\{(m,j):N-3\leq\alpha(m)+j\leq
N\\}}\|A_{mj}(\cdot,\bar{t})\|_{H_{ul}^{\theta+5}}.\end{split}$

It remains to consider the third sum in (159). Recall that for each $m\in
I_{3N}\setminus I_{N}$, $b_{m}$ is a linear combination of terms of the form

$r^{\alpha(m_{1})+\alpha(m_{2})+\alpha(m_{3})+j_{1}+j_{2}+j_{3}}A_{m_{1}j_{1}}A_{m_{2}j_{2}}A_{m_{3}j_{3}},$

with $m_{i}\in I_{N}$ and $j_{i}\in\\{1,\ldots,\tilde{\alpha}(m_{i})\\}$ for
each $i=1,2,3$. Moreover,

$\alpha(m_{1})+\alpha(m_{2})+\alpha(m_{3})+j_{1}+j_{2}+j_{3}=\alpha(m)+j\geq|m|-1+1\geq
n,$

since $m=m_{1}+m_{2}+m_{3}\in I_{3N}\setminus I_{N}$ and $j\geq 1$. For
$l=1,2$ and additionally $l=3$ if $\mu(x)\equiv 0$ and the conditions of
Proposition 5.25 are satisfied, the results in Section 5 show that all
modulation functions satisfy $A_{mj,l}(\cdot,t_{l})\in H_{ul}^{\theta+1}$, for
all $t_{l}\in[0,T_{l}]$; we refer again to Lemmas 5.9, 5.17 and 5.24. Using
Lemmas A.1 and A.3 we therefore obtain

$\begin{split}\bigg{\|}x\mapsto\sum_{m\in I_{3N}\setminus
I_{N}}b_{m}(r_{l}(t_{l})x,t_{l})\mathrm{e}^{imx}\bigg{\|}_{H_{ul}^{\theta}}&\leq\sum_{m\in
I_{3N}\setminus I_{N}}\|x\mapsto
b_{m}(r_{l}(t_{l})x,t_{l})\mathrm{e}^{imx}\|_{H_{ul}^{\theta}}\\\ &\leq
Cr_{l}(t_{l})^{n}\sup_{m\in
I_{N},j\in\\{1,\ldots,\tilde{\alpha}(m)\\}}\|x\mapsto
A_{mj,l}(r_{l}(t_{l})x,t_{l})\mathrm{e}^{imx}\|_{H_{ul}^{\theta}}\\\ &\leq
Cr_{l}(t_{l})^{n}\sup_{m\in
I_{N},j\in\\{1,\ldots,\tilde{\alpha}(m)\\}}\|A_{mj,l}(\cdot,t_{l})\|_{H_{ul}^{\theta+1}},\end{split}$

for all $t_{l}\in[0,T_{l}]$, $l=1,2$ and also for $l=3$ if $\mu(x)\equiv 0$
and the conditions of Proposition 5.25 are satisfied.

Taking a triangle inequality in (159) and using the bounds for the second and
third sums derived above yields the bound in (157). If $\mu(x)\equiv 0$, then
$\nu_{m}=0$ for all $m\in I_{N}$ in (157). It also follows from the proof of
Lemmas 5.7 and 5.9 (see equations (89) and (95) in particular) that

$\|A_{mj,l}(\cdot,t_{l})\|_{H_{ul}^{\theta_{\alpha(m)+j,N}}}\leq
C\sup_{k=1,\ldots,j-2}\|A_{\pm
1k,l}(\cdot,t_{1})\|_{H_{ul}^{\theta_{k,N-2}}},$

for all $t_{l}\in[0,T_{l}]$ and $l=1,2,3$. Combining this with (157) yields
the bound in (158). ∎

######  Remark 6.3.

In Lemma 6.2 we require $\theta_{A}=1+3(n-3)+\theta$ as opposed to lesser
requirement that $\theta_{A}=1+3(N-3)+\theta=1+3(n-4)+\theta$, which is all
that was needed in order to describe the approximation dynamics in Lemmas 3.14
and 3.16. Greater regularity is needed to bound the residual because we need
to control the norms in (160); see also Remark 4.5.

###  6.3 Estimates for the nonlinear terms

Let $F_{l}(x,t_{l})$ denote the local representation of $F(x,\bar{t})$ in
chart $\mathcal{K}_{l}$, and define

$\tilde{T}_{l}:=\min\left\\{T_{l},\sup\left\\{t_{l}>0:\|\tilde{R}_{l}(\cdot,t_{l})\|_{H_{ul}^{\theta}}\leq\tilde{M}\right\\}\right\\}$

where $\tilde{M}>M\geq\|R^{\ast}\|_{H_{ul}^{\theta}}$, for each $l=1,2,3$. We
prove the following result.

######  Lemma 6.4.

Let $\theta_{A}=1+3(N-3)+\theta$ where $\theta>1/2$. We obtain the following
estimates in charts $\mathcal{K}_{l}$:

  1. (i)

In chart $\mathcal{K}_{1}$ we have

$\begin{split}\|F_{1}(\cdot,s_{1})\|_{H_{ul}^{\theta}}\leq
C\bigg{(}\|\psi_{1}(\cdot,s_{1})\|_{H_{ul}^{\theta}}^{2}+\frac{\rho_{in}^{(\beta-1)/2}}{(1-2\varepsilon_{1}^{\ast}s_{1})^{1/4}}\|\tilde{R}_{1}(\cdot,s_{1})\|_{H_{ul}^{\theta}}&\|\psi_{1}(\cdot,s_{1})\|_{H_{ul}^{\theta}}\\\
+\frac{\rho_{in}^{\beta-1}}{(1-2\varepsilon_{1}^{\ast}s_{1})^{1/2}}&\|\tilde{R}_{1}(\cdot,s_{1})\|_{H_{ul}^{\theta}}^{2}\bigg{)}\|\tilde{R}_{1}(\cdot,s_{1})\|_{H_{ul}^{\theta}},\end{split}$

for all $t_{1}\in[0,\tilde{T}_{1}]$.

