Kuramoto model, phase synchronisation and the network’s core
Raúl J. Mondragón

School of Electronic Engineering and Computer Science

Queen Mary University of London

London E1 4NS

UK

In this note we show that for the Kuramoto model defined in a simple
undirected graph it is possible to decide which nodes form the core of the
network. The set of core–nodes is defined by its relevance to the phase
synchronisation process.

The Kuramoto model is a nonlinear system of $N$ coupled oscillators with
random frequencies $\omega_{n}$ and phases $\theta_{n}$ described by

$\dot{\theta}_{n}=\omega_{n}+K\sum_{m=1}^{N}A_{nm}\sin(\theta_{m}-\theta_{n}),$
(1)

where $A_{nm}$ is the coupling term between phase $n$ and phase $m$ with
strength $K$. If the strength $K$ is large enough the phase of, some or all,
the oscillators tend to synchronise.

If the coupling term is described by the simple–undirected graph ${\cal
G}(E,V)$ then $A_{nm}$ are the entries of the adjacency matrix ${\bf A}$.
Restrepo et. al [1] introduced the time–average approximation to estimate the
critical value $K_{c}$ when the phases start synchronising. They describe the
oscillators dynamics using local mean variable $r_{n}$

$r_{n}e^{i\psi_{n}}=\sum_{m=1}^{N}A_{nm}\langle
e^{i\theta_{m}}\rangle_{t},\quad n=1,\ldots,N,$ (2)

where $\langle\ldots\rangle_{t}$ is a time–average. Under the assumptions that
the minimum degree is large ${d}_{\rm min}\gg 1$ and that the solutions of
$r_{n}$, $\psi_{n}$ are statistically independent of $\omega_{n}$ then

$r_{n}=\sum_{|\omega_{m}|\leq
Kr_{m}}A_{nm}\sqrt{1-\left(\frac{\omega_{m}}{Kr_{m}}\right)^{2}}.$ (3)

The critical value $K_{c}$ is estimated considering that the oscillators are
not described by a particular sequence of natural frequencies
$\\{\omega_{i}\\}$ but rather by frequencies distributed according a unimodal
probability density $g(\omega)$ with $g(-\omega)=g(\omega)$ then
$K_{c}={K_{0}}/{\Lambda_{1}}$, where $K_{0}=2/(\pi g(0))$ and $\Lambda_{1}$ is
the largest eigenvalue of the adjacency matrix ${\bf A}$.

The purpose of this short communication is to define a network–core based on
the high degree nodes and their relevance to the synchronisation process. The
approach is to relate the connectivity of the high degree nodes with the
eigenvalue $\Lambda_{1}$ and hence to the critical coupling value.

Assume that the nodes are ranked in decreasing order of their degree and that
the top $m$ ranked nodes are the core of the network. An approximation to the
eigenvector which corresponds to $\Lambda_{1}$ is constructed by considering
the vector ${\bf v}(m)$ where the entry ${v}_{n}(m)$ is 1 if the node $n$
shares a link with a node in the core and zero otherwise. The entries
${u}_{n}(m)$ of ${\bf u}(m)={\bf A}{\bf v}(m)$ are the number of walks of
length 1 which start in the node $n$ and end in a node belonging to the core.
If $m=N$ then ${\bf u}_{n}(N)={d}_{n}$ is the degree of node $n$. Based on the
Rayleigh quotient an approximation $b(m)$ to the eigenvalue is

$b(m)=\frac{{\bf u}(m)^{\tt T}{\bf A}{\bf u}(m)}{{\bf u}(m)^{\tt T}{\bf
u}(m)}\leq\Lambda_{1}.$ (4)

The approximation $b(m)$ is a lower bound of $\Lambda_{1}$ as it is related to
the bound based on walks of length two and three. If ${\bf v}=(1,\ldots,1)$,
then ${\bf u}^{\tt T}(N){\bf A}{\bf u}(N)={\bf v}^{\tt T}(N){\bf A}^{3}{\bf
v}(N)=w_{3}$ is the total number of walks of length three and ${\bf u}(N)^{\tt
T}{\bf u}(N)={\bf v}^{\tt T}(N){\bf A}^{2}{\bf v}(N)=w_{2}$ is the number of
walks of length 2, then it is known that $w_{3}/w_{2}\leq\Lambda_{1}$ ([2],
Th: 10 ).

The nodes that form the core are the $m_{c}$ top ranked nodes where
$\Lambda_{1}-b(m_{c})$ is minimal so

$m_{c}=\min_{m}\left(\arg\max_{m}\left(\frac{{\bf u}(m)^{\tt T}{\bf A}{\bf
u}(m)}{{\bf u}(m)^{\tt T}{\bf u}(m)}\right)\right).$ (5)

The outer $\min$ function is to take into account that it is possible to have
two or more $\arg\max$ values, if that is the case we choose the smallest
value.

assortativity(a)$m_{c}/N$assortativity(b)$b(m)/\Lambda_{1}$assortativity(c)$\Lambda_{1}$
Figure 1:  (a) The core size decreases with increasing assortativity and (b)
the bound improves with increasing assortativity. (c) The largest eigenvalue
$\Lambda_{1}$ increases with the assortativity. The ensembles are generated
using the Barabasi–Albert model with different ${d}_{\rm min}$ (purple
${d}_{\rm min}=2$, green ${d}_{\rm min}=5$ and blue ${d}_{\rm min}=20$). The
number of nodes is $N=1000$.

Figure 1 shows an example of the number of nodes in the core relative to total
number of nodes $m_{c}/N$ and the ratio between the bound $b(m_{c})$ and the
largest eigenvalue $\Lambda_{1}$. The figure shows three sets of networks
generated using the BA model. The networks contained in a set are networks
with different density of connections between the high degree nodes but with
the same degree distribution and minimal degree ${d}_{\rm min}$. The change on
the densities is achieved by changing the assortativity of the network. A
network with positive assortativity have densely connected high degree nodes
and a disassortative network have sparse connections between high degree
nodes. Given a BA network, a new network with a target assortative is
constructed using restricted randomisation [3].

It is known that as the assortativity increases, the eigenvalue $\Lambda_{1}$
increases [4, 5] (Fig. 1(c)), the new observation is that in this case, the
core decreases (Fig. 1(a)) and the approximation to the eigenvalue improves
(Fig. 1(b)). Also the bound $b(m)$ improves with the increase of the minimal
degree.

In the Kuramoto model, high degree nodes tend to synchronise more readily than
other nodes [6], Eq. (5) tell us how many of these high degree nodes we should
consider to approximate $K_{c}$. The synchronisation of the core can be
measured by the local order parameter

$r_{c}e^{i\psi_{n}}=\sum_{n=1}^{m_{c}}\sum_{m=1}^{m_{c}}A_{nm}\langle
e^{i\theta_{m}}\rangle_{t}.$ (6)

Figure 2 compares the beginning of synchronisation for the network and its
core as a function of $K$. The network synchronisation is observed via
$r=(\sum_{n}^{N}r_{n})/(\sum_{n}^{N}{d}_{n})$, where $r_{n}$ is given by Eq.
(2), and by $r_{c}/m_{c}$ for the core, so both quantities are bound in the
interval $[0,1]$. The $x$–axis shows the normalised coupling $K/K_{c}$. For
assortative network the onset of synchronisation is well approximated by
$K_{c}={K_{0}}/{\Lambda_{1}}$ even when ${d}_{\rm min}$ is small. For
disassortative networks only when the minimal degree is large the onset of
synchronisation for the network and its core happens at similar values of the
coupling strength.

$K/K_{C}$normalised order parameter$K/K_{C}$$K/K_{C}$$K/K_{C}$normalised order
parameter$K/K_{C}$$K/K_{C}$ Figure 2:  Synchronisation transition for
assortative networks (top row, assortative coefficient $0.39$) and
disassortative networks (bottom row, assortative coefficient $-0.40$) with
different minimal degree. The behaviour of the core order parameter is shown
in purple colour and the order parameter for the whole network in green. In
both cases the networks were generated using the Barabasi–Albert model with
$N=1000$ and minimal degree, from left to right ${d}_{\rm min}=2,5$ and $20$.
The results were generated by numerically integrating the Kuramoto Model.

To summarise, in the Kuramoto model where the coupling is described by a
simple undirected network, it is possible to find the subset of high degree
nodes that form the core of the network. The core nodes are defined by their
relevance to the phase synchronisation process and these nodes can be used to
evaluate the beginning of the synchronisation for networks with positive
assortative coefficient or with large minimal degree.

## References

  * [1] Juan G Restrepo, Edward Ott, and Brian R Hunt.  Onset of synchronization in large networks of coupled oscillators.  Physical Review E, 71(3):036151, 2005.
  * [2] Hanjo Täubig and Jeremias Weihmann.  Matrix power inequalities and the number of walks in graphs.  Discrete Applied Mathematics, 176:122–129, 2014.
  * [3] Sergei Maslov and Kim Sneppen.  Specificity and stability in topology of protein networks.  Science, 296(5569):910–913, 2002.
  * [4] Piet Van Mieghem, Huijuan Wang, Xin Ge, Siyu Tang, and Fernando A Kuipers.  Influence of assortativity and degree-preserving rewiring on the spectra of networks.  The European Physical Journal B, 76(4):643–652, 2010.
  * [5] Raul J Mondragón.  Network partition via a bound of the spectral radius.  Journal of Complex Networks, 5(4):513–526, 2017.
  * [6] Tiago Pereira.  Hub synchronization in scale-free networks.  Physical Review E, 82(3):036201, 2010.


