\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 08, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2009/08\hfil Short-wave equation]
{The Cauchy problem for a short-wave equation}

\author[S. M. A. Gama, G. Smirnov\hfil EJDE-2009/08\hfilneg]
{Silvio Marques A. Gama, Gueorgui Smirnov}  % in alphabetical order

\address{Centro de Matem\'atica da Universidade do Porto
and Departamento de Matem\'atica Aplicada.
Faculdade de Ci\^encias da Universidade do Porto.
Rua do Campo Alegre, 687, 4169-007 Porto, Portugal}
\email[S. M. A. Gama]{smgama@fc.up.pt}
\email[G. Smirnov]{gsmirnov@fc.up.pt}

\thanks{Submitted July 18, 2008. Published January 6, 2009.}
\subjclass[2000]{34A12, 34A34, 35Q35, 35Q53}
\keywords{Cauchy problem; Benjamin-Bona-Mahony-Perigrine equation; 
\hfill\break\indent
short-waves}

\begin{abstract}
 We prove existence and uniqueness of solutions for the
 Cauchy problem of the simplest nonlinear short-wave equation,
 $u_{tx}=u-3u^{2}$,  with periodic boundary condition.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

In this paper we consider the Cauchy problem for the short-wave equation
\begin{equation}
u_{tx}=u-3u^{2},\label{eq:sw1d}
\end{equation}
with the boundary condition ($L>0$)
\begin{equation}
u(0,t)=u(L,t),\quad t\ge0,\label{eq:PerBoundCond}
\end{equation}
and the $L$-periodic initial condition
\begin{equation}
u(x,0)=\phi(x),\quad\forall x\in\mathbb{R}.\label{eq:InitCond}
\end{equation}
Here, $u(x,t)$ represents a small amplitude depending on one-dimensional
(fast) space variable $x$ and (slow) time $t$.

Nonlinear evolution of long waves in dispersive media with small
amplitude in shallow water is a well known subject. It has been described
by many mathematical models such as the Boussinesq equation
\cite{b,w}, the KdV equation \cite{kdv}, or the
Benjamin-Bona-Mahony-Peregrine equation (BBMP) \cite{bbm,p}.
In contrast, for short-waves, commonly called ripples, only a few
results exist \cite{mm,gkm,bkmp}. When we speak of long or
short-waves, we are referring to an underlying spacescale, $X$, to
which all space variables have been compared. Thus, for instance,
for the surface-wave motion of a fluid, the unperturbed depth
serves as a natural parameter. The shortness of the waves is
referred to this underlying parameter.

The short-wave equation \eqref{eq:sw1d} is derived in \cite{mm}
via multiple-scale perturbation theory from BBMP and governs the
leading order term of the asymptotic dynamics of short-waves
sustained by BBMP. A first study of equation \eqref{eq:sw1d} was
done in \cite{gkm}. We sketch here its derivation. Start
from BBMP
\begin{equation} \label{bbmp_eq}
  U_T+U_X-U_{XXT}=3(U^2)_X,
\end{equation}
which is the model equation for the unperturbed equation to which we will
find the short-wave limit. Here, $U(X,T)$ represents a small
amplitude depending on one-dimensional space variable $X$ and
time $T$. Its linear dispersion relation, $\omega(k)$, is real
(this means that we are not dealing with dissipative effects) and
is given by
\begin{equation}
\omega(k)=\frac{k}{1+k^2},
\end{equation}
having zero limit when $k\to \infty$. The phase and group
velocity are all bounded in the short-wave limit $k\to
\infty$. This property allows BBMP to sustain short-waves. In
fact, let us consider a short-wave with characteristic length
$\ell=\varepsilon\sim k^{-1}$, with $k\gg 1$. Define the scaled
(fast) space variable $x=\varepsilon^{-1}X$ ($\varepsilon\ll 1$).
The characteristic time associated with short-waves is given by
looking at the dispersive relation of the linear part for the time
variable. In our case,
$\omega(\varepsilon^{-1})=\varepsilon-\varepsilon^3+\varepsilon^5-\dots$.
In this way, we obtain the scaled (slow) time variable
$t=\varepsilon T$. We are lead thus to the scaled variables
$x=\varepsilon^{-1}X$ and $t=\varepsilon T$, which transforms the
$X$ and $T$ derivatives into
$\partial_X=\varepsilon^{-1}\partial_x$ and
$\partial_T=\varepsilon\partial_t$. Assume now the expansion
$U=u_0+\varepsilon u_1+\dots$. Passing to the $x$ and $t$
variables and integrating in $x$, we have the lowest order in
(\ref{bbmp_eq}) in the form
\begin{equation}
  u_{0tx}=u_0-3(u_0)^2.
\end{equation}
For simplicity, writing $u_0$ as $u$, we obtain \eqref{eq:sw1d}.

