\documentclass[reqno]{amsart}
\usepackage[notref,notcite]{showkeys}
\usepackage{hyperref}
\usepackage{amssymb}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 51, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/51\hfil Nonexistence of soliton-like solutions]
{Nonexistence of soliton-like solutions for defocusing
 generalized KdV equations}

\author[S. Kwon, S. Shao \hfil EJDE-2015/51\hfilneg]
{Soonsik Kwon, Shuanglin Shao}

\address{Soonsik Kwon \newline
Department of Mathematical Sciences\\
Korea Advanced Institute of Science and Technology\\
291 Daehak-ro Yuseong-gu, Daejeon 305-701, Korea}
\email{soonsikk@kaist.edu}

\address{Shuanglin Shao \newline
Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA}
\email{slshao@math.ku.edu}

\thanks{Submitted May 21, 2013. Published February 24, 2015.}
\subjclass[2000]{35Q53}
\keywords{Generalized KdV equation; soliton, scattering}

\begin{abstract}
 We consider the global dynamics of the defocusing generalized KdV equation
 $$
 \partial_t u + \partial_x^3 u = \partial_x(|u|^{p-1}u).
 $$
 We use Tao's theorem \cite{Tao} that the energy moves faster than the
 mass to prove a moment type dispersion estimate. As an application of
 the dispersion estimate, we show that there is no soliton-like solutions
 with a certain decaying assumption.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

In this short note, we prove a dispersion estimate of the second moment 
type for the defocusing generalized KdV equation
\begin{equation}\label{gkdv}
\partial_t u + \partial_x^3 u = \partial_x(|u|^{p-1}u),\quad
 u: \mathbb{R}\times \mathbb{R} \to \mathbb{R}.
\end{equation}
As an application, we show that there is no soliton-like solution with 
decaying condition. 

Equation \eqref{gkdv} satisfies the mass and energy conservation laws:
\begin{gather*}
M(u) = \int u^2(x) \,dx  \\
E(u) = \int \frac 12 u_x^2 + \frac{1}{p+1} |u|^{p+1} \,dx
\end{gather*}
 The local well-posedness of the Cauchy problem on the energy space $H^1(\mathbb{R})$
is well known \cite{KPV93} and the energy conservation law implies the global 
existence. 

In the focusing case, where the sign of the nonlinear term is opposite, 
there are the soliton solutions $u(t,x) = Q(x-t)$, where $Q$ is the ground 
state solution
$$ 
Q(x) = \Big(\frac{p+1}{2\cosh^2(\frac{p-1}{2}x)}  \Big)^{1/(p-1)}.  
 $$
From the Pohozaev identity, one can show that there is no such soliton 
solution of permanent form in the defocusing case. Furthermore, 
it is conjectured that the nonlinear global solution scatters to a linear 
solution. Indeed,
$$ 
 \lim_{t\to \pm \infty } \| u(t) - e^{-t\partial_x^3} u_{\pm} \|_{L^2_x} \to 0. 
$$
If it were true, as this describes a concrete asymptotic behavior, 
it implies that there is no spacially localized solutions such as $L^2$-compact 
solutions - there exists a function $x(t)$ such that for any 
$\epsilon>0$, there exists $R=R(\epsilon) >0$ such that 
$\int_{|x-x(t)|>R } u^2(t,x)\,dx < \epsilon$. But toward this direction, 
there is only a partial result \cite{KKSV}. 

The purpose of this note is to show an intermediate version. We prove the 
nonexistence of soliton-like solutions. Main ingredient is the fact 
that the energy moves faster than the mass to the left. 

\section{Results}

Define the center of mass and the center of energy
\begin{gather*}
\langle x\rangle _{M}(t) = \frac 1{M(u)}\int xu^2(t,x) \,dx, \\
\langle x\rangle _E(t) = \frac 1{E(u)}\int x (\frac 12 u_x^2 
+ \frac{1}{p+1} |u|^{p+1}) \,dx.
\end{gather*}
Tao \cite{Tao} showed the following monotonicity estimate regarding 
the center of mass and the center of energy.

\begin{theorem}[Tao \cite{Tao}]\label{th:tao}
 Let $p \ge \sqrt{3}$. We have
\begin{equation}
\partial_t\langle x\rangle _M -\partial_t\langle x\rangle _E > 0.
\end{equation}
In particular, we have the dispersion estimate: 
for any function $x(t)$,
\begin{equation}\label{tao dispersion}
\sup_{t\in \mathbb{R}} \int |x-x(t)|(\rho(t,x) + e(t,x)) \,dx =\infty\,,
\end{equation}
where $ \rho(t,x) = u^2(t,x)$ and 
$e(t,x) = \frac 12 u_x^2 + \frac{1}{p+1}|u|^{p+1} $.
\end{theorem}

