\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 119, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2010/119\hfil Ill-posedness]
{Ill-posedness for periodic nonlinear dispersive equations}
\author[J. Angulo, S. Hakkaev\hfil EJDE-2010/119\hfilneg]
{Jaime Angulo Pava, Sevdzhan Hakkaev} % in alphabetical order
\address{Jaime Angulo Pava \newline
Department of Mathematics, IME-USP\\
Rua do Mat\~ao 1010, Cidade Universit\'aria,
CEP 05508-090, S\~ao Paulo, SP, Brazil}
\email{angulo@ime.usp.br}
\address{Sevdzhan Hakkaev \newline
Faculty of Mathematics and Informatics,
Shumen University, 9712 Shumen, Bulgaria}
\email{shakkaev@fmi.shu-bg.net}
\thanks{Submitted January 26, 2010. Published August 24, 2010.}
\subjclass[2000]{76B25, 35Q51, 35Q53}
\keywords{Ill-posedness, mKdV equation; defocusing mKdV equation;
\hfill\break\indent BO equation; higher order evolutions equations}
\begin{abstract}
In this article, we establish new results about the
ill-posedness of the Cauchy problem for the modified
Korteweg-de Vries and the defocusing modified Korteweg-de
Vries equations, in the periodic case.
The lack of local well-posedness is in the sense that the
dependence of solutions upon initial data fails to be continuous.
We also develop a method for obtaining ill-posedness results in
the periodic and non-periodic cases for the equations in the
hierarchies of these equations and also in the case of the
Benjamin-Ono equation.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\section{Introduction}
The purpose of this article is to investigate ill-posedness
of the periodic Cauchy problem for some models of Korteweg-de
Vries type in the periodic Sobolev space $H_{\rm per}^{s}$.
The models that we are interested are, the modified
Korteweg-de Vries (mKdV) in \eqref{1.2}, the defocusing modified
Korteweg-de Vries (dmKdV) in \eqref{1.1} and the Benjamin-Ono
(BO) in \eqref{1.3}. Also, we develop a new technique to obtain
ill-posedness of the periodic and non periodic Cauchy problem
associated with the higher order equations in the hierarchies
of these models.
Before describing our results, it is convenient to define the
notion of well-posed\-ness (and consequently ill-posedness)
related to a general evolution equation
\begin{equation}\label{gev}
u_t=\partial_x I'(u(t))
\end{equation}
where $I(u)$ is a generic conservation law for the flow generated
by \eqref{gev}, namely, $I(u(t_1))=I(u(t_2))$ for all times $t_1, t_2$.
Here $I'$ represents the gradient of $I$, defined by
\begin{equation}\label{deriva}
\langle I'(u), v\rangle\equiv \langle
\frac{\delta I (u)}{\delta u},v\rangle
\equiv \frac{d}{d\epsilon} I(u+\epsilon v)\big |_{\epsilon=0},
\end{equation}
where $\langle \cdot, \cdot\rangle $ denotes the inner product
on $L^2$.
Throughout this paper we shall say that a Cauchy problem
associated to \eqref{gev} is locally well-posed (also
called $C^0$-well-posed) in some normed function space
$\mathcal{X}$ if, for any initial data $u_0\in \mathcal{X}$ there
exist a time $T=T(\|u_{0}\|_{\mathcal{X}})>0$, a function space $Y$
continuously embedded in $C([-T,T]; \mathcal{X})$ and a unique
solution $u(t)$ such that
\begin{enumerate}
\item $u\in C([-T, T]; \mathcal{X})\cap Y\equiv Z_{T}$
\item the mapping data-solution $u_{0} \to u$ from
$\mathcal{X}$ to $Z_{T}$ is continuous.
\end{enumerate}
A Cauchy problem associated to \eqref{gev} is globally
well-posed in $\mathcal{X}$ if $T$ above can be chosen
as $T=+\infty$. Finally, a Cauchy problem will be said to
be ill-posed if it is not $C^0$-well-posed.
Here is a more precise description of the problems of local
well-posedness and global well-posedness in the periodic case for
the models mKdV, dmKdV and BO. In \cite{KPV1} the local
well-posedness for the mKdV (and defocusing mKdV)
was obtained for $s\geq \frac{1}{2}$, and in \cite{CKSTT1, CKSTT2}
it was shown that it is globally well posed for
$s\geq {\frac{1}{2}}$. If one
strengthens our notion of well-posedness, requiring that the
mapping data-solution is smooth, Bourgain showed in \cite{Bo2}
that the known results on mKdV ($s>\frac14$ in the line, $s\geq
\frac12$ in the periodic case) are optimal in the sense of this
map to be of class $C^3$. In Christ, Colliander and Tao \cite{CCT},
ill-posedness for the defocusing mKdV is obtained for $s\in
(-1,1/2)$. Regarding the BO equation, L. Molinet in \cite{Mo1}
proved global well-posedness in $H_{\rm per}^{s}$ for $s\geq 0$ and
also showed that the mapping data-solution can not be of class
$C^{1+\alpha}, \; \alpha>0$, from $\dot{H}_{\rm per}^{s}$ into
$\dot{H}_{\rm per}^{s}$ for $s<0$ where
$\dot{H}_{\rm per}^{s}=(-\triangle)^{-s/2}L_{\rm per}^2$.
In the non-periodic case the ill-posedness for some classical
non-linear dispersive equations (for instance, Korteweg-de Vries
equation (KdV), cubic Schr\"odinger equation, complex KdV, mKdV, and BO
equations) is studied in \cite{BL, BPS, Bo1, Bo2, BKPSV, CCT,
CKSTT1, KPV1, KPV2}. The approach in \cite{BL}, \cite{BKPSV}, and
\cite{BPS} uses the existence and good properties of the solitary
wave solutions associated to the equations. In particular, a good
behavior of its Fourier transforms is required.
In this paper we extend the technique developed in \cite{BPS} to
the periodic case and to higher order evolutions equations.
Our approach is based on the theory of Jacobian
elliptic functions, the Poisson summation formula, the Floquet theory and on
techniques coming from integrable systems. Our method can be used
for studying the ill-posedness of the periodic and non periodic
Cauchy problem associated with higher order equations.
The first objective of this work is to apply our approach to the
study of the ill-posedness for the mKdV, dmKdV and BO
equations and to show that the solutions cannot depend
continuously on their initial data in the Sobolev spaces
$H_{\rm per}^{s}$ for $s<-1/2$. In other words, we construct
a sequence converging (strongly) to a specific data in
$H_{\rm per}^{s}$ and then we show that the corresponding sequence of
solutions does not converge (strongly) in $H_{\rm per}^{s}$. The
specific data will be the Dirac delta periodic distribution. The
main point in the analysis is the construction of explicit smooth
curves of periodic traveling waves solutions for the mKdV, dmKdV
and BO equations with a fixed minimal period and a specific
behavior of the associated Fourier transform. To construct such
solutions we shall use the theory of elliptic functions,
the Poisson summation formula and the implicit function theorem.
To obtain the ill-posedness results we shall use the ideas in Birnir,
Ponce and Svenstedt \cite{BPS}. Our results extend the ill-posed
results of Christ, Colliander and Tao \cite{CCT} concerning to
the mKdV and dmKdV in the periodic case.
The second objective of this paper is to show that the approach for
obtaining ill-posedness for the mKdV, dmKdV and BO equations can be
applied to the higher order evolution equations in the hierarchies
of these models. So we obtain similar results of local ill-posedness
in the spaces $H_{\rm per}^{s}$ for $s<-1/2$. Indeed, from the
ideas of Lax in \cite{lax} we develop a general scheme which
will imply that the profile given by the
periodic (or solitary) travelling wave solutions associated with
the mKdV, dmKdV and BO equations, and with a specific
speed-wave will be a periodic (or solitary) travelling wave
solutions \emph{for every equation} from the mKdV, dmKdV and
BO hierarchies respectively. For instance, we consider the
BO equation
\begin{equation}\label{1.3}
u_{t}+uu_{x}-\mathcal{H}u_{xx}=0, \quad u=u(x,t)\in
\mathbb{R},
\end{equation}
where $\mathcal{H}$ denotes the Hilbert transform on
$2l$-periodic functions, $f$, defined by
$$
\mathcal{H}f(x)={\frac{1}{2l}}p.v. \int_{-l}^{l}{\cot [
{\frac{\pi (x-y)}{2l}}] f(y)}\,dy.
$$
So, we obtain via the Fourier transform that
$\widehat{\mathcal{H}f}(k)=-i\operatorname{sgn}(k)\widehat{f}(k)$,
$k\in \mathbb{Z}$. Next, let $I(u)$ be
a generic conserved quantity for the BO equation and consider
the associated hierarchy equation \eqref{gev}, then there is
a spectral parameter $\lambda_{I,c}$ such that
$u_c(x,t)=\chi_c(x+\lambda_{I,c} t)$ is a periodic travelling wave
for \eqref{gev}, provided that $\chi_c(x-ct)$ is a periodic
travelling wave for \eqref{1.3}. The existence of the
speed-wave $\lambda_{I,c}$ is deduced from the property that
the kernel of the pseudo-differential operator
$$
\mathcal{L}_{BO}=\frac{d}{dx}\mathcal{H}- \chi_c+c.
$$
is one-dimensional and generated by $\frac{d}{dx} \chi_c$.