In the next section, under certain conditions, we prove the
existence and uniqueness of solutions for
\eqref{eq:sw1d}-(\ref{eq:InitCond}).

\section{Main result}

Let $u=u(x,t)$ be a classical solution to the Cauchy problem, that is,
 a twice continuously differentiable
function satisfying \eqref{eq:sw1d}-(\ref{eq:InitCond}).
Integrating the left-hand side of \eqref{eq:sw1d}
in $x$, from $0$ to $L$, and using \eqref{eq:PerBoundCond}, we get
\[
\frac{d}{dt}\int_{0}^{L}u_{x}(x,t)dx=\frac{d}{dt}\left(u(L,t)-u(0,t)\right)=0.
\]
 Therefore, from \eqref{eq:sw1d}, we have
\begin{equation}
0=\frac{d}{dt}\int_{0}^{L}u_{x}(x,t)dx=\int_{0}^{L}
\left(u(x,t)-3u^{2}(x,t)\right)dx.
\label{eq:OneRelation}
\end{equation}
Thus, it is natural to consider only initial conditions satisfying (\ref{eq:OneRelation}).

Note also that the $L_2$-norm of $u_x(\cdot, t)$ is a constant.
Indeed, multiplying both sides of \eqref{eq:sw1d} by $u_{x}$ and
integrating in $x$, from $0$ to $L$, we obtain
\begin{equation} \label{uxconst}
\begin{aligned}
\frac{1}{2}\frac{d}{dt}|u_x(\cdot, t)|_2^2
&=\frac{d}{dt}\int_{0}^{L}\frac{u_{x}^{2}(x,t)}{2}dx\\
&=\int_{0}^{L}\big(u(x,t)-3u^{2}(x,t)\big)u_{x}(x,t)dx \\
&=\int_{0}^{L}\frac{\partial}{\partial x}
 \big(\frac{u^{2}(x,t)}{2}-u^{3}(x,t)\big)dx\\
&=\big(\frac{u^{2}(L,t)}{2}-u^{3}(L,t)\big)
 -\big(\frac{u^{2}(0,t)}{2}-u^{3}(0,t)\big)=0.
\end{aligned}
\end{equation}
This observation is of importance in the proof of a global existence.

We will seek for solutions to problem \eqref{eq:sw1d}-\eqref{eq:InitCond}
in a generalized sense.
Namely, consider a formal Fourier series
\begin{equation}
u(x,t)=\sum_{n=-\infty}^{\infty}u_{n}(t)
e^{i2\pi nx/L},\quad u_{-n}=\overline{u_{n}},
\label{eq:Fourierexpansion}
\end{equation}
with coefficients depending on $t$. Assume that
\[
u(x,0)=\phi(x),\quad  x\in \mathbb{R},
\]
where $\phi$ is an $L$-periodic function. It is assumed that
$u_{-n}=\overline{u_{n}}$ or, equivalently, $u(x,t)\in\mathbb{R}$.
Formally substituting Fourier series \eqref{eq:Fourierexpansion}
in the differential equation we obtain
a system of ordinary differential
equations
\begin{equation}
\frac{du_{n}(t)}{dt}=-\frac{iL}{2\pi n}
\Big(u_{n}(t)-3\sum_{{\alpha+\beta=n,\, n\in\mathbb{Z}}}u_{\alpha}(t)
u_{\beta}(t)\Big),\quad  n\neq0.\label{eq:SWfourier}
\end{equation}
(Denote $u_{n}(t)$ simply by $u_{n}.$) Note that, for $n=0$, we do
not obain a differential equation for $u_{0}$, but a constraint
relating $u_{0}$ to all the others Fourier modes. Since $u_{0}$ is
the real function $u$ average value over the domain of
periodicity,  we obtain the equation
\begin{equation}
\label{u0}
u_{0}-3u_{0}^{2}=3\sum_{n\in\mathbb{Z},\, n\ne0}|u_{n}|^{2}.
\end{equation}
This equation admits real solutions
\begin{equation}
u_{0}=\frac{1}{6}\Big(1\pm\sqrt{1-36\sum_{n\in\mathbb{Z},\, n\ne 0}
|u_n|^{2}}\Big),
\label{eq:u0}
\end{equation}
only if $\sum_{n\in\mathbb{Z},\, n\ne0}|u_n|^{2}\le1/36$.
For definiteness assume
from now on that the sign in formula \eqref{eq:u0} is plus, for
example. The other choice is essentially the same, the major
difference being the fact that it results in waves travelling in
the opposite direction \cite{gkm}.