 This theorem shows that the center of energy moves faster than the center of mass. 
This behavior is intuitive. From the stationary phase of the linear equation 
$ u_t + u_{xxx} =0$, one can observe that the group velocity is $ -3\xi^2$,
 where $\xi$ is the frequency of the wave. Group velocity is negative 
definite and so every wave moves to the left. Moreover, the higher 
frequency waves move faster than low frequency waves. 
Since the energy is more weighted on high frequencies than mass, 
the center of energy moves faster to the left. 
The second part of Theorem~\ref{th:tao} is a result from the fact that 
the distance between $\langle x\rangle _M $ and $\langle x\rangle _E $ 
goes to infinity. 
We use this property to study a dispersion estimate of moment type.

\begin{theorem}\label{th:dispersion}
Let $p\ge \sqrt{3}$. Let $u(t,x)$ be a nonzero global Schwartz solution 
to \eqref{gkdv}. Then for any function $x(t)$,
\begin{equation}\label{eq:dispertion}
\sup_{t\in \mathbb{R}} \int (x-x(t))^2 u^2(t,x) \,dx = \infty.
\end{equation}
\end{theorem}

This can be seen as an improvement of \eqref{tao dispersion}, since we 
use solely the mass density. Roughly speaking, Theorem~\ref{th:dispersion} 
tells that the mass cannot be localized around the center of mass 
(or any $x(t)$), but has to spread out in time, while Theorem \ref{th:tao} 
tells that the center of mass and the center of energy cannot coexist in 
a moving local region. Usually, such a dispersion behavior is characterized 
as a time decay of solutions or the boundedness of space-time norms, 
such as the Strichartz estimates. Theorem \ref{th:dispersion} provides 
another form of dispersion estimate.

As a corollary, we observe that there is no soliton-like solution under 
decaying assumption.

\begin{corollary}\label{co:nonexistence}
Assume that $u(t,x)$ is a global soliton-like solution in the sense that 
there exists $x(t) \in \mathbb{R} $ such that for any $R>0$,
$$  
\sup_{t\in \mathbb{R}} \int_{|x-x(t)|>R} u^2(t,x)\,dx \lesssim \frac{1}{R^{2+\epsilon}}.
$$
Then $ u\equiv 0$.
\end{corollary}

 There are some works of this type. de Bouard and Martel \cite{deBouard-Martel} 
showed for the KP-II equation  the nonexistence of $L^2$- compact solutions 
under certain positivity condition on $x'(t)$. Their work can be written 
for the defocusing gKdV equation with $x'(t) > 0$ condition. 
This can read that there is no soliton-like solution moving to the right, 
as a real soliton solution moves to the right. Here, we do not specify a direction. 
In \cite{Martel-Merle}, Martel and Merle assume a similar decaying condition, 
and show the nonexistence of minimal mass blow-up solutions for 
critical gKdV equation (p=5).  

In the rest of the note, we provide the proof of Theorem~\ref{th:dispersion} 
and Corollary~\ref{co:nonexistence}.