In general, to determine the exact value of $\lambda_{I,c}$ can
be difficult and tedious. Naturally, our general scheme is
applicable to the case of travelling waves of solitary wave type
and so we can also obtain ill-posedness results for higher order
evolution equations in the hierarchies of the models above in
Sobolev space $H^s(\mathbb{R})$. We do not find an effective
algorithm which give the parameter $\lambda_{I,c}$ for every
conserved quantity $I$ given. Here we calculate it explicitly only
in the cases of the second equation from the mKdV, dmKdV and BO
hierarchies (see \eqref{5mkdv}, \eqref{1.4} and \eqref{3bo},
respectively). Of course, with a little more of work, one can
yield an ill-posedness result for the third equation from the
hierarchy of these models and so on (see \cite{C}, \cite{MM} and
Remarks after Theorem \ref{t5mkdv} below).
\section{Notation}
For $s\in\mathbb{R}$, the Sobolev space $H^s_{\rm per}([0, \ell])$
consists of all periodic distributions $f$ such that
$\|f\|_{H^s}^2= \ell \sum_{k=-\infty}^{\infty} \sum
(1+k^2)^s|\widehat{f}(k)|^2<\infty$.
For simplicity, we will use the notation $H^s_{\rm per}$ in
several places and $H^0_{\rm per}=L^2_{\rm per}$. We denote
$\|f\|_{L^2}=\|f\|$ and $\langle
f,g\rangle_{L^2}=\int_{0}^{\ell}f(x)g(x)dx=\langle f,g\rangle$.
$[H^s_{\rm per}]'$, the topological dual of $H^s_{\rm per}$, is
isometrically isomorphic to $H^{-s}_{\rm per}$ for all
$s \in \mathbb{R}$. The duality is implemented concretely by
the pairing
$$
(f,g) = \ell\sum_{k=-\infty}^{\infty}
\widehat{f}(k)\overline{\widehat{g}(k)}, \quad\text{for }
f \in H^{-s}_{\rm per}, \; g \in H^s_{\rm per}.
$$
Thus, if $f \in L^2_{\rm per}$ and
$g \in H^s_{\rm per} $ with $s\geq 0$,
it follows that $(f,g) = \langle f,g\rangle$.
The normal elliptic integral of first type is defined by
\begin{equation*}
\int_0^y\frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}}=
\int_0^{\varphi}\frac{d\theta}{\sqrt{1-k^2\sin^2\theta}}=F(\varphi,k)
\end{equation*}
where $y=\sin \varphi$ and $k\in (0,1)$. $k$ is called the modulus
and $\varphi$ the argument. When $y=1$, we denote
$F(\pi/2,k)$ by $K=K(k)$. The three basic Jacobian elliptic
functions are denoted by $\operatorname{sn}(u;k)$,
$\operatorname{cn}(u;k)$
and $\operatorname{dn}(u;k)$ (called; snoidal, cnoidal and dnoidal,
respectively),
and are defined via the previous
elliptic integral. More precisely, let
\begin{equation}\label{jef3}
u(y;k):=u=F(\varphi,k)
\end{equation}
then $y=\sin\varphi:=\operatorname{sn}(u;k)=\operatorname{sn}(u)$ and
\begin{equation}\label{jef4}
\begin{gathered}
\operatorname{cn}(u;k):=\sqrt{1-y^2}=\sqrt{1- \operatorname{sn}^2(u;k)}\\
\operatorname{dn}(u;k):=\sqrt{1-k^2y^2}=\sqrt{1-k^2 \operatorname{sn}^2(u;k)}.
\end{gathered}
\end{equation}
The following asymptotic formulas are obtained:
$\operatorname{sn}(x;1)=\tanh (x)$,
$\operatorname{cn}(x;1)=\operatorname{sech}(x)$ and
$\operatorname{dn}(x;1)=\operatorname{sech}(x)$.
\section{Ill-posedness for the mKdV}
We start this section by presenting some results about periodic
travelling wave solutions associated to the mKdV equation,
\begin{equation}\label{1.2}
u_{t}+3u^{2}u_{x}+u_{xxx}=0,\quad u=u(x,t)\in \mathbb{R},
\end{equation}
which are essential in our analysis.
Let $u_c(x,t)=\varphi_c(x-ct)$ be a periodic travelling wave
solution for \eqref{1.2}, so after integration and by choosing
the integration constant being zero we have that $\varphi_c$
needs to satisfy the nonlinear differential equation
\begin{equation}\label{pmkdv}
\varphi_{c}''+\varphi_{c}^3-c\varphi_{c}=0.
\end{equation}
Next, by following the ideas in Angulo \cite{An} and
Angulo \& Natali \cite{AN} (see also Angulo \cite{An1}) will
obtain an explicit family of periodic solution, $c\to \varphi_c$,
for \eqref{pmkdv} via the Poisson summation formula.
The method is as follows: for $\omega>0$ we consider the
positive solitary wave solution for the mKdV equation on
$\mathbb{R}$, namely,
\begin{equation}\label{solimkdv}
\phi_{\omega}(x)=\sqrt{2\omega}\operatorname{sech} (\sqrt{\omega}x).
\end{equation}
Then $\phi_\omega$ satisfies the elliptic equation
$\phi_{\omega}''+\phi_{\omega}^3-\omega\phi_{\omega}=0$.
Now, since the Fourier transform of $\phi_{\omega}$ is given by
$$
\widehat{\phi_{\omega}}(\xi)= \sqrt{2}\pi \operatorname{sech} \big({\frac{\pi
\xi}{2\sqrt{\omega}}}\big),
$$
we obtain from Poisson summation formula the following
periodic function $\psi_{\omega}$ with a minimal period $L$,
\begin{equation}
\label{2.1} \psi_{\omega}(\xi)=\sum_{n\in\mathbb{Z}}\phi_{\omega}(\xi+nL)=
{\frac{\sqrt{2}\pi}{L}}\sum_{n=0}^{\infty}\epsilon_{n}
\operatorname{sech} \big({\frac{\pi n}{2\sqrt{\omega}L}}\big)
\cos \big( {\frac{2\pi n \xi}{L}}\big),
\end{equation}
where
\begin{equation}\label{rela}
\epsilon_{n}=\begin{cases}
1, & n=0\\
2, & n=1,2,3,\dots
\end{cases}
\end{equation}
Next, we consider the Fourier expansion of the Jacobi
elliptic dnoidal function with minimal period $L$,
$$
{\frac{2K}{L}}\operatorname{dn}\big({\frac{2K\xi}{L}};k\big)
={\frac{\pi}{L}} +{\frac{4\pi}{L}}\sum_{n=1}^{\infty}
{\frac{q^{n}}{1+q^{2n}}} \cos \big({\frac{2n\pi \xi}{L}}\big),
$$
where $K(k)$ is the complete elliptic integral of the first
kind, $q=e^{-\frac{\pi K'}{K}}$,
$K'(k)=K(\sqrt{1-k^{2}})$, $k \in (0,1)$. From here
we conclude that
$$
{\frac{q^{n}}{1+q^{2n}}}={\frac{1}{2}} \operatorname{sech} \big( {\frac{n\pi
K'}{K}}\big).
$$
Therefore,
\begin{equation}
\label{2.2} \frac{2K}{L} \operatorname{dn}\big( {\frac{2K\xi}{L}}; k\big)
={\frac{\pi}{L}}+{\frac{2\pi}{L}}\sum_{n=1}^{\infty}
\operatorname{sech}\big( {\frac{n\pi K'}{K}}\big) \cos \big({\frac{2n\pi
\xi}{L}}\big).
\end{equation}
So, from \eqref{2.1}-\eqref{2.2} let
$\varphi_{c}(\xi)=\eta \operatorname{dn}
\big( {\frac{\eta \xi}{\sqrt{2}}}; k \big)$ be a periodic
solution of \eqref{pmkdv} for $c>0$ with minimal period $L$
and $\eta>0$.
Then the following identities should be satisfied
\begin{equation}\label{2.21}
\begin{gathered}
c={\frac{\eta^{2}}{\sqrt{2}}}(1+k^{'2}), \quad
k^{'2}=1-k^{2} \\
\eta =2\sqrt{2}{\frac{K(k)}{L}}, \quad k\in (0,1), \quad
\eta \in (\sqrt{c}, \sqrt{2c}).
\end{gathered}
\end{equation}
Therefore, from the asymptotic properties of $K$, we need to have
$c\in \mathcal I=({\frac{2\pi^{2}}{L^{2}}}, +\infty)$.