#  $\mathbf{Phase}$ $\mathbf{structure}$ $\mathbf{of}$
$\mathbf{Topologically}$ $\mathbf{massive}$ $\mathbf{gauge}$ $\mathbf{theory}$
$\mathbf{with}$ $\mathbf{fermion}$

Yuichi Hoshino
Kushiro National College of Technology,
Otanoshike-nishi 2-32-1,Kushro 084-0916,Hokkaido,Japan

###### Abstract

Using Bloch-Nordsieck approximation fermion propagator in 3-dimensional gauge
theory with topological mass is studied.Infrared divergence of Chern-Simon
term is soft,which modifes anomalous dimension.In unquenched QCD with
2-component spinor anomalous dimension has fractional value,where order
parmeter is divergent.

##  1  Introduction

The Lagrangeans of Topologically massive gauge theory with fermion are[1]

$L=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{4}\theta\epsilon^{\mu\nu\rho}F_{\mu\nu}A_{\rho}+\overline{\psi}(i\gamma\cdot(\partial-
ieA)-m)\psi+\frac{1}{2d}(\partial\cdot A)^{2},$ (1) $\displaystyle L$
$\displaystyle=\frac{1}{4g^{2}}tr(F_{\mu\nu}F^{\mu\nu})-\frac{\theta}{4g^{2}}\epsilon^{\mu\nu\rho}tr(F_{\mu\nu}A_{\rho}-\frac{2}{3}A_{\mu}A_{\nu}A\rho)$
$\displaystyle+\overline{\psi}(i\gamma\cdot(\partial-
ieA)-m)\psi+\frac{1}{2d}(\partial\cdot A)^{2}$ (2)

,where $\theta=(g^{2}/4\pi)n,(n=0,\pm 1,\pm 2...)$ with $\gamma$ matrix for
4-comonent fermion [2].In comparison with massless $QED_{3},QCD_{3}$
topological mass term seems to soften the infrared divergence of massive
fermion near its on-shell.In Minkowski metric we have 5 $\gamma$ matrices
$\\{\gamma_{\mu},\gamma_{\nu}\\}=2g_{\mu\nu},(\mu=0,1,2)$

$\displaystyle\gamma_{0}$
$\displaystyle=\left(\begin{array}[]{cc}\sigma_{3}&0\\\
0&-\sigma_{3}\end{array}\right),\gamma_{1,2}=-i\left(\begin{array}[]{cc}\sigma_{1,2}&0\\\
0&-\sigma_{1,2}\end{array}\right),\gamma_{4}=\left(\begin{array}[]{cc}0&I\\\
I&0\end{array}\right),\gamma_{5}=\left(\begin{array}[]{cc}0&-iI\\\
iI&0\end{array}\right),$ (11) $\displaystyle\tau$
$\displaystyle\equiv\frac{-i}{2}[\gamma_{4},\gamma_{5}]=diag\left(I,-I\right),\tau_{\pm}=\frac{1\pm\tau}{2}.$
(12)

There are two redundant matrices $\gamma_{4\text{ }}$and $\gamma_{5}$ which
anticommutes with other three $\gamma$ matrices.There exists two kind of
chiral transformation
$\psi\rightarrow\exp(i\alpha\gamma_{4})\psi,\psi\rightarrow\exp(\alpha\gamma_{5})\psi.$The
matrices $\\{\gamma_{4},\gamma_{5},I_{4},\tau\\}$ generate a $U(2)$ chiral
symmetry containing massless spinor fields.This $U(2)$ symmetry is broken down
to $U(1)\times U(1)$ by a spinor mass term $m_{e}\overline{\psi}\psi$ and
parity violating mass $m_{o}\overline{\psi}\tau\psi,$where
$\overline{\psi}\tau\psi$ is a spin density.Here after we take 4-component
spinors to study chiral symmetry breaking by $m^{e}\overline{\psi}\psi$ in
pure QED${}_{3},$where $\overline{\psi}\tau\psi$ is invariant under chiral
transformation.After that the effects of Chern-Simon term will be studied by
2-component spinors.

##  2 Fermion spectral function

The spectral function of 4-component fermion and photon propagator[3] are
defined as

$S_{F}(x)=S_{F}^{0}(x)\exp(F(x)),S_{F}^{0}(x)=-(i\gamma\cdot\partial+m)\frac{\exp(-m\sqrt{-x^{2}})}{4\pi\sqrt{x^{2}}}.$
(13)
$D_{F}^{0}(k)=-i(\frac{g_{\mu\nu}-k_{\mu}k_{\nu}/k^{2}-i\theta\epsilon_{\mu\nu\rho}k^{\rho}/k^{2}}{k^{2}-\theta^{2}+i\epsilon})+id\frac{k_{\mu}k_{\nu}}{k^{4}},$
(14)

where $F$ is an $O(e^{2})$ matrix element $|T_{1}|^{2}$ for the process
electron $(p+k)\rightarrow$electron $(p)+$photon $(k)$ as

$\displaystyle T_{1}$
$\displaystyle=-ie\frac{\epsilon_{\mu}(k,\lambda)}{\gamma\cdot(p+k)-m}\gamma^{\mu}\exp(i(p+k)\cdot
x)U(p,s),$ (15) $\displaystyle\sum_{\lambda,S}T_{1}\overline{T_{1}}$
$\displaystyle=-\frac{\gamma\cdot p+m}{2m}e^{2}[\frac{m^{2}}{(p\cdot
k)^{2}}+\frac{1}{p\cdot k}+\frac{(d-1)}{k^{2}}]-\frac{\gamma\cdot
p}{m}\frac{e^{2}}{4\theta}\frac{m}{p\cdot k},$ (16)
$F=\int\frac{d^{3}k}{(2\pi)^{2}}\exp(ikx)\theta(k_{0})\delta(k^{2}-\theta^{2})\sum_{\lambda,S}T_{1}\overline{T_{1}}.$
(17)

Here we use the retarded propagator to derive the function $F$

$D_{+}(x)=\int\frac{d^{3}k}{i(2\pi)^{2}}\exp(ik\cdot
x)\theta(k_{0})\delta(k^{2}-\theta^{2})=\frac{\exp(-\theta\sqrt{-x^{2}})}{8\pi
i\sqrt{-x^{2}}}$ (18)

The function $F$ is evaluated by $\alpha$ integration for pure QED3[4, 3].

$\displaystyle F$ $\displaystyle=ie^{2}m^{2}\int_{0}^{\infty}\alpha d\alpha
D_{F}(x+\alpha p)-e^{2}\int_{0}^{\infty}d\alpha D_{F}(x+\alpha
p)-i(d-1)e^{2}\frac{\partial}{\partial\theta^{2}}D_{F}(x,\theta^{2})$
$\displaystyle=\frac{e^{2}}{8\pi}[\frac{\exp(-\theta|x|)-\theta|x|E_{1}(\theta|x|)}{\theta}-\frac{E_{1}(\theta|x|)}{m}+\frac{(d-1)\exp(-\theta\left|x\right|))}{2\theta}].$
(19)

It is well known that function $F$ has linear and logarithmic infrared
divergence with respect to $\theta$ where $\theta$ is a bare photon mass.Here
we notice the followings.(1) $\exp(F)$ includes all infrared divergences.(2)
quenhed propagator has linear and logarithmic infrared divergences.Linear
divegence is absent in a special gauge.In unquenched case $\theta$ dependence
of $\exp(F)$ is modified by dressed boson spectral function to $\int
ds\rho_{\gamma}(s)\exp(F(x,\sqrt{s}),$where
$\pi\rho_{\gamma}(s)=-\Im(s-\Pi(s))^{-1}.$

##  3 Phase strucutures

For short and long distance we have the approximate form of the function $F$

$F_{S}\sim
A-\theta|x|+(D+C|x|)\ln(\theta\left|x\right|))-\frac{(d+1-2\gamma)e^{2}|x|}{16\pi},(\theta|x|\ll
1),F_{L}\sim 0,(1\ll\theta|x|)..$ (20)

From the above formulae we have

$\displaystyle\exp(F)$ $\displaystyle=A(\theta|x|)^{D+C|x|}(\theta|x|\ll
1),A=\exp(\frac{e^{2}(1+d)}{16\pi\theta}+\frac{e^{2}\gamma}{8\pi m}),$
$\displaystyle C$ $\displaystyle=\frac{e^{2}}{8\pi},D=\frac{e^{2}}{8\pi m},$
(21)

where $m$ is the physical mass and $\gamma$ is an Euler’s constant.Here we see
$D$ acts to change the power of $|x|$.For $D=1,S_{F}(0)$ is finite and we have
$\left\langle\overline{\psi}\psi\right\rangle\neq 0$ [4, 6].Thus if we require
$D=1,$we obtain $m=e^{2}/8\pi.$In the same way we add the conrtribution of
Chern-Simon term.For simplicity we consider the 2-component spinors in(7).In
the condensed phase we have the modified anomalous dimension for $\theta>0$

$D=\frac{e^{2}}{8\pi
m}+\frac{e^{2}}{32\pi\theta}=1,m^{{}^{\prime}}=\frac{e^{2}}{8\pi}/(1-\frac{e^{2}}{32\pi\theta}),\Delta
m=\frac{e^{2}}{8\pi}\frac{e^{2}}{32\pi\theta}/(1-\frac{e^{2}}{32\pi\theta}).$
(22)