Rewrite \eqref{eq:SWfourier}, in the integral form
\begin{equation}
\label{int}
u_{n}(t)=\phi_n-\frac{iL}{2\pi n}\int_{0}^{t}\Big(u_{n}(s)
-3\sum_{\alpha+\beta=n,\, n\in\mathbb{Z}}u_{\alpha}(s)
u_{\beta}(s)\Big)ds,\quad n\neq0,
\end{equation}
Denote by $H$ the space of complex sequences
$v=\{v_{n}\}_{n\in\mathbb{Z}}$ with the norm
\[
|v|=\Big(|v_{0}|^{2}+\sum_{n\in\mathbb{Z},\, n\ne0}
n^{2}|v_{n}|^{2}\Big)^{1/2}.
\]
The space of $L$-periodic functions $u$ with Fourier coefficients
$\{u_{n}\}_{n=-\infty}^{\infty}\in H$, we shall also denote
by $H$.  Let
\[
\phi(x)=\sum_{n=-\infty}^{\infty}\phi_{n}e^{i2\pi nx/L}\in H,
\]
with $\phi_{-n}=\overline{\phi_{n}}$. We say that  a function
$u\in C\big([0,\infty),H\big)$,
$$
t\to u(t)=\sum_{n=-\infty}^{\infty}u_{n}(t)e^{i2\pi nx/L},\quad
u_{-n}=\overline{u_{n}},
$$
is a solution to problem \eqref{eq:sw1d}-\eqref{eq:InitCond},
if $\dot{u}\in L_{\infty}\big([0,\infty),H\big)$,
and the Fourier coefficients $u_n$ satisfy
(\ref{eq:u0}), (\ref{int}), and $u_n(0)=\phi_n$, for all $n$.


Now we are in a position to formulate the main result of this paper.

\begin{theorem}\label{thm1}
If $\phi\in H$ satisfies
\[
\sum_{n\in\mathbb{Z},\, n\ne0}n^{2}|\phi_{n}|^{2}<1/72
\quad\text{and}\quad
\int_{0}^{L}\big(\phi(x)-3\phi^{2}(x)\big)dx=0,
\]
then  problem \eqref{eq:sw1d}-\eqref{eq:InitCond}  has one and
only one solution.
For all $t\geq 0$, Fourier series
\eqref{eq:Fourierexpansion} converges uniformly in $x$. Its sum is
differentiable in $x$ for almost all $x\in [0,L]$. The derivative
satisfies the conditions $u_x(\cdot,t)\in L_2([0,L],R)$ and
$u_x(x,\cdot)\in C([0,\infty[,R)$. Moreover, $u_x$ is
differentiable in $t$ and \eqref{eq:sw1d} holds for almost all
$x\in [0,L]$.
\end{theorem}

\noindent\textbf{Remark.}
The uniform convergence of Fourier series \eqref{eq:Fourierexpansion}
implies that $u(\cdot ,t)$ is a continuous $L$-periodic function.