In this point we are ready to build a solution of \eqref{pmkdv}
from the formula \eqref{2.1}. Indeed, for $c\in \mathcal I$
and by choosing the speed-wave, $\omega$, of the solitary
wave $\phi_\omega$ as being
$$
\omega (c)={\frac{c}{16(2-k^{2})K^{'2}(k)}},
$$
we obtain from \eqref{2.1},
\eqref{2.2} and \eqref{2.21} that
\begin{equation}\label{poso}
\varphi_{c}(\xi)=\psi_{\omega (c)}(\xi).
\end{equation}
Finally, from \cite{An, AN} we obtain by using the implicit function
theorem that for every $c>{\frac{2\pi^{2}}{L^{2}}}$ there is a
unique $\eta=\eta (c)\in (\sqrt{c}, \sqrt{2c})$ such that the
fundamental period of the
solution $\varphi_{c}$ in \eqref{poso} is $L$ and the mapping
$c\in \mathcal I \to \varphi_{c} \in H_{\rm per}^{n}([0,L])$
is a smooth function.
From \eqref{pmkdv}, $\varphi_{c}$ satisfies the first-order
equation
\begin{equation}\label{1order}
[\varphi_c']^2=\frac12[-\varphi_c^4+2c\varphi_c^2+4B_{\varphi_c}]
\end{equation}
where $B_{\varphi_c}$ is an integration constant determined
uniquely as follows: For $c\in ({\frac{2\pi^{2}}{L^{2}}},\infty)$
there is a unique $\eta=\eta(c)\in (\sqrt{c},\sqrt{2c})$ such that
for $\beta^2\equiv 2c-\eta^2$ and
$$
4B_{\varphi_c}=-\eta^2\beta^2,
$$
we have that $\eta$ and $\beta$ are the positive zeros of the
even polynomial $F_{\varphi_c}(t)=-t^4+2c t^2+4B_{\varphi_c}$.
We also note from the first and third relations in \eqref{2.21} that
$\eta (c) \to +\infty$ and $K(k) \to +\infty$, as
$c \to +\infty $. Hence $k(c) \to 1$,
as $c \to +\infty $. From here we conclude that $\omega
(c) \to +\infty$, as $c \to +\infty $.
Next we have the following lemmas for obtaining our
ill-posed result associated with the mKdV.
\begin{lemma}\label{l21}
The $H_{\rm per}^{s}$ norms of $u_{0}(x)=\varphi_{c}(x)$
and $u_c(x,t)=\varphi_{c}(x-ct)$ are finite for
$s<-1/2$, and
\begin{gather*}
\lim_{c\to +\infty}\|u_{0}\|_{s}=\sqrt{2}\pi \|\delta_{L}\|_{s} \\
\lim_{c\to +\infty}\|u_c(\cdot,t)\|_{s}=\sqrt{2}\pi
\|\delta_{L}\|_{s},
\end{gather*}
where $\delta_{L}$ represents the Dirac periodic distribution
centered in zero, namely, for $f\in C^{\infty}_{\rm per}([0, L])$
we have $\delta_{L}(f)=(\delta_{L}, f)=f(0)$ (see \cite {II}).
\end{lemma}
\begin{proof}
From the Parseval identity and \eqref{2.1}-\eqref{poso} we obtain
$$
\|\varphi_{c}\|_{s}^{2} ={\frac{2\pi^{2}}{L}}
\sum^{+\infty}_{n=-\infty} (1+n^{2})^{s} \operatorname{sech} ^{2}
\big( {\frac{n \pi}{2\sqrt{\omega}L}}\big)
\leq {\frac{2\pi^{2}}{L}}
\sum^{+\infty}_{n=-\infty}(1+n^{2})^{s}.
$$
Hence, if $s<-1/2$, then the series on the right-hand
side of the above inequality is uniformly convergent.
Therefore, from above analysis we obtain
$$
\lim_{c\to +\infty}\|\varphi_{c}\|_{s}^{2} =
{\frac{2\pi^{2}}{L}}\sum^{+\infty}_{n=-\infty}(1+n^{2})^{s}=2\pi^{2}
\|\delta_{L}\|_{s}^{2},
$$
since $\widehat{\delta_{L}}(n)=1/L$.
Now, for the solution $u_{c}(x,t)=\varphi_{c}(x-ct)=\tau_{ct}
\varphi_{c}$ (where $\tau_{ct}f(x)=f(x-ct)$), we have from
\eqref{2.1}-\eqref{poso},
\begin{align*}
\|u_{c}(\cdot,t)\|_{s}^{2}
&=L\sum^{+\infty}_{n=-\infty}{\frac{1}{(1+n^{2})^{-s}}}
|\widehat{u_{c}}(n)|^{2}
=L\sum^{+\infty}_{n=-\infty}{\frac{1}{(1+n^{2})^{-s}}}
|\widehat{\tau_{ct} \varphi_{c}}(n)|^{2} \\
&=L\sum^{+\infty}_{n=-\infty}{\frac{1}{(1+n^{2})^{-s}}}|
\widehat{\varphi_{c}}(n)|^{2}<
+\infty,
\end{align*}
and so
$$
\|u_{c}(\cdot,t)\|_{s}^{2} \to 2\pi^{2}
\|\delta_{L}\|_{s}^{2}, \quad \text{as } c\to +\infty.
$$
This completes the proof of the lemma.
\end{proof}
\begin{lemma}\label{l22}
The initial data $u_{0}(x)\equiv \varphi_c(x)$ converges weakly
to $\sqrt{2}\pi \delta_{L}$ as $c \to +\infty$.
\end{lemma}
\begin{proof}
Let $\phi \in C^{\infty}_{\rm per}([0, L])$
(where $C^{\infty}_{\rm per}([0, L])$ denotes the space of
smooth periodic function with period $L$). Then we have
(see \cite{II})
\begin{equation}\label{2.3}
(u_{0}, \phi)= L{\frac{\sqrt{2}\pi}{L}}
\sum^{+\infty}_{n=-\infty}
\operatorname{sech} \big( {\frac{n \pi}{2\sqrt{\omega}L}}\big)
\widehat{\phi}(n).
\end{equation}
Since
$$
\big| \operatorname{sech} \big({\frac{n \pi}{2\sqrt{\omega}L}}\big)
\widehat{\phi}(n)\big|\leq |\widehat{\phi}(n)|
$$
and the series $\sum^{+\infty}_{n=-\infty}|\widehat{\phi}(n)|$
converges, it follows from the M-Weierstrass Theorem that
\begin{align*}
\lim_{c \to +\infty} (u_{0}, \phi)
&= \sqrt{2}\pi \sum^{+\infty}_{n=-\infty}\widehat{\phi}(n)
\lim_{c \to +\infty}\operatorname{sech}\big({\frac{n \pi}{2\sqrt{\omega} L}}\big) \\
&= \sqrt{2}\pi \sum^{+\infty}_{n=-\infty}\widehat{\phi}(n)
=\sqrt{2}\pi \phi (0)=(\sqrt{2}\pi\delta_L,\phi).
\end{align*}
This shows that $u_{0}$ converges weakly to $\sqrt{2}\pi \delta_{L}$
in $H_{\rm per}^{s}$ ($s<-1/2$).
\end{proof}
We can now prove the main result of this section.
\begin{theorem}\label{princi}
The initial value problem for the mKdV is locally ill-posed
in $H^s_{\rm per}$ for $s<-12$.
\end{theorem}
\begin{proof}
From Lemma \ref{l21} the $H^s_{\rm per}$-norm of
$u_c(x,0)=\varphi_c(x)$ converges to the $H^s_{\rm per}$-norm
of $\sqrt{2}\pi \delta_L$ and by Lemma \ref{l22}, $u_c(x,0)$
converges weakly to $\sqrt{2}\pi \delta_L$. Consequently $u_c(x,0)$
converges strongly to $\sqrt{2}\pi \delta_L$ in $H^s_{\rm per}$.
Next, from \eqref{2.1}-\eqref{poso} we have that
$$
\widehat{u_c(x,0)}(n)=\widehat{\varphi_c}(n)
=\frac{\sqrt{2}\pi}{L} \operatorname{sech}\Big(\frac{n\pi}{2\sqrt{\omega}L}\Big)
\to \frac{\sqrt{2}\pi}{L}\quad\text{as } c\to +\infty.
$$
On the other hand, we have that
$\widehat{u_c(x,t)}(n)=\widehat{\tau_{ct}\varphi_c}(n)
=e^{ictn}\widehat{\varphi_c}(n)$, which not converge as
$c\to +\infty$ for all $n\neq 0$. This shows that
$u_c(x,t)$ can not converge weakly in $H^s_{\rm per}$.
