pnasresearcharticle Sharma & Wilson 2To whom correspondence should be
addressed. E-mail<EMAIL_ADDRESS>
# Origin of filaments in finite-time in Newtonian and non-Newtonian thin-films

Saksham Sharma  Department of Chemical Engineering and Biotechnology,
University of Cambridge, Philippa Fawcett Drive, Cambridge CB3 0AS, UK  D. Ian
Wilson  Department of Chemical Engineering and Biotechnology, University of
Cambridge, Philippa Fawcett Drive, Cambridge CB3 0AS, UK

###### Abstract

The sticky fluids found in pitcher plant leaf vessels can leave fractal-like
filaments behind when dewetting from a substrate. To understand the origin of
these filaments, we investigate the dynamics of a retreating thin-film of
aqueous polyethylene oxide (PEO) solutions which partially wet polydimethyl
siloxane (PDMS) substrates. Under certain conditions the retreating film
generates regularly-spaced liquid filaments. The early-stage thin-film
dynamics of dewetting are investigated to identify a theoretical criterion for
liquid filament formation. Starting with a linear stability analysis of a
Newtonian or simple non-Newtonian (power-law) thin-film, a critical film
thickness is identified which depends on the Hamaker constant for the fluid-
substrate pair and the surface tension of the fluid. When the measured film
thickness is smaller than this value, the film is unstable and forms filaments
as a result of van der Waals forces dominating its behaviour. This critical
film-height is compared with experimental measurements of film thickness
obtained for receding films of Newtonian (glycerol-water mixtures) and non-
Newtonian (PEO) solutions generated on substrates inclined at angles
$0^{\circ}$, $30^{\circ}$, and $60^{\circ}$ to the vertical. The observations
of filament and its absence show good agreement with the theory. Further
analysis of the former case, involving a stability analysis of the contact
line, yields a prediction of the spacing (wavelength) $\hat{\lambda_{f}}$
between filaments as
$\hat{\lambda}\textsubscript{f}\hat{\eta}/\hat{\gamma}\propto Ca$, where
$\hat{Ca}$ is the capillary number for contact line motion: our experiments
yield $\hat{\lambda}\textsubscript{f}\hat{\eta}/\hat{\gamma}\propto Ca^{1.08}$
and earlier studies in the literature reported
$\hat{\lambda}\textsubscript{f}\hat{\eta}/\hat{\gamma}\propto Ca^{0.945}$ .
The evolution of the thin-film shape is modelled numerically to show that the
formation of filaments arises because the thin-film equation features a
singular solution after a finite-time, hence termed a “finite-time
singularity".

###### keywords:

Thin-film $|$ Filaments $|$ Finite-time singularity $|$ Pitcher plants

###### doi:

www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX

This manuscript was compiled on

Receding thin films arise in a range of fields and are important in coating,
printing and drying applications, where the stability of the thin film near
the moving contact line will determine the uniformity of the product. Our
interest in this topic arises from observation of residual filaments left by
the evaporation of sessile droplets of the sticky digestive fluid secreted by
N. Rafflesiana pitcher plants. Fig. 1 shows that these filaments are formed in
the early stages of evaporation, where the shrinking drop forms a receding
thin film at the contact line, and can be generated artificially by sucking
liquid from the drop (Fig. 1(c)). The filaments exhibit regular spacing on
relatively smooth surfaces, indicating that the filaments arise from an
instability in the thin film rather than contact line pinning.

Pitcher plant fluids are non-Newtonian aqueous solutions containing long-chain
polysaccharides (1). Deblais et al. (2) observed similar filaments with thin
films of viscous Newtonian and non-Newtonian liquids (glycerine and synthetic
polymer solutions, respectively) generated by a blade arrangement which
allowed the initial height and contact line velocity $\hat{U}$ to be
controlled independently. As the contact line velocity $\hat{U}$ decreased,
uniform films with a straight contact line were replaced by ones with
regularly spaced cusps and rivulets: for Newtonian liquids the rivulets were
unstable and gave rise to droplets whereas the higher extensional viscosity of
the polymer solutions stabilised the filaments and gave patterns analogous to
those in Fig. 1(c)). They reported that the threshold of cusp (and filament)
formation corresponded to a critical value of the capillary number,
$Ca=\hat{\eta}\hat{U}/\hat{\gamma}$, where $\hat{\eta}$ is the apparent
viscosity and $\hat{\gamma}$ the surface tension, but did not provide a
theoretical treatment.

In the present work, we provide a theoretical explanation for the observed
filament formation. It draws on recent experimental work by Xue & Stone on a
liquid film draining down a glass slide under the action of gravity. The thin-
film non-linear PDE used to model the film considered three forces: viscous
resistance, gravity, and surface tension. A major assumption there was the
consideration of perfectly wetting (zero degree contact angle) fluids, arising
from the difficulty of incorporating partial wetting in the model, as remarked
by co-author Stone when presenting this work at the GKB 100 symposium (3).
Partial wetting is included in the present study, following the work by
Witelski and coworkers on the stability analysis and evolution of thin-films
((4), (5), (6) and (7)) by employing expressions for the van der Waals forces’
dependency on the film thickness. We demonstrate that the formation of cusps
and filaments observed during the dewetting of pitcher plant fluids has its
roots in the inherent instability of the thin-film. Strictly speaking, this
instability gives rise to a ‘finite-time singularity’.

Figure 1: Filaments formed by overnight evaporation of ground pitcher N.
Rafflesiana fluid on (a) polystyrene and (b) borosilicate glass surfaces. (c)
Forced shrinkage of a sessile drop of N. Rafflesiana on borosilicate glass,
caused by withdrawal of liquid via pipette labelled P. Evaporation in (a) and
(b) results in concentration of dissolved species in the drop, slowing
evaporation and triggering the transition from a receding film to a pinned
state and ‘coffee ring effect’ features. Liquid removal in (c) is faster and
there is little evaporation: filaments are observed once the contact line
starts to recede.

The article is organised as follows. The hydrodynamic equations for the thin-
film are presented in Sec. 1 and stability analysis is then performed to find
the critical criterion for filament formation (Sec. 2). The rationale behind
the finite-time singularity feature of the thin-film equation is discussed in
Sec. 3 along with some numerical investigations. The criterion is compared
with experimental results in Sec. 4. A scaling law for the spacing between
filaments is derived and compared with experimental data in Sec. 5. The key
findings and potential further directions for this work are discussed in Sec.
6.

##  1 Theory

Consider a thin liquid film of local thickness
$\hat{h}(\hat{x},\hat{y},\hat{t})$ on the plane $(Oxy)$ as shown in Fig.
2(a)), with height $\hat{h}$ pointing towards $Oz$. $Ox$ points horizontally
along the liquid-substrate-air contact line and $Oy$ points directly down the
slope. A standard thin-film equation for a non-Newtonian liquid (8) exhibiting
power-law behaviour with exponent $n$ is

$3\hat{\eta}\frac{\partial\hat{h}}{\partial\hat{t}}=\left(\frac{\partial}{\partial\hat{x}}\left[\hat{m}(\hat{h})\frac{\partial}{\partial\hat{x}}(\hat{p})\right]-3\hat{\rho}g^{\prime}\hat{h}^{2}\frac{\partial\hat{h}}{\partial\hat{y}}\right)^{1/n}$
(1)

Here $\hat{\eta}$ is short-form for apparent viscosity
$\hat{\eta}(\dot{\gamma})$, where $\dot{\gamma}$ is the shear-rate of the
thin-film defined at its free surface and given as $\hat{U}/\bar{h}$ (see Fig.
2(b.ii)), $\hat{\rho}$ the density, $\hat{m}(\hat{h})$ the mobility
coefficient, $\hat{p}$ is the hydrodynamic pressure $\hat{p}(\hat{x},\hat{t})$
of the thin-film, and $g^{\prime}=g\cos\alpha$; hat denotes that the term is
dimensional. The boundary condition at the film-substrate interface (on
surface $Oxy$) gives $\hat{m}(\hat{h})=\hat{h}^{k}$ where $k\in[1,3]$. The
pressure $\hat{p}(\hat{x},\hat{t})$ in the thin-film (6, see Eq. (1.3)) is
given by

$\hat{p}=\hat{\Pi}(\hat{h})-\hat{\gamma}\frac{\partial^{2}\hat{h}}{\partial\hat{x}^{2}}$
(2)

where $\hat{\Pi}(\hat{h})$ is the disjoining pressure which accounts for the
van der Waals interactions between the thin-film and the substrate. The second
term on the RHS accounts for the effect of surface tension $\hat{\gamma}$ and
curvature of the thin-film. The disjoining pressure $\hat{\Pi}(\hat{h})$ can
be written in the form

$\hat{\Pi}(\hat{h})=\frac{\hat{A}}{\hat{h}^{3}}\left[1-\frac{\hat{h}_{UTF}}{\hat{h}}\right]$
(3)

where van der Waals forces are characterised by the Hamaker constant $\hat{A}$
($\hat{A}>0$ means the interaction is hydrophobic and $\hat{A}<0$ hydrophilic)
and $\hat{h}_{UTF}$ is the height of the adsorbed precursor film in Fig.2(b))
(6). There are two timescales in (1): i) $\hat{T_{x}}$, when the pressure term
(first term on the RHS) is dominant, and ii) $\hat{T_{y}}$, when the gravity
term (second term on the RHS) dominates. A scaling analysis gives these
timescales as

$\hat{T_{x}}=\frac{3\hat{\eta}\hat{L_{x}}^{4}}{\bar{h}^{3}\hat{\gamma}};\quad\hat{T_{y}}=\frac{\hat{\eta}\hat{L_{y}}}{\hat{\rho}g^{\prime}\bar{h}^{2}}$
(4)

with $\hat{L_{x}}=\bar{h}\sqrt{\hat{\gamma}\hat{h}_{UTF}^{2}/\hat{A}}$ ((see
6, p. 016301-2)), and length scale $\hat{L_{y}}$ the length of the glass
substrate in the $Oy$ direction (25 mm in our tests). With
$\bar{h}=O(10^{-4})$ m, $\hat{A}=O(10^{-17})$ J, $\hat{h}_{UTF}=O(10^{-9})$ m
and $\hat{\gamma}=O(10^{-2})$ N/m, this gives $\hat{T_{x}}=O(10^{-8})$ s and
$\hat{T}_{y}=O(1)$ so the first term on the RHS in (1) is expected to be
dominant.

This means that early-stage dynamics of the thin-film, when surface tension
and van der Waals interactions dominate, are characterised by a time scale of
$10^{-8}$ s, compared to the intermediate stage where gravity is important.
Similar arguments about time scales have been reported previously (6, p.
016301-2). It means that, as soon as the thin film is deposited on the
substrate (or, in these experiments, the substrate is withdrawn from the
liquid pool), an interplay between surface tension and van der Waals
interaction forces begins.

The experiments reported here employed Newtonian solutions (mixtures of
glycerol and water) of different viscosity as well as non-Newtonian ones
(mixtures of polyethylene oxide, PEO, in water). Scaled viscosity is used to
label the liquids when presenting results, given by
$\hat{\eta}$/$(\hat{\rho}\hat{\gamma}^{3}/g^{\prime})^{1/4}$. The PEO
solutions exhibit shear-thinning, which can be described by the Cross model
(see Supplementary Material). The shear rate in the experimental films lie in
the power law regime for this model (see Supp. Fig. S1), which is why this
model is used in (1).

Hence, in the next section, we focus on the early-stage dynamics of the thin-
film by ignoring the gravity term in (1), viz.

$3\hat{\eta}\frac{\partial\hat{h}}{\partial\hat{t}}=\frac{\partial}{\partial\hat{x}}\left[\hat{h}^{k}\frac{\partial}{\partial\hat{x}}\left(\frac{\hat{A}}{\hat{h}^{3}}\left[1-\frac{\hat{h}_{UTF}}{\hat{h}}\right]-\hat{\gamma}\frac{\partial^{2}\hat{h}}{\partial\hat{x}^{2}}\right)\right]^{1/n}.$
(5)

(5) is non-dimensionalised by introducing scales

$\hat{h}=\bar{h}h,\quad\hat{x}=\hat{L_{x}}x,\quad\hat{t}=\hat{T_{x}}t,\quad\hat{h}_{UTF}=\bar{h}\zeta$
(6)

where the variables without hats are dimensionless and
$\zeta=\hat{h}_{UTF}/\bar{h}<1$. This yields

$\frac{\partial h}{\partial t}=\frac{\partial}{\partial
x}\left[h^{k}\frac{\partial}{\partial
x}\left(\Gamma(h)-\frac{\partial^{2}h}{\partial x^{2}}\right)\right]^{1/n}$
(7)

where $\Gamma(h)$ is the dimensionless form of $\hat{\Pi}(h)$ and can be
written as

$\Gamma(h)=\frac{\zeta^{2}}{h^{3}}\left[1-\frac{\zeta}{h}\right]$ (8)

and the dimensionless pressure term is

$p=\Gamma(h)-\frac{\partial^{2}h}{\partial x^{2}}$ (9)

which will be used in the next section to simplify the analysis of this PDE.

Figure 2: (a) Schematic of experimental setup showing the coated glass
microscope slide on which a thin-film dewets as it drains downwards. Cameras
$F$ (xiQ USB3.0) and $S$ (acA1300-200um) record the dewetting process from the
front and side, respectively. Cartesian co-ordinates indicated by axes. (b)
Side view of the receding film - (i) schematic and (ii) camera image - with
receding contact angle $\theta_{r}$, film heights $\hat{h}^{UTF}$ and
$\bar{h}$ (scale: width of glass slide is 1 mm shown by the shaded grey
region). (c) Microscope images of the substrate (i) with and (ii) without
filaments during dewetting of PEO solution. (d) Front (camera $F$) view of
substrate (i) with and (ii) without droplet formation during dewetting of
aqueous glycerol solution.