In unquenched case there are parity even and odd spectral function of gauge
boson by vacuum polarization[1].In this case we separate $D$ into parity even
and odd contribution $D^{e}=e^{2}/8\pi m,D^{O}=e^{2}/32\pi\theta.$For
Topologically Massive QCD,$\theta$ is quantized with $n(0,\pm 1,\pm
2,..).$Thus we have $D=e^{2}/(8\pi m)+1/8n=1$ for quenched case, and
$D^{e}=e^{2}/8\pi m=1,D^{O}=1/(8n)$ for unquenched case,which leads to
$\left\langle\overline{\psi}\psi\right\rangle=\infty$ for any $n\neq 0.$In the
4-component spinor free fermion propagator is decomposed into chiral
representation

$S_{F}(p)=\frac{1}{m_{\epsilon}I+m_{O}\tau-\gamma\cdot p}=\frac{(\gamma\cdot
p+m_{+})\tau_{+}}{p^{2}-m_{+}^{2}+i\epsilon}+\frac{(\gamma\cdot
p+m_{-})\tau_{-}}{p^{2}-m_{-}^{2}+i\epsilon}.$ (23)

The difference in two spectral functions is a opposite sign of each Chern-
Simon contribution for $\tau_{\pm}$.In Toplogically massive gauge theory
dynamical mass is parity even and Chern-Simon term shifts mass of different
chirality with opposite sign.However shifted mass may not strongly depend on
$\theta$ but $\left\langle\overline{\psi}\psi\right\rangle$ is proportional to
$\theta$(12)[5].Our approximation is convenient for unquenched case by the use
of gauge boson spectral function[6].

## References

  * [1] S.Deser,R.Jackiw&Templeton,Annals of Physics 281,409-449(2000).
  * [2] C.J.Burden,NuclerPhysicsB387(1992)419-446,Kei-ichi Kondo,Int.J.Mod.Phys.$\mathbf{A11}$;777-822,1996.
  * [3] R.Jackiw,L.Soloviev,Phys.Rev.173.5(1968)1485.
  * [4] Yuichi Hoshino,in CONTINUOUS ADVANCES IN QCD 2008;361-372.
  * [5] Toyoki Matsuyama,Hideko Nagahiro,Mod.Phys.Lett.A15 (2000) 2373-2386.
  * [6] C.S.Fisher,Reinhart Alkofer,T.Darm,P.Maris,Phys.Rev.D70:073007,2004.


# Revivals, or the Talbot effect, for the Airy equation

B. Pelloniπ & D.A. Smithσ πHeriot-Watt University & Maxwell Institute for the
Mathematical Sciences, Edinburgh, Scotland. σYale-NUS College & National
University of Singapore, Singapore<EMAIL_ADDRESS>σdave.smith@yale-
nus.edu.sg

###### Abstract.

We study Dirichlet-type problems for the simplest third-order linear
dispersive PDE, often referred to as the Airy equation. Such problems have not
been extensively studied, perhaps due to the complexity of the spectral
structure of the spatial operator. Our specific interest is to determine
whether the peculiar phenomenon of revivals, also known as Talbot effect, is
supported by these boundary conditions, which for third order problems are not
reducible to periodic ones. We prove that this is the case only for a very
special choice of the boundary conditions, for which a new type of weak cusp
revival phenomenon has been recently discovered [3]. We also give some new
results on the functional class of the solution for other cases.

In memory of Vassilis Dougalis, a wonderful teacher and dear friend

##  1\. Introduction

The phenomenon of revivals in linear dispersive periodic problems, originally
named Talbot effect and also known as dispersive quantisation, has been
thoroughly studied in the periodic setting, and is by now well-understood,
both from a theoretical and from an experimental point of view. An accepted
working definition of this phenomenon states that at specific values of the
time variable, called rational times, and only at these times, the solution is
a linear superposition of a finite number of translated copies of the initial
datum. This behaviour is particularly striking when the initial profile has
jump discontinuities, as these jumps then propagate in the solution at
rational times, while at any other time, jump discontinuities in the initial
datum generate a fractal solution profile. This phenomenon is known as
fractalisation. For a recent review and bibliography, see [26].

The focus of this paper is the case when the boundary conditions are not
periodic. The particular case we study is the Airy equation posed on a finite
interval, which we always specify as the interval $(0,1)$:

$\displaystyle u_{t}+u_{xxx}=0\qquad$ $\displaystyle x\in(0,1),\;0<t<\infty,$
(1) $\displaystyle u(\cdot,0)=u_{0},\quad u_{0}\in\operatorname{L}^{2}(0,1),$
$\displaystyle x\in(0,1).$ $\displaystyle 3\mbox{ boundary conditions at
}x=0\mbox{ and }x=1,\quad\;$ $\displaystyle t>0.$

The case of periodic boundary conditions was first studied in [19], and can
also be seen as a particular example of the general theory given in [8]. In
this case, both the revival property and its dual phenomenon of fractalisation
hold. However, perturbing even slightly the periodicity of the problem can
destroy the support for any form of revivals. For example, if the boundary
conditions are quasi-periodic, namely of the form

(2) $\partial_{x}^{k}u(0,t)={\rm e}^{i\theta\pi}\partial_{x}^{k}u(1,t),\qquad
k=0,1,2,\quad t>0,\quad\theta\in{\mathbb{R}},$

the revival property holds only if $\theta\in\mathbb{Q}$, see [4, 9]. This is
in marked contrast with the case of second order linear dispersive PDEs, when
the quasi-periodic boundary problem can be reduced to a periodic one and
therefore always supports the revival property.

Another natural class of boundary conditions, at least for second order
problems, is defined by the requirement that the solution vanish at the
endpoints. For second order problems, these conditions are called homogeneous
Dirichlet boundary conditions, and the resulting boundary value problem can be
formulated as a problem periodic on a larger interval, hence it always
supports revivals.

This paper is concerned with the “Dirichlet” problem for the Airy equation.
Since the equation is of third order, it is necessary to prescribe three
boundary conditions, therefore the usual Dirichlet boundary conditions
$u(0,t)=u(1,t)=0$ must be complemented with a third condition. We will call
the resulting problem of Dirichlet-type, and the question we address is
whether such a problem supports revivals in the third-order case, in
particular for the Airy equation. Here we study specifically the problem
determined by the boundary conditions

(3) $u(0,t)=u(1,t)=0,\quad u_{x}(1,t)=\beta u_{x}(0,t),\qquad
t>0,\quad\beta\in{\mathbb{R}}.$

As we argue below, without loss of generality we can restrict attention to the
case $\beta\in[0,1]$. We then show the following:

  * •

For $\beta=0$, we indicate that the problem does not support revivals, at
least in a weak sense, by showing that the $L^{2}(0,1)$ norm of the solution,
for any initial datum in $\operatorname{L}^{2}(0,1)$, decays as a function of
time as $t\to\infty$. In addition, under more stringent initial regularity
assumptions, we prove that the solution is continuous in $x$, hence cannot
support any form of revivals.

  * •

For $0<\beta<1$, we prove that the solution, in addition to the same time
decay property as the previous case, is of class
$\operatorname{C}^{\infty}[0,1]$ for all $t>0$, hence it is never
discontinuous. Therefore the problem does not support any form of revivals.

  * •

For $\beta=1$, the problem supports weak cusp revivals, as is proved in [3].
This means that the solution at the appropriate rational times can be
expressed as the sum of two functions: a continuous function, and a function
which revives any initial discontinuity as a finite number of jump
discontinuities and logarithmic cusps.

For the periodic problem for Airy on $(0,1)$, with an initial condition
$u_{0}\in\operatorname{L}^{2}(0,1)$, the periodic revival property is
completely characterised by the solution formula at rational time
$t=\frac{p}{(2\pi)^{2}}q$, given in [8] as

(4)
$u\Big{(}x,\frac{p}{4\pi^{2}q}\Big{)}=\frac{1}{q}\sum_{k,m=0}^{q-1}G_{p,q}(k,m)u_{0}\Big{(}x-2\pi\frac{k}{q}\Big{)},$

where $G_{p,q}(k,m)\in\mathbb{C}$ are explicit coefficients. In particular,
$G_{p,1}$ is periodic in $p$, hence we obtain the following relation, which
justifies the term revivals:

(5) $u\Big{(}x,\frac{np}{(2\pi)^{2}}\Big{)}=G_{p,1}u_{0}(x),\qquad
n\in{\mathbb{N}}.$

The more general notion of weak revival was first proposed in [4], and it
allows the solution to have a continuous part as well as a periodic revival
part. Cusp revivals were first described in [5] and then defined in greater
generality in [3], where the case $\beta=1$ is studied in detail. This
property implies that the solution revives any initial jump discontinuity, at
the appropriate rational times, as finitely many jumps and logarithmic cusps.
The cusps are generated by the periodic Hilbert transform of a function that
encodes the initial jump discontinuities.

Other examples of non-periodic problems are studied in [10, 20].

