\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 $$u_{tx}=u-3u^{2},\label{eq:sw1d}$$ with the boundary condition ($L>0$) $$u(0,t)=u(L,t),\quad t\ge0,\label{eq:PerBoundCond}$$ and the $L$-periodic initial condition $$u(x,0)=\phi(x),\quad\forall x\in\mathbb{R}.\label{eq:InitCond}$$ 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 $$\label{bbmp_eq} U_T+U_X-U_{XXT}=3(U^2)_X,$$ 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 $$\omega(k)=\frac{k}{1+k^2},$$ 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 $$u_{0tx}=u_0-3(u_0)^2.$$ 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 $$0=\frac{d}{dt}\int_{0}^{L}u_{x}(x,t)dx=\int_{0}^{L} \left(u(x,t)-3u^{2}(x,t)\right)dx. \label{eq:OneRelation}$$ 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 \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} 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 $$u(x,t)=\sum_{n=-\infty}^{\infty}u_{n}(t) e^{i2\pi nx/L},\quad u_{-n}=\overline{u_{n}}, \label{eq:Fourierexpansion}$$ 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 $$\frac{du_{n}(t)}{dt}=-\frac{iL}{2\pi n} \Big(u_{n}(t)-3\sum_{{\alpha+\beta=n,\, n\in\mathbb{Z}}}u_{\alpha}(t) u_{\beta}(t)\Big),\quad n\neq0.\label{eq:SWfourier}$$ (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 $$\label{u0} u_{0}-3u_{0}^{2}=3\sum_{n\in\mathbb{Z},\, n\ne0}|u_{n}|^{2}.$$ This equation admits real solutions $$u_{0}=\frac{1}{6}\Big(1\pm\sqrt{1-36\sum_{n\in\mathbb{Z},\, n\ne 0} |u_n|^{2}}\Big), \label{eq:u0}$$ 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 $$\label{int} u_{n}(t)=\phi_n-\frac{iL}{2\pi n}\int_{0}^{t}\Big(u_{n}(s) -3\sum_{\alpha+\beta=n,\, n\in\mathbb{Z}}u_{\alpha}(s) u_{\beta}(s)\Big)ds,\quad n\neq0,$$ 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 $$\phi_{0}=\frac{1}{6}\Big(1\pm\sqrt{1-36\sum_{n\in\mathbb{Z},\, n\ne0} |\phi_{n}|^{2}}\Big).\label{phi0}$$ 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 $$\sum_{n\in\mathbb{Z},\, n\ne0}n^{2}|f_{n}(v)(t)-f_{n}(w)(t)|^{2} \leq(\mathop{\rm const})T^{2}(1+\|v\|^{2}+\|w\|^{2})\|v-w\|^{2}.\label{e3}$$ 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{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} 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} \begin{thebibliography}{0} \bibitem{bbm} T. B. Benjamin, J. L. Bona and J. J. Mahony; \textit{Model Equations for Long Waves in Nonlinear Dispersive Systems}, Phil. Trans. R. Soc. 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