##  2 Stability analysis of the steady state solution of the thin-film PDE

We investigate the tendency of a thin-film to maintain (or lose its) stability
at an early stage. Since the formation of filaments happens almost
instantaneously, the phenomenon is primarily linked to the interplay of
surface tension and van der Waals interaction forces. A linear stability
analysis is performed with respect to arbitrary infinitesimal perturbations.
We start by finding a Lyapunov function for (7) - in effect, modelling it as a
dynamic system - because the existence of such a function provides an
indication of the nature of stability of the system (9). Consider the
following integral

$I[h]=\int_{0}^{1}\left(\frac{1}{2}\left|\frac{\partial h}{\partial
x}\right|^{2}-\varsigma(h)\right)dx$ (10)

where $\partial\varsigma(h)/\partial h=\Gamma(h)$. The reason behind choosing
this integral is that its first derivative w.r.t $h$, has the property

$\frac{\partial I}{\partial h}=\frac{\partial}{\partial
h}\left(\frac{1}{2}\left|\frac{\partial h}{\partial
x}\right|^{2}-\varsigma(h)\right)=\frac{\partial^{2}h}{\partial
x^{2}}-\Gamma(h)=-p.$ (11)

Such an integral is referred to as an energy functional of the system. The
rate of energy dissipation in a control volume $dV$ for this energy functional
is given by

$\frac{\partial I}{\partial t}=\frac{\partial I}{\partial h}\frac{\partial
h}{\partial t}=-p\ \nabla.\left(h^{3}\nabla p\right)$ (12)

where the derivative is written in terms of operator $\nabla$ to simplify
further analysis. Total energy dissipation over volume $V$ is

$\frac{dI}{dt}=\int_{V}-p\ \nabla.\left(h^{3}\nabla p\right)dV$ (13)

Integrating by parts gives

$\frac{dI}{dt}=-p\int_{V}\nabla.\left(h^{3}\nabla p\right)dV-\int_{V}\nabla
p\int\nabla.(h^{3}\nabla p)dV$ (14)

and applying the divergence theorem gives

$\frac{dI}{dt}=-p\int_{S}h^{3}(\nabla p.\mathbf{n})dS-\int_{S}\ h^{3}(|\nabla
p|^{2}.\mathbf{n})\ dS.$ (15)

The first term on the RHS is zero because of the boundary condition which
assumes no flux of the liquid normal to the boundary, i.e. $\nabla
p.\mathbf{n}=0$ on surface $dS$ (the liquid/vapour interface). As a result,
the expression for net energy dissipation is

$\frac{dI}{dt}=-\int_{S}\ h^{3}|\nabla p|^{2}\ dS$ (16)

which is always $\leq 0$ because $h\geq 0$. The non-increasing nature of
Lyapunov function $I[h]$ confirms the direction of stability of the steady
state solution of (7), which is towards the higher value of $h$. This approach
allows one to find all possible time-independent solutions of (7) which obey
(16), and these solutions are: (i) $h=0$ and (ii) $\nabla p=0$. The former
corresponds to the absence of any fluid on the surface and the latter means
that a uniform pressure field exists for the steady-state solution. The
steady-state solution with uniform pressure $\bar{p}$ is selected to proceed
further, such that

$\Gamma(h)-\frac{\partial^{2}h}{\partial x^{2}}=\bar{p},$ (17)

which has a uniform solution given by $h=\bar{h}$ and
$\bar{p}=\Gamma(\bar{h})=\frac{\zeta^{2}}{\bar{h}^{3}}\left[1-\frac{\zeta}{\bar{h}}\right]>0$.
It will be demonstrated in the next Section that such a solution can evolve to
approach the first solution, $h=0$, for certain parametric conditions. For
now, we are interested in finding families of solutions (apart from
$h=\bar{h}$) of (7) which can also exist if the original equation is
perturbed. To do this, we expand the pressure and the film thickness as

$\displaystyle\bar{p}=\bar{p}_{c}+\delta$ (18a) $\displaystyle
h(x)=\bar{h}+\epsilon h_{1}(x)+\epsilon^{2}h_{2}(x)+...$ (18b)

where $\epsilon,\delta<<1$ are small perturbation parameters. At
$O(\epsilon)$, (17) becomes

$\frac{3\zeta^{2}h_{1}}{\bar{h}^{4}}\left[1-\frac{\zeta}{\bar{h}}\right]+\frac{\partial^{2}h_{1}}{\partial
x^{2}}=0$ (19)

which is a second-order ODE when higher order $\epsilon^{2}$ terms are
ignored. The equation is made closed-form by considering it over a periodic
domain, $0\leq x\leq 1$, with Neumann boundary conditions:
$h_{x}(0)=h_{x}(1)=0$. Using the periodic boundary condition allows one to
explain the formation of a periodic array of filaments in the $Ox$ direction.
(19) can be written as a harmonic oscillator equation of the form

$\frac{\partial^{2}h_{1}}{\partial x^{2}}+\Lambda^{2}h_{1}=0$ (20)

where $\Lambda^{2}=3\zeta^{2}[1-\zeta/\bar{h}]/\bar{h}^{4}$. The eigenvalues
of this equation are $\Lambda_{i}=i\pi$ for $i\in\mathbb{Z^{+}}$. Solving for
$\bar{h}$, one obtains the quintic relationship

$3\frac{\zeta^{2}}{\bar{h^{4}}}\left[1-\frac{\zeta}{\bar{h}}\right]=i^{2}\pi^{2}$
(21)

with the approximate solution

$\bar{h}_{i}=3^{1/4}\sqrt{\frac{\zeta}{i\pi}}$ (22)

The solution $\bar{h}_{1}$ for $i=1$ is the primary bifurcation point which
demarcates a transition between stable and unstable solutions: above the
primary bifurcation point ($\bar{h}>\bar{h}_{1}$), the solution is stable, and
below it ($\bar{h}<\bar{h}_{1}$), it is unstable (see 4, p. 159). Higher order
bifurcations for $i\geq 2$ are not considered because they involve only
unstable modes (see 10, Theorem 5).

To find a dimensional version of this solution, dimensional analysis of (5) is
required, which yields the scaling relation
$\bar{h}\backsim\hat{\gamma}\hat{T_{x}}/\hat{\eta}$. Using $\hat{T_{x}}$ from
(4) and $\hat{L_{x}}=\bar{h}^{2}\zeta\sqrt{\hat{\gamma}/\hat{A}}$, the
critical height of the thin-film is then

$\hat{h}_{f}=3^{-5/16}\left(\frac{\zeta}{\pi}\right)^{-1/8}\zeta^{-1}\left(\frac{\hat{\gamma}}{\hat{A}}\right)^{-1/2}$
(23)

such that if $\bar{h}<\hat{h}_{f}$, the film is unstable, and if
$\bar{h}>\hat{h}_{f}$, it is stable. It should be noted that this theoretical
treatment does not predict the shape or behaviour of the unstable film: it
provides a criterion to compare the steady state film thickness with a
theoretical limit, above which the film is stable and below which it is
unstable. This analysis is independent of the rheology of the thin-film and
the shear rate associated with the motion. The physical reasoning behind this
is that the formation of filaments happens over a timescale significantly
faster (theoretically, eight orders of magnitude) than the timescale of
receding motion, and thus, the shear rate or rheology of the thin-film does
not play any role in the critical film thickness $\hat{h}_{f}$. The reader is
referred to the work of Xue and Stone (11) where the evolution of the shape of
the thin-film is considered.

An unstable filament evolves to yield the steady-state solution $h\approx 0$
(as discussed above). The dynamics of this transition are considered in the
next Section, where it is shown that the evolution of a thin-film to approach
zero film thickness in finite-time is the reason for the origin of the
filament.

##  3 Finite-time filament formation

We start with a simplified form of (7) in this section, where $n=1$, to study
the bounds or limits of the modified equation which has surface tension and
van der Waals force terms. The motivation in the section is to understand the
behaviour of the solutions and thereby predict the conditions necessary for
the solution to remain bounded or to blowup. A general form of (7) with $n=1$
(see (12)) is

$\frac{\partial h}{\partial t}=-\frac{\partial}{\partial
x}\left(M(h)\frac{\partial}{\partial
x}\left[-Q(h)+R(h)\frac{\partial^{2}h}{\partial x^{2}}\right]\right)$ (24)

with boundary condition: $h(x+L)=h(x)$. If $M(h)=f(h),R(h)=1$ and
$dQ(h)/dh=-g(h)/f(h)$, then the equation becomes

$h_{t}=-(f(h)h_{xxx})_{x}-(g(h)h_{x})_{x}$ (25)

where subscripts denote differentiation. For $f(h)=h^{k}$ ($1\leq k\leq 3$)
and $g(h)=3\frac{\zeta^{2}}{h^{4}}(1-\frac{4\zeta}{3h})$, (25) is equivalent
to (7). It was proved in (13) that the solution of (25) blows up when

$\lim_{s\rightarrow\infty}\frac{s^{2}f(s)}{g(s)}<\infty$ (26)

and when $s^{2}f(s)/g(s)\rightarrow\infty$, as $s\rightarrow\infty$, the
classical-solution is uniformly bounded and positive. In the current work,
i.e. (7) with $n=1$, $s^{2}f(s)/g(s)=s^{6+k}/(3\zeta^{2}(1-4\zeta/3s))$, which
grows to $\infty$ as $s\rightarrow\infty$. Hence, the classical-positive
solution of (7) (for $n=1$) is bounded and finite. Having established the
bounds of the equation, we next explore how the solution evolves over time.

Figure 3: Numerical solution of (5) (for $\hat{h}^{UTF}=0$), over domain
$-L_{d}\leq x\leq L_{d}$. (a) For $L_{d}=4.34$, height $\bar{h}$ approaches
zero at time $t_{f}\approx 4.79$ (finite-time singularity) after which the dry
region grows in the direction of the arrows, and the film splits to form two
filaments (not shown). (b) For $L_{d}=2.94$, height $h$ remains stable with
time and does not tend towards zero.

(5) states that the height of the film $h$ is determined by two forces: van
der Waals interactions and surface tension. For wetting liquids the former is
attractive in nature and brings the fluid in contact with the substrate; being
inversely proportional to $h$, it increases in magnitude as $h\rightarrow 0$.
The latter, on the other hand, tries to minimise the surface energy of the
fluid by reducing the area of the liquid-vapour interface; it prefers a non-
planar film to a planar one. It is expected that the combination of these will
lead to the presence of both features, namely reduced film-height and non-
planar film-geometry, in the system. To focus on the highly nonlinear terms
(van der Waals and surface tension forces) which grow very fast, we take
$\hat{h}^{UTF}=0$ (as the order of $\hat{h}^{UTF}$ is signficantly lower than
$h$) in (5), so that only the first order term in van der Waals interaction
and the surface tension is compared and evaluated in the numerical study. The
numerical solution is evaluated using Wolfram Mathematica. A finite domain
with width (in the $x$ direction) $2L_{d}$ is considered with the boundary
condition $h(-L_{d},t)=h(L_{d},t)$ and the initial condition
$h(x,0)=\bar{h}-\delta\cos{\frac{\pi x}{L_{d}}}$ such that $\bar{h}=1$ is the
average film height and $\delta=0.1$ is the amplitude of perturbation (0.1
being chosen to ease graphical visualisation). As derived previously in (7,
see Eq. (3.3)), there is a critical thickness $\sqrt{L_{d}/\pi}$ (following a
similar stability analysis to that in the previous Section) below which the
uniform film is unstable and above which the uniform film is stable to small
perturbations.

Fig. 3(a) shows the former case, for $L_{d}=4.34$, > $\pi$, where the film-
height $h$ approaches 0 at $t\approx 4.79$ (in finite-time). Near this time,
the Mathematica software suspects a singularity to exist, hence it is not
possible to find the solution at longer times. Fig. 3(b) shows an example of
the latter case, for $L_{d}=2.94$, where no such singularity is found even at
larger times; the initial sinusoidal perturbation decays to the average film-
height $\bar{h}=1$, suggesting that the film remains stable.

Figure 4: Comparison of experimental results with theory: effect of
dimensionless viscosity, $\eta_{\text{scaled}}$, on $\bar{h}/\hat{h}_{f}$ for
(a) glycerol-water mixtures, and (b) PEO solutions. Solid symbols denote
unstable film, forming singularities in the form of drops or filaments; hollow
symbols denote a stable film. Symbol colour indicates the angle of plate
inclination: black for $\alpha=0^{\circ}$; blue for $\alpha=30^{\circ}$; red
for $\alpha=60^{\circ}$. Here, $\bar{h}$ is the experimentally measured film
thickness and $\hat{h}_{f}$ is the theoretical critical film thickness.

##  4 Results

In the experiments a receding liquid film is generated on a static coated
glass slide by moving the liquid reservoir downwards. The film thickness is
measured and the initial behaviour of the receding film is recorded as
outlined in Fig. 2(b). Fig. 4 plots the ratio $\bar{h}/\hat{h}_{f}$, which
determines whether the film will remain stable or not, against the
dimensionless viscosity, $\eta_{\text{scaled}}$ (scaled with
$(\hat{\rho}\hat{\gamma}^{3}/g\sin\alpha)^{1/4})$ for glycerol-water mixtures
and aqueous PEO solutions.

Fig. 4(a) shows good agreement with the stability criterion for the Newtonian
liquids. If $\bar{h}/\hat{h}_{f}<1$, the film becomes unstable because the
thinner film gives rise to larger van der Waals forces. A more viscous liquid,
i.e. larger $\eta_{\text{scaled}}$, results in a larger film thickness
$\bar{h}$ (measured experimentally - see Supplementary Tables S3-S5), while
$\hat{h}_{f}$ does not depend on viscosity: less viscous Newtonian films are
more likley to be unstable. An unstable film touches the surface to create a
dry spot which grows in the region (7, see Fig.5) and gives rise to satellite
beads. Satellite beads have been reported previously, e.g.(14), where drops
sliding down on a partially wetting substrate emitted smaller drops from their
cusped tail.

For the non-Newtonian case, Fig. 4(b), when $\bar{h}/\hat{h}_{f}>1$, the thin-
film again remains stable and does not yield a singularity. However, for
$\bar{h}/\hat{h}_{f}<1$, the thin-film yields a singular solution and
filaments are observed. Experimentally observed zones of filament formation
and non-formation are predicted very well by the stability theory. The
crossover between the zones occurs at different $\eta_{scaled}$ values for the
two fluid types: elucidation of this aspect requires further work, as does
testing on pitcher fluids.

It is to be noted that the current theory has not been tested for the pitcher
fluids. The critical requirements to perform these experiments on pitcher
fluids or any other biological fluids, are as follows: (i) ample amount of
fluid should be available (approximately 3-5 mL) for enough number of tests
across a wide range of parameters; (ii) measurements of wetting and dispersive
properties of the fluid. We invite future studies in this direction, wherein,
the experts of pitcher fluids and that of thin-film physics could join forces
to further test this theory against pitcher fluids, belonging to different
species and locations.