\end{proof}
\section{Ill-posedness for the BO equation}
In this section we consider the ill-posedness for the Benjamin-Ono
equation \eqref{1.3}. As in the previous section, first we will
obtain a periodic solution with minimal period $2L$ for the BO
equation by using the Poisson summation formula. So, we consider
the family of solitary wave solutions for the BO equation,
$u(x,t)=\phi_{\omega}(x-\omega t)$, where $\phi_{\omega}$
satisfies the nonlocal differential equation
$$
\mathcal{H}\phi_{\omega}'+\omega \phi_{\omega}
-\frac12 \phi_{\omega}^2=0
$$
with a profile given by
\begin{equation}\label{solita}
\phi_{\omega}(x)={\frac{4\omega}{1+\omega^{2}x^{2}}}, \quad
\omega>0.
\end{equation}
So, we obtain that the Fourier transform of $\phi_{\omega}$
is given by
$$
\widehat{\phi_{\omega}}(\xi)=4\pi e^{-2\pi |\xi|/\omega}.
$$
By the Poisson summation formula, we obtain the following periodic
function $\rho_{\omega}$ with a minimal period $2L$
(see \cite{ben,An, An1, AN})
\begin{equation}\label{pbo}
\begin{aligned}
\rho_{\omega}(x)
&=\sum_{n=-\infty}^{+\infty} \phi_{\omega}(x+2Ln)
=\frac{2\pi}{L} \sum_{n=-\infty}^{+\infty}
e^{-{\frac{\pi |n|}{\omega L}}}e^{\frac{i\pi n x}{L}}\\
&={\frac{2\pi}{L}}\sum_{n=0}^{+\infty}\epsilon_{n}
e^{-{\frac{\pi |n|}{\omega L}}}
\cos \left( {\frac{n\pi x}{L}}\right)
=\frac{2\pi}{L} \operatorname{Re} \big[ \coth
\Big( {\frac{\pi}{2\omega L}}+{\frac{i\pi x}{2L}}\Big) \big]\\
&=\frac{2\pi}{L}{\frac{\sinh\left( {\frac{\pi}{\omega L}}\right)}
{\cosh \left( {\frac{\pi}{\omega L}}\right)-
\cos \left( {\frac{\pi x}{L}}\right)}},
\end{aligned}
\end{equation}
where $\epsilon_{n}$ is defined in $\eqref{rela}$. Now, let
$\chi_{c}(x-ct)$ be a smooth periodic travelling wave solution of the
BO equation, with $c>0$ and minimal period $2L$. Then by considering the
Fourier expansion series
$$
\chi_{c}(x)=\sum_{n=-\infty}^{+\infty}a_{n}e^{\frac{i\pi nx}{L}}
$$
and by substituting this expression in
$$
\mathcal{H} \chi_{c}'+c \chi_{c}-\frac12 \chi_{c}^2=0,
$$
we obtain
\begin{equation}\label{conv}
\big( {\frac{\pi |n|}{L}}+c\big)
={\frac{1}{2}}\sum_{m=-\infty}^{+\infty}a_{n-m}a_{n}.
\end{equation}
By \eqref{pbo}, we define $a_{n}={\frac{2\pi}{L}}e^{-\gamma |n|}$
with $n\in \mathbb{Z}$ and $\gamma \in \mathbb{R}$ to be chosen.
So, from equality \eqref{conv} we have the basic relation
$$
c+{\frac{\pi |n|}{L}}={\frac{2\pi}{2L}}(|n|+\coth \gamma),
$$
which implies
$$
c={\frac{\pi}{L}} \coth \gamma.
$$
Then, for $\gamma\equiv \frac{\pi}{\omega L}$ and
$c>{\frac{\pi}{L}}$ we
chose the speed-wave for the solitary wave solution $\phi_\omega$
in \eqref{solita}, $\omega=\omega (c)$, such that
$\tanh (\gamma)={\frac{\pi}{cL}}$. Therefore, from \eqref{pbo} we have
\begin{equation}\label{formubo}
\chi_{c}(x)=\rho_{\omega(c)}(x)={\frac{2\pi}{L}}\Big( {\frac{\sinh
\gamma}{\cosh \gamma - \cos{\frac{\pi x}{L}}}}\Big)
\end{equation}
is a periodic solution of the BO equation, with period $2L$ and
Fourier coefficients given by
$$
\widehat{\chi_{c}}(n)={\frac{2\pi}{L}}e^{-\gamma |n|}.
$$
Note that $\gamma \to 0$ as $c \to +\infty$.
So we have our ill-posedness result for the BO equation.
\begin{theorem}\label{t31}
The initial value problem for the BO equation is locally ill-posed
in $H_{\rm per}^{s}$ for $s<-1/2$.
\end{theorem}
\begin{proof}
As in the case of the mKdV we have
$$
\|\chi_{c}\|_{s}^{2}
={\frac{8\pi^{2}}{L}}\sum^{+\infty}_{n=0}(1+n^{2})^{s}e^{-2\gamma
|n|}\leq {\frac{8\pi^{2}}{L}}\sum^{+\infty}_{n=0}(1+n^{2})^{s}.
$$
If $s<-1/2$, then the series on the right-hand side of
the above inequality converges uniformly and therefore
$$
\|\chi_{c}\|_{s}^{2} \to 16\pi^{2} \|\delta_{2L}\|_{s}^{2}, \quad
\text{as } c\to +\infty .
$$
This shows that the $H_{\rm per}^{s}$ norm of $u_{0}(x)=\chi_{c}(x)$
converges to the $H_{\rm per}^{s}$ norm of $4\pi \delta_{2L}$.
Now, for $\phi\in C^\infty_{\rm per}([0,2L])$, we have
$$
\langle u_{0}, \phi \rangle =\int_{0}^{L}{\chi_{c}(x) \phi(x)}dx
=4\pi \sum^{+\infty}_{n=-\infty}e^{-\gamma |n|}\widehat{\phi}(n)
\to 4\pi\phi(0), \quad\text{as } c\to +\infty .
$$
From the above, we obtain that $u_{0}(x)$ converges strongly to
$4\pi \delta_{2L}$ in $H_{\rm per}^{s}$ for $s<-1/2$. On the
other hand, for $u_{c}(x,t)=\chi_{c}(x-ct)$, we have
$$
\|u_{c}(\cdot,t)\|_{s}^{2}=2L\sum^{+\infty}_{n=-\infty}(1+n^{2})^{s}
|\widehat{\chi_{c}}(n)|^{2}<+\infty,
$$
and so
$$
\|u_{c}(\cdot,t)\|_{s}^{2} \to 16\pi^{2}
\|\delta_{2L}\|_{s}^{2}.
$$
Moreover, since
$\widehat{u_c(x,t)}(n)=\widehat{\tau_{ct}\chi_c}(n)
=e^{ictn}\widehat{\chi_c}(n)$
we get that the rest of the proof is the same as the one
for Theorem \ref{princi}. This completes the proof.
\end{proof}
We remark that recently Molinet \cite{Mo2} showed ill-posedness
of the BO equation in $H^s_{\rm per}$ for $s<0$.
\section{Ill-posedness for higher order evolution equations}
In this section we develop a general scheme which
shows that every travelling wave solution (periodic or solitary wave)
for the mKdV and BO equations
\eqref{1.2} and \eqref{1.3}, respectively, is also a travelling
wave solution (with a different speed wave) of every equation
belonging to the hierarchy generated by these two basic equations.
So, we can deduce ill-posedness results in the periodic and
non-periodic cases, for instance, for the fifth order
modified Korteweg-de Vries equation (5-mKdV)
\begin{equation}\label{5mkdv}
u_{t}-u_{xxxxx}-30u^4u_x
-10u^2u_{xxx}-10(u_x)^3-40uu_xu_{xx}=0,
\end{equation}
and for the third order Benjamin-Ono equation (3-BO)
\begin{equation}\label{3bo}
u_{t}-4u_{xxx}+ 3u^2u_{x}- 3(u\mathcal{H} u_{x})_x -3\mathcal{H}(uu_x)_x=0.
\end{equation}
\subsection{The Method}
Initially we set an abstract hamiltonian system of the form
\begin{equation}\label{hamil}
u_t=\partial_x E'(u(t)),
\end{equation}
where $E$ is a conserved quantity for \eqref{hamil} with $E''(u)$
being a self-adjoint linear operator. We assume that \eqref{hamil}
is invariant under the symmetry of translation. More specifically,
let $\{T(\gamma)\}_{\gamma\in \mathbb{R}}$ be the one-paramenter
group of unitary operators on $L^2$ defined for $\gamma\in
\mathbb{R}$ as
$$
T(\gamma)f(x)=f(x+\gamma).
$$
So, for $u(\cdot,t)$ being a solution of \eqref{hamil} with
initial data $u(x,0)=u_0(x)$ we obtain that for $\gamma\in
\mathbb{R}$, $T(\gamma)u(\cdot,t)=u(\cdot+\gamma,t)$ is solution
of \eqref{hamil} with initial data
$T(\gamma)u(x,0)=u_0(x+\gamma)$. Next we denote by $T'(0)$ the
infinitesimal generator of the group of translations, then
$T'(0)=\frac{d}{dx}$.