  2. (ii)

In chart $\mathcal{K}_{2}$ we have

$\begin{split}\|F_{2}(\cdot,s_{2})\|_{H_{ul}^{\theta}}\leq
C\bigg{(}\|\psi_{2}(\cdot,s_{2})\|_{H_{ul}^{\theta}}^{2}+r_{2}^{\beta-1}\|\tilde{R}_{2}(\cdot,s_{2})\|_{H_{ul}^{\theta}}&\|\psi_{2}(\cdot,s_{2})\|_{H_{ul}^{\theta}}\\\
+r_{2}^{2\beta-2}&\|\tilde{R}_{2}(\cdot,s_{2})\|_{H_{ul}^{\theta}}^{2}\bigg{)}\|\tilde{R}_{2}(\cdot,s_{2})\|_{H_{ul}^{\theta}},\end{split}$

for all $t_{2}\in[0,\tilde{T}_{2}]$.

  3. (iii)

If $\mu(x)\equiv 0$ and the conditions of Proposition 5.25 are satisfied then
in chart $\mathcal{K}_{3}$ we have

$\begin{split}\|F_{3}(\cdot,s_{3})\|_{H_{ul}^{\theta}}\leq
C\bigg{(}\|\psi_{3}(\cdot,s_{3})\|_{H_{ul}^{\theta}}^{2}+\frac{{r_{3}^{\ast}}^{\beta-1}}{(1+2\zeta
s_{3})^{1/4}}\|\tilde{R}_{3}(\cdot,s_{3})\|_{H_{ul}^{\theta}}&\|\psi_{3}(\cdot,s_{3})\|_{H_{ul}^{\theta}}\\\
+\frac{{r_{3}^{\ast}}^{2\beta-2}}{(1+2\zeta
s_{3})^{1/2}}&\|\tilde{R}_{3}(\cdot,s_{3})\|_{H_{ul}^{\theta}}^{2}\bigg{)}\|\tilde{R}_{3}(\cdot,s_{3})\|_{H_{ul}^{\theta}},\end{split}$

for all $t_{3}\in[0,\tilde{T}_{3}]$.

###### Proof.

By Propositions 5.11, 5.18 and 5.25 we have that $\psi_{l}(\cdot,t_{l})\in
H_{ul}^{\theta}$ for all $t_{l}\in[0,\tilde{T}_{l}]$ (although only if
$\mu(x)\equiv 0$ and the conditions of Proposition 5.25 are satisfied in case
$l=3$). Since $H_{ul}^{\theta}$ is closed under multiplication and
$\tilde{R}_{l}(\cdot,t_{l})\in H_{ul}^{\theta}$ for all
$t_{l}\in[0,\tilde{T}_{l}]$ by the definition of $\tilde{T}_{l}$, it follows
that for any fixed integers $n_{1},n_{2}\geq 1$ there is a constant $C>0$ such
that

$\|\psi_{l}(\cdot,s_{l})^{n_{1}}\tilde{R}_{l}(\cdot,s_{l})^{n_{2}}\|_{H_{ul}^{\theta}}\leq
C\|\psi_{l}(\cdot,s_{l})\|_{H_{ul}^{\theta}}^{n_{1}}\|\tilde{R}_{l}(\cdot,s_{l})\|_{H_{ul}^{\theta}}^{n_{2}},\qquad
t_{l}\in[0,\tilde{T}_{l}].$

Thus we obtain the following after taking the norm of (156):

$\begin{split}\|F_{l}(\cdot,s_{l})\|_{H_{ul}^{\theta}}\leq
C\bigg{(}\|\psi_{l}(\cdot,s_{l})\|_{H_{ul}^{\theta}}^{2}+r_{l}(s_{l})^{\beta-1}\mathrm{e}^{I_{l}(s_{l})}\|\tilde{R}_{l}(\cdot,s_{l})\|_{H_{ul}^{\theta}}&\|\psi_{l}(\cdot,s_{l})\|_{H_{ul}^{\theta}}\\\
+r_{l}(s_{l})^{2\beta-2}\mathrm{e}^{2I_{l}(s_{l})}&\|\tilde{R}_{l}(\cdot,s_{l})\|_{H_{ul}^{\theta}}^{2}\bigg{)}\|\tilde{R}_{l}(\cdot,s_{l})\|_{H_{ul}^{\theta}},\end{split}$

where $I_{l}(s_{l})$ denotes the local representation of $I(\bar{s})$ in chart
$\mathcal{K}_{l}$. The expressions in Assertions (i), (ii) and (iii) follow
from the explicit solution formulae for $r_{l}(t_{l})$ (see Lemmas 5.2, 5.19
and recall that $r_{2}(t_{2})=r_{2}$) together with

$\mathrm{e}^{I_{1}(s_{1})}=\frac{1}{(1-2\varepsilon_{1}^{\ast}s_{1})^{\beta/4}},\qquad\mathrm{e}^{I_{2}(s_{2})}=1,\qquad\mathrm{e}^{I_{3}(s_{3})}=\frac{1}{(1+2\zeta
s_{3})^{\beta/4}}.$

∎

###  6.4 Proof of Theorem 3.20

In this section we use the estimates derived in Sections 6.1, 6.2 and 6.3
above to prove Theorem 3.20. We begin by deriving estimates for the (weighted)
error function $R_{l}$ in local coordinate charts $\mathcal{K}_{l}$.

####  6.4.1 Error estimates in charts

Using (155) and the results obtained above we can estimate the error in each
chart $\mathcal{K}_{l}$.

######  Proposition 6.5.

Let $\theta_{A}=1+3(n-3)+\theta$ where $\theta\geq 1$, and $\beta=n-2$. Then
for $l=1,2$ we have that $R_{l}(\cdot,t_{l})\in H_{ul}^{\theta}$ for all
$t_{l}\in[0,T_{l}]$, with

$\|R_{1}(\cdot,T_{1})\|_{H_{ul}^{\theta}}\leq
C\left(1+\|R_{1}^{\ast}\|_{H_{ul}^{\theta}}\mathrm{e}^{-\tilde{c}_{1}/2\varepsilon_{1}^{\ast}}\right),\qquad\|R_{2}(\cdot,T_{2})\|_{H_{ul}^{\theta}}\leq
C\left(1+\|R_{2}^{\ast}\|_{H_{ul}^{\theta}}\right),$

where $\tilde{c}_{1}=c_{1}-2C\tilde{C}(\tilde{K},\tilde{M})\in(0,1)$ can be
chosen arbitrarily close to $1$ by fixing $K,M>0$ sufficiently small. If
additionally $\mu(x)\equiv 0$ then the following assertions are true:

  1. (i)

In chart $\mathcal{K}_{1}$ we have

$\|R_{1}(\cdot,T_{1})\|_{H_{ul}^{\theta}}\leq
C\left(\|R_{1}^{\ast}\|_{H_{ul}^{\theta}}+\max_{k=1,\ldots,N-2}\|A_{k,1}^{\ast}\|_{H_{ul}^{\theta_{k,N-2}}}\right)\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}.$