##  5 Stability analysis of moving contact line

In previous sections, the condition for formation of droplets or filaments was
determined using stability analysis of a young thin-film. We now seek to
determine how the spacing between such filaments depends on the contact line
velocity. The theory developed in (15) is used to derive the scaling-relation
for the size and spacing of filaments, which is then compared with our
experiments and experimental data reported in the literature.
The dimensionless lubrication equation for a moving contact line of a thin-
film of height $h(x,y,t)$ with slip at the fluid-solid interface is given by
(16)

$\displaystyle\partial_{t}h+\nabla.(h\ \textbf{U})=0$ (27a)
$\displaystyle\nabla\kappa+\textbf{e}_{y}+\frac{3(-Ca\
\textbf{e}_{y}-\mathbf{U})}{h(h+3l_{s})}=0$ (27b)

where $\textbf{U}(x,y,t)=U_{x}\textbf{e}_{x}+U_{y}\textbf{e}_{y}$ is the
depth-averaged fluid velocity inside the film, $l_{s}$ is the slip length to
ensure Navier-slip along the substrate, $Ca$ is the capillary number given by
$\hat{\eta}(\dot{\gamma})\hat{U}/\hat{\gamma}$, $\hat{U}$ is the contact line
velocity, and $\nabla=\textbf{e}_{x}\partial_{x}+\textbf{e}_{y}\partial_{y}$.
All the terms are dimensionless. A linear stability analysis of (27) yields

$\displaystyle h(x,y,t)=h_{0}(y)+\epsilon h;_{1}(y)e^{-\sigma t+iqx}$ (28a)
$\displaystyle\kappa(x,y,t)=\kappa_{0}(y)+\epsilon\kappa_{1}(y)e^{-\sigma
t+iqx}$ (28b)

where $\kappa$ is two times the mean curvature of the interface
(nondimensionalised by $1/\bar{h}$). $h_{0}(z)$ and $\kappa_{0}(z)$ denote the
shape of the film as a result of balance between capillary forces and gravity;
$\hat{\sigma}$ is the growth rate of perturbations; $\hat{q}$ is the
wavenumber of the disturbance in the $x$ direction (Fig. 5(a)). The detailed
stability analysis of (28) is given in the Appendix of (15), wherein the
following analytical result for $\hat{\sigma}$ is derived

$\hat{\sigma}=\frac{|\hat{q}|\hat{\gamma}}{\hat{\eta}}\frac{\tan\theta_{r}}{\cos\theta_{r}}\left(\frac{d\tan\theta_{r}}{dCa}\right)^{-1}$
(29)

where $\theta_{r}$ is the receding contact angle of the thin film, estimated
using the Cox-Voinov law (17),

$\theta_{r}^{3}=\theta_{e}^{3}-9\ Ca\ \ln(y/l_{s})$ (30)

where $\theta_{e}$ is the equilibrium contact angle, $y$ is the distance from
the contact line (Fig. 5(a)) and $l_{s}$ is the slip length. Differentiating
w.r.t. $Ca$ gives

$3\theta_{r}^{2}\ \frac{d\theta_{r}}{dCa}=-9\
\ln(y/l_{s})\frac{d(y/l_{s})}{dCa}$ (31)

such that $\frac{d(y/l_{s})}{dCa}\propto\hat{T}_{Y}$ because $l_{s}$ is
dependent only on molecular factors (self-diffusion coefficient and molecular
size) (18). Since $\theta_{r}$ is small, $\tan\theta_{r}\approx\theta_{r}$ and
(29) becomes

$\hat{\sigma}=-|\hat{q}|\frac{\hat{\gamma}}{\hat{\eta}}\frac{\theta_{r}}{1-\theta_{r}^{2}/2}\left(\frac{3}{\theta_{r}^{2}}\ln\left(\frac{y}{l_{s}}\right)\hat{T}_{Y}^{-1}\right)\approx-|\hat{q}|\frac{\hat{\gamma}}{\hat{\eta}}\frac{\theta_{r}^{3}}{3\ln(y/l_{s})\hat{T}_{Y}}$
(32)

Figure 5: Comparison of the experimental data from literature and experiments
in the current work on filament spacing, $\hat{\lambda}$f with predictions.
(a) Schematic of a moving contact line with disturbance wavelength
$\hat{\lambda}$ in the $x$ direction. $g^{\prime}$ is the component of gravity
acting along the plane of the substrate; (b) Effect of receding contact line
velocity $\hat{U}$ on spacing between filaments
$\hat{\lambda}\textsubscript{f}$ for experiments performed by us and reported
in the literature (2); (c) Effect of capillary number $Ca$ on
$\frac{\hat{\lambda}\textsubscript{f}\hat{\eta}}{\hat{\gamma}}$ for PEO
solutions; symbol colour indicates angle of inclination: black for
$\alpha=0^{\circ}$; blue for $\alpha=30^{\circ}$; red for $\alpha=60^{\circ}$.
Star symbols denote the experimental values reported in the literature (2).

Ultimately, the expression for wavelength $\hat{\lambda}$ of the contact line
disturbance, using (32), is given by

$\hat{\lambda}=\frac{2\pi}{|\hat{q}|}=\frac{2\pi\hat{\gamma}}{\hat{\sigma}\hat{\eta}}\frac{-\theta_{r}^{3}}{3\ln(y/l_{s})\hat{T}_{Y}}$
(33)

Substituting (30) in (33) gives

$\hat{\lambda}=\frac{2\pi\hat{\gamma}}{\hat{\sigma}\hat{\eta}\hat{T}_{Y}}\left[3Ca-\frac{\theta_{e}^{3}}{3\ln(y/l_{s})}\right]$
(34)

with the first term on the RHS given by
$\frac{6\pi\hat{\gamma}}{\hat{\sigma}\hat{T}_{Y}\hat{\eta}}Ca=\frac{6\pi}{\hat{\sigma}\hat{T}_{Y}}\hat{U}$.
This predicts that $\hat{\lambda}$ is dependent on the first power of
$\hat{U}$ for a given growth rate $\hat{\sigma}$. It should be noted that
growth rate in (29) is positive, because receding angle $\theta_{r}$ increases
with $Ca$ (15). Hence, any small perturbation $\epsilon$ on $h$ will
eventually die with a relaxation rate $\hat{\sigma}$, which can be assumed to
be constant for a given disturbance $\epsilon$. Interestingly, a similar kind
of stability analysis is performed while deriving the growth rate of a
Plateau-Rayleigh instability; the major difference being that the wavelength
of the most unstable mode is analytically derived in that case, because of
imaginary values of growth rate $\hat{\sigma}$. In the present case, we have
shown that wavelength $\hat{\lambda}$ is dependent on the first power of
$\hat{U}$.

In §3, it was shown that the thin-film, with a thickness lower than the
critical film thickness $\hat{h}_{f}$, can break apart, self-similarly, to
form filaments. The origin of filament formation is then followed by the
motion of contact line and formation of fluctuation-induced crests and troughs
of wavelength $\hat{\lambda}$ (Fig. 5(a)). Filaments are experimentally
observed to have a regular spacing between them, given by $\hat{\lambda}_{f}$.

We now compare the above theory with: (i) experimental data collected using
PEO solutions in the setup in Figure 2, and (ii) the experimental data
reported for aqueous polyacrylamide solutions in (2). For the former, the
spacing between filaments $\hat{\lambda}_{f}$ varies with $\hat{U}$ as:
$\hat{\lambda}_{f}\propto\hat{U}^{0.72}$ (Fig. 5(b)) , whereas, for the
latter, it varies as $\hat{\lambda}_{f}\propto\hat{U}^{0.80}$ (Fig. 5(c)). In
terms of $Ca$, for the former (i.e. our experiments), plot of
$\hat{\lambda}\textsubscript{f}\hat{\eta}/\hat{\gamma}$ versus $Ca$ shows the
following relation:
$\hat{\lambda}\textsubscript{f}\hat{\eta}/\hat{\gamma}\propto Ca^{1.08}$ in
our experiments and
$\hat{\lambda}\textsubscript{f}\hat{\eta}/\hat{\gamma}\propto Ca^{0.945}$ in
the literature. There is therefore good agreement between the experiments
reported here, those of Deblais et al. (2), result obtained here from
stability analysis.

##  6 Discussion and conclusions

The ability to predict regimes of filament/drop formation by a thin-film is
likely the result of simplifying the geometry of this problem and focusing on
one single question: what drives the formation of these thin-film
instabilities? The answer to this question inherently lies in the finite-time
singularity feature of thin-film PDE which is a 4th order parabolic degenerate
PDE. The stability analysis of this equation allows us to find an analytical
criterion to predict the conditions which makes the thin-film unstable. Once
it happens, the thin-film dynamically evolves to touch the substrate and
creates a dry spot region which grows in size and divides the film into
multiple filaments. The spacing between filaments is explained by performing
stability analysis on the moving contact line, to find out how
$\hat{\lambda}_{f}$ depends on $\hat{U}$.

After numerically indicating the finite-time rupture feature of thin-film
using sinusoidal perturbation. By applying sinusoidal perturbation to the
initial film height in the thin-film equation, the numerical results show that
the thin-film tends towards a rupture ($h=0$) in a finite-time. After this, an
important question to consider is whether this rupture occurs for any
arbitrary perturbation, which is smooth and belongs to $C^{\infty}$ class of
function (i.e., can be differentiated infinite times). Pinching-singularities
for different thin-film equations have been proposed previously, but only some
have been proven to date. One recent example is the Hele-Shaw case, with
$f(h)=h,g(h)=0$ in (25), for which it was numerically shown by (19) that
$h\rightarrow 0$ in finite-time, and was proved in 2018 by (20) that the thin-
film height $h$ reaches a minimum after a finite-time, which is zero. In the
similar spirit, the following question remains unsolved:

  * •

For (25) with $f(h)=h^{3}$ and $g(h)=h^{-1}$, prove that for all profiles of
initial data, the thin-film height $h$ after a finite-time reaches a minima,
i.e. zero, giving rise to finite-time singularity.

One possible way to prove finite-time singularity for the above problem is to
embed this form of (25) (hereafter, referred to as the filament equation) on a
Turing machine. The techniques on how to do this for Euler equations were
reported recently (21). If the same can be done for the filament equation,
then it means that for a general initial perturbation, the solution to the
equation is computable or, in other words, Turing-complete. A Turing-complete
solution can be used to program the fluid flow according to the requirement of
the problem, as mentioned in (22). The physical interpretation of a Turing-
complete flow is that the path of particles are undecidable 111An undecidable
problem is a problem for which it is proved to be impossible construct a
function/algorithm that always yield a correct ‘yes’ or ‘no’ answer. in a
sense that nothing a priori can be said about their future trajectory; this
feature is also known as universality of thin-film equation. Dimensional
analysis of the filament equation (4) shows that the solution behaves in a
self-similar way, because of its scalar-invariant nature. Combining the
university feature with the scalar-invariant feature, the computable solution
of the thin-film equation should evolve in a self-similar fashion for an
arbitrary initial datum. The height of the thin-film reduces over time,
following the trend shown in Supplementary Figure S4, and then it eventually
blows up in a finite-time $t_{f}$ at height $h=0$, for any arbitrary initial
datum. The physical manifestation of this singularity is in the origin of the
filament formation. Detailed discussion of this point is given in
Supplementary Note 2.

Apart from the theoretical-computational treatment as the next step in this
work, it is also possible to solve (25) using an operator-learner based PDE
solver. A physics-informed neural network (PINN) technique was recently
employed to find a smooth self-similar solution (scalar invariant) for
Boussinesq equations (waves of wavelength $\lambda$ in a shallow water pool of
depth $d$; $\lambda\gg d$) (23). Whilst the PINN technique can be used to
solve (25) numerically, the problem of dealing with a singularity requires a
resolution-invariant tool. Attention-based operator learners are reported to
have this capability, hence any blowup of the solution, if it occurs, will not
be dependent on the resolution of the software (24). This is essential if one
needs to use a numerically-computed self-similar solution of thin-film PDE,
which blows up in a resolution-invariant way, to devise a computer-assisted
proof of finite-time singularity formation from an initial datum (25).

##  7 Materials and Methods

###  7.1 Experimental setup

The experimental setup consists of a coated glass slide, placed on a stand as
shown in Fig. 2(a). A bath consisting of the test liquid is raised vertically
upwards to wet the initially dry glass slide, and then brought downwards
rapidly to its initial position to generate a thin-film which drains downwards
under the action of gravity g. Camera $F$ is used to capture the thin-film in
the $xy$ plane, and side camera $S$ is used to capture the macroscopic
receding contact angle $\theta_{r}$ of the thin-film in the $yz$ plane. Movies
of the receding film are recorded using a Ximea xiQ USB3.0 ($F$) camera at 17
fps and a Basler acA1300-200um USB3.0 ($S$) camera at 50 fps. Subsequent image
analysis is performed using ImageJ software. All experiments are performed at
a lab temperature of $21^{\circ}$C.

Borosilicate glass slides with dimensions $25\times 75$ mm2 and thickness 1 mm
were coated with PDMS (polydimethylsiloxane) solution. The PDMS solution was
prepared by dissolving SylgardTM 184 silicone elastomer and curing agent in
the weight ratio 10:1. Slides were prepared using the following protocol:
cleansing with surface active cleaning agent (Decon®90 at $2\%$), rinsing with
pure water, drying with an air gun, and further cleaning using an air plasma
cleaner. PDMS solution was then coated on the glass slide by spin-coating at
3000 rpm for 30 s.

###  7.2 Solution preparation

PEO solutions (3500 to 11000 ppm) were prepared by adding the required amount
of PEO powder (mol. wt. $8\times 10^{6}$, Acros Organics) to deionised water
at room temperature and stirred at 800 rpm for 48 hours to obtain a uniform
solution. Glycerol-water mixtures (80-100 $\%$ v/v) were stirred for shorter
periods.

###  7.3 Material characterisation

This section describes the characterisation of the liquids employed in the
current work. These can be broadly classified as (i) Newtonian, glycerol-water
mixtures, (ii) non-Newtonian, aqueous PEO solutions.

