%\documentclass{article}
\documentclass{beamer}

% Just comment out:
\usetheme{Outremont}
%\usetheme{Warsaw}

\setbeamercovered{transparent}

\nonstopmode
\usepackage[english]{babel}
\usepackage[latin1]{inputenc}
\usepackage{amssymb,bm}
\usepackage{graphicx,color}
\usepackage{mathrsfs}
\usepackage{amsthm}
%%%%\usepackage{times}





%\newcommand\pawse{{\vskip -15pt\textcolor{red}{$\hskip -20pt \scriptstyle{\mathbb{PAUSE}}$}}}
%\qquad\mathbb{PAUSE}\qquad\mathbb{PAUSE}\qquad\mathbb{PAUSE}$}}
%\textwidth=5.2in\textheight=2in
%\gdef\pause{}\renewcommand\cc[1]{}

%\newcommand\noframe[1]{}
%\providecommand\frame[1]{#1 \newpage}
%\renewcommand\frame[1]{#1 \newpage}
%\providecommand\frametitle[1]{{\large\bf #1}}
%\renewcommand\frametitle[1]{{\large\bf #1}}



\newcommand\pawse{\pause}
\renewcommand{\baselinestretch}{1.2}

\font\ccfont=cmss10 at 15pt
\definecolor{mybrown}{rgb}{1,0,1}
\definecolor{kommentcolor}{rgb}{0,0.4,0}
\definecolor{dred}{rgb}{0.95,0,0}
\definecolor{dblue}{rgb}{0,0,0.8}
\definecolor{dgreen}{rgb}{0,0.6,0}
\definecolor{ddgreen}{rgb}{0,0.5,0}
\definecolor{dddgreen}{rgb}{0,0.2,0}

\newcommand\range{\mathop{\rm Range}}
\newcommand\image{\mathop{\rm Image}}
\newcommand\Spec{\mathop{\rm Spec}}
\newcommand\br{\hfill$\phantom{\int}$\\}


%\theoremstyle{plain}
%\newtheorem{theorem}{Theorem}
%\newtheorem{hypothesis}[theorem]{Hypothesis}
%\newtheorem{lemma}{Lemma}[section]
%\newtheorem{corollary}[lemma]{Corollary}
%\newtheorem{proposition}{Proposition}
%\newtheorem{claim}{Claim}[section]
% Roman ``theorems''
%\theoremstyle{definition}
%\newtheorem{definition}[lemma]{Definition}
%\newtheorem{assumption}{Assumption}

% Humble things: remarks and examples.
%\theoremstyle{remark}
\newtheorem{remark}{Remark}
%\newtheorem{example}[lemma]{Example}

\newenvironment{komment}
  {\list{}{
     \listparindent 0.0cm%
  %\itemindent 1cm
  %\parindent 1cm
  \baselineskip 0pt
  \setlength{\leftmargin}{0in}
  \setlength{\rightmargin}{-1.3in}
  }%
  \item\relax}
  {\endlist}

\setlength{\parindent}{0cm}
%\font\Titlefont=cmssbx10 at 30pt
%\font\titlefont=cmssbx10 at 16pt

\newcommand\then{\ \textcolor{ddgreen}{\Rightarrow}\ }

\newcommand\rk{\mathop{\rm rk}}
\newcommand\dist{\mathop{\rm dist}}
\newcommand\rank{\mathop{\rm rank}}
\newcommand\supp{\mathop{\rm supp}}
\newcommand\p{\partial}
\newcommand{\at}[1]{\vert\sb{\sb{#1}}}
\newcommand{\At}[1]{\Big\vert\sb{#1}}
\def\R{{\mathbb R}}
\def\hvar{{\hbar}}
\def\hvar{{\varepsilon}}
\newcommand\C{{\mathbb C}}

\newcommand\sphere{{\mathbb S}}
\newcommand\N{{\mathbb N}}\newcommand\Z{{\mathbb Z}}

\newcommand{\Abs}[1]{\left\vert#1\right\vert}
\newcommand{\abs}[1]{\vert #1 \vert}
\newcommand{\Norm}[1]{\left\Vert #1 \right\Vert}
\newcommand{\norm}[1]{\Vert #1 \Vert}
\newcommand\const{\mathop{const}}
\newcommand\Const{{C{\hskip -1.5pt}onst}\,}
\newcommand\sothat{\,\,;\,\,}