  2. (ii)

In chart $\mathcal{K}_{2}$ we have

$\|R_{2}(\cdot,T_{2})\|_{H_{ul}^{\theta}}\leq
C\left(\|R_{2}^{\ast}\|_{H_{ul}^{\theta}}+\max_{k=1,\ldots,N-2}\|A_{k,2}^{\ast}\|_{H_{ul}^{\theta_{k,N-2}}}\right).$

  3. (iii)

If the conditions of Proposition 5.25 are satisfied, in chart
$\mathcal{K}_{3}$ we have that $R_{3}(\cdot,t_{3})\in H_{ul}^{\theta}$ for all
$t_{3}\in[0,T_{3}]$ and

$\|R_{3}(\cdot,T_{3})\|_{H_{ul}^{\theta}}\leq
C{r_{3}^{\ast}}^{n-2}\left(\|R_{3}^{\ast}\|_{H_{ul}^{\theta}}+\mathrm{e}^{-(\tilde{\sigma}_{0}-\tilde{\sigma})\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}}\right)\mathrm{e}^{\tilde{c}_{3}\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}},$

where $\tilde{c}_{3}:=c_{3}+2C\tilde{C}(\tilde{K},\tilde{M})>1$ can be set
arbitrarily close to $1$ by fixing sufficiently small $K,M>0$.

###### Proof.

In chart $\mathcal{K}_{1}$ we have

$\begin{split}\tilde{R}_{1}(\cdot,t_{1})=\mathrm{e}^{t_{1}\Lambda_{1}}R_{1}^{\ast}&+\int_{0}^{t_{1}}\mathrm{e}^{(t_{1}-s_{1})\Lambda_{1}}F_{1}(\cdot,s_{1})ds_{1}\\\
&+\int_{0}^{t_{1}}\mathrm{e}^{(t_{1}-s_{1})\Lambda_{1}}r_{1}(s_{1})^{-\beta-2}\mathrm{e}^{-I_{1}(s_{1})}\textup{Res}(r_{1}(s_{1})\psi_{1}(\cdot,s_{1}))ds_{1},\end{split}$

for $t_{1}\in[0,\tilde{T}_{1}]$. Using the bound for the evolution family in
Lemma 6.1, we obtain

$\begin{split}\|\tilde{R}_{1}(\cdot,t_{1})\|_{H_{ul}^{\theta}}\leq
C\mathrm{e}^{-c_{1}t_{1}}\|R_{1}^{\ast}\|_{H_{ul}^{\theta}}&+C\int_{0}^{t_{1}}\mathrm{e}^{-c_{1}(t_{1}-s_{1})}\|F_{1}(\cdot,s_{1})\|_{H_{ul}^{\theta}}ds_{1}\\\
+\
C\int_{0}^{t_{1}}\mathrm{e}^{-c_{1}(t_{1}-s_{1})}&r_{1}(s_{1})^{-\beta-2}\mathrm{e}^{-I_{1}(s_{1})}\|\textup{Res}(r_{1}(s_{1})\psi_{1}(\cdot,s_{1}))\|_{H_{ul}^{\theta}}ds_{1},\end{split}$
(161)

for all $t_{1}\in[0,\tilde{T}_{1}]$. The right-most integral can be bounded
using Lemma 6.2 and the fact that
$r_{1}(s_{1})^{-\beta-2}\mathrm{e}^{-I_{1}(s_{1})}=1/(\rho_{in}^{\beta/2}r_{1}(s_{1})^{2})$.
We obtain

$\begin{split}\int_{0}^{t_{1}}\mathrm{e}^{-c_{1}(t_{1}-s_{1})}r_{1}(s_{1})^{-\beta-2}\mathrm{e}^{-I_{1}(s_{1})}&\|\textup{Res}(r_{1}(s_{1})\psi_{1}(\cdot,s_{1}))\|_{H_{ul}^{\theta}}ds_{1}\\\
\leq\frac{C}{\rho_{in}^{\beta/2}}\int_{0}^{t_{1}}\mathrm{e}^{-c_{1}(t_{1}-s_{1})}&r_{1}(s_{1})^{n-2}\left(\delta_{n,4}\max_{m\in
I_{N}}|\nu_{m}|+\sup_{m,j}\|A_{mj,1}(\cdot,s_{1})\|_{H_{ul}^{\theta_{\alpha(m)+j,N}}}\right)ds_{1},\end{split}$
(162)

for all $t_{1}\in[0,T_{1}]$, where the supremum is taken with respect to
modulation functions with $m\in I_{N}$ and
$j\in\\{1,\ldots,\tilde{\alpha}(m)\\}$. By Lemma 5.9, all such modulation
functions in chart $\mathcal{K}_{1}$ satisfy $A_{mj,1}(\cdot,s_{1})\in
H_{ul}^{\theta+1}$ for all $s_{1}\in[0,T_{1}]$. It follows that

$\begin{split}\frac{C}{\rho_{in}^{\beta/2}}\int_{0}^{t_{1}}\mathrm{e}^{-c_{1}(t_{1}-s_{1})}r_{1}(s_{1})^{n-2}\left(\delta_{n,4}\max_{m\in
I_{N}}|\nu_{m}|+\sup_{m,j}\|A_{mj,1}(\cdot,s_{1})\|_{H_{ul}^{\theta_{\alpha(m)+j,N}}}\right)ds_{1}&\leq\\\
\frac{C}{\rho_{in}^{\beta/2}}\int_{0}^{t_{1}}\mathrm{e}^{-c_{1}(t_{1}-s_{1})}&r_{1}(s_{1})^{n-2}ds_{1}.\end{split}$

The right-most integral can be evaluated directly using the expression for
$r_{1}(s_{1})$ in Lemma 5.2 together with the integral identity and gamma
function properties in Lemma A.5. We obtain