####  7.3.1 Shear rheology

The shear rheology of the liquids was studied using a Kinexus controlled shear
rheometer (Malvern Instruments, UK) fitted with a Couette cell. The apparent
viscosity, $\eta_{app}$, was measured using controlled stress ramps.
Supplementary Fig. S1(a) shows that the glycerol-water mixtures were
Newtonian. The PEO solutions exhibit shear-thinning behaviour, particularly in
the experimentally-relevant shear rate range of $0.1-60\,s^{-1}$.
Supplementary Fig. S1(b) shows the fit of the PEO data to the Crosss model,
$\frac{\hat{\eta}-\hat{\eta}_{\infty}}{\hat{\eta}_{0}-\hat{\eta}_{\infty}}=\frac{1}{1+(\lambda_{c}\dot{\gamma})^{m}}$.
The model parameters are reported in Table S1 and the values of
$\hat{\eta}_{0}$ and $\lambda_{c}$ for different PEO concentrations are
plotted in Supplementary Fig. 2.

####  7.3.2 Receding contact angle and surface tension

The receding angle $\theta_{r}$ of the thin-film was measured using the side
camera $S$, as shown in the schematic, Fig. 2(d). The surface tension
$\hat{\gamma}$ of the liquids was measured using pendant droplet tensiometry
(see (26)). The values obtained are reported in Supplementary Table S1.

We wish to thank Prof. Walter Federle and Dr Victor Kang for their preliminary
experiments on filament formation with pitcher plant fluids which inspired
this work. The experimental work benefitted from insights provided by Victor
Kang, Sagnik Middya, Ratul Das, Sundeep Vema, and Suraj Pavagada on rig
development and surface coating techniques. Discussions with Dr A.J.D.
Shaikeea, Dr Shuvrangsu Das and a summer school organised by UMass Amherst on
Soft Solids and Complex Fluids provided inputs to the theoretical side of this
work.

### Author Affiliations

Department of Chemical Engineering and Biotechnology, University of Cambridge,
Philippa Fawcett Drive, Cambridge CB3 0AS, UK

### Bibliography

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$\displaystyle\left|\int_{\mathbb{R}^{2}}\partial_{x}\partial_{y}G(x-s,y-t)\partial_{y}h_{2}(s,t)dsdt\right|\leq
C\varepsilon^{-\frac{1}{2}}r^{-\frac{3}{2}}\|h_{2}\|_{***}.$ (4.75)

The above three inequalities (4.78), (4.79) and (4.80) give that

$\|(1+r)^{\frac{3}{2}}\partial_{x}\partial_{y}\phi(x,y)\|_{L^{\infty}(\mathbb{R}^{2})}\leq
C\varepsilon^{-\frac{1}{2}}\left(\|h_{1}\|_{**}+\|h_{2}\|_{***}\right).$
(4.76)

For the term $\partial_{x}^{2}\partial_{y}\phi$, we have

$\displaystyle\partial_{x}^{2}\partial_{y}\phi(x,y)=$
$\displaystyle~{}-6(\sqrt{2}-\varepsilon^{2})\int_{\mathbb{R}^{2}}\partial_{x}\partial_{y}G(x-s,y-t)\partial_{x}^{2}(\partial_{x}q\partial_{x}\phi)(s,t)dsdt$
(4.77)
$\displaystyle-\int_{\mathbb{R}^{2}}\partial_{x}\partial_{y}G(x-s,y-t)\partial^{2}_{x}h_{1}(s,t)dsdt$
$\displaystyle-\int_{\mathbb{R}^{2}}\partial_{x}\partial_{y}G(x-s,y-t)\partial_{x}\partial_{y}h_{2}(s,t)dsdt.$

For the first term on the right hand side of (4.77), by Theorem 3.3, (4.41),
(4.47) and (4.52), following almost the same arguments in (4.54) we get

$\displaystyle\left|\int_{\mathbb{R}^{2}}\partial_{x}\partial_{y}G(x-s,y-t)\partial_{x}^{2}(\partial_{x}q\partial_{x}\phi)(s,t)dsdt\right|\leq\frac{C\varepsilon^{-\frac{1}{2}}}{r^{\frac{3}{2}}}\left(\|h_{1}\|_{**}+\|h_{2}\|_{***}\right),$
(4.78)

For the other two terms on the right hand side of (4.77), as (4.60)-(4.61) we
have

$\displaystyle\left|\int_{\mathbb{R}^{2}}\partial_{x}\partial_{y}G(x-s,y-t)\partial_{x}^{2}h_{1}(s,t)dsdt\right|\leq
C\varepsilon^{-\frac{1}{2}}r^{-\frac{3}{2}}\|h_{1}\|_{**},$ (4.79)

and

$\displaystyle\left|\int_{\mathbb{R}^{2}}\partial_{x}\partial_{y}G(x-s,y-t)\partial_{x}\partial_{y}h_{2}(s,t)dsdt\right|\leq
C\varepsilon^{-\frac{1}{2}}r^{-\frac{3}{2}}\|h_{2}\|_{***}.$ (4.80)

The above three inequalities give that

$\|(1+r)^{\frac{3}{2}}\partial_{x}^{2}\partial_{y}\phi(x,y)\|_{L^{\infty}(\mathbb{R}^{2})}\leq
C\varepsilon^{-\frac{1}{2}}\left(\|h_{1}\|_{**}+\|h_{2}\|_{***}\right).$
(4.81)

Concerning $\partial_{x}\partial_{y}^{2}\phi$, using integration by parts we
have

$\displaystyle\partial_{x}\partial_{y}^{2}\phi=$
$\displaystyle-6(\sqrt{2}-\varepsilon^{2})\int_{\mathbb{R}^{2}}\partial_{y}^{2}G(x-s,y-t)\partial_{x}^{2}(\partial_{x}q\partial_{x}\phi)(s,t)dsdt$
(4.82)
$\displaystyle-\int_{\mathbb{R}^{2}}\partial_{y}^{2}G(x-s,y-t)\partial_{x}^{2}h_{1}(s,t)dsdt$
$\displaystyle-\int_{\mathbb{R}^{2}}\partial_{y}^{2}G(x-s,y-t)\partial_{x}\partial_{y}h_{2}(s,t)dsdt.$

Using (4.41), (4.47), (4.52) and Theorem 3.3, we get

$\displaystyle\left|\int_{\mathbb{R}^{2}}\partial_{y}^{2}G(x-s,y-t)\partial_{x}^{2}(\partial_{x}q\partial_{x}\phi)(s,t)dsdt\right|$
(4.83)
$\displaystyle\leq~{}\frac{C}{r^{7/2}}\varepsilon^{-\frac{1}{2}}\left(\|h_{1}\|_{**}+\|h_{2}\|_{***}\right)\int_{B_{r/2}(x,y)}|\partial_{y}^{2}G(x-s,y-t)|dsdt$
$\displaystyle\quad+\frac{C}{r^{3/2}}\varepsilon^{-\frac{3}{2}}\left(\int_{\mathbb{R}^{2}\setminus
B_{r/2}(x,y)}|\partial_{x}q|^{2}dsdt\right)^{\frac{1}{2}}\left(\int_{\mathbb{R}^{2}\setminus
B_{r/2}(x,y)}|\partial_{x}^{3}\phi|^{2}dsdt\right)^{\frac{1}{2}}$
$\displaystyle\quad+\frac{C}{r^{3/2}}\varepsilon^{-\frac{3}{2}}\left(\int_{\mathbb{R}^{2}\setminus
B_{r/2}(x,y)}|\partial_{x}^{2}q|^{2}dsdt\right)^{\frac{1}{2}}\left(\int_{\mathbb{R}^{2}\setminus
B_{r/2}(x,y)}|\partial_{x}^{2}\phi|^{2}dsdt\right)^{\frac{1}{2}}$
$\displaystyle\quad+\frac{C}{r^{3/2}}\varepsilon^{-\frac{3}{2}}\left(\int_{\mathbb{R}^{2}\setminus
B_{r/2}(x,y)}|\partial_{x}^{3}q|^{2}dsdt\right)^{\frac{1}{2}}\left(\int_{\mathbb{R}^{2}\setminus
B_{r/2}(x,y)}|\partial_{x}\phi|^{2}dsdt\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\varepsilon^{-\frac{3}{2}}r^{-\frac{3}{2}}\left(\|h_{1}\|_{**}+\|h_{2}\|_{***}\right),$

and as (4.60)-(4.61) we get that

$\displaystyle\left|\int_{\mathbb{R}^{2}}\partial_{y}^{2}G(x-s,y-t)\partial_{x}^{2}h_{1}(s,t)dsdt\right|\leq
C\varepsilon^{-\frac{3}{2}}r^{-\frac{3}{2}}\|h_{1}\|_{**},$ (4.84)
$\displaystyle\left|\int_{\mathbb{R}^{2}}\partial_{y}^{2}G(x-s,y-t)\partial_{x}\partial_{y}h_{2}(s,t)dsdt\right|\leq
C\varepsilon^{-\frac{3}{2}}r^{-\frac{3}{2}}\|h_{2}\|_{***}.$ (4.85)

Combining (4.83), (4.84) and (4.85) we have

$\|(1+r)^{\frac{3}{2}}\partial_{x}\partial_{y}^{2}\phi(x,y)\|_{L^{\infty}(\mathbb{R}^{2})}\leq
C\varepsilon^{-\frac{3}{2}}\left(\|h_{1}\|_{**}+\|h_{2}\|_{***}\right).$
(4.86)

Following almost the same arguments from (4.82) to (4.85) we have

$\|(1+r)^{\frac{3}{2}}\partial_{x}^{2}\partial_{y}^{2}\phi(x,y)\|_{L^{\infty}(\mathbb{R}^{2})}\leq
C\varepsilon^{-\frac{5}{2}}\left(\|h_{1}\|_{**}+\|h_{2}\|_{***}\right).$
(4.87)

Step 4. In this step we estimate $\partial_{x}^{-1}\partial_{y}^{2}\phi$ and
$\partial_{x}^{-1}\partial_{y}^{3}\phi$, for the former one, using Green’s
representation formula, we have

$\displaystyle\partial_{x}^{-1}\partial_{y}^{2}\phi={}$
$\displaystyle-6(\sqrt{2}-\varepsilon^{2})\int_{\mathbb{R}^{2}}\partial_{y}^{2}G(x-s,y-t)(\partial_{x}q\partial_{x}\phi)(s,t)dsdt$
(4.88)
$\displaystyle-\int_{\mathbb{R}^{2}}\partial_{y}^{2}G(x-s,y-t)h_{1}(s,t)dsdt$
$\displaystyle-\int_{\mathbb{R}^{2}}\partial^{2}_{y}G(x-s,y-t)\partial_{x}^{-1}\partial_{y}h_{2}(s,t)dsdt.$

For the first one on the right hand side of (4.88), we have

$\displaystyle\left|\int_{\mathbb{R}^{2}}\partial_{y}^{2}G(x-s,y-t)(\partial_{x}q\partial_{x}\phi)(s,t)dsdt\right|$
(4.89) $\displaystyle\leq
Cr^{-7/2+\delta}\left(\|h_{1}\|_{**}+\|h_{2}\|_{***}\right)\int_{B_{r/2}(x,y)}|\partial_{y}^{2}G(x-s,y-t)|dsdt$
$\displaystyle\quad+C\varepsilon^{-\frac{3}{2}}r^{-\frac{3}{2}}\left(\int_{\mathbb{R}^{2}\setminus
B_{r/2}(x,y)}|\partial_{x}q|^{2}dsdt\right)^{\frac{1}{2}}\left(\int_{\mathbb{R}^{2}\setminus
B_{r/2}(x,y)}|\partial_{x}\phi|^{2}dsdt\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\varepsilon^{-\frac{3}{2}}r^{-\frac{3}{2}}\left(\|h_{1}\|_{**}+\|h_{2}\|_{***}\right).$

For the second term on the right hand side of (4.88) we get that

$\displaystyle\left|\int_{\mathbb{R}^{2}}\partial_{y}^{2}G(x-s,y-t)h_{1}(s,t)dsdt\right|$
(4.90) $\displaystyle\leq
Cr^{-\frac{5}{2}+\delta}\|h_{1}\|_{**}\int_{B_{r/2}(x,y)}|\partial_{y}^{2}G(x-s,y-t)|dsdt$
$\displaystyle\quad+C\varepsilon^{-\frac{3}{2}}r^{-\frac{3}{2}}\int_{\mathbb{R}^{2}\setminus
B_{r/2}(x,y)}|h_{1}(s,t)|dsdt$ $\displaystyle\leq
C\varepsilon^{-\frac{3}{2}}r^{-\frac{3}{2}}\|h_{1}\|_{**},$

while for the third term on the right hand side of (4.88), by (4.24) we have

$\displaystyle\left|\int_{\mathbb{R}^{2}}\partial_{y}^{2}G(x-s,y-t)\partial_{x}^{-1}\partial_{y}h_{2}(s,t)dsdt\right|$
(4.91) $\displaystyle\leq
Cr^{-2+\delta}\|h_{2}\|_{***}\int_{B_{r/2}(x,y)}|\partial_{y}^{2}G(x-s,y-t)|dsdt$
$\displaystyle\quad+C\varepsilon^{-\frac{3}{2}}r^{-\frac{3}{2}}\|h_{2}\|_{***}\int_{B_{r/2}(0)}\frac{1}{(1+\sqrt{s^{2}+t^{2}})^{2-\delta}}dsdt$
$\displaystyle\quad+C\varepsilon^{-\frac{3}{2}}\|h_{2}\|_{***}\int_{\mathbb{R}^{2}\setminus(B_{r/2}(x,y)\cup
B_{r/2}(0))}\frac{C(1+\sqrt{s^{2}+t^{2}})^{-2+\delta}}{(\sqrt{(x-s)^{2}+(y-t)^{2}})^{3/2}}dsdt$
$\displaystyle\leq
C\varepsilon^{-\frac{3}{2}}r^{-\frac{3}{2}+\delta}\|h_{2}\|_{**}.$

Then from (4.89)-(4.91) we get that

$\|(1+r)^{\frac{3}{2}-\delta}\partial_{x}^{-1}\partial_{y}^{2}\phi\|_{L^{\infty}(\mathbb{R}^{2})}\leq
C\varepsilon^{-\frac{3}{2}}\left(\|h_{1}\|_{**}+\|h_{2}\|_{***}\right).$
(4.92)