The proof of Theorem \ref{thm1} is divided in several steps.
First note that the condition
\[
\int_{0}^{L}(\phi(x)-3\phi^{2}(x))dx=0,
\]
implies
\[
\phi_{0}=3|\phi_{0}|^{2}+3\sum_{n\in\mathbb{Z},\, n\ne0}|\phi_{n}|^{2}.
\]
From this, we get
\begin{equation}
\phi_{0}=\frac{1}{6}\Big(1\pm\sqrt{1-36\sum_{n\in\mathbb{Z},\, n\ne0}
|\phi_{n}|^{2}}\Big).\label{phi0}\end{equation}
Since
$$
\sum_{n\in\mathbb{Z},\, n\ne0}|\phi_{n}|^{2}\leq
\sum_{n\in\mathbb{Z},\, n\ne0}n^2|\phi_{n}|^{2}
<1/72,
$$
it follows that $\phi_0$ is well defined.
Let $v(\cdot)\in L_{\infty}([0,T],H)$. The norm in this
space we shall denote by $\|v\|$. Define an operator
$f:L_{\infty}([0,T],H)\to L_{\infty}([0,T],H)$
as follows:
\begin{gather}
  f_{n}(v(\cdot))(t)=\phi_{n}-\frac{iL}{2\pi n}\int_{0}^{t}\Big(v_{n}(s)-
3\sum_{k=-\infty}^{\infty}v_{k}(s)v_{n-k}(s)\Big)ds,\quad
 n\neq0,\label{fn}\\
  f_{0}(v(\cdot))(t)=\frac{1}{6}
\Big(1+\sqrt{1-36\sum_{n\in\mathbb{Z},\, n\ne0}|f_{n}(v(\cdot))(t)|^{2}}\Big).
\label{f0}
\end{gather}
Let $M>0$. Denote by $\Phi\in L_{\infty}([0,T],H)$ the
constant function $\Phi(t)\equiv\phi$ and consider a complete metric
space
\[
V_{TM}=\{v(\cdot)\in L_{\infty}([0,T],H):\|v-\Phi\|\leq M\}
\]
with the metric induced by $L_{\infty}([0,T],H)$. We need
the following auxiliary results.

\begin{proposition} \label{p1}
If
$\sum_{n\neq0}n^{2}|\phi_{n}|^{2}<1/72$
 and $T$ is sufficiently small, then $f$ is well defined and is
a contractive map from $V_{TM}$ into $V_{TM}$.
\end{proposition}