\def\hardy{{\bm h}}
%% For article.sty:
\def\varDelta{{\mathit\Delta}}
\def\varPi{\bm{\mathit\Pi}}
\def\varGamma{\bm{\mathit\Gamma}}
\def\varOmega{\mathit\Omega}
\def\varSigma{\mathit\Sigma}
\def\varLambda{\mathit\Lambda}
\def\varXi{\mathit\Xi}
\newcommand{\os}{\mathop{\rm o}}
\newcommand{\ol}{\mathop{\rm O}}

\newcommand\by[2]{\textcolor{dddgreen}{{\it #1}\,\,{\rm [#2]}}}
\newcommand\pby[1]{\textcolor{dddgreen}{{#1} }}

%[Instability of critical traveling wave in gKdV]
\title
{Nonlinear instability of a critical traveling wave in the generalized Korteweg -- de Vries equation}
\author{Andrew Comech
\\
Texas A\&M University
}

\date{November 3, 2006}


\begin{document}

\title{\bf Nonlinear instability of a critical traveling wave
in the generalized Korteweg -- de Vries equation}


%\setcounter{page}{0}

\frame{

\begin{center}
{\bf 
Nonlinear instability of a critical traveling wave in the generalized Korteweg -- de Vries equation}
\\\ \\
Andrew Comech, {\it Texas A\&M University}
%\thanks
\\
{\small 
\qquad \qquad (Partial support:
%Max-Planck Institute,
MPI-Leipzig,
NSF Grant
% DMS-0434698 and 
DMS-0621257)}
\\
Scipio Cuccagna,
{\it University of Modena and Reggio Emilia}
\\
Dmitry Pelinovsky, {\it McMaster University}
\end{center}

\bigskip

\noindent
We prove the instability of a ``critical'' solitary wave,
%of the generalized KdV equation, 
the one with
the speed at the border of the stability region.

The instability is ``purely nonlinear'':
the linearization at a critical soliton does not have
eigenvalues with positive real part.

Essentially, the
instability is caused by
higher algebraic degeneracy of
zero eigenvalue of the linearized system.
\\
\ \\\ \\\ 
%We prove that critical solitons correspond generally to
%the saddle-node bifurcation of two branches of solitons.

%\vfill
%\noindent
%\bigskip

%Joint work with Scipio Cuccagna,
%University of Modena and Reggio Emilia,
%and
%Dmitry Pelinovsky, McMaster University.
%\end{abstract}

}
\frame{
\frametitle{Generalized KdV}

$
\p\sb{t}{\bm u}
=\p\sb{x}\left(-\p\sb{x}^2 {\bm u}+f({\bm u})\right),
$
\hfill
$
{\bm u}={\bm u}(x,t)\in\R,\quad x\in\R
$

$f\in C\sp\infty$
real-valued,
$
f(0)=f'(0)=0.
$

\begin{definition}
Solitary waves:
solutions of the form
${\bm u}(x,t)=\bm\phi\sb{c}(x-ct)$.
\end{definition}

Generically, exist for speeds $c$
from intervals of $\R$
 (depending on $f$).

\bigskip

For given $f$,
solitons with certain speeds are orbitally stable with
respect to perturbations of the initial data;
others are unstable.

\bigskip

\cite{MR759907}:
gKdV is well-posed
in $H\sp{s}\cap L\sb{\mu}\sp 2$ for $s\ge 2$, $\mu>0$.

}
\frame{
\frametitle{Example: Classical KdV, gKdV-k}

Classical KdV:
\quad
$f({\bm u})=-3{\bm u}^2$,
\quad
$
\p\sb{t}{\bm u}+\p\sb{x}\sp{3}{\bm u}+6{\bm u}\p\sb{x}{\bm u}=0
$

Solitary waves:
\[
{\bm u}\sb{c}(x,t)
=\bm\phi\sb{c}(x-ct)
=\frac{c}{2
\cosh\sp{2}\big((x-ct)\sqrt{c}/2\big)},
\qquad c>0.
\]