##  2\. Dirichlet-type boundary value problems for the Airy equation

We consider the time evolution equation associated with the third-order
operator

$T_{\beta}=(-i\partial_{x})^{3}:{\mathcal{D}_{\beta}}\longrightarrow\operatorname{L}^{2}(0,1),\qquad\beta\in{\mathbb{R}},$

where

(6) ${\mathcal{D}_{\beta}}=\\{v\in H^{3}(0,1):\;v(0)=v(1)=0,\;v_{x}(1)=\beta
v_{x}(0)\\}.$

Noting that formally $u_{xxx}=-iT_{\beta}u$, we associate to this operator the
following Dirichlet-type boundary value problem for the function $u(x,t)$:

$\displaystyle u_{t}+u_{xxx}=0\quad$ $\displaystyle x\in(0,1),\;0<t<\infty,$
(7) $\displaystyle u(0,t)=u(1,t)=0,\quad u_{x}(1,t)=\beta u_{x}(0,t)\quad$
$\displaystyle 0<t<\tau,$ $\displaystyle u(\cdot,0)=u_{0},\quad
u_{0}\in\operatorname{L}^{2}(0,1),$ $\displaystyle x\in(0,1).$

The well-posedness of the problem in $\operatorname{L}^{2}(0,1)$ for
$\beta\neq 0$, follows from general results in spectral theory, see [7]. In
the case $\beta=0$, for which there is no eigenfunction basis, this problem is
studied in [23].

#### Functional estimates and the range of $\beta$

The real parameter $\beta$ gives the coupling of the values of first
derivative at the end points of the interval. The cases of particular interest
are $\beta=0$, $0<\beta<1$ and $\beta=1$, which we consider separately. If
$\beta=0$, the third boundary condition is $u_{x}(1,t)=0$, and there is no
coupling. If $0<\beta<1$ the boundary conditions are coupled, but in general
the problem is not self-adjoint and indeed the associated eigenvalues are
complex. If $\beta=1$, so that $u_{x}(1,t)=u_{x}(0,t)$, the problem is self-
adjoint [3]. We will review these three cases, that illustrate the possible
behaviour of any well-posed Dirichlet-type problem for this third order PDE.

Assume the equation admits a classical solution $u(x,t)$. Multiplying the
equation by $2u$, we find after a formal integration by parts that

(8)
$\|u(x,t)\|^{2}_{\operatorname{L}^{2}(0,1)}=\|u_{0}(x)\|^{2}_{\operatorname{L}^{2}(0,1)}-(1-\beta)\int_{0}^{t}u_{x}^{2}(0,s)ds.$

Hence, if $\beta>1$,

$\|u(x,t)\|^{2}_{\operatorname{L}^{2}(0,1)}\geq\|u_{0}(x)\|^{2}_{\operatorname{L}^{2}(0,1)},$

and the $\operatorname{L}^{2}$ norm of the solution is not necessarily bounded
as a function of $t$, hence the problem may be ill-posed.111 Indeed, it can be
shown that for $\beta>1$ the problem is ill-posed, with the solution becoming
unbounded for arbitrarily small times, [24, 25].

Hence we assume that $\beta\leq 1$. In this case, as is easily verified, both
$T_{\beta}$ and its adjoint $T_{\beta}^{*}$ are dissipative, namely

$\langle T_{\beta}f,f\rangle\leq 0,\quad\langle T_{\beta}^{*}f,f\rangle\leq
0\qquad\forall f\in\operatorname{L}^{2}(0,1),$

and the associated boundary value problem (7) admits a unique solution in
$\operatorname{L}^{2}(0,1)$ [2].

This problem, specifically in operator-theoretic terms, has been studied in
detail in [23]. The study of the adjoint operator $T_{\beta}^{\star}$ is
equivalent to reversing the direction of time and mapping $\beta\mapsto
1/\beta$, i.e. in this case to considering the PDE $u_{t}-u_{xxx}=0$. Because
of this identification, it is enough to consider $\beta\in[0,1]$.222The case
$\beta<0$ is entirely analogous, yielding a well -poseed problems for
$-1\leq\beta<0$ for which similar results hold.

It is straightforward to establish a regularising estimate for the operator
$T_{\beta}$, with $0\leq\beta<1$, a variation on, and slight generalisation
of, the one given in [2] for the case that $\beta=0$.333In [2], the equation
studied includes the lower order term $u_{x}$.

######  Proposition 1.

Let $u_{0}\in\operatorname{L}^{2}(0,1)$ be given. For all $t\geq 0$, and
$0\leq\beta<1$, the solution of $u_{t}=iT_{\beta}u$ with $u(x,0)=u_{0}(x)$
satisfies

(9)
$\displaystyle\|u(\cdot,t)\|_{\operatorname{L}^{2}(0,1)}^{2}\leq\|u_{0}\|_{\operatorname{L}^{2}(0,1)}^{2},$
(10)
$\displaystyle\int_{0}^{t}u^{2}_{x}(0,s)ds\leq\frac{1}{1-\beta}\|u_{0}\|_{\operatorname{L}^{2}(0,1)}^{2},$
(11)
$\displaystyle\int_{0}^{1}xu^{2}(x,t)dx+3\int_{0}^{t}\int_{0}^{1}u_{x}^{2}(x,s)dxds\leq\frac{1}{1-\beta}\|u_{0}\|_{\operatorname{L}^{2}(0,1)}^{2}.$
(12)
$\displaystyle\lim_{t\to\infty}\|u(\cdot,t)\|^{2}_{\operatorname{L}^{2}(0,1)}=0.$

###### Proof.

Assuming that $u$ is a strong (smooth) solution (a limiting argument, as in
[2] then gives the result for weak solutions), integration of
$uu_{t}=-uu_{xxx}$ with the given boundary conditions yields (8), and, since
$\beta<1$, this identity implies the two bounds (9) and (10).

On the other hand, integration by parts of $2xuu_{t}=-2xuu_{xxx}$ using the
given boundary conditions gives

$\displaystyle\int_{0}^{1}xu^{2}(x,t)dx$
$\displaystyle+3\int_{0}^{t}\int_{0}^{1}u^{2}_{x}(x,s)dxds=\int_{0}^{1}xu_{0}^{2}(x)dx+\beta\int_{0}^{t}u_{x}^{2}(0,s)ds$
$\displaystyle\leq\int_{0}^{1}xu_{0}^{2}(x)dx+\frac{\beta}{1-\beta}\int_{0}^{1}u_{0}^{2}(x)dx\leq\frac{1}{1-\beta}\|u_{0}\|^{2}_{\operatorname{L}^{2}(0,1)},$

which is (11). By (10),
$f(t):=\|u(\cdot,t)\|_{\operatorname{L}^{2}(0,1)}\in\operatorname{L}^{2}(0,\infty)$.
In addition, by integration by parts, $f^{\prime}(t)=-u^{2}_{x}(0,t)$, so that
$f^{\prime}(t)\in\operatorname{L}^{2}(0,\infty)$ by (10). This implies that
$\displaystyle{\lim_{t\to\infty}f(t)=0}$ so that (12) must hold. ∎

These estimates imply that the function

$t\mapsto\|u(\cdot,t)\|_{H^{1}(0,1)}$

is in $\operatorname{L}^{2}(0,t)$, so that for almost all $t>0$,
$u(\cdot,t)\in H^{1}(0,1)$. In particular, for almost all $t>0$, the solution
admits a well defined weak first derivative.

######  Remark 1.

Heuristically, fractalisation is a necessary consequence of the revival of
initial discontinuities at every rational value of the time, see [18].
Therefore, if the problem supported a form of revival, it would also present
the dual property of fractalisation. This property would imply that the
solution is nowhere differentiable, which cannot be the case given that
$u(\cdot,t)\in H^{1}(0,1)$ for almost all $t>0$.

This remark, as well as the time decay of the $L^{2}(0,1)$ norm of the
solution, suggests that for $0\leq\beta<1$, revivals are not supported.

The main result of this paper is the precise formulation and proof of this
statement.

#### The spectral structure of $T_{\beta}$

Despite the formal similarity of the two problems obtained by choosing
$\beta=0$ or $\beta\neq 0$, the respective spectral structure is fundamentally
different. Indeed, it can be shown that for the case $\beta=0$, although the
operator $T_{\beta}$ has infinitely many discrete eigenvalues, the associated
eigenfunctions do not form a Riesz basis, and that there is no generalised
Fourier series representation for the solution of this problem [14, 22, 23].

On the other hand, if $0<\beta\leq 1$, there is an infinite family of
discrete, generally complex, eigenvalues, and the associated eigenfunctions
and their adjoints form a complete biorthogonal basis. Hence it is possible to
represent the solution of the boundary value problem as a generalised Fourier
series.

The case $\beta=1$ is the only one resulting in a self-adjoint problem, for
which the eigenvalues are purely real. On the basis of this spectral
structure, we distinguish three separate cases:

  * (a)

$0<\beta<1$: there is a generalised Fourier series representation for the
solution. We prove that the solution is smooth in $x$ and its $L^{2}(0,1)$
norm decays in $t$. Hence no revival property holds.

  * (b)

$\beta=0$: there is no generalised Fourier series representation for the
solution. The $L^{2}(0,1)$ norm of the solution decays in $t$ and, under
slightly higher regularity assumptions, we prove that the solution is
continuous, hence no revival property can hold.

  * (c)

$\beta=1$: there is a generalised Fourier series representation for the
solution, and a weak cusp revival property holds, as proved in [3].

### The case $0<\beta<1$

We prove in Theorem 1 that, for any integrable initial condition, in addition
to the time decay of its $L^{2}(0,1)$, the solution is actually of class
$\operatorname{C}^{\infty}$, for all $t>0$. This implies that this problem
cannot support any type of revivals. Indeed, any form of the revival property
implies that the solution is not continuous at countably many values of $t$.