\begin{proof}
Since
\begin{align*}
f_{n}(v)(t)-f_{n}(w)(t)
&=-\frac{iL}{2\pi n} \int_{0}^{t}\Big[(v_{n}(s)-w_{n}(s))
 +3\! \sum_{k=-\infty}^{\infty}((v_{k}(s)-w_{k}(s))v_{n-k}(s)\\
&\quad +w_{k}(s)(v_{n-k}(s)-w_{n-k}(s))\Big]ds,\quad  n\neq 0,
\end{align*}
we have
\begin{align*}
&\sum_{n\in\mathbb{Z},\, n\ne0}n^{2}|f_{n}(v)(t)-f_{n}(w)(t)|^{2}\\
&\leq(\mathop{\rm const})\sum_{n\in\mathbb{Z},\, n\ne0}
 \Big[\int_{0}^{t}\Big[|v_{n}(s)-w_{n}(s)|\\
&\quad + \sum_{k=-\infty}^{\infty}(|v_{k}(s)-w_{k}(s)||v_{n-k}(s)|
 +|w_{k}(s)||v_{n-k}(s)-w_{n-k}(s)|\Big]ds\Big]^{2}\\
&\leq(\mathop{\rm const})t \sum_{n\in\mathbb{Z},\, n\ne0}\int_{0}^{t}
 \Big[|v_{n}(s)-w_{n}(s)|\\
&\quad +\sum_{k=-\infty}^{\infty}|v_{k}(s)-w_{k}(s)|(|v_{n-k}(s)|+|w_{n-k}(s)|)
 \Big]^{2}ds\\
&\leq(\mathop{\rm const})t \sum_{n\in\mathbb{Z},\, n\ne0}\int_{0}^{t}
 \Big[|v_{n}(s)-w_{n}(s)|^{2}\\
&\quad +\Big(\sum_{k=-\infty}^{\infty}|v_{k}(s)-w_{k}(s)|(|v_{n-k}(s)|
 +|w_{n-k}(s)|)\Big)^{2}\Big]ds\\
&\leq(\mathop{\rm const})t \sum_{n\in\mathbb{Z},\, n\ne0}\int_{0}^{t}
 \Big[|v_{n}(s)-w_{n}(s)|^{2}+|v_{0}(s)-w_{0}(s)|^{2}(|v_{n}(s)|^{2}
 +|w_{n}(s)|^{2})\\
&\quad + \Big(\sum_{k\neq0}\frac{1}{k^{2}}\Big)
 \sum_{k=-\infty}^{\infty}k^{2}|v_{k}(s)-w_{k}(s)|^{2}(|v_{n-k}(s)|^{2}
 +|w_{n-k}(s)|^{2})\Big]ds\\
&\leq(\mathop{\rm const})t\int_{0}^{t}
 \Big[\sum_{n\in\mathbb{Z},\, n\ne0}|v_{n}(s)-w_{n}(s)|^{2}
 +|v_{0}(s)-w_{0}(s)|^{2}\sum_{n\in\mathbb{Z},\, n\ne0}(|v_{n}(s)|^{2}\\
&\quad  +|w_{n}(s)|^{2})
 \sum_{k=-\infty}^{\infty}k^{2}|v_{k}(s)-w_{k}(s)|^{2}
 \sum_{n\in\mathbb{Z},\, n\ne0}(|v_{n}(s)|^{2}+|w_{n}(s)|^{2})\Big]ds\\
&\leq(\mathop{\rm const})t\int_{0}^{t}
 \Big[1+\sum_{n\in\mathbb{Z},\, n\ne0}(|v_{n}(s)|^{2}
 +|w_{n}(s)|^{2})\Big]ds\|v-w\|^{2}\\
&\leq(\mathop{\rm const})T^{2}(1+\|v\|^{2}+\|w\|^{2})\|v-w\|^{2}.
\end{align*}
 We have thus proved the  inequality
\begin{equation}
\sum_{n\in\mathbb{Z},\, n\ne0}n^{2}|f_{n}(v)(t)-f_{n}(w)(t)|^{2}
\leq(\mathop{\rm const})T^{2}(1+\|v\|^{2}+\|w\|^{2})\|v-w\|^{2}.\label{e3}
\end{equation}
We also have
\begin{align}
&|f_{0}(v)(t)-f_{0}(w)(t)|^{2}\nonumber \\