For $\ f({\bm u})=-{\bm u}^p$, $p>1$,
\ gKdV-$k$ equations: \hfill $k=p-1$
\[
\p\sb{t}{\bm u}+\p\sb{x}\sp{3}{\bm u}+\p\sb{x}({\bm u}^{p})=0.
\]
Solitons
of subcritical gKdV,
$1<p<5$, are orbitally stable:
\mbox{\cite{MR0338584,MR0386438,MR886343,MR887857}}

%\begin{remark}
%For gKdV-k,
%$\bm\phi\sb{c}(x)=c\sp{\frac{1}{p-1}}\bm\phi\sb 1(c\sp{\frac 1 2}x)$,
%$\mathscr{N}(\bm\phi\sb{c})
%=\mathop{\rm const} c\sp{\frac{5-p}{2(p-1)}}$
%so that $\frac{d}{dc}\mathscr{N}(\bm\phi\sb{c})>0$ for $p<5$
%in agreement with the stability criterion {\rm (\ref{stability-criterion})}
%derived in \cite{MR897729}.
%\end{remark}

}
\frame{
\frametitle{Hamiltonian structure of KdV}

$\dot{\bm u}=JE'({\bm u})$, where:

\[
J=\p\sb x,
\qquad
E({\bm u})
=\int\sb{\R}
\Big(
\frac{1}{2}(\p\sb{x}{\bm u})^2+F({\bm u})
\Big)\,dx
\]
$F({\bm u})$: antiderivative of $f({\bm u})$ such that $F(0)=0$.

\bigskip
\bigskip

Two more invariants of motion:

\qquad\quad
mass
$
\mathcal{I}({\bm u})=\displaystyle\int\sb{\R} {\bm u}\,dx
$,
\quad
momentum
$
\mathscr{N}({\bm u})
=\displaystyle\int\sb{\R}\frac{{\bm u}^2}{2}\,dx
$

\bigskip

Denote
$\mathscr{N}\sb{c}=\mathscr{N}(\bm\phi\sb{c})$,
$\mathcal{I}\sb{c}=\mathcal{I}(\bm\phi\sb{c})$.

}
\frame{
\frametitle{Orbital stability of gKdV}

%\begin{definition}\label{def-stability}
%$\bm\phi\sb{c}(x-ct)$ is orbitally stable if
%$\forall\epsilon>0$ $\exists\delta>0$
%$\forall\bm u\sb 0$,
%$\norm{\bm u\sb 0-\bm\phi\sb{c}}\sb{H\sp 1}\le\delta$
%${\bm u}(t)$ with ${\bm u}(0)={\bm u}\sb 0$
%$\Rightarrow$
%$\sup\limits\sb{t\ge 0}\inf\limits\sb{s\in\R}
%\norm{{\bm u}(x,t)-\bm\phi(x-s)}\sb{H\sp 1}<\epsilon
%$
%\end{definition}

\cite{MR897729}:
%for gKdV,
$\bm\phi\sb{c}(x-ct)$
is orbitally stable if
$
\mathscr{N}\sb{c}'
%=\frac{d}{dc}\mathscr{N}\sb{c}
%=\frac{d}{dc}\mathscr{N}(\bm\phi\sb{c})
>0
$

%\begin{remark}
%$\bm\phi\sb{c}(0)$
%is monotonically increasing with $c$;
%$\mathscr{N}\sb c$ does not have to.
%\end{remark}

\begin{figure}[htbp]
\begin{center}
\input kdvform-n-vs-c.tex
\end{center}
\vskip -30pt
\caption{
Stable and unstable regions.
%on a possible graph of $\mathscr{N}\sb c$ vs. $c$.
Stars denote critical waves
}
%\label{fig-n-vs-c}
\end{figure}

\cite{MR901236}:
similar result for
${\bf U}(1)$-invariant systems


}
\frame{
\frametitle{``Counterexample''}

%\cite{MR897729}:
Critical traveling wave:
$\bm\phi\sb{c\sb\star}(x)$,
$\mathscr{N}\sb{c\sb\star}'=0$.
\hfill
Unstable???