Similarly, we could derive that

$\|(1+r)^{\frac{3}{2}-\delta}\partial_{x}^{-1}\partial_{y}^{3}\phi\|_{L^{\infty}(\mathbb{R}^{2})}\leq
C\varepsilon^{-\frac{7}{2}}\left(\|h_{1}\|_{**}+\|h_{2}\|_{***}\right).$
(4.93)

Step 5. In the final step we estimate the terms $\partial_{y}^{4}\phi$ and
$\partial^{-1}_{x}\partial_{y}^{4}\phi$. For the former one, using (4.47),
(4.57), (4.69), (4.87) and equation (4.26), we get

$\varepsilon^{4}\|(1+r)^{\frac{3}{2}}\partial_{y}^{4}\phi\|_{L^{\infty}(\mathbb{R}^{2})}\leq
C\varepsilon^{-\frac{3}{2}}\left(\|h_{1}\|_{**}+\|h_{2}\|_{***}\right).$
(4.94)

Concerning the last term $\partial_{x}^{-1}\partial_{y}^{4}\phi$, we take
$\partial_{x}^{-1}$ on both sides of (4.26), and we get that

$\varepsilon^{4}\|(1+r)^{\frac{3}{2}-\delta}\partial_{x}^{-1}\partial_{y}^{4}\phi\|_{L^{\infty}(\mathbb{R}^{2})}\leq
C\varepsilon^{-\frac{3}{2}}\left(\|h_{1}\|_{**}+\|h_{2}\|_{***}\right).$
(4.95)

Then we finish the whole proof. ∎

##  5\.  The nonlinear system and proof of Theorem 1.1

In section 2, we have seen the leading terms of both equations (2.2) and (2.3)
can be reduced to the following single equation

$f_{1}=\frac{\sqrt{2}}{2}\partial_{x}g_{1}-\frac{g_{1}^{2}}{2}.$ (5.1)

Once $g_{1}$ is given, then we are able to solve $O(1)$ terms in both (2.2)
and (2.3) by finding $f_{1}$ through (5.1). To derive a solution to the
original GP equation (1.2), we need to solve the equations (2.30) and (2.12):

$\displaystyle\partial_{x}^{4}g_{1}-(2\sqrt{2}-\varepsilon^{2})\partial_{x}^{2}g_{1}-3(\sqrt{2}-\varepsilon^{2})\partial_{x}((\partial_{x}g_{1})^{2})-2\partial_{y}^{2}g_{1}+2\varepsilon^{2}\partial_{x}^{2}\partial_{y}^{2}g_{1}+\varepsilon^{4}\partial_{y}^{4}g_{1}$
(5.2) $\displaystyle=P_{1}+P_{2}+P_{3},$

and

$(\sqrt{2}-\varepsilon^{2})\partial_{x}f_{2}+2g_{1}f_{2}=\partial_{y}^{2}g_{1}+\partial_{x}f_{1}-(f_{1}+\varepsilon^{2}f_{2})^{2}g_{1},$
(5.3)

where $P_{1},~{}P_{2}$ and $P_{3}$ are given in (2.27)-(2.29).

We look for solutions of (5.2) such that $g_{1}=q+\phi$, with $q$ defined in
(2.32), then equation (5.2) could be written as

$\displaystyle\partial_{x}^{4}\phi-(2\sqrt{2}-\varepsilon^{2})\partial_{x}^{2}\phi-6(\sqrt{2}-\varepsilon^{2})\partial_{x}(q\phi)-2\partial_{y}^{2}\phi+2\varepsilon^{2}\partial_{x}^{2}\partial_{y}^{2}\phi+\varepsilon^{4}\partial_{y}^{4}\phi$
(5.4)
$\displaystyle=-\partial_{x}^{4}q+(2\sqrt{2}-\varepsilon^{2})\partial_{x}^{2}q+3(\sqrt{2}-\varepsilon^{2})\partial_{x}((\partial_{x}q)^{2})+2\partial_{y}^{2}q-2\varepsilon^{2}\partial_{x}^{2}\partial_{y}^{2}q-\varepsilon^{4}\partial_{y}^{4}q$
$\displaystyle\quad+3(\sqrt{2}-\varepsilon^{2})\partial_{x}((\partial_{x}\phi)^{2})+P_{1}+P_{2}+P_{3}.$

We set

$\Gamma_{q}=-\partial_{x}^{4}q+(2\sqrt{2}-\varepsilon^{2})\partial_{x}^{2}q+3(\sqrt{2}-\varepsilon^{2})\partial_{x}((\partial_{x}q)^{2})+2\partial_{y}^{2}q-2\varepsilon^{2}\partial_{x}^{2}\partial_{y}^{2}q-\varepsilon^{4}\partial_{y}^{4}q.$
(5.5)

It is not difficult to see that $\Gamma_{q}\in\mathcal{H}_{ox}$ and

$|\Gamma_{q}|+|\partial_{x}\Gamma_{q}|\leq C\varepsilon^{2}(1+r)^{-5}.$ (5.6)

As a consequence,

$\partial_{x}^{-1}\Gamma_{q}=-\int_{x}^{\infty}\Gamma_{q}(s,y)ds\quad\mbox{is
well defined and we have}~{}\|\partial_{x}^{-1}\Gamma_{q}\|_{**}\leq
C\varepsilon^{2}.$ (5.7)

The estimation (5.7) suggests that we look for solutions of (5.4) in the
following space

$\mathcal{F}_{\phi}=\\{\phi\in\mathcal{F}_{1}\mid\|\phi\|_{*}\leq
C\varepsilon^{2}\\},$ (5.8)

with $C$ sufficiently large. On the other hand, it is not difficult to check
that $\|(\partial_{x}\phi)^{2}\|_{**}\leq C\varepsilon^{2}$. Therefore we can
absorb $\Gamma_{q}$ and
$3(\sqrt{2}-\varepsilon^{2})\partial_{x}((\partial_{x}\phi)^{2})$ into
$P_{1}$.

In the following, we shall analyze (5.3). For a given $g_{1}=q+\phi$ with
$\phi\in\mathcal{F}_{\phi}$, we solve $f_{2}$ of (5.3) in the following space

$\mathcal{F}_{f_{2}}:=\\{f\mid f\in\mathcal{H}_{e},~{}\|f\|_{****}<C\\},$
(5.9)

where

$\displaystyle\|f\|_{****}={}$
$\displaystyle\|(1+r)^{\frac{3}{2}-\delta}f\|_{L^{\infty}(\mathbb{R}^{2})}+\|(1+r)^{\frac{3}{2}-\delta}\partial_{x}f\|_{L^{\infty}(\mathbb{R}^{2})}$
(5.10)
$\displaystyle+\|(1+r)^{\frac{3}{2}-\delta}\partial_{x}^{2}f\|_{L^{\infty}(\mathbb{R}^{2})}+\varepsilon\|(1+r)^{\frac{3}{2}-\delta}\partial_{x}^{3}f\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle+\varepsilon^{2}\|(1+r)^{\frac{3}{2}-\delta}\partial_{y}f\|_{L^{\infty}(\mathbb{R}^{2})}+\varepsilon^{2}\|(1+r)^{\frac{3}{2}-\delta}\partial_{x}\partial_{y}f\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle+\varepsilon^{4}\|(1+r)^{\frac{3}{2}-\delta}\partial_{y}^{2}f\|_{L^{\infty}(\mathbb{R}^{2})}+\varepsilon^{4}\|(1+r)^{\frac{3}{2}-\delta}\partial_{x}\partial_{y}^{2}f\|_{L^{\infty}(\mathbb{R}^{2})}.$

######  Lemma 5.1.

Let $\phi\in\mathcal{F}_{\phi}$. Then for sufficiently small $\varepsilon$
there exists a solution $f_{2,\phi}\in\mathcal{F}_{f_{2}}$ of (5.3). In
addition, for any two functions $\phi_{1},~{}\phi_{2}\in\mathcal{F}_{\phi}$,
we have the corresponding solutions $f_{2,\phi_{1}}$ and $f_{2,\phi_{2}}$
satisfying

$\displaystyle\|f_{2,\phi_{1}}-f_{2,\phi_{2}}\|_{****}\leq
C\varepsilon^{-\frac{3}{2}}\|\phi_{1}-\phi_{2}\|_{*}.$ (5.11)

###### Proof.

We recall that $f_{1}$ is determined by $g_{1}$ by (5.1). For each given
$g_{1},$ we shall solve (5.3) by a perturbation argument. To this aim, for
each fixed $y\in\mathbb{R},$ let us consider the non-homogeneous equation

$\left(\sqrt{2}-\varepsilon^{2}\right)\partial_{x}F_{\mathfrak{h}}+2qF_{\mathfrak{h}}=\mathfrak{h}.$
(5.12)

It is known the homogeneous equation

$\left(\sqrt{2}-\varepsilon^{2}\right)\partial_{x}F_{0}+2qF_{0}=0$

has a solution of the form

$F_{0}\left(x,y\right)=\exp\left(-\int_{-\infty}^{x}\frac{2q\left(s,y\right)}{\sqrt{2}-\varepsilon^{2}}ds\right).$

Using the expression

$q(x,y)=-\left(\frac{2\sqrt{2}}{2\sqrt{2}-\varepsilon^{2}}\right)^{2}\dfrac{\sqrt{8-2\sqrt{2}\varepsilon^{2}}x}{\frac{2\sqrt{2}-\varepsilon^{2}}{2\sqrt{2}}x^{2}+\frac{(2\sqrt{2}-\varepsilon^{2})^{2}}{4\sqrt{2}}y^{2}+\frac{3}{2\sqrt{2}}},$

we derive that

$\int_{-\infty}^{x}q\left(s,y\right)ds\quad\mbox{is close to}\quad-
D_{\varepsilon}\log\left(\frac{2\sqrt{2}-\varepsilon^{2}}{2\sqrt{2}}x^{2}+\frac{(2\sqrt{2}-\varepsilon^{2})^{2}}{4\sqrt{2}}y^{2}+\frac{3}{2\sqrt{2}}\right),$

where

$D_{\varepsilon}=\left(\frac{2\sqrt{2}}{2\sqrt{2}-\varepsilon^{2}}\right)^{3}\sqrt{2-\frac{\sqrt{2}}{2}\varepsilon^{2}}.$

As a consequence,

$F_{0}\quad\mbox{is close
to}\quad\left(x^{2}+\sqrt{2}y^{2}+\frac{3}{2\sqrt{2}}\right)^{2}.$ (5.13)

In particular, $F_{0}$ is not decaying at infinity and even in $x$. Applying
the variation of parameter formula, we can write the solution
$F_{\mathfrak{h}}$ of (5.12) as the following form

$F_{\mathfrak{h}}=F_{0}\left(x,y\right)\int_{-\infty}^{x}\frac{\mathfrak{h}\left(s,y\right)}{F_{0}\left(s,y\right)}ds,$

where the integral makes sense due to the boundedness of $\mathfrak{h}$ and
the decay property of $F_{0}^{-1}$, see (5.13).

For a given $\phi\in\mathcal{F}_{\phi}$ and $f_{2}\in\mathcal{F}_{f_{2}}$, we
write equation (5.3) as

$\displaystyle(\sqrt{2}-\varepsilon^{2})\partial_{x}f_{2}+2qf_{2}=$
$\displaystyle-2\phi
f_{2}+\partial_{y}^{2}(q+\phi)+\partial_{x}\left(\frac{\sqrt{2}}{2}\partial_{x}(q+\phi)-\frac{1}{2}(q+\phi)^{2}\right)$
(5.14)
$\displaystyle-\left[\left(\frac{\sqrt{2}}{2}\partial_{x}(q+\phi)-\frac{1}{2}(q+\phi)^{2}\right)+\varepsilon^{2}f_{2}\right]^{2}(q+\phi).$

Denoting the right hand side of (5.14) by $E_{\phi}$ and we define the map
$f_{2}\to F_{E_{\phi}}$ by $\mathcal{M}(f_{2})$.

We shall first derive a solution to (5.14) in the following space

$\displaystyle\widetilde{\mathcal{F}}_{f_{2}}:=\left\\{f\mid
f\in\mathcal{H}_{e},~{}\|f\|_{*****}<C\right\\},$ (5.15)

where

$\displaystyle\|f\|_{*****}={}$
$\displaystyle\|(1+r)^{\frac{3}{2}-\delta}f\|_{L^{\infty}(\mathbb{R}^{2})}+\|(1+r)^{\frac{3}{2}-\delta}\partial_{x}f\|_{L^{\infty}(\mathbb{R}^{2})}$
(5.16)
$\displaystyle+\varepsilon^{2}\|(1+r)^{\frac{3}{2}-\delta}\partial_{y}f\|_{L^{\infty}(\mathbb{R}^{2})}+\varepsilon^{4}\|(1+r)^{\frac{3}{2}-\delta}\partial_{y}^{2}f\|_{L^{\infty}(\mathbb{R}^{2})}.$

We claim that

$\mathcal{M}(f_{2})\in\widetilde{\mathcal{F}}_{f_{2}}.$ (5.17)

Let us analyze the terms defined in $\|\cdot\|_{*****}$ term by term. At
first, we write

$\displaystyle\mathcal{M}(f_{2})=$ $\displaystyle-
F_{0}(x,y)\int_{x}^{\infty}\frac{E_{\phi}(s,y)}{F_{0}(s,y)}ds$ (5.18)
$\displaystyle=$
$\displaystyle~{}\partial_{x}^{-1}E_{\phi}(x,y)-F_{0}(x,y)\int_{x}^{\infty}\frac{\partial_{x}^{-1}E_{\phi}(s,y)}{F_{0}^{2}(s,y)}\partial_{x}F_{0}(s,y)ds.$

For $f_{2}\in\mathcal{F}_{f_{2}}$. By the definition of $E_{\phi}$,
$\|\phi\|_{*}\leq C\varepsilon^{2}$ and $\|f_{2}\|_{*****}\leq C,$ it is not
difficult to check that

$\displaystyle|\partial_{x}^{-1}E_{\phi}(x,y)|\leq{}$ $\displaystyle
C(1+r)^{-\frac{3}{2}+\delta}\left(1+\log\frac{1}{\varepsilon}\|f\|_{*****}\|\phi\|_{*}+\varepsilon^{-\frac{3}{2}}\|\phi\|_{*}\right)$
(5.19) $\displaystyle\leq{}$ $\displaystyle
C(1+r)^{-\frac{3}{2}+\delta}(1+\varepsilon^{-\frac{3}{2}}\|\phi\|_{*}).$