&=\frac{1}{36}\Big|\sqrt{1-36\sum_{n\in\mathbb{Z},\, n\ne0}
  |f_{n}(v)(t)|^{2}}
 -\sqrt{1-36\sum_{n\in\mathbb{Z},\, n\ne0}|f_{n}(w)(t)|^{2}}\Big|^{2}
\label{e4} \\
&\leq\frac{(\mathop{\rm const})\sum_{n\in\mathbb{Z},\, n\ne0}
  (|f_{n}(v)(t)|^{2}+|f_{n}(w)(t)|^{2})}
 {\big|\sqrt{1-36\sum_{n\in\mathbb{Z},\, n\ne0}|f_{n}(v)(t)|^{2}}
      +\sqrt{1-36\sum_{n\in\mathbb{Z},\, n\ne0}|f_{n}(w)(t)|^{2}}\big|^{2}}
\nonumber\\
&\quad \times\sum_{n\in\mathbb{Z},\, n\ne0}|f_{n}(v)(t)-f_{n}(w)(t)|^{2}.
\nonumber
\end{align}
The inclusion $v\in V_{TM}$ implies
$\|v\|^{2}\leq(\|\Phi\|+\|\Phi-v\|)^{2}\leq(\|\Phi\|+M)^{2}$.
Since $\Phi=f(0)$, from \eqref{e3} we get
\[
\sum_{n\in\mathbb{Z},\, n\ne0}n^{2}|f_{n}(v)(t)-\phi_{n}|^{2}
\leq(\mathop{\rm const})T^{2}(1+(\|\Phi\|+M)^{2})^{2}.
\]
 Therefore
\begin{align*}
\sum_{n\in\mathbb{Z},\, n\ne0}|f_{n}(v)(t)|^{2}
&\leq 2\sum_{n\in\mathbb{Z},\, n\ne0}n^{2}|\phi_{n}|^{2}
 +2\sum_{n\in\mathbb{Z},\, n\ne0}n^{2}|f_{n}(v)(t)-\phi_{n}|^{2}\\
&\leq2\sum_{n\in\mathbb{Z},\, n\ne0}n^{2}|\phi_{n}|^{2}
 +(\mathop{\rm const})T^{2}(1+(\|\Phi\|+M)^{2})^{2}\\
&\leq\sigma<\frac{1}{36},
\end{align*}
 whenever $T>0$ is small enough. Thus the map $f$ is well defined
(see (\ref{fn}) and (\ref{f0})). From \eqref{e3} and (\ref{e4})
we obtain
\begin{align*}
|f_{0}(v)(t)-f_{0}(w)(t)|^{2}
&\leq(\mathop{\rm const})  \sum_{n\in\mathbb{Z},\, n\ne0}|f_{n}(v)(t)
  -f_{n}(w)(t)|^{2}\\
&\leq(\mathop{\rm const})\sum_{n\in\mathbb{Z},\, n\ne0}n^{2}|f_{n}(v)(t)
  -f_{n}(w)(t)|^{2}\\
&\leq(\mathop{\rm const})T^{2}(1+\|v\|^{2}+\|w\|^{2})\|v-w\|^{2}.
\end{align*}
 Invoking again \eqref{e3}, we get
\begin{equation}
\begin{aligned}
\|f(v)-f(w)\|^{2}
&\leq(\mathop{\rm const})T^{2}(1+\|v\|^{2}+\|w\|^{2})\|v-w\|^{2}\\
&\leq(\mathop{\rm const})T^{2}(1+(\|\Phi\|+M)^{2})\|v-w\|^{2}.\label{6}
\end{aligned}
\end{equation}
In particular, we have
\[
\|f(v)-\Phi\|^{2}\leq(\mathop{\rm const})T^{2}(1+(\|\phi\|+M)^{2})^{2}\leq M^{2},
\]
for small $T>0$. Thus we see that $f:V_{TM}\to V_{TM}$
and from (\ref{6}) it follows that $f$ is a contraction, whenever
$T>0$ is small enough.
\end{proof}