The set 
$S=\{c\sothat \bm\phi\sb{c}{\rm \ is\ stable}\}$
does not have to be open:

\begin{figure}[htbp]
\begin{center}
\input instability-fig-counterexample.tex
\caption{
%Planar flow.
Stationary states: $\R$.
\ \ Stable states: $S=(-\infty,0\bm ]$}
\label{instability-fig-counterexample}
\end{center}
\end{figure}

}
\frame{
\frametitle{Instability of critical solitary waves}

\begin{theorem}
Assume:
\begin{enumerate}
\item
$\mathscr{N}\sb{c\sb\star}'=0$,
$\mathcal{I}\sb{c\sb\star}'\ne 0$
\item
%$\mathscr{N}'\sb{c}=0$ at $c=c\sb\star$,
$\mathscr{N}'\sb{c}<0$ for $c>c\sb\star$
\hfill (or for $c<c\sb\star$ or both)
\item
$J\mathcal{H}\sb{c\sb\star}$
has no $L\sp 2$-eigenvalues except $\lambda=0$
\end{enumerate}
Then
%the critical traveling wave 
$\bm\phi\sb{c\sb\star}(x)$ is
orbitally unstable
\quad
{\rm \cite{kdvform-sima}}
\end{theorem}


{\bf Remarks:}

No linear instability;
slow growth of perturbations

Better control
via asymptotic stability \cite{MR1289328}

\smallskip

$L\sp 2$-critical KdV, $p=5$:
%$\mathscr{N}\sb{c}=\const$.
blow-up
% \cite{MR1824989},
\cite{MR1829643}
%\cite{MR1888800},
%\cite{MR1896235}.

\smallskip

Analogous result for gNLS: \cite{MR1995870}

}
\frame{
\frametitle{Linearization}

Write ${\bm u}(x,t)=\bm\phi\sb{c}(x-ct)+\textcolor{ddgreen}{\bm\rho}(x-ct,t)$;
\[
\dot{\textcolor{ddgreen}{\bm\rho}}
=
%\p\sb{x}\bigl(-\p\sb{x}\sp{2}\textcolor{ddgreen}{\bm\rho}
%+f'(\bm\phi\sb{c})\textcolor{ddgreen}{\bm\rho}+c\textcolor{ddgreen}{\bm\rho}\bigr)
%\equiv
J\mathcal{H}\sb{c}\textcolor{ddgreen}{\bm\rho},
\qquad
%J=\p\sb{x},
%\quad
\mathcal{H}\sb{c}=-\p\sb{x}^2+f'(\bm\phi\sb{c})+c,
\qquad
\sigma\sb{ess}(J\mathcal{H}\sb{c})=i\R
\]


%$\lambda=0$
%is embedded into the essential spectrum,
%$\sigma\sb{ess}(J\mathcal{H}\sb{c})=i\R$

%(with $\p\sb x\bm\phi\sb c$ being the corresponding eigenvector).

\bigskip

%}
%\frame{
%\frametitle{Exponentially weighted spaces}

Consider $J\mathcal{H}\sb{c}$ in
$
L\sb\mu\sp{2}=\left\{\bm\psi
\sothat
e\sp{\mu x}\bm\psi(x)\in L\sp{2}\right\},
$
$\mu\ge 0$.
%We also denote $L\sb\mu\sp{2}=H\sb\mu\sp{0}.$

%Let
%$A\sp{\mu}\sb{c}=e^{\mu x}\circ J\mathcal{H}\sb{c}\circ e^{-\mu x}$;
%$\sigma\sb{ess}(\mathscr{A}\sp{\mu}\sb{c})$ is:

\begin{figure}[htbp]
\begin{center}
\input kdvform-spectrum.tex
\end{center}
\vskip -30pt
\caption{
$\sigma\sb{ess}(J\mathcal{H}\sb c\at{L\sp 2\sb\mu})
=\sigma\sb{ess}(e^{\mu x}\cdot J\mathcal{H}\sb{c}\cdot e^{-\mu x})$
}
%Essential spectrum of $J\mathcal{H}\sb{c}$,
%$c=1$
%in the exponentially weighted space $L\sb\mu\sp{2}$
%for $\mu=0.1<\sqrt{c/3}$ (solid) and $\mu=0.65>\sqrt{c/3}$ (dashed).
%}
\end{figure}