$\begin{split}\mathrm{e}^{-c_{1}t_{1}}\int_{0}^{t_{1}}\mathrm{e}^{c_{1}s_{1}}r_{1}(s_{1})^{n-2}ds_{1}&=C\rho_{in}^{n-2}\mathrm{e}^{-c_{1}(t_{1}-1/2\varepsilon_{1}^{\ast})}{\varepsilon_{1}^{\ast}}^{(n-2)/4}\Gamma\left(\frac{n+2}{4},c_{1}\left(\frac{1}{2\varepsilon_{1}^{\ast}}-s_{1}\right)\right)\bigg{|}_{0}^{t_{1}}\\\
&\leq C\rho_{in}^{n-2}{\varepsilon_{1}^{\ast}}^{(n-2)/4},\end{split}$

for all $t_{1}\in[0,T_{1}]$. Combining this with (161) leads to

$\|\tilde{R}_{1}(\cdot,t_{1})\|_{H_{ul}^{\theta}}\leq
C\left(\mathrm{e}^{-c_{1}t_{1}}\|R_{1}^{\ast}\|_{H_{ul}^{\theta}}+\rho_{in}^{n-\beta-2}{\varepsilon_{1}^{\ast}}^{(n-2)/4}\right)+C\int_{0}^{t_{1}}\mathrm{e}^{-c_{1}(t_{1}-s_{1})}\|F_{1}(\cdot,s_{1})\|_{H_{ul}^{\theta}}ds_{1},$

for $t_{1}\in[0,\tilde{T}_{1}]$ (at this point the estimates in Lemma 6.4 are
only shown to hold for $t_{1}\in[0,\tilde{T}_{1}]$). For our present purposes
it suffices to note that by Lemma 6.4 Assertion (i) we have

$\|F_{1}(\cdot,s_{1})\|_{H_{ul}^{\theta}}\leq\tilde{C}(\tilde{M},\tilde{K})\|\tilde{R}_{1}(\cdot,s_{1})\|_{H_{ul}^{\theta}}$

for all $s_{1}\in[0,\tilde{T}_{1}]$, where the constant
$\tilde{C}(\tilde{K},\tilde{M})>0$ can be chosen arbitrarily small by choosing
$K,M>0$ sufficiently small (we only require that $\tilde{K}>K$ and
$\tilde{M}>M$). Thus

$\|\tilde{R}_{1}(\cdot,t_{1})\|_{H_{ul}^{\theta}}\leq
C\left(\mathrm{e}^{-c_{1}t_{1}}\|R_{1}^{\ast}\|_{H_{ul}^{\theta}}+\rho_{in}^{n-\beta-2}{\varepsilon_{1}^{\ast}}^{(n-2)/4}\right)+C\tilde{C}(\tilde{K},\tilde{M})\int_{0}^{t_{1}}\mathrm{e}^{-c_{1}(t_{1}-s_{1})}\|\tilde{R}_{1}(\cdot,s_{1})\|_{H_{ul}^{\theta}}ds_{1},$

for all $t_{1}\in[0,\tilde{T}_{1}]$. Applying the Grönwall inequality in Lemma
A.4 yields

$\begin{split}\|\tilde{R}_{1}(\cdot,t_{1})\|_{H_{ul}^{\theta}}&\leq
C\left(\rho_{in}^{n-\beta-2}{\varepsilon_{1}^{\ast}}^{(n-2)/4}+\|R_{1}^{\ast}\|_{H_{ul}^{\theta}}\right)\mathrm{e}^{-(c_{1}-2C\tilde{C}(\tilde{K},\tilde{M}))t_{1}}\\\
&+{\varepsilon_{1}^{\ast}}^{(n-2)/4}\rho_{in}^{n-\beta-2}\int_{0}^{t_{1}}\mathrm{e}^{-(c_{1}-2C\tilde{C}(\tilde{K},\tilde{M}))(t_{1}-s_{1})}ds_{1}\\\
&=C\left(\rho_{in}^{n-\beta-2}{\varepsilon_{1}^{\ast}}^{(n-2)/4}+\|R_{1}^{\ast}\|_{H_{ul}^{\theta}}\right)\mathrm{e}^{-(c_{1}-2\tilde{C})t_{1}}\\\
&+C\rho_{in}^{n-\beta-2}{\varepsilon_{1}^{\ast}}^{(n-2)/4}\left(\frac{1-\mathrm{e}^{-(c_{1}-2\tilde{C})t_{1}}}{c_{1}-2C\tilde{C}(\tilde{K},\tilde{M})}\right),\end{split}$

for all $t_{1}\in[0,\tilde{T}_{1}]$. Choosing $K,M>0$ sufficiently small
allows us to choose $\tilde{C}(\tilde{K},\tilde{M})$ such that
$c_{1}-2C\tilde{C}(\tilde{K},\tilde{M})\in(0,1)$. In this case the right-hand
side is finite for all $t_{1}\in[0,T_{1}]$, such that $\tilde{T}_{1}=T_{1}$.
Multiplying through by the integrating factor yields

$\begin{split}\|R_{1}(\cdot,t_{1})\|_{H_{ul}^{\theta}}&=\mathrm{e}^{I_{1}(t_{1})}\|\tilde{R}_{1}(\cdot,t_{1})\|_{H_{ul}^{\theta}}\\\
&\leq\frac{C}{(1-2\varepsilon_{1}^{\ast}t_{1})^{\beta/4}}\left(\|R_{1}^{\ast}\|_{H_{ul}^{\theta}}\mathrm{e}^{-\tilde{c}_{1}t_{1}}+\rho_{in}^{n-\beta-2}{\varepsilon_{1}^{\ast}}^{(n-2)/4}\right),\end{split}$
(163)

where $\tilde{c}_{1}=c_{1}-2C\tilde{C}(\tilde{K},\tilde{M})\in(0,1)$. Since
$(1-2\varepsilon_{1}^{\ast}t_{1})^{-\beta/4}\in[1,(\zeta/\varepsilon_{1}^{\ast})^{\beta/4}]$
for $t_{1}\in[0,T_{1}]$, (163) is bounded above by a constant if we set
$\beta=n-2$, as required. Subsequent evaluation at $t_{1}=T_{1}$ yields the
desired bound.

