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\begin{document}
\title{Exponential stability in a nonlinear string equation}
\author{Simone Paleari
$\quad$ Dario Bambusi $\quad$ Sergio Cacciatori}
\email{bambusi@mat.unimi.it,} \email{simone@mat.unimi.it,}
\email{cacciatori@mi.infn.it} \address{Dipartimento di Matematica
``F. Enriques'', Via Saldini 50, 20133 Milano, Italy.}
\address{Dipartimento di fisica, Via Celoria 16, 20133 Milano, Italy.}
%\date{\today}
\begin{abstract}
We study the nonlinear wave equation
\begin{equation}
\label{equa}
u_{tt}-c^2u_{xx}=\psi(u) \qquad u(0,t)=0=u(\pi,t)
\end{equation}
with an analytic nonlinearity of the type $\psi(u)=\pm u^3 +
\sum_{k\ge 4}\alpha_k u^k$. On each small--energy surface we consider
a solution of the linearized system with initial datum having the
profile of an elliptic sinus: we show that solutions starting close to
the corresponding phase space trajectory remain close to it for times
growing exponentially with the inverse of the energy. To obtain the
result we have to compute the resonant normal form of \ref{equa}, and
we think this could be interesting in itself.
\end{abstract}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Introduction and statement
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begsection{Introduction and statement} In this paper we study small
amplitude solutions of the equation \ref{equa} concentrating on the
periodic behaviour.
Consider the solution of the {\it linear} string equation
$u_{tt}-c^2u_{xx}=0$ with the initial datum:
\begin{equation}
\label{gamma}
u_{\epsilon}(x,0)={\epsilon} \Vm {\rm sn}(\omega x|\mm) \quad \dot
u(x,0)=0 \ ,
\end{equation}
where $\sn$ is the elliptic sine \cite{abrwtz}, $\mm$, $\omega$ and
$\Vm$ are constants whose values will be fixed later in the paper (see
\ref{EK},\ref{omeg}), and $\epsilon$ is a small parameter; its phase
space trajectory is a closed curve that will be denoted by
$\Gamma_\epsilon$. The main result of the present paper is the
forthcoming theorem \ref{1}, it ensures that solutions starting close
to $\Gamma_\epsilon$ remain close to it for times exponentially long
with $\epsilon^{-1}$. In the corresponding statement we will make use
of the distance $d(.,.)$ induced in the phase space $L^2(0,\pi)\times
H^{1}_0(0,\pi) \ni (\dot u,u)$ by the norm
\begin{equation}
\norma{(\dot u,u)}^2:=\frac12\int_0^\pi(\dot u(x)^2+c^2u_x(x)
^2)dx \ .
\end{equation}.
\begin{theorem}
\label{1}
Consider the nonlinear wave equation \ref{equa} with $\psi$ analytic
in a neighbourhood of the origin, assume
$\psi(0)=\psi'(0)=\psi''(0)=0$ and $\psi'''(0)\not=0$. Then there
exist strictly positive constants $\epsilon_*,C_1,...,C_4$ such that
the following holds true: fix $\epsilon<\epsilon_*$ and consider an
initial datum $z_0=(\dot u_0,u_0)\in L^2(0,\pi)\times H^{1}_0(0,\pi)$
close to $\Gamma_\epsilon$, precisely such that
$$
d(z_0,\Gamma_\epsilon)\leq C_{1} \epsilon^2 \ ,
$$
then the corresponding solution $z(t)$ remains close to it for
exponentially long times, precisely one has
$$
d(z(t),\Gamma_\epsilon)\leq C_{2} \epsilon^2
$$
for times $\left|t\right|\leq C_{3} \epsilon^{2}{\rm exp}
\left(\frac{C_4}{\epsilon}\right)$.
\end{theorem}
This result is obtained by applying the general theory developed in
\cite{BN}, and recalled in the next section, which requires the
verification of a non degeneracy condition. From the technical point of
view the verification of the non degeneracy condition is quite
complicate, indeed as pointed out by Moser \cite{mos70} in a similar
context, non degeneracy is ''a property which can be rarely verified'';
this is a remarkable case where it is possible.