Now, we suppose that $F(u)=\frac12\|u\|^2$ is also a conserved
quantity for \eqref{hamil} and $E$ is invariant under translation.
So, we have our first hypothesis:
\begin{itemize}
\item[(H1)] (Existence of travelling wave) Suppose
the existence of travelling wave type solutions
$u_c(x,t)=\phi_c(x-ct)$ of \eqref{hamil} such that the
mapping $c\in I\subset \mathbb{R}\to \phi_c$ is smooth
and for every $c\in I$, the profile $\phi_c$ is a
critical point for the functional $H\equiv E+cF$, namely,
\begin{equation}\label{crit}
H'(\phi_c)=E'(\phi_c)+c\phi_c=0.
\end{equation}
\end{itemize}
Let us call the set $\Omega_{\phi_c}=\{T(\gamma)\phi_c:
\gamma\in\mathbb{R}\}$ the $\phi_c$-orbit. Then from the
invariance of $H$ under translation and from \eqref{crit} we
obtain that every point of $\Omega_{\phi_c}$ is a critical point
of $H$, $H'(T(\gamma)\phi_c)=0$ for all $\gamma\in\mathbb{R}$.
Therefore,
\begin{equation}\label{ker}
0=\frac{d}{d\gamma}H'(T(\gamma)\phi_c)\big|_{\gamma=0}
=H''(\phi_c)(T'(0)\phi_c)=H''(\phi_c)\Big(\frac{d}{dx}\phi_c\Big).
\end{equation}
Hence $\frac{d}{dx}\phi_c$ belongs to the kernel of the
unbounded and self-adjoint linear operator
\begin{equation}\label{linear}
\mathcal{L}_c= E''(\phi_c)+c.
\end{equation}
Next we have our second hypothesis:
\begin{itemize}
\item[(H2)] (One-dimensional kernel) The operator $\mathcal{L}_c$
has kernel spanned by $T'(0)\phi_c=\frac{d}{dx}\phi_c$.
\end{itemize}
Now, from \eqref{hamil} we consider the linear variational equation
\begin{equation}\label{varili}
v_t=V(u)v,
\end{equation}
where $V(u)$ denotes the derivative of
$K(u)\equiv\partial_x E'(u)$ (see \eqref{deriva}), namely, $V(u)v=K'(u)v$
is a linear function of $v$ given by
$$
V(u)v=\partial_x (E''(u)(v)).
$$
Let $I(u)$ be any conservation law
for \eqref{hamil} with derivative $G(u)$, namely,
$$
I'(u)(v)=\langle G(u),v\rangle.
$$
Therefore, we obtain that for any solution
$u(t)$ of \eqref{hamil} and $v(t)$ of \eqref{varili}
\begin{equation}\label{vari}
\frac{d}{dt} \langle G(u(t)), v(t)\rangle=0.
\end{equation}
Then for $u(t)$ being a travelling wave of the model in
\eqref{hamil} we get from \eqref{vari} that
$$
\langle G(\phi_c(x-ct)), v(t)\rangle =\langle G(\phi_c(x)),
v(x+ct,t)\rangle
$$
is independent of $t$. Therefore, for $w(x,t)\equiv v(x+ct,t)$,
we obtain that for every $t$,
$$
0=\langle G(\phi_c), w_t(t)\rangle =\langle G(\phi_c),
[c\partial_x+V(\phi_c)]w \rangle
=\langle [-c\partial_x+(V(\phi_c))^*]G(\phi_c),w(t)\rangle ,
$$
where $V^*$ represents the adjoint operator associated to $V$. So,
since the value $w(0)$ can be arbitrary we have
\begin{equation}\label{1rela}
0=[-c\partial_x+(V(\phi_c))^*]G(\phi_c)
= -[c+E''(\phi_c)] (\partial_x I'(\phi_c))
=-\mathcal{L}_c \partial_x I'(\phi_c).
\end{equation}
By hypothesis (H2) above we obtain from \eqref{1rela}
that there is $\lambda=\lambda(c, I)$ such that
\begin{equation}\label{main}
\partial_x I'(\phi_c)=\lambda \frac{d}{dx}\phi_c.
\end{equation}
Relation \eqref{main} contains the most important information
in our study. Indeed, if we consider the evolution equation
\begin{equation}\label{hierar}
z_t=\partial_x I'(z(t)),
\end{equation}
then $z_\lambda(x,t)=\phi_c(x+\lambda t)$ is a travelling wave
solution for \eqref{hierar}. In general, the value of $\lambda$
depending on $I$
and $c$, is not easy to find out. Note that from
\eqref{main} we have $I'(\phi_c)=\lambda \phi_c +\beta$, with $\beta$
being an integration constant. If $\phi_c$ is a solitary wave
solution ($\lim_{|\xi|\to \infty}\phi(\xi)=0$) then $\beta=0$. In
the case of periodic travelling waves solutions we will also
assume $\beta=0$. \smallskip
\noindent\textbf{Remark.}
Hypothesis (H2), it is a delicate issue to be verified in
the periodic setting, but the techniques developed
in Angulo and Natali \cite{AN,AN1} can be useful for this purpose.
Next, we apply the foregoing to the 3-BO and 5-mKdV equations
in \eqref{3bo} and \eqref{5mkdv}, respectively.
\subsection{The 3-BO case}
Let $\psi$ denote the travelling wave solutions of type
solitary wave or periodic wave for the BO \eqref{1.3}. This
equation can be written in the Hamiltonian form
$$
u_t=\partial_xE'_{BO}(u(t))
$$
with
$$
E_{BO}(u)=\int u\mathcal{H} u_x-\frac16 u^3\,dx.
$$
So, the hypothesis (H1) above follows from Section 4
(formulas \eqref{solita} and \eqref{formubo}).
Now, for verifying hypothesis (H2) we need to study
the kernel of the pseudo-differential operator
$$
\mathcal{L}_{BO}=\frac{d}{dx}\mathcal{H}-\psi+c.
$$
For $\psi$ being the solitary wave solution in \eqref{solita} or
the periodic travelling wave in \eqref{formubo}, the works of Bennett
\emph{et al.} \cite{BBBSS}, Albert \cite{A1}, Albert \emph{et al.}
\cite{AB} and Angulo \emph{et al.} \cite{AN}, show that
$\ker(\mathcal{L}_{BO})=[\frac{d}{dx}\psi]$
(see Angulo \cite{An1} for a summary of these results).
Thus, there is a constant $\lambda_{BO}$ such that
$\psi(x+\lambda_{BO} t)$ is a travelling wave solution of
\eqref{3bo} according to the results established in Subsection 5.1.
Next, we obtain the exact value of $\lambda_{BO}$. We start
establishing some nontrivial facts about the solutions of the
pseudo-differential equation
\begin{equation} \label{trabo}
\mathcal{H}\psi'+c \psi-\frac12 \psi^2=0,\quad c>0.
\end{equation}
In Albert \cite{A2} it was shown an alternative method of proof of
uniqueness of the solitary waves solutions for the intermediate
long wave equation (ILW) and the BO (see \eqref{solita}), which does
not use complex
analysis (see \cite{AT}, \cite{AmT}). His method make use of
positive-operator theory and suitable identities associated to the
dispersion operator in the ILW and BO equations. In the case of
the BO was used the well-known product formula
\begin{equation}\label{bofor}
fg+\mathcal{H}(f\cdot\mathcal{H} g +g\cdot\mathcal{H} f)
-\mathcal{H} f\cdot
\mathcal{H} g=0,
\end{equation}
valid for $f,g\in L^2(\mathbb{R})$. Inspecting his proof, one can
observe that the key equality in Lemma 3 in \cite{A2},
established on the line for $N=N_H$ being the ``dispersion
operator'' defined by $\widehat{N_Hf}(\xi)=\xi coth(\xi
H)\widehat{f}(\xi)$, $\xi\in \mathbb{R}$, it is also true in the
periodic setting with $N$ replaced by $M=\partial_x \mathcal{H}$.
Indeed, since the formula \eqref{bofor} is true for $f,g\in
L_{\rm per}^2$, differentiation of \eqref{bofor} yields the main
equality
$$
f'g+fg'+M\Big[ f\Big(\int_0^xM g\Big)+ g\Big(\int_0^xM f\Big)\Big]
-M f\Big(\int_0^xM g\Big)-M g\Big(\int_0^xM f\Big)=0.
$$
Hence following the ideas in \cite{A2}, we obtain that every
positive periodic solution $\psi$ of \eqref{trabo} satisfies
$$
\mathcal{H}\psi'=-2 \Big(\frac{\psi'}{\psi}\Big)'.
$$
Therefore, from \eqref{trabo} the following ordinary
differential equation holds:
\begin{equation}
\label{ordbo} \Big(\frac{\psi'}{\psi}\Big)'=\frac{c}{2}
\psi-\frac14 \psi^2.