In order get a better bound when $\mu(x)\equiv 0$, we start again with (162)
except with $\nu_{m}=0$ for all $m\in I_{N}$. We use the improved bound (6.2)
in Lemma 6.2, which applies when $\mu(x)\equiv 0$. Substituting this for
right-most bound into (162) leads to the upper bound

$\begin{split}C\mathrm{e}^{-\tilde{c}t_{1}}\max_{k=1,\ldots,N-2}\|A_{k,1}^{\ast}\|_{H_{ul}^{\theta_{k,N-2}}}\int_{0}^{t_{1}}\mathrm{e}^{(c_{1}-\tilde{c})s_{1}}(1-2\varepsilon_{1}^{\ast}s_{1})^{(n-3)/4}ds_{1}&\leq\\\
C\mathrm{e}^{-\tilde{c}t_{1}}&\max_{k=1,\ldots,N-2}\|A_{k,1}^{\ast}\|_{H_{ul}^{\theta_{k,N-2}}},\end{split}$

which we obtain by using $(1-2\varepsilon_{1}^{\ast}s_{1})^{(n-3)/4}\leq 1$
and setting $c_{1}=\tilde{c}$. Thus we have shown that

$\begin{split}\|\tilde{R}_{1}(\cdot,t_{1})\|_{H_{ul}^{\theta}}\leq
C\mathrm{e}^{-c_{1}t_{1}}&\left(\|R_{1}^{\ast}\|_{H_{ul}^{\theta}}+\max_{k=1,\ldots,N-2}\|A_{k,1}^{\ast}\|_{H_{ul}^{\theta_{k,N-2}}}\right)\\\
&\qquad\qquad+C\tilde{C}(\tilde{K},\tilde{M})\int_{0}^{t_{1}}\mathrm{e}^{-c_{1}(t_{1}-s_{1})}\|\tilde{R}_{1}(\cdot,s_{1})\|_{H_{ul}^{\theta}}ds_{1}.\end{split}$

Applying the Grönwall inequality in Lemma A.4 gives

$\|\tilde{R}_{1}(\cdot,t_{1})\|_{H_{ul}^{\theta}}\leq
C\left(\|R_{1}^{\ast}\|_{H_{ul}^{\theta}}+\max_{k=1,\ldots,N-2}\|A_{k,1}^{\ast}\|_{H_{ul}^{\theta_{k,N-2}}}\right)\mathrm{e}^{-(c_{1}-2C\tilde{C}(\tilde{K},\tilde{M}))t_{1}}.$

Multiplying through by the integrating factor and evaluating at $t_{1}=T_{1}$
we obtain

$\|R_{1}(\cdot,T_{1})\|_{H_{ul}^{\theta}}\leq
C\left(\|R_{1}^{\ast}\|_{H_{ul}^{\theta}}+\max_{k=1,\ldots,N-2}\|A_{k,1}^{\ast}\|_{H_{ul}^{\theta_{k,N-2}}}\right)\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}},$

as required.

In chart $\mathcal{K}_{2}$ we have

$R_{2}(\cdot,t_{2})=\mathrm{e}^{t_{2}\Lambda_{2}}R_{2}^{\ast}+\int_{0}^{t_{2}}\mathrm{e}^{(t_{2}-s_{2})\Lambda_{2}}F_{2}(\cdot,s_{2})ds_{2}\\\
+\int_{0}^{t_{2}}\mathrm{e}^{(t_{2}-s_{2})\Lambda_{2}}r_{2}^{-\beta-2}\textup{Res}(r_{2}\psi_{2}(\cdot,s_{2}))ds_{2},$
(164)

for $t_{2}\in[0,\tilde{T}_{2}]$ (recall that $\mathrm{e}^{I_{2}(t_{2})}=1$ so
that we can work directly with $R_{2}=\tilde{R}_{2}$). Using the bound for the
evolution family in Lemma 6.1 and the bound for the residual in Lemma 6.2, we
obtain

$\begin{split}\|R_{2}(\cdot,t_{2})\|_{H_{ul}^{\theta}}\leq&C\|R_{2}^{\ast}\|_{H_{ul}^{\theta}}+C\int_{0}^{t_{2}}\|F_{2}(\cdot,s_{2})\|_{H_{ul}^{\theta}}ds_{2}\\\
&+Cr_{2}^{n-\beta-2}\int_{0}^{t_{2}}\left(\delta_{n,4}\max_{m\in
I_{N}}|\nu_{m}|+\sup_{m,j}\|A_{mj,2}(\cdot,s_{2})\|_{H_{ul}^{\theta_{\alpha(m)+j,N}}}\right)ds_{2}.\end{split}$

If we set $\beta=n-2$ and use Lemma 6.4 together with the fact that each
$A_{mj,2}(\cdot,s_{2})\in H_{ul}^{\theta_{\alpha(m)+j,N}}$ for all
$s_{2}\in[0,T_{2}]$ (this follows from Lemma 5.16), we obtain

$\|R_{2}(\cdot,t_{2})\|_{H_{ul}^{\theta}}\leq
C\left(1+\|R_{2}^{\ast}\|_{H_{ul}^{\theta}}\right)+C\tilde{C}(\tilde{M},\tilde{K})\left(1+r_{2}^{n-3}+r_{2}^{2(n-3)}\right)\int_{0}^{t_{2}}\|R_{2}(\cdot,s_{2})\|_{H_{ul}^{\theta}}ds_{2},$

for all $t_{2}\in[0,\tilde{T}_{2}]$, where the constant
$\tilde{C}(\tilde{M},\tilde{K})>0$ can again be chosen as small as necessary
by choosing sufficiently small $K,M>0$. Applying Grönwall’s inequality yields

$\|R_{2}(\cdot,t_{2})\|_{H_{ul}^{\theta}}\leq
C\left(1+\|R_{2}^{\ast}\|_{H_{ul}^{\theta}}\right)\exp\left(C\tilde{C}(\tilde{M},\tilde{K})\left(1+r_{2}^{n-3}+r_{2}^{2(n-3)}\right)t_{2}\right).$

It follows that $R_{2}(\cdot,t_{2})\in H_{ul}^{\theta}$ for all
$t_{2}\in[0,T_{2}]$ (recall that $T_{2}$ is finite).

Similarly to the $\mathcal{K}_{1}$ analysis above, the bound on the residual
integral can be improved if $\mu(x)\equiv 0$ because we can use the bound
(158) in Lemma 6.2. Using this result to improve the bound on the right-most
integral in (164) leads to

$\begin{split}\|R_{2}(\cdot,t_{2})\|_{H_{ul}^{\theta}}\leq
C&\left(\|R_{2}^{\ast}\|_{H_{ul}^{\theta}}+\max_{k=1,\ldots,N-2}\|A_{k,2}^{\ast}\|_{H_{ul}^{\theta_{k,N-2}}}\right)+\\\
&\qquad\qquad
C\tilde{C}(\tilde{M},\tilde{K})\left(1+r_{2}^{n-3}+r_{2}^{2(n-3)}\right)\int_{0}^{t_{2}}\|R_{2}(\cdot,s_{2})\|_{H_{ul}^{\theta}}ds_{2},\end{split}$

which holds for all $t_{2}\in[0,\tilde{T}_{2}]$. The bound in Assertion (ii)
follows after an application of Grönwall’s inequality.