We think that besides its interest for the dynamics of the string
equation, our result could be interesting as a first natural
application of the theory of \cite{BN}.
We recall the work \cite{LS} where existence of some periodic solutions
for the equation
$$
u_{tt}-u_{xx}+u^3=0
$$
was obtained, but as far as we know no stability result are known for
these solutions.
Finally we recall that the results usually obtained by KAM theory (see
\cite{Kuk94,cra93,Bourga,poe96a}) or Nekhoroshev theory
\cite{Bourga,bam98} do not apply to \ref{equa} since it is a
perturbation of a completely resonant system, and even the result of
\cite{bam97??} does not apply since it requires the existence of
an integrable first order normal form.
\finsection
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% General setting and Main result
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begsection{General setting}
First we rescale $u$ by introducing a variable $u'$ defined by
$\epsilon u'=u$. We recall that the equation for the rescaled variable
is hamiltonian with hamiltonian
\begin{equation}
\label{h.1}
\H(p',u')=h_\omega(p',u')+ \epsilon^2f(u')+ \epsilon^3f_1(u') \ ,
\end{equation}
where
$$
h_\omega(p',u')=\frac12\int_0^\pi\left(p^{'2}(x)+c^2u_x^{'2}(x)\right)dx \ ,
$$
the main part of the perturbation given by
\begin{equation}
\label{f1}
f(u')=\pm \frac14 \int_0^\pi u'(x)^4dx \ ,
\end{equation}
and higher order corrections given by
$$
f_1(u')=\sum_{k\geq4}\frac{\alpha_k\epsilon^{k-4}}{k+1}
\int_0^\pi u^{'k+1}(x)dx \ .
$$
To fix ideas we will always consider the case where the sign in front
of the integral in \ref{f1} is plus. In the minus case nothing
changes.
The idea of \cite{BN} is to use averaging theory to transform
\ref{h.1} into a system of the form
\begin{equation}
\label{ave}
\H(p',u')=h_\omega(p',u')+ \epsilon^2 \mfmed(p',u')+ \epsilon^3
Z_\epsilon(p',u')+\exp\left(-\frac{C}{\epsilon}\right)
\R_\epsilon(p',u') \ ,
\end{equation}
where \fmed\ is the average of $f$ with respect the unperturbed flow
(for a precise definition see equation \ref{defmed}),
$Z$ is a part of the Hamiltonian which commutes with $h_{\omega}$, and
$\R$ is an exponentially small remainder.
Consider then the {\it simplified system} obtained from \ref{ave} by
neglecting $\R$, namely
\begin{equation}
\label{simp}
\H_s(p',u')=h_\omega(p',u')+ \epsilon^2 \mfmed(p',u')+
\epsilon^3 Z_\epsilon(p',u') \ .
\end{equation}
Then $h_\omega$ is an integral of motion independent of the energy for
\ref{simp}. The critical points of $\H_s$ constrained to the surface
of constant $h_\omega=1$ are invariant sets. It easy to realize that
if they are non degenerate in the transversal directions they are
trajectories of periodic solutions of \ref{simp}. Moreover, if such
non degenerate critical points are also extrema, the corresponding
periodic orbits are stable.
Taking into account the remainder $\R$ one obtains that the above
extrema are no more invariant sets, but, due to the very slow change
in time of both $\H_{s}$ and $h_\omega$ for the complete dynamics, it
turns out that solutions starting close to extrema remain close to
them for exponentially long times.
Finally, remark that usually one can hope to compute \fmed\ but not the
exact expression of $\H_{s}$, so we will look for non degenerate
critical points of $\H_s\vert_{h_\omega^{-1}(1)}$ as perturbations of
non degenerate critical points of $\mfmed\vert_{h_\omega^{-1}(1)}$.