\end{equation}
Then, it is easy to see that $\phi_c$ in \eqref{solita}
(with $\omega=c>0$) and $\chi_c$ defined in \eqref{formubo},
satisfy \eqref{ordbo}. Now, from \eqref{ordbo} it follows that
$\psi$ satisfies
\begin{equation}
\label{ordbo2} [\psi']^2=\psi^2[c\psi-\frac14 \psi^2 +D],
\end{equation}
where $D$ is the constant of integration. The value of this
constant in the case $\psi=\phi_c$ in \eqref{solita} it is $D=0$, and
for $\psi=\chi_c$ in \eqref{formubo}, it is given by
$D=-\frac{\pi^2}{L^2}$.
Now, by denoting $G(u)=4u_{xx}+3\mathcal{H}(uu_x)+3u\mathcal{H}u_x-u^3$,
we can write \eqref{3bo} in the hamiltonian form
$$
u_t=\partial_x G(u).
$$
Here $G(u)=I'(u)$ with $I(u)$ being the
following conservation law for the BO,
$$
I(u)=\int 2(u_x)^2-\frac32 u^2 \mathcal{H} u_x+\frac14 u^4\,dx.
$$
Then for $u_c(x,t)=\psi(x+\lambda_{BO}t)$ with
$\lambda_{BO}=D-3c^2$, we obtain after some computations based on
relations \eqref{trabo} and \eqref{ordbo2} that
$$
G(\psi)=\lambda_{BO}\psi.
$$
Hence, $u_c(x,t)$ is a travelling wave solution
(solitary or periodic) for the 3-BO \eqref{3bo}.
Finally, from the approach in \cite{BPS}, Section 3, and Section 4 above,
we obtain the following result.
\begin{theorem}\label{ill3bo}
The initial value problem for the third order BO
equation \eqref{3bo} is locally ill-posed in $H^s(\mathbb{R})$ and
in $H_{\rm per}^{s}$
for $s<-1/2$.
\end{theorem}
\subsection{The 5-mKdV case}
We start with the following scaling for $u$ being solution of the
mKdV \eqref{1.2}. For $v(x,t)=\frac{\sqrt{2}}{2} u(x,t)$ we have that
\begin{equation}
\label{6mkdv} v_t+6v^2v_x+v_{xxx}=0.
\end{equation}
So, we have that \eqref{5mkdv} is the second equation from the
mKdV \eqref{6mkdv} hierarchy. Now, for $v(x,t)=\zeta_c(x-ct)$ we
have
\begin{equation}
\label{5perio} \zeta_c''+2\zeta_c^3-c\zeta_c=0,\;\;\text{and}\;\;\;
[\zeta_c']^2=-\zeta_c^4+c\zeta_c^2+B_c,
\end{equation}
with $B_c$ the integration constant. Then for $\phi_c$ in
\eqref{solimkdv} ($c=\omega>0$) we have that
$\zeta_c=\frac{\sqrt{2}}{2}\phi_c$ satisfies \eqref{5perio} with
$B_c=0$ and with $B_c=B_{\varphi_c}$ for
$\zeta_c=\frac{\sqrt{2}}{2}\varphi_c$, and $\varphi_c$ being the
dnoidal wave solution satisfying \eqref{1order} for
$c>\frac{2\pi^2}{L^2}$.
Equation \eqref{6mkdv} can be written in the Hamiltonian form as
$$
v_t=\partial_xE'_{mKdV}(v(t))
$$
with
$$
E_{mKdV}(v)=\frac12\int
(v_x)^2 -v^4\,dx.
$$
Moreover, the family of travelling wave $\zeta_c$ satisfies
$E_{mKdV}'(\zeta_c)+c\zeta_c=0$. So, we obtain the hypothesis
(H1) in Subsection 5.1.
For obtaining hypothesis (H2) we study the kernel of the
second order differential operator
$$
\mathcal{L}_{mKdV}=-\frac{d^2}{dx^2}-6\zeta_c^2+c.
$$
For $\zeta_c$ being a solitary wave solution we have that an
elementary application of the Oscillation theory of the
Sturm-Liouville theory implies that zero is a simple eigenvalue
with eigenfunction $\frac{d}{dx}\zeta_c$. For
$\zeta_c=\frac{\sqrt{2}}{2}\varphi_c$ and $\varphi_c$ being the
dnoidal wave solution defined in Section 3, the analysis is more
delicate. In this case, the Floquet theory can be used (see Angulo
\cite{An} or the proof of Theorem \ref{spdmkdv} below) for
obtaining the desired property for $\mathcal{L}_{mKdV}$. We note
in this point that by using the new technique in Angulo\&Natali
\cite{AN} (see also Angulo \cite{An1}) which is based in positive
properties of the Fourier transform of $\varphi_c$, we can also
deduce hypothesis (H2).
Then for $G(u)=u_{xxxx}+6u^5+10u^2u_{xx}+10u(u_x)^2$, one can write
\eqref{5mkdv} in the form
$$
u_t=\partial_xG(u)
$$
with $G(u)=M'(u)$ and $M$ being the following conservation
law for the mKdV in \eqref{6mkdv},
$$
M(u)=\int \frac12 (u_{xx})^2+u^6-5u^2(u_x)^2\,dx.
$$
Therefore, for $u_c(x,t)=\zeta_c(x+\lambda_{mKdV}t)$ with
$$
\lambda_{mKdV}=c^2-2B_{c},
$$
we obtain after some computations based in the relations
in \eqref{5perio}, that
$G(\zeta_c)=\lambda_{mKdV}\zeta_c$. Hence, $u_c(x,t)$ represents a
travelling wave solution for the 5-mkdv \eqref{5mkdv}.
From Section 3 in \cite{BPS} and Section 3 above we
obtain the following result.
\begin{theorem}\label{t5mkdv}
The initial value problem for the fifth order modified
Korteweg-de Vries equation \eqref{5mkdv} is not locally
well-posed in $H^s(\mathbb{R})$ and
$H_{\rm per}^{s}$ for any $s<-1/2$.
\end{theorem}
\subsection*{Remarks}
\noindent\textbf{(a)}
Recently Kwon in \cite{Kwon} showed that the
initial value problem (IVP) for the following general
fifth order mKdV equation
$$
u_{t}-u_{xxxxx}+c_0u^4u_x
+c_1(u^3)_{xxx}+c_2uu_xu_{xx}+c_4u^2u_{xxx}=0,
$$
with $c_j$ constants, is local well-posedness in $H^s(\mathbb{R})$ for
$s\geq 3/4$ and that the solution map from data to the solutions,
fails to be uniformly continuous for $s\in (-\frac{7}{24}, \frac34)$.
Theorem \ref{t5mkdv} above shows that the IVP is also ill-posed in
$H^s(\mathbb{R})$ for $s<-1/2$.
\noindent\textbf{(b)}
The third equation from the mKdV \eqref{6mkdv} hierarchy
is given by
\begin{equation}\label{3mkdv}
\begin{aligned}
& u_{t}-\partial_x^7u-84u\partial_x u\partial_x^4u-560
u^3\partial_x u\partial_x^2u
-14u^2\partial_x^5u-140u\partial_x^2u\partial_x^3u\\
\\
&-126\partial_x^3u(\partial_x u)^2-182\partial_x u(\partial^2_x
u)^2 -70u^4\partial_x^3u-420u^2(\partial_x u)^3-140u^6\partial_x
u=0,
\end{aligned}
\end{equation}
which is coming from the conservation law
$$
N(u)=\int-\frac12
(u_{xxx})^2-35u^4(u_x)^2+7u^2(u_{xx})^2-\frac72(u_x)^4+\frac52u^8\,dx.
$$
Then we have that \eqref{3mkdv} has the hamiltonian form
$u_t=\partial_xN'(u)$. Therefore we obtain that $
u_c(x,t)=\zeta_c(x+\lambda_{N,c}t)$
with
$$
\lambda_{N,c}=c^3-6cB_c,
$$
is a travelling wave to \eqref{3mkdv}. Then we get that the IVP
for the seventh order modified Korteweg-de Vries equation
\eqref{3mkdv} is ill-posed in $H^s(\mathbb{R})$ and
$H^s_{\rm per}$ for $s<-1/2$.
\noindent\textbf{(c)}
The third equation from the BO hierarchy is
\begin{equation}\label{4bo}
u_t=\partial_xS(u)
\end{equation}
where
\begin{align*}
S(u)&=u^4-4u^2\mathcal{H}u_x-4u\mathcal{H}(uu_x) +2(\mathcal{H}
u_x)^2+4\mathcal{H}(u\mathcal{H}u_x)_x\\
&\quad -6(u_x)^2-12uu_{xx}+8\mathcal{H}u_{xxx}.