#### Spectral structure

If $\beta\neq 0$, the operator is associated to a biorthogonal basis for
$\operatorname{L}^{2}(0,1)$, composed of the eigenfunctions of $T_{\beta}$ and
those of the adjoint $T_{\beta}^{\star}$ [7, 14]. This statement holds
provided all eigenvalues are simple, which we henceforth assume to be the
case, based on the argument given below that shows that all but finitely many
of them are simple. Hence we can study the generalised Fourier series
representation of the solution with a view to verify the conjecture that the
revival property does not hold in this case.

As shown in [22], the eigenvalues are given by $k_{n}^{3}$, where the
$k_{n}$’s are the zeros of

$\Delta(k)=\beta[{\rm e}^{-ik}+\alpha{\rm e}^{-i\alpha k}+\alpha^{2}{\rm
e}^{-i\alpha^{2}k}]+{\rm e}^{ik}+\alpha{\rm e}^{i\alpha k}+\alpha^{2}{\rm
e}^{i\alpha^{2}k},\quad\alpha={\rm e}^{2\pi i/3}.$

While this transcendental equation cannot be solved exactly, the position of
finitely many $k_{n}$ can be found numerically using an algorithm based on the
argument principle. For example, if $\beta=10^{-6}$, then the first few lie at
the crosses in figure 1.

Figure 1.  The zeros of $\Delta$ for $\beta=10^{-6}$. Also shown are line
segments and rays along which the zeros lie, approximately.

It appears that they lie approximately along the dashed blue line segment and
the solid red rays, and their lighter colored rotations about the origin by
$2\pi/3$, $4\pi/3$. The rotational symmetry is immediately confirmed by the
identity $\Delta(\alpha k)=\alpha^{2}\Delta(k)$, and implies that we only need
to characterise the location of the solid red horizontal rays and dashed blue
vertical line segment. To this end, following the approach of [17], consider
$k$ with $\operatorname{Re}(k)\gg\lvert\operatorname{Im}(k)\rvert$, and
observe that the dominant terms in $\Delta(k)$ are $\beta\alpha e^{-i\alpha
k}$ and $\alpha^{2}e^{i\alpha^{2}k}$, while all other terms are
$\mathcal{O}(1)$ in $\operatorname{Re}(k)$. Imposing that the leading order
terms approximately cancel requires that
$k_{n}\approx\left(2n-\frac{1}{3}\right)\pi+i\log(\beta)$ for $n\to\infty$.
The existence of zeros close to $-\left(2n-\frac{1}{3}\right)\pi+i\log(\beta)$
is similarly justified. Note that since the zeros of $\Delta^{\prime}$ are
asymptotically located at $2n\pi+i\log(\beta)$, all but finitely many of the
zeros of $\Delta$ must be simple. The zeros close to the dashed blue line
segment arise when $\beta$ is very small so that, for $\lvert k\rvert$ not too
large, the terms in $\Delta(k)$ with factor $\beta$ are negligible. Using the
same kind of asymptotic argument as above, choosing a region of $\mathbb{C}$
such that two terms dominate the remaining one and requiring that those
leading order terms cancel, we find that there is a finite number of zeros
around $i\left(2n-1\right)\frac{\pi}{\sqrt{3}}$, for $n$ positive but not too
large. Note that, although $0$ is a zero of $\Delta$, it is not the cube root
of an eigenvalue, because no quadratic expression satisfies the boundary
conditions.

The above arguments show that on the horizontal (red) rays,

(13) $\displaystyle k_{n}=\kappa_{n}+i\log\beta+\epsilon_{n},$
$\displaystyle\qquad\mbox{where, as }|n|\to\infty,$
$\displaystyle\kappa_{n}=\pi\left(2n-\frac{\operatorname{sgn}(n)}{3}\right),$
$\displaystyle\qquad\epsilon_{n}=\mathcal{O}\left(e^{-\sqrt{3}\pi\lvert
n\rvert}\right).$

The corresponding eigenfunctions are given by

(14)
$X_{n}=\sum_{j=0}^{2}e^{-i\alpha^{j}k_{n}x}(e^{-i\alpha^{j+1}k_{n}}-e^{-i\alpha^{j+2}k_{n}}).$

It is not difficult to show that the adjoint operator has eigenvalues
$k_{n}^{\star}=\overline{k_{n}}$ and eigenfunctions $X_{n}^{\star}$ given by
the same formula as above for $X_{n}$ but with $k_{n}^{\star}$ replacing
$k_{n}$.

#### General solution formula

We can express the solution of this boundary value problem as a generalised
Fourier series, namely

(15) $u(x,t)=\sum_{n}e^{-ik_{n}^{3}t}X_{n}(x)\frac{\langle
u_{0},X_{n}^{\star}\rangle}{\langle X_{n},X_{n}^{\star}\rangle}.$

We can now prove the following regularity result for the solution, which in
particular confirms that no revival property can be supported by boundary
value problems in this class.

######  Theorem 1.

Suppose $u_{0}\in\operatorname{L}^{2}(0,1)$. Then, for all $t>0$, the solution
$u$ of the problem (7) with $0<\beta<1$ satisfies
$u({}\cdot{},t)\in\operatorname{C}^{\infty}[0,1]$.

###### Proof.

Using the formula (14) for the eigenfunctions, we can compute

$\displaystyle\left\langle X_{n},X_{n}^{\star}\right\rangle$
$\displaystyle\hskip
40.00006pt=\sum_{j,r=0}^{2}\int_{0}^{1}e^{-i\alpha^{j}k_{n}x}\overline{e^{-i\alpha^{r}\overline{k_{n}}x}}dx\left(e^{-i\alpha^{j+1}k_{n}}-e^{-i\alpha^{j+2}k_{n}}\right)\overline{\left(e^{-i\alpha^{r+1}\overline{k_{n}}}-e^{-i\alpha^{r+2}\overline{k_{n}}}\right)}$
$\displaystyle=\sum_{j=0}^{2}\left(e^{-i\alpha^{j+1}k_{n}}-e^{-i\alpha^{j+2}k_{n}}\right)\left(e^{i\alpha^{j+2}k_{n}}-e^{i\alpha^{j+1}k_{n}}\right)$
$\displaystyle\hskip 10.00002pt+\sum_{\begin{subarray}{c}j,r=0:\\\
\alpha^{j}\neq\alpha^{-r}\end{subarray}}^{2}\left[\frac{e^{-ik_{n}(\alpha^{j}-\alpha^{-r})}-1}{-ik_{n}(\alpha^{j}-\alpha^{-r})}\right]\left(e^{-i\alpha^{j+1}k_{n}}-e^{-i\alpha^{j+2}k_{n}}\right)\left(e^{i\alpha^{-(r+1)}k_{n}}-e^{i\alpha^{-(r+2)}k_{n}}\right).$

Recalling expression (13) for the asymptotic behaviour of the eigenvalues, we
find that the first sum, as $n\to\pm\infty$, is $e^{\sqrt{3}\lvert
k_{n}\rvert}+\mathcal{O}(e^{\sqrt{3}\pi\lvert n\rvert})$.

The terms in the second sum each take one of the forms

$\frac{e^{-ik_{n}(\alpha^{\ell}-\alpha^{m})}}{-ik_{n}(\alpha^{j}-\alpha^{-r})},\qquad\frac{e^{-ik_{n}(\alpha^{j}(1+\alpha^{\ell})-\alpha^{-r}(1+\alpha^{m}))}}{-ik_{n}(\alpha^{j}-\alpha^{-r})},$

for $\ell,m\in\\{0,1,2\\}$ in the first form and $\ell,m\in\\{1,2\\}$ in the
second form. Therefore, these terms are all dominated by the leading order
term from the first sum. In summary, we obtain

$\left\langle X_{n},X_{n}^{\star}\right\rangle=e^{\sqrt{3}\lvert
k_{n}\rvert}+\mathcal{O}\left(\frac{e^{2\sqrt{3}\pi\lvert
n\rvert}}{n}\right).$

By Hölder’s inequality,

$\langle u_{0},X_{n}^{\star}\rangle\leq\left\lVert
u_{0}\right\rVert_{\operatorname{L}^{1}(0,1)}\left\lVert
X_{n}^{\star}\right\rVert_{\operatorname{L}^{\infty}(0,1)}\lesssim\left\lVert
u_{0}\right\rVert_{\operatorname{L}^{1}(0,1)}e^{\frac{\sqrt{3}}{2}\lvert\kappa_{n}\rvert},$

with constant uniform in $n$. Therefore

$\left\lVert\frac{\left\langle u_{0},X_{N}^{\star}\right\rangle}{\left\langle
X_{n},X_{n}^{\star}\right\rangle}X_{n}\right\rVert_{\operatorname{L}^{\infty}(0,1)}$

is bounded as $n\to\pm\infty$ and, for the $m$-th derivative, we obtain

$\left\lVert\frac{\left\langle u_{0},X_{N}^{\star}\right\rangle}{\left\langle
X_{n},X_{n}^{\star}\right\rangle}X_{n}^{(m)}\right\rVert_{\operatorname{L}^{\infty}(0,1)}=\mathcal{O}\left(n^{m}\right).$

On the other hand, computing the imaginary part of the eigenvalues yields

$\displaystyle\operatorname{Im}(k_{n}^{3})$
$\displaystyle=\operatorname{Im}\left(\kappa_{n}^{3}+3i\log(\beta)\kappa_{n}^{2}-3[\log(\beta)]^{2}\kappa_{n}-i[\log(\beta)]^{3}+\mathcal{O}\left(n^{2}e^{-\sqrt{3}\pi
n}\right)\right)$ $\displaystyle=3\log(\beta)\kappa_{n}^{2}+\mathcal{O}(1).$

It follows that

$e^{-ik_{n}^{3}t}=\mathcal{O}\left(e^{3\kappa_{n}^{2}t\log(\beta)}\right)$

and, since $t\log(\beta)<0$ for $0<\beta<1$, this factor is decaying
exponentially. Hence, for all $t>0$, the series in equation (15) for $u(x,t)$
converges absolutely uniformly in $x\in[0,1]$, and so do its term-by-term
spatial derivatives. From the fact that
$X_{n}\in\operatorname{C}^{\infty}[0,1]$, it follows that, for all $t>0$,
$u({}\cdot{},t)\in\operatorname{C}^{\infty}[0,1]$. ∎

### The case $\beta=0$

For this case, we know that the operator has infinitely many real eigenvalues,
but that the eigenfunctions do not form a Riesz basis for the associated
$\operatorname{L}^{2}$ space [23].