Actually, applying the theory of \cite{BN} (see theorem 6.1), in the
improved formulation by \cite{nek} one obtains the following theorem
(stated in non rescaled variables), which gives a more precise
description than theorem \ref{1}.
\begin{theorem}
\label{teo}
Consider the nonlinear wave equation \ref{equa} with $\psi$ analytic
in a neighbourhood of the origin, assume
$\psi(0)=\psi'(0)=\psi''(0)=0$ and $\psi'''(0)\not=0$. There exist
strictly positive constants $\epsilon_*,C_4,...,C_9$ and a family of
curves $\left\{\gamma_\epsilon\right\}_{0\leq\epsilon<\epsilon_*}$
with the following properties:
\\
1) For any initial datum $(\dot
u_0,u_0)\equiv z_0\in \P$ satisfying $d_0=d(z_0,\gamma_\epsilon)\leq
C_5 \epsilon$ the corresponding solution $z(t)$ satisfies
$$
d\left(z(t),\gamma_\epsilon\right)\leq \sqrt{C_6 d_0^2 +
C_7\epsilon^2 \vert t\vert {\rm
exp}\left(-\frac{C_4}{\epsilon}\right)} \ ,
$$
$\perogni t$ such that $|t|\leq C_8{\rm exp}
\left(\frac{C_4}{\epsilon}\right)$;
\\
2) $\gamma_\epsilon$ depends smoothly on $\epsilon$ and is close to
$\Gamma_\epsilon$, precisely one has
$$
\sup_{z\in\Gamma_\epsilon}d\left(\gamma_\epsilon,z\right)
\leq C_9 \epsilon^2 \ .
$$
3) If $(\dot u,u)\in\gamma_{\epsilon}$ then both $\dot u$ and $u$ are
$C^{\infty}$ functions.
\end{theorem}
Theorem \ref{1} is an obvious corollary of this result. The proof of
theorem \ref{teo} requires the computation of \fmed, of the extrema of
$\mfmed|_{h_\omega^{-1}(1)}$ and the verification that they are
non degenerate. This occupy the rest of the paper.
\begin{rem}
>From the estimate of the point 2) of the theorem it follows that the
ratio between the distance $d(\Gamma_\epsilon,\gamma_\epsilon)$ and
the size of the energy surface tends to zero as $\epsilon\to0$.
\end{rem}
\finsection
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Normal Form
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begsection{Normal form}
In this section we give the explicit form of the average \fmed\ of the
perturbation and give the precise statement of the normal form theorem
for the case of eq.~\ref{equa}.
First we recall the precise definition of \fmed, which is the average
of $f$ with respect to the linear flow; denoting by $\Phi^t(p,u)$ the
flow of the linear string equation with initial data $\dot
u(x,0)=p(x),\ u(x,0)=u(x)$, we define
\begin{equation}
\label{defmed}
\mfmed(p,u):=\frac{1}{T}\int_{-T/2}^{T/2} f(\Phi^t(p,u))dt \ .
\end{equation}
In order to be able to do explicit calculation, one needs good
coordinates. To introduce them we proceed as follows. First we extend
both $u'$ and $p'\equiv \dot u'$ to $[-\pi,\pi]$ by skew symmetry, and
define $P(x)$ as the unique function with zero average such that
$$
p'(x)=cP_x(x)\ ;
$$
remark that $P(x)$ is an even function. Define $\Pt\subset
H^1(\toro^1)$ as the space of periodic functions of period $2\pi$ with
zero average. Then the map
$$
\P\ni(p,u')\mapsto V:=u'+P\in\Pt
$$
is an isomorphism. Remark that, under this isomorphism the linear flow $\Phi^t$
takes the form
$$
(\Phi^t V)(x)=V(x+ct) \ .
$$
Using $V$ to parameterize phase points, the hamiltonian $f$ of the main
part of the perturbation takes the form
$$
f(V)=\frac18 \int_{-\pi}^\pi
\left(\frac{V(x+y)-V(-x+y)}{2}\right)^4dxdy \ .