\end{align*}
Here $S(u)=W'(u)$ for $W$ being the following conservation law
for the BO equation
$$
W(u)=\int\frac15 u^5-\frac43 u^3\mathcal{H}u_x-u^2\mathcal{H}(uu_x)+
2u(\mathcal{H}u_x)^2+6u(u_x)^2+4u_{xx}\mathcal{H}u_x\,dx.
$$
Therefore we obtain that $u_c(x,t)=\psi(x+\lambda_{W,c}t)$ with
$$
\lambda_{W,c}=4c^3-4cD,
$$
and $D$ given in \eqref{ordbo2}, it is a
travelling wave to \eqref{4bo} satisfying $S(\psi)=\lambda_{W,c} \psi$. Then
we deduce that the IVP for \eqref{4bo} is ill-posed in
$H^s(\mathbb{R})$ and $H^s_{\rm per}$ for $s<-1/2$.
\noindent\textbf{(d)}
The role of the index $s=-1/2$ in Theorems
\ref{ill3bo} and \ref{t5mkdv} for the spaces $H^s(\mathbb{R})$, can be
explained via a scaling argument, that is, if $u(x,t)$ solves the
IVP \eqref{5mkdv}, then $u_\lambda(x,t)=\lambda u(\lambda
x,\lambda^5 t)$, $\lambda>0$, solves the same equation with data
$u_{0,\lambda}(x)=\lambda u_0(\lambda x)$. Then for $D^s$ defined by
$\widehat{D^s f}(\xi)=|\xi|^s\widehat{f}(\xi)$, we obtain that the equality
$$
\|D^s u_{0,\lambda}\|=\lambda^{s+\frac12}\|D^s u_{0}\|
$$
implies that this norm is independent of $\lambda$ only when
$s=-1/2$. A similar analysis is carried on with \eqref{3bo} via
the scaling
$$
u_\lambda(x,t)=\lambda u(\lambda x,\lambda^3 t).
$$
\section{Ill-posedness for the dmKdV and the fifth order dmKdV}
In this section we develop a theory of ill-posedness in the
periodic case for the defocusing modified Korteweg-de Vries
equation (dmKdV)
\begin{equation}\label{1.1}
v_{t}+6v^2v_{x}-v_{xxx}=0,\quad v=v(x,t)\in \mathbb{R},
\end{equation}
and for the fifth order defocusing modified Korteweg-de Vries
equation (5-dmKdV)
\begin{equation}\label{1.4}
v_t-v_{xxxxx}-30v^4v_x
+10v^2v_{xxx}+10(v_x)^3+40vv_xv_{xx}=0.
\end{equation}
The use of the theory of elliptic functions and the Floquet
theory associated to the Lam\'e equation will be basic in
our analysis.
\subsection{The dmKdV case}
In this section we focus to the ill-posedness result for the
defocusing mKdV \eqref{1.1}. We start obtaining a family of
periodic travelling wave solutions of \eqref{1.1} in the form
$$
v(x,t)=Q_c(x-ct).
$$
So, if we substitute this specific solution in the defocusing mKdV and
consider the integration constant equal to zero then $Q=Q_c$
satisfies the ordinary differential equation
\begin{equation}\label{dmkdv0}
Q''+cQ-2Q^3=0.
\end{equation}
From this we obtain the first order differential equation
(the associated quadrature form)
\begin{equation}\label{dmkdv1}
[Q']^2=Q^4-cQ^2+A,
\end{equation}
where $A$ is the integration constant and which need to be
different of zero for obtaining periodic profile solutions. Let us
suppose that the fourth order polynomial $F(t)=t^4-ct^2+A$ has the
positive roots $\eta_1>\eta_2>0$. From \eqref{dmkdv1} it follows
that
\begin{equation}\label{dmkdv2}
\begin{gathered}
{}[Q']^2=(Q^2-\eta_1^2)(Q^2-\eta_2^2),\quad
-\eta_2\leq Q\leq \eta_2,\\
\eta_1^2+\eta_2^2=c>0,\quad
\eta_1^2\eta_2^2=A>0.
\end{gathered}
\end{equation}
Next, we normalize $Q$ by letting $\varphi=Q/{\eta_2}$, so that
\eqref{dmkdv2} becomes
\begin{equation}\label{dmkdv3}
[\varphi']^2=\eta_1^2(k^2\varphi^2-1)(\varphi^2-1)
\end{equation}
with $k^2=\eta_2^2/{\eta_1^2}$. Next, by letting now
$\varphi(\xi)=sin(\psi(\xi))$ with $\psi(0)=0$ and $\psi$
continuous, the substitution of it into \eqref{dmkdv3},
yields the equation
$$
[\psi']^2=\eta_1^2(1-k^2sin^2\psi).
$$
We may solve for $\psi$ implicitly to obtain
\begin{equation}\label{dmkdv4}
F(\psi;k) =\int^{\psi(\xi)}_0 \frac{dt}{\sqrt{1-k^2\sin^2
t}}=\eta_1\,\xi.
\end{equation}
The left-hand side of \eqref{dmkdv4} is just the standard elliptic
integral of the first kind and so, for fixed $k$, the elliptic
function snoidal $\operatorname{sn}(\xi;k)$ is defined in
terms of the inverse of
the mapping $\psi\longmapsto F(\psi;k)$. Hence, \eqref{dmkdv4}
implies that
$$
\operatorname{sn} (\eta_1 \xi;k)=\sin\psi(\xi).
$$
Therefore, we obtain the snoidal periodic profile for \eqref{dmkdv0}
$$
Q_c(\xi)=\eta_2\operatorname{sn} (\eta_1 \xi;k),
$$
which is determined by the elliptic modulus
$k^2=\eta_2^2/{\eta_1^2}\in (0,1)$.
Next we establish the main information for obtaining a
smooth curve, $c\to Q_c$, of periodic travelling waves for
\eqref{dmkdv0} with minimal period $L$. Since $\operatorname{sn}$
has minimal period $4K(k)$ then the minimal period
of $Q_c$, $T_{Q_c}$, is given by $T_{Q_c}=4K(k)/{\eta_1}$.
Moreover, from the relations in \eqref{dmkdv2} it follows
$$
0<\eta_2<\sqrt{c/2}<\eta_1<\sqrt{c},
$$
and $k$ and $T_{Q_c}$ can be
seen as functions of $c$ and $\eta_1$, namely,
$$
k^2(\eta_1, c)=\frac{c-\eta_1^2}{\eta_1^2},\ \
T_{Q_c}(\eta_1,c)=\frac{4}{\eta_1}K(k(\eta_1, c)).
$$
Therefore, from the properties of $K(k)$ and from the implicit
function theorem we have the following result
(see Angulo \cite{An, An1}).
\begin{theorem}\label{tdmkdv}
Let $L>0$ be a fixed number and $n$ any positive integer. Then
there exists a smooth branch of snoidal waves,
$c\in (\frac{4\pi^2}{L^2},+\infty)\mapsto Q_c
\in H^n_{\rm per}([0,L])$,
such that
\begin{equation}
Q_c''(\xi)+cQ_c(\xi)-2Q^3(\xi)=0,\quad\text{for all }\xi\in \mathbb{R},
\end{equation}
where
\begin{equation}\label{sndmkdv}
Q_c(\xi)=\eta_2\operatorname{sn} (\eta_1 \xi;k).
\end{equation}
Here, $\eta_1$, $\eta_2$ and $k$ are smooth functions of $c$,
satisfying the relations
\begin{equation}
\begin{gathered}
\eta_1\in (\sqrt{c/2},\sqrt{c}),\quad
\eta_2=\sqrt{c-\eta_1^2},\quad
k^2=\frac{\eta_2^2}{\eta_1^2}\\
\frac{4K(k(c))}{\eta_1(c)}=L,\quad\text{for all }c>\frac{4\pi^2}{L^2}.
\end{gathered}
\end{equation}
Moreover, $k(c)\to 1$ as $c\to \infty$.
\end{theorem}
The following theorem is the main piece in our study of
ill-posedness for the defocusing mKdV.
\begin{theorem}\label{fodmkdv}
The Fourier coefficients $\{\widehat{Q_c}(m)\}_{m\in\mathbb Z}$
for $Q_c$ defined in \eqref{sndmkdv} satisfy
$$
\lim_{c\to +\infty} \widehat{Q_c}(m)=\frac{4\pi}{L},\quad
\text{for all }m\geq 0.
$$
\end{theorem}
\begin{proof}
From \cite{BF} we have for $q=e^{-\pi K'/K}$ that the
Fourier series of $sn$ is
$$
\operatorname{sn}u=\frac{2\pi}{kK}\sum_{m=0}^\infty
\frac{q^{m+\frac12}}{1-q^{2m+1}}\sin
\Big[(2m+1)\frac{\pi}{2}\frac{u}{K}\Big].
$$
Therefore, from the relations $\eta_1=4K/L$ and $\eta_2=\eta_1k$
we obtain
$$
Q_c(\xi)=\frac{4\pi}{L}\sum_{m=0}^\infty
\operatorname{sech}\Big[(2m+1)\frac{\pi}{2}\frac{K'}{K}\Big]
\sin\Big[(2m+1)\frac{2\pi \xi }{L}\Big].
$$
Hence from Theorem \ref{tdmkdv} we finish the proof.