Together with (5.18) we get that

$\|(1+r)^{\frac{3}{2}-\delta}\mathcal{M}(f_{2})\|_{L^{\infty}(\mathbb{R}^{2})}\leq
C(1+\varepsilon^{-\frac{3}{2}}\|\phi\|_{*}).$ (5.20)

Concerning the term $\partial_{x}f_{2}$, by (5.18) we have

$\partial_{x}\mathcal{M}(f_{2})=-\partial_{x}F_{0}(x,y)\int_{x}^{\infty}\frac{E_{\phi}(s,y)}{F_{0}(s,y)}ds+E_{\phi}(x,y).$
(5.21)

For the right hand side of (5.14) we have

$|E_{\phi}(x,y)|\leq
C(1+r)^{-\frac{3}{2}}\left(1+\|f_{2}\|_{*****}\|\phi\|_{*}+\varepsilon^{-\frac{3}{2}}\|\phi\|_{*}\right).$
(5.22)

Then it is not difficult to verify that

$\|(1+r)^{\frac{3}{2}-\delta}\partial_{x}\mathcal{M}(f_{2})\|_{L^{\infty}(\mathbb{R}^{2})}\leq
C(1+\varepsilon^{-\frac{3}{2}}\|\phi\|_{*}).$ (5.23)

Next, we study the terms $\partial_{y}\mathcal{M}(f)$ and
$\partial_{y}^{2}\mathcal{M}(f)$, by direct computation

$\displaystyle\partial_{y}\mathcal{M}(f_{2})=$
$\displaystyle-\partial_{y}F_{0}(x,y)\int_{x}^{\infty}\frac{E_{\phi}(s,y)}{F_{0}(s,y)}ds+F_{0}(x,y)\int_{x}^{\infty}\frac{E_{\phi}(s,y)\partial_{y}F_{0}(s,y)}{F_{0}^{2}(s,y)}ds$
(5.24) $\displaystyle-
F_{0}(x,y)\int_{x}^{\infty}\frac{\partial_{y}E_{\phi}(s,y)}{F_{0}(s,y)}ds.$

Using (5.22) we can estimate the first two terms of (5.24) by

$\displaystyle\left|\partial_{y}F_{0}(x,y)\int_{x}^{\infty}\frac{E_{\phi}(s,y)}{F_{0}(s,y)}ds\right|+\left|F_{0}(x,y)\int_{x}^{\infty}\frac{E_{\phi}(s,y)\partial_{y}F_{0}(s,y)}{F_{0}^{2}(s,y)}ds\right|$
(5.25) $\displaystyle\leq
C(1+r)^{-\frac{3}{2}}\left(1+\|f_{2}\|_{*****}\|\phi\|_{*}+\varepsilon^{-\frac{3}{2}}\|\phi\|_{*}\right).$

While for the last term on the right hand side of (5.24), as (5.18) we write
it as

$\displaystyle
F_{0}(x,y)\int_{x}^{\infty}\frac{\partial_{y}E_{\phi}(s,y)}{F_{0}(s,y)}ds={}$
$\displaystyle
F_{0}(x,y)\int_{x}^{\infty}\frac{\partial_{x}^{-1}\partial_{y}E_{\phi}(s,y)}{F_{0}^{2}(s,y)}\partial_{x}F_{0}(s,y)ds$
(5.26) $\displaystyle-\partial_{x}^{-1}\partial_{y}E_{\phi}(x,y).$

Using the condition $\|\phi\|_{*}\leq C\varepsilon^{2}$ and
$\|f_{2}\|_{*****}\leq C$ we get that

$|\partial_{x}^{-1}\partial_{y}E_{\phi}(x,y)|\leq
C(1+r)^{-\frac{3}{2}+\delta}\left(1+\varepsilon^{-2}\log\frac{1}{\varepsilon}\|f\|_{****}\|\phi\|_{*}+\varepsilon^{-\frac{7}{2}}\|\phi\|_{*}\right).$
(5.27)

Together with (5.24), (5.25) and (5.26) we have

$\|(1+r)^{\frac{3}{2}-\delta}\partial_{y}\mathcal{M}(f_{2})\|_{L^{\infty}(\mathbb{R}^{2})}\leq
C(1+\varepsilon^{-\frac{7}{2}}\|\phi\|_{*}).$ (5.28)

Concerning $\partial_{y}^{2}\mathcal{M}(f_{2}),$

$\displaystyle\partial_{y}^{2}\mathcal{M}(f_{2})=$
$\displaystyle-\partial_{y}^{2}F_{0}(x,y)\int_{x}^{\infty}\frac{E_{\phi}(s,y)}{F_{0}(s,y)}ds+2\partial_{y}F_{0}(x,y)\int_{x}^{\infty}\frac{E_{\phi}(s,y)\partial_{y}F_{0}(s,y)}{F_{0}^{2}(s,y)}ds$
(5.29)
$\displaystyle-2\partial_{y}F_{0}(x,y)\int_{x}^{\infty}\frac{\partial_{y}E_{\phi}(s,y)}{F_{0}(s,y)}ds+2F_{0}(x,y)\int_{x}^{\infty}\frac{\partial_{y}E_{\phi}(x,y)\partial_{y}F_{0}(s,y)}{F_{0}^{2}(s,y)}ds$
$\displaystyle+F_{0}(x,y)\int_{x}^{\infty}\frac{E_{\phi}(s,y)(\partial_{y}^{2}F_{0}(s,y)F_{0}(s,y)-2(\partial_{y}F_{0}(s,y))^{2})}{F_{0}^{3}(s,y)}ds$
$\displaystyle-
F_{0}(x,y)\int_{x}^{\infty}\frac{\partial_{y}^{2}E_{\phi}(s,y)}{F_{0}(s,y)}ds.$

For the previous five terms on the right hand side of (5.29), we could use
(5.22) and (5.27) to bound them by

$C(1+r)^{-\frac{3}{2}+\delta}\left(1+\varepsilon^{-2}\log\frac{1}{\varepsilon}\|f\|_{*****}\|\phi\|_{*}+\varepsilon^{-\frac{7}{2}}\|\phi\|_{*}\right).$
(5.30)

While for the sixth term, we write it as

$\displaystyle
F_{0}(x,y)\int_{x}^{\infty}\frac{\partial_{y}^{2}E_{\phi}(s,y)}{F_{0}(s,y)}ds={}$
$\displaystyle
F_{0}(x,y)\int_{x}^{\infty}\frac{\partial_{x}^{-1}\partial_{y}^{2}E_{\phi}(s,y)}{F_{0}^{2}(s,y)}\partial_{x}F_{0}(s,y)ds$
(5.31) $\displaystyle-\partial_{x}^{-1}\partial_{y}^{2}E_{\phi}(x,y).$

Using the condition $\|\phi\|_{*}\leq C\varepsilon^{2}$ and
$\|f_{2}\|_{*****}\leq C$,

$|\partial_{x}^{-1}\partial_{y}^{2}E_{\phi}(x,y)|\leq
C(1+r)^{-\frac{3}{2}+\delta}\left(1+\varepsilon^{-4}\log\frac{1}{\varepsilon}\|f\|_{*****}\|\phi\|_{*}+\varepsilon^{-\frac{11}{2}}\|\phi\|_{*}\right).$
(5.32)

Combined with (5.29)-(5.32) we get that

$\|(1+r)^{\frac{3}{2}-\delta}\partial_{y}^{2}\mathcal{M}(f_{2})\|_{L^{\infty}(\mathbb{R}^{2})}\leq
C(1+\varepsilon^{-\frac{11}{2}}\|\phi\|_{*}).$ (5.33)

Using (5.20), (5.23), (5.28), (5.33) we get

$\|\mathcal{M}(f_{2})\|_{*****}\leq
C(1+\varepsilon^{-\frac{3}{2}}\|\phi\|_{*}).$ (5.34)

It proves (5.17).

Next we shall show that $\mathcal{M}$ is a contraction map in
$\widetilde{\mathcal{F}}_{f_{2}}$. For any
$f_{2,1},f_{2,2}\in\widetilde{\mathcal{F}}_{f_{2}}$, we have

$\mathcal{M}(f_{2,1})-\mathcal{M}(f_{2,2})=-F_{0}(x,y)\int_{x}^{\infty}\frac{H(f_{2,1},f_{2,2})}{F_{0}(s,y)}ds,$

where

$\displaystyle
H(f_{2,1},f_{2,2})=-2\phi(f_{2,1}-f_{2,2})-2\varepsilon^{2}f_{1}g_{1}(f_{2,1}-f_{2,2})-\varepsilon^{4}g_{1}(f_{2,1}-f_{2,2})(f_{2,1}+f_{2,2}).$

By the condition $\|\phi\|_{*}\leq C\varepsilon^{2}$, we could show that

$\displaystyle\|(1+r)^{\frac{3}{2}-\delta}\partial_{x}^{-1}H(f_{2,1},f_{2,2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq C\varepsilon^{\frac{3}{2}}\|f_{2,1}-f_{2,2}\|_{*****},$
(5.35)
$\displaystyle\|(1+r)^{\frac{3}{2}-\delta}H(f_{2,1},f_{2,2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq C\varepsilon^{\frac{3}{2}}\|f_{2,1}-f_{2,2}\|_{*****},$
$\displaystyle\|(1+r)^{\frac{3}{2}-\delta}\partial_{y}H(f_{2,1},f_{2,2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq C\|f_{2,1}-f_{2,2}\|_{*****},$
$\displaystyle\|(1+r)^{\frac{3}{2}-\delta}\partial_{y}^{2}H(f_{2,1},f_{2,2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq C\varepsilon^{-2}\|f_{2,1}-f_{2,2}\|_{*****}.$

From the above inequality (5.35) one can prove that

$\|\mathcal{M}(f_{2,1})-\mathcal{M}(f_{2,2})\|_{*****}\leq
C\varepsilon^{\frac{3}{2}}\|f_{2,1}-f_{2,2}\|_{*****}.$ (5.36)

Then using the contraction mapping principle we deduce the existence of
solution $f_{2}\in\widetilde{\mathcal{F}}_{f_{2}}$.

In order to show that $\mathcal{M}(f_{2})\in\mathcal{F}_{f_{2}}$, we have to
obtain the corresponding estimation on the following terms
$\partial_{x}^{2}\mathcal{M}(f_{2}),~{}\partial_{x}^{3}\mathcal{M}(f_{2})$,
$\partial_{x}\partial_{y}\mathcal{M}(f_{2})$ and
$\partial_{x}\partial_{y}^{2}\mathcal{M}(f_{2})$. Since the argument is almost
similar, we shall use $\partial_{x}\partial_{y}\mathcal{M}(f_{2})$ as an
example to explain how to derive the estimation. We have already proved the
existence of a solution to (5.14) in $\widetilde{\mathcal{F}}_{f_{2}}$, we
denote the solution by $f_{2}$ and it satisfies (5.14). We differentiate
(5.14) with respect to $y$, we get that

$\displaystyle(\sqrt{2}-\varepsilon^{2})\partial_{x}\partial_{y}f_{2}=$
$\displaystyle-2q\partial_{y}f_{2}-2\partial_{y}qf_{2}-2\partial_{y}\phi
f_{2}-2\phi\partial_{y}f_{2}+\partial_{y}^{3}(q+\phi)$ (5.37)
$\displaystyle+\partial_{x}\partial_{y}\left(\frac{\sqrt{2}}{2}\partial_{x}(q+\phi)-\frac{1}{2}(q+\phi)^{2}\right)$
$\displaystyle-\partial_{y}\left(\left[\left(\frac{\sqrt{2}}{2}\partial_{x}(q+\phi)-\frac{1}{2}(q+\phi)^{2}\right)+\varepsilon^{2}f_{2}\right]^{2}(q+\phi)\right).$

Using $\phi\in\mathcal{F}_{\phi}$ and $f_{2}\in\widetilde{F}(f_{2})$, we see
the right hand side of (5.37) is bounded by

$C(1+r)^{-\frac{3}{2}+\delta}\left(1+\varepsilon^{-2}\|f_{2}\|_{*****}+\varepsilon^{-2}\|\phi\|_{*}\|f_{2}\|_{*****}+\varepsilon^{-\frac{7}{2}}\|\phi\|_{*}\right).$
(5.38)

As a consequence,

$\varepsilon^{2}\|(1+r)^{\frac{3}{2}-\delta}\partial_{x}\partial_{y}f_{2}\|_{L^{\infty}(\mathbb{R}^{2})}\leq
C(1+\|f_{2}\|_{*****}+\varepsilon^{-\frac{3}{2}}\|\phi\|_{*}).$ (5.39)

Similarly, we can show that

$\displaystyle\|(1+r)^{\frac{3}{2}-\delta}\partial_{x}^{2}f_{2}\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq
C(1+\|f_{2}\|_{*****}+\varepsilon^{-\frac{3}{2}}\|\phi\|_{*}),$ (5.40)
$\displaystyle\varepsilon\|(1+r)^{\frac{3}{2}-\delta}\partial_{x}^{3}f_{2}\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq
C(1+\|f_{2}\|_{*****}+\varepsilon^{-\frac{3}{2}}\|\phi\|_{*}),$
$\displaystyle\varepsilon^{4}\|(1+r)^{\frac{3}{2}-\delta}\partial_{x}\partial_{y}^{2}f_{2}\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq
C(1+\|f_{2}\|_{*****}+\varepsilon^{-\frac{3}{2}}\|\phi\|_{*}).$

From (5.34), (5.39) and (5.40) we have seen the solution
$f_{2}\in\mathcal{F}_{f_{2}}$.