$$
In the following we will use the notation
$$
[g]=\frac{1}{2\pi}\int_{-\pi}^{\pi}g(x)dx
$$
for the average of a
function.
\begin{proposition}
\label{nf}
There exist open domains $U_2\subset U_1$, containing the
sphere of radius 1 in the phase space, constants $C_4$ and an
analytical canonical transformation $T:U_2\to U_1$ such that $H\circ
T$ takes the form
$$
h_\omega(V)+\epsilon^2\mfmed (V)+\epsilon^3 Z_\epsilon(V) + {\rm exp}
\left(-\frac{C_4}{\epsilon}\right)\R_\epsilon(V) \ ,
$$
with
\begin{equation}
\label{med}
\mfmed(V)=
\frac{\pi}{32}\left(\left[V^4\right] + 3 \left[V^2\right]^2 \right)
\ ,
\end{equation}
\end{proposition}
and with $Z_\epsilon$, $\R_\epsilon$, and their vector fields, bounded
on $U_2$, uniformly with respect to $\epsilon$.
\begin{rem}
We recall the paper \cite{ver87,kro89} where the computation of the
normal form of a finite dimensional approximation of \ref{equa} was
done. In that paper the authors also studied some features
of the dynamics of the so obtained system.
\end{rem}
\begin{proof} This is an application of Lemma 8.1 of ref. \cite{BN}.
The only point to be proved is the explicit formula \ref{med}. This is obtained
by a
calculation that we are going to summarize. Using the changes of
variables $y=ct$ one has
\begin{equation*}
\begin{split}
\mfmed (V)=&\frac{1}{2\pi}\int_{-\pi}^{\pi} \frac14
\left( \int_0^\pi u'(x,t)^4dx \right) dt=
\\
=&\frac{1}{2\pi} \int_{-\pi}^{\pi} \frac18
\int_{-\pi}^\pi \left(\frac{V(x+y)-V(-x+y)}{2}\right)^4dxdy \ .
\end{split}
\end{equation*}
Making the change of variables $x+y=\xi,\;
-x+y=\eta$ and using the periodicities to double the
interval of integration the above quantity takes the form
\begin{equation*}
\begin{split}
=&\frac{1}{256\pi} \frac12 \int_{-2\pi}^{2\pi} \int_{-2\pi}^{2\pi}
\left[ V^4(\xi)+V^4(\eta)+6V^2(\xi)V^2(\eta)-4V^3(\xi)V(\eta)+\right.
\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\left. -4V(\xi)V^3(\eta) \right] \frac{d\xi d\eta}{2}\ ,
\end{split}
\end{equation*}
which, taking into account that $V$ has zero average, is easily seen to
coincide with \ref{med}.
\end{proof}
%%%
\begin{rem}
With these variables one can easily check that every nonlinearity
$\psi(u)=u^{2n}$ has zero average, so the theory of \cite{BN} does
not apply directly to the case of even power.
\end{rem}
%%%
\finsection
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%% Critical Points
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begsection{Critical Points}
Using the explicit expression of \fmed, we can look for its
critical points constrained to the constant harmonic energy surface
$\S_1$, defined by $\S_E=\{ V\in \Pt \;\vert\; h_\omega(V)=E \} $ with
$E=1$.
Actually we study the critical points of \fmed\ constrained to the
surface $\S_E$ with an arbitrary $E$. Such critical points are
expressed in terms of elliptic functions (for definitions and
properties of elliptic functions and elliptic integrals see
\cite{abrwtz}). In what follows we will denote by $\KK=\KK(\mm)$ the
complete elliptic integral of the first kind and by $\EE=\EE(\mm)$ the
complete elliptic integral of second kind; moreover we fix the value
of the
parameter $\mm$ as the only solution of the equation
\begin{equation}
\label{EK}
\frac{\EE}{\KK}=\frac{7+\mm}{6} \ .