This boundary value problem was studied in [2], where a version of Proposition
1 is proved. In that paper, it is also reported how, when the initial datum is
in the domain of the operator, $u_{0}\in{\mathcal{D}_{0}}$, there exist a
unique classical solution
$u\in\operatorname{C}([0,\infty);H^{3}(0,1))\cap\operatorname{C}^{1}([0,\infty);\operatorname{L}^{2}(0,1))$
of the boundary value problem. When $u_{0}\in\operatorname{L}^{2}(0,1)$, there
is a unique solution in the same class.

#### Decay in time

We now show how the assumption of revivals leads to a contradiction. This
contradiction is reached if the solution is assumed to satisfy a specific form
of weak revival property, including an additional time decay property of its
continuous component.

More precisely, suppose that the solution $u(x,t)$ of (7) with $\beta=0$ has
the following weak revival property:

At rational times $t=\frac{p}{(2\pi)^{2}}$, $u(x,t)$ can be written as
$u(x,t)=U_{\mathcal{R}}(x,t)+U_{\mathcal{C}}(x,t)$, where
$U_{\mathcal{C}}(x,t)$ is continuous and decays as $t\to\infty$, while
$U_{\mathcal{R}}(x,t)$ satisfies the periodic revival property (4), possibly
with $u_{0}$ modified by multiplication with a $C^{\infty}$ function (this
does not change the location and number of the initial jump discontinuities).

The decay of $\|u({}\cdot{},t)\|_{\operatorname{L}^{2}(0,1)}$ as $t\to\infty$,
given by proposition 1, implies that no such weak revival property can hold.

Indeed, if it did, then for each fixed $p\in\mathbb{N}$ there would be
arbitrarily large times $t_{n}=\frac{np}{(2\pi)^{2}}$, $n\in{\mathbb{N}}$,
where the revival part of solution $U_{\mathcal{R}}(x,t)$, according to (5),
revives the initial condition, up to a fixed multiple, hence for infinitely
many values of the time, its norm is equal to a multiple of
$\|u_{0}\|_{\operatorname{L}^{2}(0,1)}$. Therefore it cannot hold that
$\displaystyle{\|U_{\mathcal{R}}({}\cdot{},t)\|_{\operatorname{L}^{2}(0,1)}\to_{t\to\infty}0}$.

In particular, the solution of this problem cannot have the periodic revival
property ($U_{\mathcal{C}}(x,t)=0$) or a weak version of the type described in
theorem 3 below for the case $\beta=1.$ Indeed, in the latter case, the
continuous part of the solution is an exponentially decaying function of $t$
as $t\to\infty$.

#### Continuity properties of the solution

Using the unified transform method of Fokas [1, 6, 11, 12], it can be shown
that the boundary value problem corresponding to $\beta=0$, for a smooth
$u_{0}$, has a unique solution $u(x,t)$, and that solution admits the explicit
contour integral representation

(16) $u(x,t)=\frac{1}{2\pi}\int_{\mathbb{R}}{\rm
e}^{ikx+ik^{3}t}\widehat{u_{0}}(k)dk\\\ +\left\\{\int_{\Gamma^{+}}{\rm
e}^{ikx+ik^{3}t}\frac{\zeta^{+}(k)}{\Delta(k)}dk+\int_{\Gamma^{-}}{\rm
e}^{ik(x-1)+ik^{3}t}\frac{\zeta^{-}(k)}{\Delta(k)}dk\right\\}.$

In this expression, $\Gamma^{\pm}$ are contours in ${\mathbb{C}}^{\pm}$,
defined as the lines where the exponential ${\rm e}^{ik^{3}t}$ is purely
oscillatory. Namely, the contours $\Gamma^{\pm}$ are the positively oriented
boundaries of sectors $k\in{\mathbb{C}}^{\pm}$ with
$\operatorname{Re}(ik^{3})>0$,

(17) $\displaystyle\Delta(k)={\rm e}^{-ik}+\alpha{\rm e}^{-i\alpha
k}+\alpha^{2}{\rm e}^{-i\alpha^{2}k},$ (18)
$\displaystyle\zeta^{-}(k)=\widehat{u_{0}}(k)+\alpha\widehat{u_{0}}(\alpha
k)+\alpha^{2}\widehat{u_{0}}(\alpha^{2}k),$ (19)
$\displaystyle\zeta^{+}(k)={\rm
e}^{-ik}\zeta^{-}(k)-\widehat{u_{0}}(k)\Delta(k)$
$\displaystyle=\alpha\left({\rm e}^{-ik}\widehat{u_{0}}(\alpha k)-{\rm
e}^{-i\alpha k}\widehat{u_{0}}(k)\right)+\alpha^{2}\left({\rm
e}^{-ik}\widehat{u_{0}}(\alpha^{2}k)-{\rm
e}^{-i\alpha^{2}k}\widehat{u_{0}}(k)\right).$

The zeros of $\Delta$ are the cube roots of the eigenvalues of the spatial
differential operator, and it can be shown (as we detail in the proof of
Theorem 2 below) that they are not on the integration contours. Hence the
integrands in the last two integrals are analytic functions on the contours,
and the integrals are well defined.

This integral representation is not equivalent to a discrete generalised
Fourier series. Nevertheless, we are able to establish continuity of
$u(\cdot,t)$ for all $t>0$, provided the initial condition, while still
discontinuous, has specific piecewise regularity. The continuity of $u$ as a
function of $x$ implies that no revival property can be supported.

######  Theorem 2.

Suppose that $u_{0}\in\operatorname{C}^{2}[0,1]$ piecewise. Let $u(x,t)$ be
defined by equation (16). Then, for all $t>0$,
$u({}\cdot{},t)\in\operatorname{C}(0,1)$.

###### Proof of Theorem 2.

We begin by arguing that some components of the contour integrals in equation
(16) yield, for all $t>0$, functions $u({}\cdot{},t)$ which are
$\operatorname{C}^{\infty}[0,1]$. The final part of the proof, given as a
separate proposition, justifies continuity of the remaining components.

As shown in [22], the zeros of $\Delta$ lie exactly on the rays
$-i\alpha^{j}{\mathbb{R}}^{+}$ and asymptotically at
$-i\alpha^{j}(2n+\frac{1}{3})\pi/\sqrt{3}$ for $n\in{\mathbb{N}}$ and
$j\in\\{0,1,2\\}$, with exponentially decaying error. Moreover, because the
zeros of $\Delta^{\prime}$ lie asymptotically at
$-i\alpha^{j}(2n+\frac{1}{3})\pi/\sqrt{3}$, at most finitely many zeros of
$\Delta$ are nonsimple. Careful bounds can be obtained to show that all
nonzero zeros are simple. There is a removable singularity of
$\zeta^{\pm}/\Delta$ at $0$. We denote the nonzero zeros of $\Delta$ lying
along $-i{\mathbb{R}}^{+}$ by $k_{n}$ for $n\in{\mathbb{N}}$.

Consider $k\to\infty$ with $-2\pi/3\leq\arg(k)\leq-\pi/3$. Away from zeros of
$\Delta$,

$\Delta(k)={\rm e}^{i\frac{k}{2}}\left(\alpha{\rm
e}^{-i\frac{\sqrt{3}ik}{2}}+\alpha^{2}{\rm
e}^{i\frac{\sqrt{3}ik}{2}}\right)+O\left({\rm e}^{-\frac{\left\lvert
k\right\rvert\sqrt{3}}{2}}\right).$

Let $\epsilon>0$ be sufficiently small that both $u_{0}\rvert_{[0,\epsilon]}$
and $u_{0}\rvert_{[1-\epsilon,1]}$ are $\operatorname{C}^{1}$ functions.
Integrating by parts, for $k$ away from the zeros of $\Delta$,

$\displaystyle\alpha^{j}\widehat{u_{0}}(\alpha^{j}k)$
$\displaystyle=\frac{i}{k}\left(\left[{\rm
e}^{-i\alpha^{j}kz}u_{0}(z)\right]_{z=0}^{z=\epsilon}+\left[{\rm
e}^{-i\alpha^{j}kz}u_{0}(z)\right]_{z=1-\epsilon}^{z=1}\right.$
$\displaystyle\hskip
40.00006pt-\left.\left\\{\int_{0}^{\epsilon}+\int_{1-\epsilon}^{1}\right\\}{\rm
e}^{-i\alpha^{j}kz}u^{\prime}_{0}(z)\mathrm{d}z\right)+\alpha^{j}\int_{\epsilon}^{1-\epsilon}{\rm
e}^{-i\alpha^{j}kz}u_{0}(z)\mathrm{d}z$
$\displaystyle=O\left(k^{-1}\Delta(k)\right),$

so also is $\zeta^{-}(k)$. Define the sectors

$E^{\pm}=\\{k\in{\mathbb{C}}^{\pm}\mbox{ such that
}\operatorname{Re}(ik^{3})<0\\}.$