It remains to prove Assertion (iii). We work in chart $\mathcal{K}_{3}$, set
$\mu(x)\equiv 0$ and assume that the hypotheses of Proposition 5.25 are
satisfied. We have

$\begin{split}\tilde{R}_{3}(\cdot,t_{3})=\mathrm{e}^{t_{3}\Lambda_{3}}R_{3}^{\ast}&+\int_{0}^{t_{3}}\mathrm{e}^{(t_{3}-s_{3})\Lambda_{3}}F_{3}(\cdot,s_{3})ds_{3}\\\
&+\int_{0}^{t_{3}}\mathrm{e}^{(t_{3}-s_{3})\Lambda_{3}}r_{3}(s_{3})^{-\beta-2}\mathrm{e}^{-I_{3}(s_{3})}\textup{Res}(r_{3}(s_{3})\psi_{3}(\cdot,s_{3}))ds_{3},\end{split}$

for $t_{3}\in[0,\tilde{T}_{3}]$. Using the bound for the evolution family in
Lemma 6.1, we obtain

$\begin{split}\|\tilde{R}_{3}(\cdot,t_{3})\|_{H_{ul}^{\theta}}\leq
C\mathrm{e}^{c_{3}t_{3}}\|R_{3}^{\ast}\|_{H_{ul}^{\theta}}&+C\int_{0}^{t_{3}}\mathrm{e}^{c_{3}(t_{3}-s_{3})}\|F_{3}(\cdot,s_{3})\|_{H_{ul}^{\theta}}ds_{3}\\\
+C\int_{0}^{t_{3}}\mathrm{e}^{c_{3}(t_{3}-s_{3})}&r_{3}(s_{3})^{-\beta-2}\mathrm{e}^{-I_{3}(s_{3})}\|\textup{Res}(r_{3}(s_{3})\psi_{3}(\cdot,s_{3}))\|_{H_{ul}^{\theta}}ds_{3},\end{split}$

for all $t_{3}\in[0,\tilde{T}_{3}]$. The right-most integral can be bounded
using Lemma 6.2 and the fact that
$r_{3}(s_{1})^{-\beta-2}\mathrm{e}^{-I_{3}(s_{1})}=1/({r_{3}^{\ast}}^{\beta}r_{3}(s_{3})^{2})$.
We obtain

$\begin{split}\int_{0}^{t_{3}}\mathrm{e}^{c_{3}(t_{3}-s_{3})}r_{3}(s_{3})^{-\beta-2}\mathrm{e}^{-I_{3}(s_{3})}&\|\textup{Res}(r_{3}(s_{3})\psi_{3}(\cdot,s_{3}))\|_{H_{ul}^{\theta}}ds_{3}\\\
\leq&\frac{C}{{r_{3}^{\ast}}^{\beta}}\int_{0}^{t_{3}}\mathrm{e}^{c_{3}(t_{3}-s_{3})}r_{3}(s_{3})^{n-2}\sup_{k=1,\ldots,N-2}\|A_{k,3}(\cdot,s_{3})\|_{H_{ul}^{\theta_{k,N-2}}}ds_{3},\end{split}$

for all $t_{1}\in[0,T_{3}]$. Under the assumptions of Proposition 5.25 we have
that

$\sup_{k=1,\ldots,N-2}\|A_{k,3}(\cdot,s_{3})\|_{H_{ul}^{\theta+1}}\leq
C\mathrm{e}^{-(\tilde{\sigma}_{0}-\tilde{\sigma})\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}},$

for all $t_{3}\in[0,T_{3}]$. Using this and Lemma A.5, we obtain

$\begin{split}\frac{C}{{r_{3}^{\ast}}^{\beta}}&\int_{0}^{t_{3}}\mathrm{e}^{c_{3}(t_{3}-s_{3})}r_{3}(s_{3})^{n-2}\sup_{k=1,\ldots,N-2}\|A_{k,3}(\cdot,s_{3})\|_{H_{ul}^{\theta_{k,N-2}}}ds_{3}\\\
&\leq
C{r_{3}^{\ast}}^{n-\beta-2}\mathrm{e}^{-(\tilde{\sigma}_{0}-\tilde{\sigma})\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}}\mathrm{e}^{c_{3}t_{3}}\int_{0}^{t_{3}}\mathrm{e}^{-c_{3}s_{3}}(1+2\zeta
s_{3})^{(n-2)/4}ds_{3}\\\ &\leq
C{r_{3}^{\ast}}^{n-\beta-2}\mathrm{e}^{-(\tilde{\sigma}_{0}-\tilde{\sigma})\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}}\mathrm{e}^{c_{3}t_{3}}\Gamma\left(\frac{n+2}{4},c_{3}\left(\frac{1}{2\zeta}+s_{3}\right)\right)\bigg{|}_{0}^{t_{3}}\\\
&\leq
C\mathrm{e}^{-(\tilde{\sigma}_{0}-\tilde{\sigma})\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}}\mathrm{e}^{c_{3}t_{3}},\end{split}$

where we set $\beta=n-2$ and used the gamma function asymptotics (166) in
Appendix A to derive the last inequality. We shall also use

$\|F_{3}(\cdot,s_{3})\|_{H_{ul}^{\theta}}\leq\tilde{C}(\tilde{K},\tilde{M})\|\tilde{R}_{3}(\cdot,s_{3})\|_{H_{ul}^{\theta}},\qquad
t_{3}\in[0,\tilde{T}_{3}],$

which follows from Lemma 6.4. Applying the Grönwall inequality in Lemma A.4 to

$\|\tilde{R}_{3}(\cdot,t_{3})\|_{H_{ul}^{\theta}}\leq
C\mathrm{e}^{c_{3}t_{3}}\left(\|R_{3}^{\ast}\|_{H_{ul}^{\theta}}+\mathrm{e}^{-(\tilde{\sigma}_{0}-\tilde{\sigma})\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}}\right)+C\tilde{C}(\tilde{K},\tilde{M})\int_{0}^{t_{3}}\mathrm{e}^{c_{3}(t_{3}-s_{3})}\|\tilde{R}_{3}(\cdot,s_{3})\|_{H_{ul}^{\theta}}ds_{3},$

yields the estimate

$\|\tilde{R}_{3}(\cdot,t_{3})\|_{H_{ul}^{\theta}}\leq
C\left(\|R_{3}^{\ast}\|_{H_{ul}^{\theta}}+\mathrm{e}^{-(\tilde{\sigma}_{0}-\tilde{\sigma})\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}}\right)\mathrm{e}^{(c_{3}+2C\tilde{C}(\tilde{K},\tilde{M}))t_{3}},$

for all $t_{3}\in[0,\tilde{T}_{3}]$. Multiplying through by the integrating
factor gives