It remains to prove (5.11), we shall first show that

$\|f_{2,\phi_{1}}-f_{2,\phi_{2}}\|_{*****}\leq
C\varepsilon^{-\frac{3}{2}}\|\phi_{1}-\phi_{2}\|_{*}.$ (5.41)

We notice that the function $f_{2,\phi_{1}}-f_{2,\phi_{2}}$ satisfies

$\displaystyle(\sqrt{2}-\varepsilon^{2})\partial_{x}(f_{2,\phi_{1}}-f_{2,\phi_{2}})+2q(f_{2,\phi_{1}}-f_{2,\phi_{2}})$
(5.42)
$\displaystyle=-2\phi_{1}(f_{2,\phi_{1}}-f_{2,\phi_{2}})-2(\phi_{1}-\phi_{2})f_{2,\phi_{2}}+\partial_{y}^{2}(\phi_{1}-\phi_{2})$
$\displaystyle\quad+\frac{\sqrt{2}}{2}\partial_{x}^{2}(\phi_{1}-\phi_{2})-\partial_{x}(q\phi_{1}-q\phi_{2})-\frac{1}{2}\partial_{x}(\phi_{1}^{2}-\phi_{2}^{2})$
$\displaystyle\quad-\left(\frac{\sqrt{2}}{2}\partial_{x}(q+\phi_{1})-\frac{1}{2}(q+\phi_{1})^{2}\right)^{2}(q+\phi_{1})$
$\displaystyle\quad+\left(\frac{\sqrt{2}}{2}\partial_{x}(q+\phi_{2})-\frac{1}{2}(q+\phi_{2})^{2}\right)^{2}(q+\phi_{2})$
$\displaystyle\quad-\varepsilon^{2}(\sqrt{2}\partial_{x}(q+\phi_{1})-(q+\phi_{1})^{2})(q+\phi_{1})(f_{2,\phi_{1}}-f_{2,\phi_{2}})$
$\displaystyle\quad+\varepsilon^{2}\left(\frac{\sqrt{2}}{2}\partial_{x}(q+\phi_{1})^{2}-(q+\phi_{1})^{3}-\frac{\sqrt{2}}{2}\partial_{x}(q+\phi_{2})^{2}+(q+\phi_{2})^{3}\right)f_{2,\phi_{2}}$
$\displaystyle\quad-\varepsilon^{4}(q+\phi_{1})(f_{2,\phi_{1}}^{2}-f_{2,\phi_{2}}^{2})-\varepsilon^{4}f_{2,\phi_{2}}^{2}(\phi_{1}-\phi_{2})$
$\displaystyle=\Gamma_{1}(f_{2,\phi_{1}},f_{2,\phi_{2}})+\Gamma_{2}(\phi_{1},\phi_{2}),$

where

$\displaystyle\Gamma_{1}(f_{2,\phi_{1}},f_{2,\phi_{2}})=$
$\displaystyle-2\phi_{1}(f_{2,\phi_{1}}-f_{2,\phi_{2}})-\varepsilon^{4}(q+\phi_{1})(f_{2,\phi_{1}}^{2}-f_{2,\phi_{2}}^{2})$
$\displaystyle-\varepsilon^{2}(\sqrt{2}\partial_{x}(q+\phi_{1})-(q+\phi_{1})^{2})(q+\phi_{1})(f_{2,\phi_{1}}-f_{2,\phi_{2}}),$

and $\Gamma_{2}(\phi_{1},\phi_{2})$ collects the left terms in (5.42). As
(5.35) we get that

$\displaystyle\|(1+r)^{\frac{3}{2}-\delta}\partial_{x}^{-1}\Gamma_{1}(f_{2,\phi_{1}},f_{2,\phi_{2}})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq
C\varepsilon^{\frac{3}{2}}\|f_{2,\phi_{1}}-f_{2,\phi_{2}}\|_{*****},$ (5.43)
$\displaystyle\|(1+r)^{\frac{3}{2}-\delta}\Gamma_{1}(f_{2,\phi_{1}},f_{2,\phi_{2}})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq
C\varepsilon^{\frac{3}{2}}\|f_{2,\phi_{1}}-f_{2,\phi_{2}}\|_{*****},$
$\displaystyle\|(1+r)^{\frac{3}{2}-\delta}\partial_{y}\Gamma_{1}(f_{2,\phi_{1}},f_{2,\phi_{2}})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq
C\log\frac{1}{\varepsilon}\|f_{2,\phi_{1}}-f_{2,\phi_{2}}\|_{*****},$
$\displaystyle\|(1+r)^{\frac{3}{2}-\delta}\partial_{y}^{2}\Gamma_{1}(f_{2,\phi_{1}},f_{2,\phi_{2}})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq
C\varepsilon^{-2}\log\frac{1}{\varepsilon}\|f_{2,\phi_{1}}-f_{2,\phi_{2}}\|_{*****},$

and

$\displaystyle\|(1+r)^{\frac{3}{2}-\delta}\partial_{x}^{-1}\Gamma_{2}(\phi_{1},\phi_{2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq C\varepsilon^{-\frac{3}{2}}\|\phi_{1}-\phi_{2}\|_{*},$
(5.44)
$\displaystyle\|(1+r)^{\frac{3}{2}-\delta}\Gamma_{2}(\phi_{1},\phi_{2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq C\varepsilon^{-\frac{3}{2}}\|\phi_{1}-\phi_{2}\|_{*},$
$\displaystyle\|(1+r)^{\frac{3}{2}-\delta}\partial_{y}\Gamma_{2}(\phi_{1},\phi_{2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq C\varepsilon^{-\frac{7}{2}}\|\phi_{1}-\phi_{2}\|_{*},$
$\displaystyle\|(1+r)^{\frac{3}{2}-\delta}\partial_{y}^{2}\Gamma_{2}(\phi_{1},\phi_{2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq C\varepsilon^{-\frac{11}{2}}\|\phi_{1}-\phi_{2}\|_{*},$

Using (5.43) and (5.44) we could argue as from (5.18) to (5.34) to conclude
that

$\|f_{2,\phi_{1}}-f_{2,\phi_{2}}\|_{*****}\leq
C\varepsilon^{\frac{3}{2}}\|f_{2,\phi_{1}}-f_{2,\phi_{2}}\|_{*****}+C\varepsilon^{-\frac{3}{2}}\|\phi_{1}-\phi_{2}\|_{*},$
(5.45)

and it implies that

$\|f_{2,\phi_{1}}-f_{2,\phi_{2}}\|_{*****}\leq
C\varepsilon^{-\frac{3}{2}}\|\phi_{1}-\phi_{2}\|_{*}.$ (5.46)

Next we use equation (5.42) and argue as (5.37)-(5.40) to derive that

$\displaystyle\|(1+r)^{\frac{3}{2}-\delta}\partial_{x}^{2}(f_{2,\phi_{1}}-f_{2,\phi_{2}})\|_{L^{\infty}(\mathbb{R}^{2})}\leq
C(\|f_{2,\phi_{1}}-f_{2,\phi_{2}}\|_{*****}+\varepsilon^{-\frac{3}{2}}\|\phi_{1}-\phi_{2}\|_{*}),$
(5.47)
$\displaystyle\varepsilon\|(1+r)^{\frac{3}{2}-\delta}\partial_{x}^{3}(f_{2,\phi_{1}}-f_{2,\phi_{2}})\|_{L^{\infty}(\mathbb{R}^{2})}\leq
C(\|f_{2,\phi_{1}}-f_{2,\phi_{2}}\|_{*****}+\varepsilon^{-\frac{3}{2}}\|\phi_{1}-\phi_{2}\|_{*}),$
$\displaystyle\varepsilon^{2}\|(1+r)^{\frac{3}{2}-\delta}\partial_{y}\partial_{x}(f_{2,\phi_{1}}-f_{2,\phi_{2}})\|_{L^{\infty}(\mathbb{R}^{2})}\leq
C(\|f_{2,\phi_{1}}-f_{2,\phi_{2}}\|_{*****}+\varepsilon^{-\frac{3}{2}}\|\phi_{1}-\phi_{2}\|_{*}),$
$\displaystyle\varepsilon^{4}\|(1+r)^{\frac{3}{2}-\delta}\partial_{y}^{2}\partial_{x}(f_{2,\phi_{1}}-f_{2,\phi_{2}})\|_{L^{\infty}(\mathbb{R}^{2})}\leq
C(\|f_{2,\phi_{1}}-f_{2,\phi_{2}}\|_{*****}+\varepsilon^{-\frac{3}{2}}\|\phi_{1}-\phi_{2}\|_{*}).$

Together with (5.46) we get (5.11) holds. ∎

Now we are able to give the proof to Theorem 1.1.

###### Proof of Theorem 1.1..

With Lemma 5.1 we have already seen that for a given function $\phi$, we could
find a solution to (5.3). Therefore, the original problem is equivalent to
finding a solution to (5.4). We write (5.4) as

$\displaystyle\partial_{x}^{4}\phi-(2\sqrt{2}-\varepsilon^{2})\partial_{x}^{2}\phi-6(\sqrt{2}-\varepsilon^{2})\partial_{x}(q\phi)-2\partial_{y}^{2}\phi+2\varepsilon^{2}\partial_{x}^{2}\partial_{y}^{2}\phi+\varepsilon^{4}\partial_{y}^{4}\phi$
(5.48) $\displaystyle=\hat{P}_{1}+P_{2}+P_{3},$

where

$\hat{P}_{1}=P_{1}+\Gamma_{q}+3(\sqrt{2}-\varepsilon^{2})\partial_{x}((\partial_{x}\phi)^{2}).$

We shall find a solution of (5.48) in (5.8) via fixed point argument.

For any $\phi\in\mathcal{F}_{\phi}$, we first claim that

$\displaystyle\|\hat{P}_{1}\|_{**}\leq C\varepsilon^{2}.$ (5.49)

To prove the above claim, we have to check all the terms in $\hat{P}_{1}$ are
bounded by $C\varepsilon^{2}$ in the norm of $\|\cdot\|_{**}.$ For example, we
use $\partial_{x}(\partial_{x}\phi)^{2}$ and
$\varepsilon^{2}\partial_{x}(\partial_{y}g_{1})^{2}$ as examples to explain,
while the other terms can be handled similarly, we leave the details to the
interested reader. For $\partial_{x}(\partial_{x}\phi)^{2}$,

$\displaystyle\|(1+r)^{\frac{5}{2}-\delta}(\partial_{x}\phi)^{2}\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq
C\|(1+r)^{\frac{3}{2}-\delta}\partial_{x}\phi\|^{2}_{L^{\infty}(\mathbb{R}^{2})}\leq
C\|\phi\|_{*}^{2},$ (5.50)
$\displaystyle\|(1+r)^{\frac{5}{2}-\delta}\partial_{x}(\partial_{x}\phi)^{2}\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq
C\|\phi\|_{*}\|(1+r)^{\frac{3}{2}}\partial_{x}^{2}\phi\|_{L^{\infty}(\mathbb{R}^{2})}\leq
C\|\phi\|_{*}^{2},$
$\displaystyle\|(1+r)^{\frac{5}{2}-\delta}\partial_{x}^{2}(\partial_{x}\phi)^{2}\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq
C\|\phi\|_{*}\|(1+r)^{\frac{3}{2}}\partial_{x}^{3}\phi\|_{L^{\infty}(\mathbb{R}^{2})}+C\|\phi\|_{*}^{2}$
$\displaystyle\leq C\varepsilon^{-\frac{1}{2}}\|\phi\|_{*}^{2}.$

Thus, we have

$\displaystyle\|(\partial_{x}\phi)^{2}\|_{**}\leq C\varepsilon^{2}.$ (5.51)

While for the later one, we can check that

$\displaystyle\varepsilon^{2}\|(1+r)^{\frac{5}{2}-\delta}(\partial_{y}g_{1})^{2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq
C\varepsilon^{2}+C\varepsilon^{\frac{3}{2}}\|\phi\|_{*}+C\varepsilon\|\phi\|_{*}^{2},$
(5.52)
$\displaystyle\varepsilon^{2}\|(1+r)^{\frac{5}{2}-\delta}\partial_{x}(\partial_{y}g_{1})^{2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq
C\varepsilon^{2}+C\varepsilon^{\frac{3}{2}}\|\phi\|_{*}+C\varepsilon\|\phi\|_{*}^{2},$
$\displaystyle\varepsilon^{2}\|(1+r)^{\frac{5}{2}-\delta}\partial_{x}^{2}(\partial_{y}g_{1})^{2}\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq
C\varepsilon^{2}+C\varepsilon^{\frac{3}{2}}\|\phi\|_{*}+C\varepsilon\|\phi\|_{*}^{2}.$

It implies that

$\|(\partial_{y}g_{1})^{2}\|_{**}\leq C\varepsilon^{2}.$ (5.53)

As (5.49), we can prove that

$\|P_{2}\|_{***}\leq C\varepsilon^{2}.$ (5.54)

Concerning $P_{3}$, we could show that each term decays faster than $r^{-3}$,
then $\partial_{x}^{-1}P_{3}$ is well defined. Subsequently we could absorb
$P_{3}$ into $\hat{P}_{1}$ and prove that $\|\partial_{x}^{-1}P_{3}\|_{**}\leq
C\varepsilon^{2}$. For example, we shall use the term
$g_{1}\partial_{y}^{2}(g_{1}^{2})$ to illustrate this point. It is easy to see
that

$\displaystyle\varepsilon^{2}\|(1+r)^{\frac{7}{2}-\delta}g_{1}\partial_{y}^{2}(g_{1}^{2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq
C\varepsilon^{2}+C\varepsilon^{\frac{1}{2}}\|\phi\|_{*}+C\varepsilon^{-1}\|\phi\|_{*}^{2}+C\varepsilon^{-2}\|\phi\|_{*}^{3},$
(5.55)
$\displaystyle\varepsilon^{2}\|(1+r)^{\frac{7}{2}-\delta}\partial_{x}(g_{1}\partial_{y}^{2}(g_{1}^{2}))\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq
C\varepsilon^{2}+C\varepsilon^{\frac{1}{2}}\|\phi\|_{*}+C\varepsilon^{-1}\|\phi\|_{*}^{2}+C\varepsilon^{-2}\|\phi\|_{*}^{3}.$

Then one can easily verify that

$\|\partial_{x}^{-1}(g_{1}\partial_{y}^{2}(g_{1}^{2}))\|_{**}\leq
C\varepsilon^{2}.$ (5.56)

Repeating the same process for the other terms in $P_{3}$ we derive that

$\|\partial_{x}^{-1}(P_{3})\|_{**}\leq C\varepsilon^{2}.$ (5.57)

With (5.49), (5.54) and (5.57) and Proposition 4.2, we find out a solution
(denoted by $\mathcal{N}(\phi)$) to (5.48) such that

$\|\mathcal{N}(\phi)\|_{*}\leq
C(\|\hat{P}_{1}\|_{**}+\|P_{2}\|_{***}+\|\partial_{x}^{-1}P_{3}\|_{**})\leq
C\varepsilon^{2}.$ (5.58)

Therefore we could define a map $\phi\to\mathcal{N}(\phi)$ which sends
$\mathcal{F}_{\phi}$ to itself provided $C$ is large enough.