}
\frame{
\frametitle{Spectral decomposition at $\lambda=0$}

%For $c\in\varSigma$,
%\quad
$
J\mathcal{H}\sb{c}\underbrace{(-\p\sb{x}\bm\phi\sb{c})}\sb{{\bm e}\sb{1,c}}
=0,
\quad
$
$
J\mathcal{H}\sb{c}\underbrace{\p\sb{c}\bm\phi\sb{c}}\sb{{\bm e}\sb{2,c}}
=\underbrace{-\p\sb{x}\bm\phi\sb{c}}\sb{{\bm e}\sb{1,c}}.
$
\quad
%Is there ${\bm e}\sb{3,c}$ so that
$J\mathcal{H}\sb c\underbrace{\textcolor{dred}{{\bm e}\sb{3,c}}}\sb{\exists???}
=\textcolor{blue}{{\bm e}\sb{2,c}}$


\[
\langle
\textcolor{blue}{{\bm e}\sb{2,c}},
\!\!\!
\underbrace{\bm\phi\sb c}
\sb{\ker(\mathcal{H}\sb{c}J)}
\!\!\!
\rangle
=
\langle\p\sb c\bm\phi\sb{c},\bm\phi\sb c\rangle
=\p\sb c\langle\bm\phi\sb{c},\bm\phi\sb c\rangle/2
=\mathscr{N}\sb c'
\]
%$\Rightarrow$


\begin{lemma}
$\mathscr{N}\sb{c\sb\star}'=0$
%, $\mathcal{I}\sb{c\sb\star}'\ne 0$
$\quad \Rightarrow\quad$
$\exists\textcolor{dred}{{\bm e}\sb{3,c\sb\star}}$,
$\ N\sb g(J\mathcal{H}\sb{c\sb\star})
=\mathop{\rm span}
\langle
{\bm e}\sb{1,c\sb\star},{\bm e}\sb{2,c\sb\star},
\textcolor{dred}{{\bm e}\sb{3,c\sb\star}}
\rangle$

$\exists$ extension
$
\ c\mapsto {\bm e}\sb{3,c}
%\in H\sb\mu\sp\infty
\ $
of $\textcolor{dred}{{\bm e}\sb{3,c\sb\star}}$ so that

$X\sb c=\mathop{\rm span}
\langle
{\bm e}\sb{1,c},{\bm e}\sb{2,c},{\bm e}\sb{3,c}
\rangle$
is the invariant subspace of $J\mathcal{H}\sb{c}$;

\bigskip 

$
J\mathcal{H}\sb{c}\at{
X\sb{c}
}
=
{\scriptsize
\left[\begin{array}{ccc}0&1&0
\\0&0&1
\\0&0&\lambda\sb{c}\end{array}
\right]
},
\quad
\lambda\sb{c}
=-\frac{\mathscr{N}\sb{c}'}{\langle\bm\phi\sb{c},{\bm e}\sb{3,c}\rangle},
\quad
\langle\bm\phi\sb{c},{\bm e}\sb{3,c}\rangle>0
$
\end{lemma}

${\bm e}\sb{3,c}\sim P\sb{X\sb c}
\textcolor{dred}{{\bm e}\sb{3,c\sb\star}}$,
\quad
$P\sb{X\sb c}
=
e^{-\mu x}
\cdot \frac{1}{2\pi i}\oint\sb\gamma
\frac{d\lambda}{\lambda-e^{\mu x}J\mathcal{H}\sb c e^{-\mu x}}
\cdot
e^{\mu x}$

}
\frame{
\frametitle{Center manifold reduction}

Approximate ${\bm u}(x,t)$ by $\bm\phi\sb{c(t)}$:

\mbox{
$
{\bm u}(x,t)
=\bm\phi\sb{c(t)}\bm (x-\xi(t)-
\int\limits\sb 0\sp t c(s)\,ds\bm )
+\textcolor{ddgreen}{\bm\rho}\bm (x-\xi(t)-\int\limits\sb 0\sp t c(s)\,ds,\,t\bm )
$
}

\bigskip

Decompose
$
\textcolor{ddgreen}{\bm\rho}(x,t)=\zeta(t){\bm e}\sb{3,c(t)}(x)
+\textcolor{blue}{\bm\upsilon}(x,t)
$

\bigskip

Choose
\ 
$\xi(t)$,
\ $c(t)=c\sb\star+\eta(t)$,
\ $\zeta(t)$
\quad
so that
$P\sb{X\sb{c(t)}}\textcolor{blue}{\bm\upsilon}(t)=0$:

\bigskip

\ \hfill
$\textcolor{blue}{\bm\upsilon}(t)$
$\leadsto$
$\sigma\sb{ess}(J\mathcal{H}\sb{c(t)})$

\[
\p\sb t
\left[
\begin{array}{c}
\eta
\\
\zeta
\end{array}
\right]
=
\left[
\begin{array}{ccc}
0&1
\\
0&\lambda\sb c
\end{array}
\right]
\left[
\begin{array}{c}
\eta
\\
\zeta
\end{array}
\right]
+
\left[
\begin{array}{c}
R\sb 1(\eta,\zeta,\textcolor{blue}{\bm\upsilon})
\\
R\sb 2(\eta,\zeta,\textcolor{blue}{\bm\upsilon})
\end{array}
\right]
\]

}
\frame{
\frametitle{Nonlinear estimates}

%We adapt the analysis from \cite{MR1289328}.

Energy conservation:
%\begin{lemma}
$
\norm{\textcolor{ddgreen}{\bm\rho}(t)}\sb{H\sp{1}}
\le
\left(
\abs{\eta(t)}
+\abs{\zeta(t)}+\norm{\textcolor{blue}{\bm\upsilon}(t)}\sb{H\sb\mu\sp{1}}
\right)
$
%\end{lemma}

\bigskip

Dissipation:
%\begin{lemma}[\cite{MR1289328}]\label{lemma-dispersion-0}
%$
%\norm{e\sp{A\sp{\mu}\sb{c}t} Q\sp{\mu}\sb{c}\bm\upsilon}\sb{H\sp{1}}
%\le
%a t\sp{-1/2} e\sp{-b t}\norm{\bm\upsilon}\sb{L\sp{2}}.
%$
%\end{lemma}
%\begin{lemma}\label{lemma-dispersion}
$
\norm{\textcolor{blue}{\bm\upsilon}(t)}\sb{H\sb\mu\sp{1}}
\le
\bm\sup\limits\sb{0\le s\le t}
\left[\zeta^2(s)+\abs{\zeta(s)}\norm{\textcolor{ddgreen}{\bm\rho}(s)}\sb{H\sp{1}}
\right]
$
%\end{lemma}

%\begin{proof}
%\[\dot{\bm\upsilon}-J\mathcal{H}\sb{c}\bm\upsilon
%=-\sum\sb{j=1}\sp{3}R\sb j{\bm e}\sb{j,c}
%-\zeta(\zeta+R\sb 2)\p\sb{c}{\bm e}\sb{3,c}
%+R\sb 1\p\sb{x}\bm\rho+J{\bm N}\dots
%\]
%where $c=c(t)=c\sb\star+\eta(t)$, $\zeta=\zeta(t)$.
%\end{proof}

\bigskip

Maximal functions:
$
\eta\sb M(t)=\bm\sup\limits\sb{0\le s\le t}\eta(s),
$
\quad
$
\zeta\sb M(t)=\bm\sup\limits\sb{0\le s\le t}\abs{\zeta(s)}
$

\begin{lemma}
$\norm{\textcolor{ddgreen}{\bm\rho}}\sb{H\sp{1}}
\le F(\eta\sb M,\zeta\sb M)$,
$
\ \norm{\textcolor{blue}{\bm\upsilon}}\sb{H\sb\mu\sp{1}}
\le \zeta\sb M F(\eta\sb M,\zeta\sb M);
$
$
\ \ \abs{R\sb j(\eta,\zeta,\textcolor{blue}{\bm\upsilon})}
\le
\zeta\sb M^2
$
\end{lemma}

\bigskip

$
\left\{
\begin{array}{l}
\dot\eta=\zeta+R\sb 1
\\
\dot\zeta=\lambda(\eta)\zeta+R\sb 2
\end{array}
\right.
$
\quad$\Rightarrow$\quad
$
\frac{d\zeta}{d\eta}
\approx\lambda(\eta),
$
\quad
$\zeta\approx\Lambda(\eta)$,
\quad
$\dot\eta\approx\Lambda(\eta)$

}

\bibliographystyle{talk}
%\bibliographystyle{amsalpha}
\bibliography{kdv}

\end{document}