$\|R_{3}(\cdot,t_{3})\|_{H_{ul}^{\theta}}\leq\frac{C}{(1+2\zeta
t_{3})^{(n-2)/4}}\left(\|R_{3}^{\ast}\|_{H_{ul}^{\theta}}+\mathrm{e}^{-(\tilde{\sigma}_{0}-\tilde{\sigma})\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}}\right)\mathrm{e}^{(c_{3}+2C\tilde{C}(\tilde{K},\tilde{M}))t_{3}}.$

Evaluating this at $t_{3}=T_{3}$ yields the estimate in Assertion (iii). ∎

####  6.4.2 Proof of Theorem 3.20

We are now in a position to prove Theorem 3.20. The fact that
$u(\cdot,t)-\Psi(\cdot,t)\in[0,\widehat{T}]$, where $\widehat{T}=T_{mid}$ for
general $\mu(x)$ and $\widehat{T}=T$ if $\mu(x)\equiv 0$, follows from the
definition of the weighted error function $R$ in (155) and the fact that
$R_{l}(\cdot,t_{l})\in H_{ul}^{\theta}$ for all $t_{l}\in[0,T_{l}]$ by
Proposition 6.5, for each $l=1,2,3$ (assuming the additional conditions
including $\mu(x)\equiv 0$ are satisfied in case $l=3$).

In order to prove the bounds in Assertions (i) and (ii), we need to relate
estimates for the (weighted) error functions $R_{l}$ in different charts
$\mathcal{K}_{l}$. This can be done with the following change of coordinate
transformations, which are derived from the form of the maps $\kappa_{ij}$ in
(46):

$R_{1}=(-v_{2})^{-\beta/2}R_{2},\ v_{2}<0,\qquad
R_{2}=\varepsilon_{3}^{-\beta/4}R_{3},\ \varepsilon_{3}>0.$

In particular, we have

$R_{1}(\cdot,T_{1})=\zeta^{-\beta/4}R_{2}^{\ast},\qquad
R_{2}(\cdot,T_{2})=\zeta^{-\beta/4}R_{3}^{\ast}.$

We start with the general case where $\mu(x)$ is given by (5), i.e. we first
prove Assertion (i). We need to consider the size of the (weighted) error
function $R$ as it evolves forward in time from an initial condition in
$\Sigma_{1}^{in}$ in chart $\mathcal{K}_{1}$ up to the exit section
$\Sigma^{out}_{2}$ in chart $\mathcal{K}_{2}$. At
$\Sigma_{2}^{in}=\kappa_{12}(\Sigma_{1}^{out})$ we have

$\|R_{2}^{\ast}\|_{H_{ul}^{\theta}}=\zeta^{\beta/4}\|R_{1}(\cdot,T_{1})\|_{H_{ul}^{\theta}}\leq
C\left(1+\|R_{1}^{\ast}\|_{H_{ul}^{\theta}}\mathrm{e}^{-\tilde{c}_{1}/2\varepsilon_{1}^{\ast}}\right),$

and therefore

$\|R_{2}(\cdot,T_{2})\|_{H_{ul}^{\theta}}\leq
C\left(1+\|R_{1}^{\ast}\|_{H_{ul}^{\theta}}\mathrm{e}^{-\tilde{c}_{1}/2\varepsilon_{1}^{\ast}}\right),$

by Proposition 6.5. This implies that

$r_{2}^{\beta}\|R_{2}(\cdot,T_{2})\|_{H_{ul}^{\theta}}=\|u(x,T_{mid})-\Psi(x,T_{mid})\|_{H_{ul}^{\theta}}\leq
Cr_{2}^{\beta}\left(1+\|R_{1}^{\ast}\|_{H_{ul}^{\theta}}\mathrm{e}^{-\tilde{c}_{1}/2\varepsilon_{1}^{\ast}}\right).$

Applying the blow-down transformations
$\varepsilon_{1}^{\ast}=\varepsilon/\rho_{in}^{2}$, $r_{2}=\varepsilon^{1/4}$
and setting $\beta=n-2$ yields

$\|u(\cdot,T_{mid})-\Psi(\cdot,T_{mid})\|_{H_{ul}^{\theta}}\leq
C\varepsilon^{(n-2)/4}\left(1+\|R_{1}^{\ast}\|_{H_{ul}^{\theta}}\mathrm{e}^{-\tilde{c}_{1}/2\varepsilon_{1}^{\ast}}\right)\leq
C\varepsilon^{(n-2)/4},$

thereby proving Assertion (i) in Theorem 3.20.

It remains to prove Assertion (ii), for which we set $\mu(x)\equiv 0$. In this
case, the error is exponentially small in chart $\mathcal{K}_{1}$, and
Proposition 6.5 implies the following at bound at
$\Sigma_{2}^{in}=\kappa_{12}(\Sigma_{1}^{out})$:

$\|R_{2}^{\ast}\|_{H_{ul}^{\theta}}=\zeta^{\beta/4}\|R_{1}(\cdot,T_{1})\|_{H_{ul}^{\theta}}\leq
C\left(\|R_{1}^{\ast}\|_{H_{ul}^{\theta}}+\max_{k=1,\ldots,N-2}\|A_{k,1}^{\ast}\|_{H_{ul}^{\theta_{k,N-2}}}\right)\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}\leq
C\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}.$

Applying Proposition 6.5 again, this time at
$\Sigma_{3}^{in}=\kappa_{23}(\Sigma_{2}^{out})$, we obtain

$\|R_{3}^{\ast}\|_{H_{ul}^{\theta}}=\zeta^{\beta/4}\|R_{2}(\cdot,T_{2})\|_{H_{ul}^{\theta}}\leq
C\left(\|R_{2}^{\ast}\|_{H_{ul}^{\theta}}+\max_{k=1,\ldots,N-2}\|A_{k,2}^{\ast}\|_{H_{ul}^{\theta_{k,N-2}}}\right).$

Using the bound for $\|R_{2}^{\ast}\|_{H_{ul}^{\theta}}$ above and the bounds
for $\|A_{k,2}^{\ast}\|_{H_{ul}^{\theta}}$ in (148), which apply when
$\mu(x)\equiv 0$, we have that