Next, we shall prove the map $\phi\to\mathcal{N}(\phi)$ is a contraction map.
We claim that for any two functions $\phi_{1},\phi_{2}\in\mathcal{F}_{\phi}$,

$\displaystyle\|\hat{P}_{1}(\phi_{1})-\hat{P}_{1}(\phi_{2})\|_{**}$
$\displaystyle\leq C\varepsilon^{\frac{1}{2}}\|\phi_{1}-\phi_{2}\|_{*},$
(5.59) $\displaystyle\|P_{2}(\phi_{1})-P_{2}(\phi_{2})\|_{***}$
$\displaystyle\leq C\varepsilon^{\frac{1}{2}}\|\phi_{1}-\phi_{2}\|_{*},$
$\displaystyle\|\partial_{x}^{-1}P_{3}(\phi_{1})-\partial_{x}^{-1}P_{3}(\phi_{2})\|_{**}$
$\displaystyle\leq C\varepsilon^{\frac{1}{2}}\|\phi_{1}-\phi_{2}\|_{*}.$

We shall pick out one term from $\hat{P}_{1}$, $P_{2}$ and $P_{3}$ (see
(2.27)-(2.29)) respectively as examples to prove (5.59). To stress the
dependence of $f_{1},f_{2},g_{1}$ on $\phi$, we replace them by
$f_{1,\phi},f_{2,\phi},g_{1,\phi}.$ For $\hat{P}_{1}$, we take
$\varepsilon^{4}\partial_{x}(f_{1}f_{2})$ into consideration, we have

$\displaystyle\varepsilon^{4}(f_{1,\phi_{1}}f_{2,\phi_{1}}-f_{1,\phi_{2}}f_{2,\phi_{2}})$
(5.60)
$\displaystyle=\varepsilon^{4}(f_{1,\phi_{1}}-f_{1,\phi_{2}})f_{2,\phi_{1}}+\varepsilon^{4}f_{1,\phi_{2}}(f_{2,\phi_{1}}-f_{2,\phi_{2}})$
$\displaystyle=\varepsilon^{4}\left(\left(\frac{\sqrt{2}}{2}\partial_{x}(q+\phi_{1})-\frac{1}{2}(q+\phi_{1})^{2}\right)-\left(\frac{\sqrt{2}}{2}\partial_{x}(q+\phi_{2})-\frac{1}{2}(q+\phi_{2})^{2}\right)\right)f_{2,\phi_{1}}$
$\displaystyle\quad+\varepsilon^{4}f_{1,\phi_{2}}(f_{2,\phi_{1}}-f_{2,\phi_{2}})$
$\displaystyle\leq
C\varepsilon^{4}(1+r)^{-\frac{5}{2}+\delta}\|(1+r)^{\frac{3}{2}-\delta}\partial_{x}(\phi_{1}-\phi_{2})\|_{L^{\infty}(\mathbb{R}^{2})}\|(1+r)^{\frac{3}{2}-\delta}f_{2,\phi_{1}}\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\quad+C\varepsilon^{4}(1+r)^{-\frac{5}{2}+\delta}\|(1+r)^{\frac{3}{2}-\delta}(\phi_{1}-\phi_{2})\|_{L^{\infty}(\mathbb{R}^{2})}\|(1+r)^{\frac{3}{2}-\delta}f_{2,\phi_{1}}\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\quad+C\varepsilon^{4}(1+r)^{-\frac{5}{2}+\delta}\|(1+r)^{\frac{3}{2}-\delta}(\phi_{1}^{2}-\phi_{2}^{2})\|_{L^{\infty}(\mathbb{R}^{2})}\|(1+r)^{\frac{3}{2}-\delta}f_{2,\phi_{1}}\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\quad+C\varepsilon^{4}(1+r)^{-\frac{5}{2}+\delta}(1+\|\phi\|_{*})\|(1+r)^{\frac{3}{2}-\delta}(f_{2,\phi_{1}}-f_{2,\phi_{2}})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq
C(1+r)^{-\frac{5}{2}+\delta}\varepsilon^{\frac{1}{2}}\|\phi_{1}-\phi_{2}\|_{*},$

where we used Lemma 5.1. Similarly, one can also show that

$\displaystyle\|(1+r)^{\frac{5}{2}-\delta}\partial_{x}(f_{1,\phi_{1}}f_{2,\phi_{1}}-f_{1,\phi_{2}}f_{2,\phi_{2}})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq C\varepsilon^{\frac{1}{2}}\|\phi_{1}-\phi_{2}\|_{*},$
(5.61)
$\displaystyle\|(1+r)^{\frac{5}{2}-\delta}\partial_{x}^{2}(f_{1,\phi_{1}}f_{2,\phi_{1}}-f_{1,\phi_{2}}f_{2,\phi_{2}})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq C\varepsilon^{\frac{1}{2}}\|\phi_{1}-\phi_{2}\|_{*}.$

Using (5.60) and (5.61) we have

$\varepsilon^{4}\|f_{1,\phi_{1}}f_{2,\phi_{1}}-f_{1,\phi_{2}}f_{2,\phi_{2}}\|_{**}\leq
C\varepsilon^{\frac{1}{2}}\|\phi_{1}-\phi_{2}\|_{*}.$ (5.62)

For $P_{2}$, we consider
$\varepsilon^{2}\partial_{y}(\partial_{x}g_{1}\partial_{y}g_{1})$, by direct
computation, we have

$\displaystyle|\partial_{x}g_{1,\phi_{1}}\partial_{y}g_{1,\phi_{1}}-\partial_{x}g_{1,\phi_{2}}\partial_{y}g_{1,\phi_{2}}|$
(5.63)
$\displaystyle=|\partial_{x}(q+\phi_{1})\partial_{y}(q+\phi_{1})-\partial_{x}(q+\phi_{2})\partial_{y}(q+\phi_{2})|$
$\displaystyle\leq|\partial_{x}q\partial_{y}(\phi_{1}-\phi_{2})|+|\partial_{y}q\partial_{x}(\phi_{1}-\phi_{2})|+|\partial_{x}\phi_{1}\partial_{y}\phi_{1}-\partial_{x}\phi_{2}\partial_{y}\phi_{2}|$
$\displaystyle\leq
C\varepsilon^{-\frac{1}{2}}(1+r)^{-3+\delta}\|\phi_{1}-\phi_{2}\|_{*}(1+\|\phi_{1}\|_{*}+\|\phi_{2}\|_{*})$
$\displaystyle\leq
C\varepsilon^{-\frac{1}{2}}(1+r)^{-3+\delta}\|\phi_{1}-\phi_{2}\|_{*}.$

In a similar way as (5.63), one can check that

$\displaystyle\varepsilon^{2}\|(1+r)^{3-\delta}\partial_{y}(\partial_{x}g_{1,\phi_{1}}\partial_{y}g_{1,\phi_{1}}-\partial_{x}g_{1,\phi_{2}}\partial_{y}g_{1,\phi_{2}})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq C\varepsilon^{\frac{1}{2}}\|\phi_{1}-\phi_{2}\|_{*},$
(5.64)
$\displaystyle\varepsilon^{2}\|(1+r)^{3-\delta}\partial_{x}\partial_{y}(\partial_{x}g_{1,\phi_{1}}\partial_{y}g_{1,\phi_{1}}-\partial_{x}g_{1,\phi_{2}}\partial_{y}g_{1,\phi_{2}})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq C\varepsilon^{\frac{1}{2}}\|\phi_{1}-\phi_{2}\|_{*}.$

As a consequence of (5.63) and (5.64) we get

$\varepsilon^{2}\|\partial_{x}g_{1,\phi_{1}}\partial_{y}g_{1,\phi_{1}}-\partial_{x}g_{1,\phi_{2}}\partial_{y}g_{1,\phi_{2}}\|_{***}\leq
C\varepsilon^{\frac{1}{2}}\|\phi_{1}-\phi_{2}\|_{*}.$ (5.65)

While for $P_{3},$ we study the term
$\varepsilon^{4}\partial_{y}g_{1}\partial_{y}(g_{1}^{2})$,

$\displaystyle\|(1+r)^{\frac{7}{2}-\delta}\left(\partial_{y}g_{1,\phi_{1}}\partial_{y}(g_{1,\phi_{1}}^{2})-\partial_{y}g_{1,\phi_{2}}\partial_{y}(g_{1,\phi_{2}}^{2})\right)\|_{L^{\infty}(\mathbb{R}^{2})}$
(5.66)
$\displaystyle=\|(1+r)^{\frac{7}{2}-\delta}(\partial_{y}(q+\phi_{1})\partial_{y}((q+\phi_{1})^{2})-\partial_{y}(q+\phi_{2})\partial_{y}((q+\phi_{2})^{2}))\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq
C\|(1+r)^{\frac{7}{2}-\delta}(\partial_{y}q)^{2}(\phi_{1}-\phi_{2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\quad+C\|(1+r)^{\frac{7}{2}-\delta}q\partial_{y}q\partial_{y}(\phi_{1}-\phi_{2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\quad+C\|(1+r)^{\frac{7}{2}-\delta}\partial_{y}q(\partial_{y}\phi_{1}\phi_{1}-\partial_{y}\phi_{2}\phi_{2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\quad+C\|(1+r)^{\frac{7}{2}-\delta}q((\partial_{y}\phi_{1})^{2}-(\partial_{y}\phi_{2})^{2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\quad+C\|(1+r)^{\frac{7}{2}-\delta}(\phi_{1}(\partial_{y}\phi_{1})^{2}-\phi_{2}(\partial_{y}\phi_{2})^{2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq
C\varepsilon^{-\frac{1}{2}}\|\phi_{1}-\phi_{2}\|_{*}+C\varepsilon^{-1}\|\phi_{1}-\phi_{2}\|^{2}_{*}+C\varepsilon^{-1}\|\phi_{1}-\phi_{2}\|_{*}^{3},$

and

$\displaystyle\|(1+r)^{\frac{5}{2}-\delta}\partial_{x}\left(\partial_{y}g_{1,\phi_{1}}\partial_{y}(g_{1,\phi_{1}}^{2})-\partial_{y}g_{1,\phi_{2}}\partial_{y}(g_{1,\phi_{2}}^{2})\right)\|_{L^{\infty}(\mathbb{R}^{2})}$
(5.67)
$\displaystyle=\|(1+r)^{\frac{5}{2}-\delta}\partial_{x}(\partial_{y}(q+\phi_{1})\partial_{y}((q+\phi_{1})^{2})-\partial_{y}(q+\phi_{2})\partial_{y}((q+\phi_{2})^{2}))\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq
C\|(1+r)^{\frac{5}{2}-\delta}\partial_{x}((\partial_{y}q)^{2})(\phi_{1}-\phi_{2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\quad+C\|(1+r)^{\frac{5}{2}-\delta}(\partial_{y}q)^{2}\partial_{x}(\phi_{1}-\phi_{2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\quad+C\|(1+r)^{\frac{5}{2}-\delta}\partial_{x}(q\partial_{y}q)\partial_{y}(\phi_{1}-\phi_{2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\quad+C\|(1+r)^{\frac{5}{2}-\delta}q\partial_{y}q\partial_{x}\partial_{y}(\phi_{1}-\phi_{2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\quad+C\|(1+r)^{\frac{5}{2}-\delta}\partial_{x}(\partial_{y}q)(\partial_{y}\phi_{1}\phi_{1}-\partial_{y}\phi_{2}\phi_{2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\quad+C\|(1+r)^{\frac{5}{2}-\delta}\partial_{y}q\partial_{x}(\partial_{y}\phi_{1}\phi_{1}-\partial_{y}\phi_{2}\phi_{2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\quad+C\|(1+r)^{\frac{5}{2}-\delta}\partial_{x}q((\partial_{y}\phi_{1})^{2}-(\partial_{y}\phi_{2})^{2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\quad+C\|(1+r)^{\frac{5}{2}-\delta}q\partial_{x}((\partial_{y}\phi_{1})^{2}-(\partial_{y}\phi_{2})^{2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\quad+C\|(1+r)^{\frac{5}{2}-\delta}\partial_{x}(\phi_{1}(\partial_{y}\phi_{1})^{2}-\phi_{2}(\partial_{y}\phi_{2})^{2})\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\leq
C\varepsilon^{-\frac{1}{2}}\|\phi_{1}-\phi_{2}\|_{*}+C\varepsilon^{-1}\|\phi_{1}-\phi_{2}\|^{2}_{*}+C\varepsilon^{-1}\|\phi_{1}-\phi_{2}\|_{*}^{3}.$

From (5.66) ad (5.67) we derive that

$\varepsilon^{4}\|\partial_{x}^{-1}\left(\partial_{y}g_{1,\phi_{1}}\partial_{y}((g_{1,\phi_{1}}^{2}))-\partial_{y}g_{1,\phi_{2}}\partial_{y}((g_{1,\phi_{2}}^{2}))\right)\|_{**}\leq
C\varepsilon^{\frac{1}{2}}\|\phi_{1}-\phi_{2}\|_{*}.$ (5.68)

The other terms in $\hat{P}_{1},~{}P_{2}$ and $P_{3}$ can be treated in a
similar way and we leave the computations to the interested reader.

Using the claim (5.59) and Proposition 4.2, we derive that

$\|\mathcal{N}(\phi_{1})-\mathcal{N}(\phi_{2})\|_{*}\leq
C\varepsilon^{\frac{1}{2}}\|\phi_{1}-\phi_{2}\|_{*}.$ (5.69)

Then we deduce the existence of a solution $\phi$ to (5.48) by contraction
principle. Thereby the original existence result is established. ∎

Acknowledgement

Y. Liu is partially supported by NSFC No. 11971026 and The Fundamental
Research Funds for the Central Universities WK3470000014. Z. Wang is partially
supported by NSFC No.11871386 and NSFC No.11931012. J. Wei is partially
supported by NSERC of Canada. W. Yang is partially supported by NSFC
No.11801550, No.11871470 and No.12171456.

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