\end{proof}
So, following the method in Section 3 we obtain the main result of
this Subsection.
\begin{theorem}\label{illdmkdv}
The Cauchy problem for the defocusing mKdV is not locally
well-posed in $H_{\rm per}^{s}$, $s<-1/2$, in the sense
that the mapping data-solution $u_{0} \to u$ is not continuous.
\end{theorem}
Then, from Christ, Colliander and Tao \cite[Theorem 8 ]{CCT} and from
Theorem \ref{illdmkdv} above we obtain the following
sharp ill-posedness type result for \eqref{1.1}.
\begin{theorem}\label{shdmkdv}
The Cauchy problem for the defocusing mKdV is not locally
well-posed in $H_{\rm per}^{s}$, $s<1/2$, in the sense
that the mapping data-solution $u_{0} \to u$ is not
uniformly continuous.
\end{theorem}
\subsection{The fifth order dmKdV case}
In this Subsection we focus to the ill-posed\-ness result for the
5-dmKdV \eqref{1.4}. We start writing the dmKdV equation \eqref{1.1}
in the hamiltonian form
$$
v_t=\partial_x E'_{dmKdV}(v(t))
$$
with
$$
E_{dmKdV}(v)=\frac12\int (v'(x))^2+v^4(x)\,dx.
$$
Therefore, the snoidal wave solution $Q_c$ in \eqref{sndmkdv} is a
critical point of the functional $H=E_{dmKdV}-cF$ for
$F(u)= \|u\|^2/2$ and for every $c>4\pi^2/{L^2}$. Hence from
Subsection 5.1 we need to show that the kernel of the linear operator
$H''(Q_c)=\mathcal{L}_{dmKdV}$,
\begin{equation}\label{specd}
\mathcal{L}_{dmKdV}=-\frac{d^2}{dx^2}+6Q_c^2-c,
\end{equation}
is generated by $\frac{d}{dx}Q_c$. So, we establish the next
periodic spectral problem for
$\mathcal{L}_{sn} \equiv \mathcal{L}_{dmKdV}$, namely,
\begin{equation}
\begin{gathered}
\mathcal{L}_{sn}\chi=\lambda\chi, \\
\chi(0)=\chi(L),\quad \chi'(0)=\chi'(L),
\end{gathered} \label{spec}
\end{equation}
and the following result is obtained in this context.
\begin{theorem}\label{spdmkdv}
Let $Q_c$ be the snoidal wave given in Theorem \ref{tdmkdv} for
$c\in (\frac{4\pi^2}{L^2},+\infty)$. Let
\[
\lambda_0\leq\lambda_1\leq\lambda_2\leq\lambda_3\leq\lambda_4\leq
\cdot\cdot\cdot,
\]
connote the eigenvalues of the problem \eqref{spec}. Then
$$
\lambda_0< \lambda_1=0<\lambda_2<\lambda_3<\lambda_4
$$
are all simple whilst, for $j\geq 5$, the $\lambda_j$ are double
eigenvalues. The $\lambda_j$ only accumulate at $+\infty$.
\end{theorem}
\begin{proof}
Theorem \ref{spdmkdv} is a consequence of Floquet
theory (Magnus and Winkler \cite{MW}) together with some
particular facts about the periodic eigenvalue problem associated
to the Lam\'e equation,
\begin{equation}
\begin{gathered}
\frac{d^2}{dx^2}\Lambda+[\rho-6k^2sn^2(x;k)]\Lambda=0\\
\Lambda(0)=\Lambda(4K(k)),\;\;\Lambda'(0)=\Lambda'(4K(k)).
\end{gathered} \label{lame}
\end{equation}
Indeed, we certainly know that $\lambda_0<\lambda_1\leq
\lambda_2$. Since $\mathcal{L}_{sn}\frac{d}{dx}Q_c=0$ and
$\frac{d}{dx} Q_c$ has 2 zeros in $[0,L)$, it follows that $0$
is either $\lambda_1$ or $\lambda_2$. We will show that
$0=\lambda_1<\lambda_2$. First, we perform the change of variable
$\Lambda(x)\equiv\chi( x/{\eta_1})$. Then, using the explicit form
\eqref{sndmkdv} for $Q_c$ and that $\eta_1^2k^2=\eta_2^2$, the problem
\eqref{spec} is equivalent to the eigenvalue problem \eqref{lame}
with
\begin{equation}\label{rho}
\rho=\frac{c+\lambda}{\eta_1^2}=\frac{\eta_1^2+\eta_2^2+\lambda}{\eta_1^2}
=1+k^2+\frac{\lambda}{\eta_1^2}.
\end{equation}
Next, from Floquet theory the Lam\'e equation in \eqref{lame}
(see Angulo \cite{An, An1}) with
boundary conditions $\Lambda^{j}(0)=\Lambda^{j}(2K(k))$, $j=0,1$,
has exactly three instability intervals, so the first five
eigenvalues for \eqref{lame}, $\{\rho_j: 0\leq j \leq 4\}$, are
simple and for $j\geq 5$, the $\rho_j$ are double eigenvalues. In
Angulo \cite{An} the explicit values of that simple eigenvalues
and its eigenfunctions are given. Indeed, the eigenvalues are:
\begin{equation}
\begin{gathered}
\rho_0= 2[1+k^2-\sqrt{1-k^2+k^4}],\quad
\rho_1= 1+k^2,\quad \rho_2=1+4k^2,\\
\rho_3=4+k^2,\quad \rho_4=2[1+k^2+\sqrt{1-k^2+k^4}].
\end{gathered} \label{eigen}
\end{equation}
Then from \eqref{rho} we obtain the following relations
\begin{equation}
\begin{gathered}
\rho_0 \mapsto \lambda_0<0,\quad
\rho_1\mapsto \lambda_1=0 ,\quad
\rho_2\mapsto \lambda_2>0,\\
\rho_3\mapsto \lambda_3>\lambda_2,\quad
\rho_4\mapsto \lambda_4>\lambda_3.
\end{gathered} \label{eigenb}
\end{equation}
This completes the proof.
\end{proof}
Now, for $P(v)=v_{xxxx}+6v^5 -10v(v_{x})^2-10v^2v_{xx}$ we write
\eqref{1.4} in the form
$$
v_t=\partial_xP(v)
$$ with $P(v)=R'(v)$ and
$R$ being the following conservation law for the dmKdV
equation \eqref{1.1},
$$
R(v)=\int \frac12 (v_{xx})^2+v^6+5v^2(v_x)^2\,dx.
$$
Then for $v_c(x,t)=Q_c(x+\lambda_{R,c} t)$ with
$$
\lambda_{R,c}=c^2+2A,
$$
we obtain from \eqref{dmkdv1} that
$P(Q_c)=\lambda_{R,c} Q_c$. Hence, $v_c(x,t)$ represents a travelling
wave solution for the 5-dmKdV \eqref{1.4}.
Then following the method in Section 3 we obtain the following
result.
\begin{theorem}
The Cauchy problem for the fifth order
defocusing mKdV \eqref{1.4} is not locally
well-posed in $H_{\rm per}^{s}$, $s<-1/2$, more
precisely, the mapping data-solution $u_{0} \to u$ fails to be
continuous with respect to the $H_{\rm per}^{s}$.
\end{theorem}
\noindent\textbf{Remark:} Theorem \ref{spdmkdv} and the
property of concavity of the function
$$
d(c)=E_{dmKdV}(Q_c)-c\frac12 \int Q_c^2(x)\,dx
$$
($d''(c)<0$) give us the basic information which could give
the initial steps for obtaining a instability theory of the
orbit generated by the snoidal wave $Q_c$, namely,
$\Omega_{Q_c}=\{Q_c(\cdot+r):r\in \mathbb{R}\}$, by the
periodic flow generated by the defocusing mKdV. We note
that the classical stability theories in \cite{GSS} and
\cite{BSS} do not give a light for obtaining a conclusive
answer about this issue. We plan to discuss this in a subsequent
paper.
\subsection*{Acknowledgements}
J. Angulo was supported partially by grant
from CNPq/Brazil and by Edital Universal MCT/CNPq, 14/2009.
S. Hakkaev was supported by FAPESP S\~ao Paulo/SP Brazil.
The second author would like to express his thanks to the
Institute of Mathematics and Statistic (IME) at the
University of S\~ao Paulo for its hospitality.
The authors are grateful to the reviewers for their fruitful
remarks.
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\end{document}