$\|R_{3}^{\ast}\|_{H_{ul}^{\theta}}\leq
C\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}.$

Hence, Assertion (iii) of Proposition 6.5 implies that

$\begin{split}\|R_{3}(\cdot,t_{3})\|_{H_{ul}^{\theta}}&\leq
C{r_{3}^{\ast}}^{n-2}\left(\|R_{3}^{\ast}\|_{H_{ul}^{\theta}}+\mathrm{e}^{-(\tilde{\sigma}_{0}-\tilde{\sigma})\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}}\right)\mathrm{e}^{(c_{3}+2C\tilde{C}(\tilde{K},\tilde{M}))t_{3}}\\\
&\leq
C{r_{3}^{\ast}}^{n-2}\left(\mathrm{e}^{-\kappa/2\varepsilon_{1}^{\ast}}+\mathrm{e}^{-(\tilde{\sigma}_{0}-\tilde{\sigma})\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}}\right)\mathrm{e}^{(c_{3}+2C\tilde{C}(\tilde{K},\tilde{M}))t_{3}}.\end{split}$

We recall from the proof of Lemma 3.16 in Section 5.3 that the optimal choice
for $\tilde{\sigma}_{0}>\tilde{\sigma}>1$ is
$\tilde{\sigma}_{0}\rho_{out}^{2}/2\zeta{r_{3}^{\ast}}^{4}=\kappa/2\varepsilon_{1}^{\ast}$.
Applying this choice and $\varepsilon_{1}^{\ast}=\varepsilon/\rho_{in}^{2}$
leads to

$\|R_{3}(\cdot,T_{3})\|_{H_{ul}^{\theta}}\leq
C\exp\left(-\frac{\kappa}{2\varepsilon}\left(\rho_{in}^{2}-\frac{\tilde{\sigma}}{\kappa}\rho_{out}^{2}-\frac{\tilde{c}_{3}}{\kappa}\rho_{out}^{2}\right)\right),$

where $\tilde{c}_{3}:=c_{3}+2C\tilde{C}(\tilde{K},\tilde{M})>c_{3}>1$ can be
chosen arbitrarily close to $c_{3}$ (and thus to $1$) if $K,M>0$ are chosen
sufficiently small. Since the same is true of $\tilde{\sigma}$, we can choose
$\tilde{c}_{3}=\tilde{\sigma}$, leading to

$\|R_{3}(\cdot,t_{3})\|_{H_{ul}^{\theta}}\leq
C\exp\left(-\frac{\kappa}{2\varepsilon}\left(\rho_{in}^{2}-2\frac{\tilde{\sigma}}{\kappa}\rho_{out}^{2}\right)\right),$

which is exponentially small as $\varepsilon\to 0$ as long as

$\frac{\rho_{in}}{\rho_{out}}\geq\sqrt{\frac{2\tilde{\sigma}}{\kappa}}>\sqrt{2},$

where the strict inequality on the right can be arbitrarily close to equality
if $K,M>0$ are sufficiently small. This is precisely the assumption in Theorem
3.20. It follows that

$\|u(\cdot,T)-\Psi(\cdot,T)\|_{H_{ul}^{\theta}}=r_{3}(T_{3})^{\beta}\|R_{3}(\cdot,T_{3})\|_{H_{ul}^{\theta}}\leq
C\exp\left(-\frac{\kappa}{2\varepsilon}\left(\rho_{in}^{2}-2\frac{\tilde{\sigma}}{\kappa}\rho_{out}^{2}\right)\right).$

Writing $\kappa_{-}:=\kappa$ and $\kappa_{+}:=\tilde{\sigma}/\kappa$ as in the
proof of Lemma 3.16 yields the bound in Assertion (ii). ∎

##  7 Summary

Turing instabilities have been identified as a key local mechanism for the
emergence of patterned steady states in a wide variety of different contexts
[24, 41, 66, 71]. The dynamic counterpart, i.e. the slow passage problem, is
also expected to arise in a wide variety of applications, based on the fact
that realistic models often feature system parameters which evolve slowly in
time. Nevertheless, the slow passage through a Turing bifurcation has, by
comparison, received far less attention in the literature. As notable
exceptions we refer again to [20] for a detailed numerical study of a model
with applications in vegetation patterns, and to [6] for the first rigorous
results on the local mechanisms for delay phenomena for systems with a slow
passage through an $O(2)$-symmetric Turing bifurcation using the center
manifold theory of [39, 93], which applied for systems possessing a spectral
gap. The inapplicability of reduction methods based on center manifold theory
or Lyapunov-Schmidt reduction on large or unbounded spatial domains presents a
significant analytical challenge, which arises naturally in the context of
pattern forming systems for which the spatial scale characterising the
inhomogeneity of the pattern is small in comparison to the size of the domain
itself. Modulation theory has developed in order to address this problem in
classical bifurcation theory, but has as yet, to the best of our knowledge,
not been extended to the fast-slow setting which is natural for the study of
slow passage problems.

This work constitutes a first step towards the development of modulation
theory for fast-slow and singularly perturbed systems in infinite dimensions.
Although we developed our methods in the context of the dynamic SH equation
(4), the methods themselves are intended to be general in the sense that they
can be applied to a wide variety of applications. In Section 4, we showed that
the formal modulation ansatz (10) which is known from classical modulation
theory (and can be derived for different applications according to their
‘clustered mode distribution’; recall Figure 2) admits of a fast-slow
generalisation in the form of the blow-up transformation (24). This blow-up
transformation is easy to derive if the multiple-scales ansatz from classical
modulation theory is already known. The idea is to replace the small parameter
$\delta$ in the classical ansatz with a time-dependent variable $r(\bar{t})$,
and to replace the simple rescalings $\bar{x}=\delta^{2}x$ and
$\bar{t}=\delta^{2}t$ with the time-dependent desingularizations in (15).

Applying this method to the dynamic SH equation (4), we derived a set of
modulation equations which take the form of non-autonomous GL equations posed
in the blown-up space. The modulation equations were presented in both local
and global coordinates in Section 3.1. Lemmas 3.14 and 3.16 characterise the
dynamics of the modulation equations, which are still infinite dimensional,
but dynamically simpler in the sense that they are lower order equations.
Lemma 3.14 provides rigorous asymptotics for the approximation $\Psi(x,t)$ in
the case that the source term is given by (5). In contrast to the static