\begin{proposition}\label{p2}
Let $u\in L_{\infty}([0,T],H)$ be a solution
to the equation $u=f(u)$. Assume that \[
\sum_{n\in\mathbb{Z},\, n\ne0}n^{2}|u_{n}(t)|^{2}\leq\delta<1/36.\]
 Then $u\in C([0,T],H)$ and $\dot{u}\in L_{\infty}([0,T],H)$.
\end{proposition}

\begin{proof}
Similarly to inequality (\ref{e4}) we have
\begin{align*}
|u(t_{2})-u(t_{1})|^{2}
&=|u_{0}(t_{2})-u_{0}(t_{1})|^{2}
 +\sum_{n\in\mathbb{Z},\, n\ne0}n^{2}|u_{n}(t_{2})-u_{n}(t_{1})|^{2}\\
&\leq(\mathop{\rm const})\sum_{n\in\mathbb{Z},\, n\ne0}n^{2}|u_{n}(t_{2})
  -u_{n}(t_{1})|^{2}.
\end{align*}
 From (\ref{fn}) we see that the right side of the inequality is
less than or equal to
\begin{align*}
&(\mathop{\rm const})|t_{2}-t_{1}|\sum_{n\in\mathbb{Z},\, n\ne0}
 \Big|\int_{t_{1}}^{t_{2}}\Big(|u_{n}(s)+3\sum_{k=-\infty}^{\infty}
 |u_{k}(s)||u_{n-k}(s)|\Big)^{2}ds\Big|\\
&\leq(\mathop{\rm const})|t_{2}-t_{1}|\sum_{n\in\mathbb{Z},\, n\ne0}
 \int_{t_{1}}^{t_{2}}\Big(1+|u_{0}(s)|^{2}\\
&\quad +\Big(\sum_{k\in\mathbb{Z},\, k\ne0}\frac{1}{k^{2}}\Big)
 \sum_{k\in\mathbb{Z},\, k\ne0}k^{2}|u_{k}|^{2}\Big)
 \sum_{n\in\mathbb{Z},\, n\ne0}|u_{n}(s)|^{2}ds\\
&\leq(\mathop{\rm const})|t_{2}-t_{1}|^{2}.
\end{align*}
 This proves the continuity of $u(t)$.
Since
\[
|\dot{u}_{0}|^{2}=\frac{9\Big|\sum_{n\in\mathbb{Z},\, n\ne0}
(\dot{u}_{n}u_{-n}+u_{n}\dot{u}_{-n})\Big|^{2}}
{1-36\sum_{n\in\mathbb{Z},\, n\ne0}|u_{n}|^{2}}
\leq(\mathop{\rm const})\sum_{n\in\mathbb{Z},\, n\ne0}
 |\dot{u}_{n}|^{2}\sum_{n\in\mathbb{Z},\, n\ne0}|u_{n}|^{2}
\]
 and
\[
\sum_{n\in\mathbb{Z},\, n\ne0}n^{2}|\dot{u}_{n}|^{2}
=\sum_{n\in\mathbb{Z},\, n\ne0}\big(\frac{L}{2\pi}\big)^{2}
\Big|u_{n}-3\sum_{n\in\mathbb{Z},\, n\ne0}u_{k}u_{n-k}\Big|^{2},
\]
 we have
\begin{align*}
&|\dot{u}_{0}|^{2}+\sum_{n\in\mathbb{Z},\, n\ne0}n^{2}
 |\dot{u}_{n}|^{2}\\
&\leq(\mathop{\rm const})\sum_{n\in\mathbb{Z},\, n\ne0}
 \Big|u_{n}-3\sum_{n\in\mathbb{Z},\, n\ne0}u_{k}u_{n-k}\Big|^{2}\\
&\leq(\mathop{\rm const})\sum_{n\in\mathbb{Z},\, n\ne0}
 \Big(|u_{n}|^{2}+|u_{0}|^{2}|u_{n}|^{2}
 +\Big(\sum_{{k\in\mathbb{Z},\, k\ne0}}\frac{1}{k^{2}}\Big)
 \sum_{k\in\mathbb{Z},\, k\ne0}k^{2}|u_{k}|^{2}|u_{n-k}|^{2}\Big)\\
&\leq(\mathop{\rm const})\Big(1+|u_{0}|^{2}
 +\Big(\sum_{k\in\mathbb{Z},\, k\ne0}\frac{1}{k^{2}}\Big)
 \sum_{k\in\mathbb{Z},\, k\ne0}k^{2}|u_{k}|^{2}\Big)
 \sum_{n\in\mathbb{Z},\, n\ne0}|u_{n}|^{2}
\leq(\mathop{\rm const}).
\end{align*}
 Thus $\dot{u}\in L_{\infty}([0,T],H)$.
\end{proof}

Note that we also proved that the function $u\in C([0,T],H)$
is Lipschitzian.
Now show that generalized solutions also satisfy property
(\ref{uxconst}).

\begin{proposition}\label{p3}
Assume that $u\in L_{\infty}([0,T],H)$ satisfies \eqref{eq:u0}. Then
\[
\sum_{n\in\mathbb{Z},\, n\ne0}n^{2}|u_{n}(t)|^{2}=(\mathop{\rm const}).
\]
\end{proposition}