By the asymptotic estimates established above, Jordan’s lemma, and a residue
calculation,

(20) $\int_{\partial E^{-}}{\rm
e}^{ik(x-1)+ik^{3}t}\frac{\zeta^{-}(k)}{\Delta(k)}\mathrm{d}k=\sum_{n\in{\mathbb{N}}}{\rm
e}^{ik_{n}(x-1)+ik_{n}^{3}t}\frac{\zeta^{-}(k_{n})}{\Delta^{\prime}(k_{n})}.$

From the asymptotic formula for $k_{n}$,
$\Delta^{\prime}(k_{n})=(-1)^{n+1}i{\rm
e}^{(2n+\frac{1}{3})\frac{\pi}{2\sqrt{3}}}+o(1),$ so
$\frac{\mathrm{d}^{m}}{\mathrm{d}k^{m}}\exp(ik_{n}(x-1))\zeta^{-}(k_{n})/\Delta(k_{n})=O\left(n^{m-1}\right)$
as $n\to\infty$, uniformly in $x\in[0,1]$. Because, for all $t>0$, ${\rm
e}^{ik_{n}^{3}t}$ decays exponentially as $n\to\infty$, the series in equation
(20) converges absolutely uniformly in $x$, along with all its termwise
derivatives. Hence the integral in equation (20) represents a
$\operatorname{C}^{\infty}[0,1]$ function of $x$. Consequently, equation (16)
may be reexpressed as

$2\pi u(x,t)=V_{\mathcal{C}}(x,t)-\int_{\partial E^{+}}{\rm
e}^{ikx+ik^{3}t}\frac{\zeta^{+}(k)}{\Delta(k)}\mathrm{d}k,$

in which $V_{\mathcal{C}}$ is, for all $t>0$, a $\operatorname{C}^{\infty}$
function of $x\in[0,1]$.

Let

$\rho(k)={\rm e}^{-ik}\widehat{u_{0}}(\alpha^{2}k)-{\rm
e}^{-i\alpha^{2}k}\widehat{u_{0}}(k),$

so that $\zeta^{+}(k)=\alpha^{2}\rho(k)-\alpha\rho(\alpha k)$. The ratio
$\rho(\alpha k)/\Delta(k)$ decays exponentially as $k\to\infty$ within the
sector $\frac{2\pi}{3}\leq\arg(k)\leq\pi$ away from the zeros of $\Delta$, as
does $\rho(k)/\Delta(k)$ within the sector $0\leq\arg(k)\leq\frac{\pi}{3}$.
Hence, by a similar argument to above, for all $t>0$, these terms yield
contributions to $u(x,t)$ that are $\operatorname{C}^{\infty}[0,1]$ in $x$.
Changing variables, so that the contour about the two connected components of
$E^{+}$ is replaced with the boundary of the single sector
$E_{2}^{+}=\\{k\in{\mathbb{C}}$ with $2\pi/3<\arg(k)<\pi\\}$, we find that

(21) $2\pi u(x,t)=U_{\mathcal{C}}(x,t)-\int_{\partial E^{+}_{2}}{\rm
e}^{ik^{3}t}\frac{\rho(k)}{\Delta(k)}\left(\alpha^{2}{\rm e}^{ikx}-\alpha{\rm
e}^{i\alpha^{2}kx}\right)\mathrm{d}k,$

in which $U_{\mathcal{C}}$ is, for all $t>0$, a $\operatorname{C}^{\infty}$
function of $x\in[0,1]$.

To complete the proof of the theorem, it remains to show that, for all
$x\in(0,1)$, $t>0$, $\lim_{\delta\to 0}u(x,t)-u(x+\delta,t)=0$. This is the
conclusion of Proposition 2, proved by a careful and lengthy computation,
given in appendix. ∎

It is an immediate corollary of Theorem 2 that the boundary value problem with
$\beta=0$ does not support revivals.

### The case $\beta=1$

In this case,we need to analyse the spectral structure of the operator

$T_{1}=-i\partial_{x}^{3}:{\mathcal{D}_{1}}\longrightarrow\operatorname{L}^{2}(0,1).$

It follows from general spectral theory that $A$ is self-adjoint, hence that
$iA$ is the generator of a $C_{0}$ one-parameter semigroup. This guarantees,
for any $u_{0}\in\operatorname{L}^{2}(0,1)$, the existence of a unique
solution $u(x,t)\in\operatorname{C}([0,\infty);\operatorname{L}^{2}(0,1))$ of
the associated boundary value problem (7). The problem is conservative, so the
$L^{2}(0,1)$ norm of the solution does not decay in time.

If $u_{0}\in{\mathcal{D}_{1}}$, then $u$ is a strong solution:
$u\in{\mathcal{D}_{1}}$ and

$u\in
C([0,\infty);H^{3}(0,1))\cap\operatorname{C}^{1}([0,\infty);\operatorname{L}^{2}(0,1)).$

We are interested in the case when $u_{0}$ has some jumps discontinuities, but
is otherwise smooth. While this is the case of most interest, the theorem
below is proved in [3] for a more general class of initial data.

######  Theorem 3.

Assume that $u_{0}\in\operatorname{BV}[0,1]$ is piecewise Lipschitz. Then, the
solution of the boundary value problem (7), with $\beta=1$, admits the
representation

$u(x,t)=U_{\mathcal{R}}(x,t)+U_{\mathcal{C}}(x,t),\qquad x\in(0,1),\;t>0,$

where

$U_{\mathcal{C}}({}\cdot{},t)\in\operatorname{C}(0,1)$ for all
$t\in{\mathbb{R}}$, and

(22)
$U_{\mathcal{R}}(x,t)=\sum_{n=1}^{\infty}2\operatorname{Re}\left[\widehat{G_{u_{0}}}(n){\rm
e}^{i(2n-\frac{1}{3})^{3}\pi^{3}t}{\rm e}^{i(2n-\frac{1}{3})\pi x}\right],$

where

At any time of the form $t=\frac{p}{\pi^{2}q}$, with $p,q\in{\mathbb{N}}$
coprime, the function $U_{\mathcal{R}}(x,t)$ can be written as

(23)
$U_{\mathcal{R}}\left(x,\frac{p}{q\pi^{2}}\right)=\operatorname{Re}\left\\{{\rm
e}^{-i\frac{\pi}{27}\left(9x+\frac{p}{q}\right)}[L_{1}(x)+iL_{2}(x)-L_{3}]\right\\},$

where

$\displaystyle L_{1}(x)$
$\displaystyle=\sum_{k=0}^{q-1}d_{k}^{p,q}G_{u_{0}}\left(x+\frac{p}{3q}-\frac{k}{q}\right),$
$\displaystyle L_{2}(x)$
$\displaystyle=\sum_{k=0}^{q-1}d_{k}^{p,q}[{\mathcal{H}}G_{u_{0}}]\left(x+\frac{p}{3q}-\frac{k}{q}\right),$
$\displaystyle L_{3}$ $\displaystyle=\int_{0}^{1}G_{u_{0}}(y)\mathrm{d}y,$

$\mathcal{H}$ denotes the $1$-periodic Hilbert transform, and

(24) $d_{k}^{p,q}=\frac{1}{q}\sum_{m=0}^{q-1}{\rm e}^{2\pi
i\left(mk+\left[4m^{3}-2m^{2}\right]p\right)/q}.$

The conclusion of this theorem is that, up to a continuous perturbation, the
solution of (7) with $\beta=1$ at rational times $t=\frac{p}{\pi^{2}q}$ is
given by a finite superposition of copies of the initial datum and of the
1-periodic Hilbert transform of the initial datum. Since the periodic Hilbert
transform maps every jump discontinuity to a logarithmic cusp, the profile of
the solution at rational times, when the initial datum has jump
discontinuities, is the superposition of a continuous function with a finite
number of jumps and logarithmic cusps, namely it has the _weak cusp revival
property_ discussed in [3].

### Numerical simulations

Figure 2.  Figure 3.  Figure 4.

We used Julia to plot the solution, approximated by the truncated
eigenfunction series. In the $\beta=1$ case, because the problem is
selfadjoint, we used the ApproxFun package [21] to compute the eigenvalues and
eigenfunctions directly and evaluate the inner products in the solution
representation. When $0<\beta<1$, we instead calculate the $k_{n}$ as roots of
$\Delta$, then use expression (14) for the eigenfunctions. For $k_{n}$ close
to $0$, we use the Julia implementation [13] of the GRPF algorithm [15, 16],
based on the argument principle and, for $\lvert k_{n}\rvert\gg 0$, we use the
leading order asymptotic term in the expression for $k_{n}$ given in equation
(13). We use the first 614 eigenvalues for the selfadjoint problem where
$\beta=1$ and the first 120 for problems with $0<\beta<1$. In the case
$\beta=0$, as may be inferred from the delicacy of the asymptotic analysis in
the proof of theorem 2, numerical evaluation of the solution representation
via highly oscillatory contour integrals (16) is somewhat delicate, so is
reserved for future study.