\begin{proof}[Proof of Theorem~\ref{th:dispersion}] 
As $\langle x\rangle _M = \frac{1}{M(u)} \int xu^2(x)\,dx$ is a critical point of
$$ 
f(a) = \int (x-a)^2u^2(x)\,dx, 
$$ 
$\int (x-x(t))^2 u^2(t,x) \,dx $ is minimized at $x(t) = \langle x\rangle _M $. 
So, it suffices to show
\begin{equation}\label{dispersion}
\sup_{t\in \mathbb{R}} \int (x-\langle x\rangle _M)^2 u^2(t,x) \,dx = \infty.
\end{equation}
This simple observation allows us to compute the moment explicitly. 
We use  equation \eqref{gkdv} and integration by parts to compute
\begin{align*}
&\frac{d}{dt}\int (x-\langle x\rangle _M)^2 u^2(t,x)\,dx \\
&= -\int 2(x-\langle x\rangle _M)  u^2 \,dx\cdot \frac{d}{dt}\langle x\rangle _M 
 + \int (x-\langle x\rangle _M)^2 2uu_t \, dx \\
&= 0 + \int (x-\langle x\rangle _M)^2 2u(-u_{xxx} + \partial_x(|u|^{p-1}u) )\,dx \\
&\ge -6 \int u_x^2(x-\langle x\rangle _M) \,dx 
 -4 \int (x-\langle x\rangle _M)|u|^{p+1}\,dx \\
&\quad + \frac{4}{p+1} \int (x-\langle x\rangle _M) |u|^{p+1} \,dx \\
&=-12\int \Big(\frac 12 u_x^2 + \frac{1}{p+1}|u|^{p+1}\Big)(x-\langle x\rangle _M) \,dx 
 - \frac{4p-12}{p+1} \int |u|^{p+1}(x-\langle x\rangle _M) \,dx \\
&= -12E(u) \Big( \langle x\rangle _E - \langle x\rangle _M \Big) 
   - \frac{4p-12}{p+1} \int|u|^{p+1}(x-\langle x\rangle _M) \,dx
\end{align*}
The second term is bounded because of the Sobolev embedding and conservation laws:
\begin{align*}
 \int|u|^{p+1}(x-\langle x\rangle _M)\,dx 
&\le \|u\|^{p-1}_{L^\infty}\Big( \int u^2(t,x)(x-\langle x\rangle _M)^2 \,dx +M(u)
 \Big) \\
&\le  2(E(u)+M(u))(C+ M(u)) \le C_1.
\end{align*}
We show \eqref{dispersion} by contradiction, assuming that
$$ 
\sup_{t\in \mathbb{R}} \int (x-\langle x\rangle _M)^2 u^2(t,x) \,dx < C.
$$
We have
$$ 
\int_{|x-\langle x\rangle _M| =O(1)} u^2(t,x) \,dx  \ge c, 
$$ 
and so
$$ 
\int_{|x-\langle x\rangle _M| =O(1)} |u|^{p+1}(t,x) \,dx  \ge c_1.
$$ 
Then as the argument in Tao \cite{Tao} (reviewing the proof of Theorem 1),
 we obtain
$$ 
\partial_t \langle x\rangle _M -\partial_t\langle x\rangle _E  \ge c_2. 
$$
Since $\langle x\rangle _E-\langle x\rangle _M$ monotonically decreases, 
we have eventually
$$
 \frac{d}{dt}\int (x-\langle x\rangle _M)^2 u^2(t,x)\,dx 
\ge -12E(u)(\langle x\rangle _E-\langle x\rangle _M) -C_1 > 0. 
$$ 
This makes a contradiction.
\end{proof}

\begin{proof}[Proof of Corollary~\ref{co:nonexistence}] 
We simply estimate
\begin{align*}
\int (x-x(t))^2 u^2(t,x)\,dx 
&\le M(u) + \sum_{k=0}^\infty \int_{\{2^{k+1}> |x-x(t)|\ge 2^{k}\}}
 (x-x(t))^2 u^2(t,x)\,dx \\
& \lesssim M(u) + \sum_{k=0}^\infty  2^{2(k+1)}\cdot 2^{(-2-\epsilon)k} < \infty.
\end{align*}
Hence, by Theorem~\ref{th:dispersion}, $u\equiv 0$.
\end{proof}


\subsection*{Acknowledgements}
We want to thank Stefan Steinerberger for pointing out an error in the first draft. 
S.K. is partially supported by NRF(Korea) grant 2010-0024017. 
S.S. is partially supported by DMS-1160981.

\begin{thebibliography}{99}

\bibitem{deBouard-Martel} 
A. de Bouard, Y. Martel;
 \emph{Non existence of $L^2$-compact solutions of the Kadomtsev-Petviashvili 
II equation}, Math. Ann. 328 (2004), 525--544.

\bibitem{KPV93} C. E. Kenig, G. Ponce, L. Vega;
\emph{Well-posedness and scattering results for the generalized Korteweg-de
 Vries equation via the contraction principle}, 
Comm. Pure Appl. Math. 46 (1993), no. 4, 527--620.

\bibitem{KKSV} R. Killip, S. Kwon, S. Shao, M. Visan;
\emph{On the mass-critical generalized KdV equation}, Discrete Contin. 
Dyn. Syst. 32 (2012), no. 1, 191--221.

\bibitem{Martel-Merle} Y. Martel, F. Merle;
\emph{Nonexistence of blow-up solution with minimal $L^2$-mass for the critical 
gKdV equation}, Duke Math. J. 115 (2002), no. 2, 385--408.

\bibitem{Tao} T. Tao;
 \emph{Two remarks on the generalised Korteweg-de Vries equation}, 
Discrete Contin. Dyn. Syst. 18 (2007), no. 1, 1--14.

\end{thebibliography}

\end{document}