\begin{proof}
Indeed, we have
\begin{align*}
&\frac{d}{dt} \sum_{n=-\infty}^\infty
\big(\frac{2\pi n}{L}\big)^2 |u_n|^2\\
&= \sum_{n=-\infty}^\infty
\big(\frac{2\pi n}{L}\big)^2 (\dot{u}_n u_{-n} + u_n\dot{u}_{-n})\\
&= \frac{2\pi i}{L} \sum_{n=-\infty}^\infty n
\Big[u_n\Big( u_{-n}-3 \sum_{k=-\infty}^\infty
u_ku_{-n-k}\Big)-u_{-n}\Big( u_{n}-3 \sum_{k=-\infty}^\infty
u_ku_{n-k}\Big)\Big] \\
&= - \frac{6\pi i}{L}S,
\end{align*}
where
$$
S=\sum_{n=-\infty}^\infty n \Big[u_n
\sum_{k=-\infty}^\infty u_ku_{-n-k}
-u_{-n} \sum_{k=-\infty}^\infty u_ku_{n-k}\Big].
$$
Observe that
$$
S=\sum_{n,k=-\infty}^\infty n u_n u_k u_{-n-k}
- \sum_{n,k=-\infty}^\infty n u_{-n} u_k u_{n-k}
= 2\sum_{n,k=-\infty}^\infty n u_n u_k u_{-n-k}.
$$
On the other hand, introducing a new summation index $m=n-k$,
we can rewrite $S$ in the  form
$$
S= \sum_{n,k=-\infty}^\infty n u_n u_k u_{-n-k}
-\sum_{m,k=-\infty}^\infty (m+k) u_{-m-k} u_k u_{m}
=-\sum_{m,k=-\infty}^\infty k u_{-m-k} u_k u_{m}.
$$
Combining this with the previous equality, we get $S=-S/2$. Thus $S=0$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1}]
From Proposition \ref{p1} we see that the problem under consideration
has one and only one solution $u\in L_{\infty}([0,T],H)$,
whenever $T>0$ is small enough. By Proposition \ref{p2}
$u\in C([0,T],H)$ and $\dot{u}\in L_{\infty}([0,T],H)$.
Finally, Proposition \ref{p3} implies the existence of the solution for
all $t\geq0$.

Show that, $u(x,t)$, the sum of Fourier series
\eqref{eq:Fourierexpansion} satisfies \eqref{eq:sw1d}. From the inequality
$$
\sum_{n\in\mathbb{Z},\, n\ne0}|u_n(t)|\leq
\sqrt{\Big(\sum_{n\in\mathbb{Z},\, n\ne0}\frac{1}{n^2}\Big)
\sum_{n\in\mathbb{Z},\, n\ne0}n^2|u_n(t)|^2}=(\mathop{\rm const})
$$
we see that Fourier series \eqref{eq:Fourierexpansion} converges
uniformly in $x$ for all $t\geq 0$. The inequality
$$
\sum_{n=-\infty}^{\infty}\Big|
\sum_{k=-\infty}^{\infty}u_k(t)u_{n-k}(t)\Big|
\leq \sum_{k=-\infty}^{\infty}|u_k(t)|
\sum_{n=-\infty}^{\infty}|u_n(t)|
$$
implies that the series
$$
\sum_{n=-\infty}^{\infty}\Big(\sum_{k=-\infty}^{\infty}u_k(t)u_{n-k}(t)\Big)
e^{i2\pi nx/L}
$$
converges for all $t\geq 0$. Multiplying (\ref{int}) by
$e^{i2\pi nx/L}$ and adding the obtained equalities, we get
\begin{align*}
\sum_{n=-\infty}^{\infty}i\frac{2\pi}{L}n u_{n}(t)e^{i2\pi nx/L}
&=\sum_{n=-\infty}^{\infty}i\frac{2\pi}{L}n \phi_n e^{i2\pi nx/L}
 + \sum_{n\in\mathbb{Z},\, n\ne0}
\int_{0}^{t}\Big(u_{n}(s)\\
&\quad -3\sum_{{\alpha+\beta=n,\, n\in\mathbb{Z}}}
u_{\alpha}(s)u_{\beta}(s)\Big) e^{i2\pi nx/L}ds
\end{align*}
From the Lebesgue dominated convergence theorem and the above estimates
we have
$$
u_x(x,t)=\phi_x(x)+\int_{0}^{t} \sum_{n\in\mathbb{Z},\, n\ne0}
\Big(u_{n}(s)-3\sum_{{\alpha+\beta=n,\, n\in\mathbb{Z}}}
 u_{\alpha}(s)u_{\beta}(s)\Big) e^{i2\pi nx/L}ds.
$$
Combining this with (\ref{u0}), we obtain
$$
u_x(x,t)=\phi_x(x)+\int_{0}^{t}(u(x,s)-3u^2(x,s))ds.
$$
This completes the proof.
\end{proof}

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\end{document}