A simplified version of the GPRF algorithm to find all roots of an analytic
function within a bounded region proceeds as follows. Select sample points on
the region’s boundary, evaluate the principal value of the argument of the
function at each sample point. Proceeding anticlockwise about the boundary,
calculate the difference between the arguments at adjacent sample points and
count the number times this jumps from $\pi^{-}$ to $-\pi^{+}$, subtracting
any jumps in the opposite direction. By the argument principal, this is equal
to the number of zeros interior to the region. Now bisect the polygon, and
repeat on each subregion. Excluding from further consideration any subregions
containing no roots, recursively subdivide and count roots to the desired
accuracy.

We present plots of the solution starting from a piecewise constant initial
datum for both cases. Specifically, in these plots the initial datum is the
function

$u_{0}(x)=\begin{cases}1&\frac{1}{3}<x<\frac{2}{3},\\\
0&\mbox{otherwise}.\end{cases}$

The plots for $\beta=0.01$ and $\beta=0.9$ clearly show the solution starting
from this discontinuous initial datum to be smooth (and decaying) at all
positive times. The plots for $\beta=1$ show the cusps (which are infinite
logarithmic cusps, though they are shown as finite by the graphic interface)
generated by the discontinuities at rational times, and the fractalisation
phenomenon at the generic time $\displaystyle{t=\frac{1}{\pi^{2}}+0.001}$.

## Acknowledgements

BP was partially supported by a Leverhulme Research Fellowship. DAS gratefully
acknowledges support from the Quarterly Journal of Mechanics and Applied
Mathematics Fund for Applied Mathematics.

## Appendix

######  Proposition 2.

Let $u(x,t)$ be defined by equation (21). Then, for all $x\in(0,1)$ and all
$t>0$,

$\lim_{\delta\to 0}u(x+\delta,t)=u(x,t).$

###### Proof.

To prove the claim, we need to evaluate the $\delta\to 0$ limit of

$\int_{\partial E^{+}_{2}}{\rm
e}^{ik^{3}t}\frac{\rho(k)}{\Delta(k)}\left(\alpha^{2}{\rm e}^{ikx}\left[1-{\rm
e}^{ik\delta}\right]-\alpha{\rm e}^{i\alpha^{2}kx}\left[1-{\rm
e}^{ik\alpha^{2}\delta}\right]\right)\mathrm{d}k.$

Consider the integral along the negative real axis, which can be rewritten as
$J_{1}+\alpha^{2}J_{2}-\alpha^{2}J_{3}-\alpha J_{4}$, for

$\displaystyle J_{1}$
$\displaystyle=\int_{0}^{\left\lvert\delta\right\rvert^{\frac{-1}{3}}}{\rm
e}^{-ik^{3}t}\frac{\rho(-k)}{\Delta(-k)}\left(\alpha^{2}{\rm
e}^{-ikx}\left[1-{\rm e}^{-ik\delta}\right]-\alpha{\rm
e}^{-i\alpha^{2}kx}\left[1-{\rm
e}^{-ik\alpha^{2}\delta}\right]\right)\mathrm{d}k,$ $\displaystyle J_{2}$
$\displaystyle=\int_{\left\lvert\delta\right\rvert^{\frac{-1}{3}}}^{\infty}{\rm
e}^{-ikx-ik^{3}t}\frac{\rho(-k)}{\Delta(-k)}\mathrm{d}k,$ $\displaystyle
J_{3}$
$\displaystyle=\int_{\left\lvert\delta\right\rvert^{\frac{-1}{3}}}^{\infty}{\rm
e}^{-ik(x+\delta)-ik^{3}t}\frac{\rho(-k)}{\Delta(-k)}\mathrm{d}k,$
$\displaystyle J_{4}$
$\displaystyle=\int_{\left\lvert\delta\right\rvert^{\frac{-1}{3}}}^{\infty}{\rm
e}^{-ik^{3}t}\frac{\rho(-k)}{\Delta(-k)}{\rm e}^{-i\alpha^{2}kx}\left[1-{\rm
e}^{-ik\alpha^{2}\delta}\right]\mathrm{d}k.$

We shall argue that each has limit $0$ as $\delta\to 0$. The argument for the
integral along the other ray of $\partial E_{2}^{+}$ is analogous, so shall be
omitted.

Let

$m=\sup_{x\in[0,\infty)}\left\lvert\frac{\rho(-k)}{\Delta(-k)}\right\rvert,$

which, from our earlier analysis of the asymptotic behaviour of these
functions and the locus of the zeros of $\Delta$, is finite. Then

$\displaystyle\left\lvert J_{1}\right\rvert$ $\displaystyle\leq
m\left(\int_{0}^{\left\lvert\delta\right\rvert^{\frac{-1}{3}}}\left\lvert
1-{\rm
e}^{-ik\delta}\right\rvert\mathrm{d}k+\int_{0}^{\left\lvert\delta\right\rvert^{\frac{-1}{3}}}\left\lvert
1-{\rm e}^{-ik\alpha^{2}\delta}\right\rvert\mathrm{d}k\right)$
$\displaystyle\leq
m\left(\int_{0}^{\left\lvert\delta\right\rvert^{\frac{-1}{3}}}k\left\lvert\delta\right\rvert\mathrm{d}k+\int_{0}^{\left\lvert\delta\right\rvert^{\frac{-1}{3}}}2k\left\lvert\delta\right\rvert\mathrm{d}k\right),$

where the inequality in the first integrand is bounding the chord by the
arclength, and the second follows from the fact that

$\left\lvert 1-{\rm e}^{-i\alpha^{2}k}\right\rvert^{2}=\left(1-{\rm
e}^{-i\alpha^{2}k}\right)\left(1-{\rm e}^{i\alpha
k}\right)=k^{2}-k^{3}3\sqrt{3}+O\left(k^{4}\right)$

as $\left\lvert k\right\rvert\to 0$. Hence $\left\lvert J_{1}\right\rvert\to
0$ as $\delta\to 0$. We also bound

$\left\lvert J_{4}\right\rvert\leq
2m\int_{\left\lvert\delta\right\rvert^{\frac{-1}{3}}}^{\infty}{\rm
e}^{-kx\sqrt{3}/2}\mathrm{d}k,$

which has limit $0$ as $\delta\to 0$.

Integrating by parts twice in the numerator, we find that

(25) $\frac{\rho(-k)}{\Delta(-k)}=\frac{i\alpha}{k}\sum_{j=1}^{n-1}{\rm
e}^{iky_{j}}\delta_{j}-\frac{\sqrt{3}u_{0}(1)}{k}{\rm
e}^{iky_{n}}-\frac{i\alpha u_{0}(0)}{k}{\rm
e}^{iky_{0}}+O\left(k^{-2}\right),$

in which $[0=y_{0},y_{1},\ldots,y_{n}=1]$ is a partition such that each
$u_{0}\rvert_{(y_{j},y_{j+1})}$ is extensible to a
$\operatorname{C}^{2}[y_{j},y_{j+1}]$ function, and
$\delta_{j}=u_{0}(y_{j}^{+})-u_{0}(y_{j}^{-})$. We split each of $J_{2}$ and
$J_{3}$ into $n+2$ terms: one for each of the oscillatory behaviours ${\rm
e}^{iky_{j}}$ on the right of equation (25), and one for the asymptotic term.
The asymptotic term yields an absolutely convergent improper integral for each
of $J_{2}$ and $J_{3}$; in the case of $J_{3}$, the convergence is uniform in
the $\delta$ appearing in the integrand. Hence, in the limit $\delta\to 0$, we
have convergence to $0$ of the part of each of $J_{2}$ and $J_{3}$ resulting
from the $O\left(k^{-2}\right)$ term.

Let $\phi_{\delta}(k)=k^{3}t+k(x+\delta)$. Then there exists an $M>0$ such
that, for sufficiently large $a>0$, all $b>a$, and all sufficiently small
$\delta>0$, we have that $\phi_{\delta}^{\prime\prime}(k)>0$ and
$\phi_{\delta}^{\prime}(k)-y_{j}>M>0$. Therefore,

$\displaystyle\left\lvert\int_{a}^{b}{\rm
e}^{i(y_{j}k-\phi(k))}\mathrm{d}k\right\rvert$
$\displaystyle=\left\lvert\frac{{\rm
e}^{i(y_{j}b-\phi_{\delta}(b))}}{i(y_{j}-\phi_{\delta}^{\prime}(b))}-\frac{{\rm
e}^{i(y_{j}a-\phi_{\delta}(a))}}{i(y_{j}-\phi_{\delta}^{\prime}(a))}-\int_{a}^{b}\frac{\phi^{\prime\prime}(k){\rm
e}^{i(y_{j}k-\phi_{\delta}(k))}}{i(y_{j}-\phi_{\delta}^{\prime}(k))^{2}}\mathrm{d}k\right\rvert$
$\displaystyle\leq\frac{1}{\phi_{\delta}^{\prime}(b)-y_{j}}+\frac{1}{\phi_{\delta}^{\prime}(a)-y_{j}}+\int_{a}^{b}\frac{\phi_{\delta}^{\prime\prime}(k)}{(\phi_{\delta}^{\prime}(k)-y_{j})^{2}}\mathrm{d}k$
$\displaystyle=\frac{2}{\phi_{\delta}^{\prime}(a)-y_{j}}<\frac{2}{M}.$

Hence, by Dirichlet’s test for convergence of improper integrals, each of the
other $n+1$ constituents of each of $J_{2}$ and $J_{3}$ converges. Taking the
limit $\delta\to 0$, these $n+1$ constituents of each of $J_{2}$ and $J_{3}$
converge to $0$. So also do the whole of $J_{2}$ and $J_{3}$ converge to $0$
in the $\delta\to 0$ limit. ∎

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