\end{equation}
It is not difficult to verify that $\mm\approx -0.2554$.
\begin{proposition}
\label{crit}
Consider the function
\begin{equation}
\label{v}
V(x)=\Vm \; {\rm sn}(\omega\; x\vert \mm)
\end{equation}
with
\begin{equation}
\label{omeg}
\omega:=\frac{2\KK}{\pi}\ ,\quad
\Vm^2:=\frac{9\pi^2}{2}\,\frac{\mm}{(\mm^2+14\mm+1)\KK^3}\;\,
\frac{E}{c^2} \ ;
\end{equation}
then the critical points of $\mfmed|_{\S_E}$ are
$$
V_n(x):=\frac{1}nV(nx) \quad n=1,2,\dots\ .
$$
\end{proposition}
\begin{proof}
Defining the $L^2$--gradient $\nabla f$ of a function $f$ by
$$
\left\langle\nabla f (V); h\right\rangle_{L^2}= df(V)h
$$
and using the method of Lagrange multipliers one obtains that the equation
for the critical points of $\mfmed\vert_{\S_E}$ is given by
\begin{equation}
\label{eid}
-\lambda c^2 V_{xx}(x)+\frac{1}{8}V^3(x)+
\frac{3}{16\pi}\left(\int_{-\pi}^{\pi}V^2(x)dx\right)V(x)=\kappa \ .
\end{equation}
with the following conditions:
$$
h_\omega(V)=E \ ,
$$
\begin{equation}
\label{condiz}
V(-\pi)=V(\pi) \ , \quad V_x(-\pi)=V_x(\pi) \ ,
\end{equation}
\begin{equation}
\label{aver}
\int_{-\pi}^{\pi}V(x)dx=0 \ .
\end{equation}
The above problem can be written in the following form
\begin{gather}
mV_{xx}=\kappa -\mu V-\frac18 V^3 \ ,\label{sdnl}
\\
E=\frac{c^2}{4}\int_{-\pi}^{\pi}V_x^2(x)dx \ , \label{con.en}
\\
\mu=\frac{3}{16\pi}\int_{-\pi}^{\pi}V^2(x)dx \ , \label{con.mu}
\end{gather}
together with the boundary conditions \ref{condiz} and the condition
of zero average \ref{aver}. The unknown being $(V,\mu,E,m,\kappa)$.
\begin{rem}
\label{risc}
If $(V,\mu,E,m,\kappa)$ is a solution of the above system, then, for
any $\rho>0$, also $(\rho V,\rho^2\mu,\rho^2E,\rho^2 m,\rho^3\kappa)$
is a solution of the same system.
\end{rem}
So, we will remove the condition \ref{con.en} and look for a
solution of the system \ref{sdnl}, \ref{con.mu}, \ref{condiz},
\ref{aver}, with the unknowns $(V,\mu,m,\kappa)$.
Remark that it is possible to find by quadratures the solution of
\ref{sdnl}: define $\Phi_{\kappa}(s):=-\kappa s+\frac12 \mu
s^2+\frac1{32}s^4$, then $V(x)$ is obtained by
\begin{equation}
\label{solu}
x=\int_{0}^{V}\frac{ds}{\sqrt{\frac2m \left(E'-\Phi_\kappa(s)\right)
}} \ ,
\end{equation}
where $E'$ is a parameter representing the mechanical energy of the
nonlinear oscillator \ref{sdnl}.
We state now a lemma which
will be useful in the following. Its proof is deferred to the appendix.
%%%
\begin{lem}
\label{dopo}
The equation \ref{sdnl} with the condition \ref{aver} has solutions
only if $\kappa=0$. In such case condition \ref{aver} is implied by
\ref{condiz}.
\end{lem}
%%%
So we are left with the system \ref{sdnl},
\ref{condiz}, \ref{con.mu}, with $\kappa=0$, for the unknowns $(V,\mu,m)$.
From
$$
x=\int_0^V \frac{dV}{\sqrt{\frac2m\left(E'-\Phi(V)\right)}} \ ,
$$
with $\Phi:=\Phi_0$, one gets
$$
x=\frac{4\sqrt{m}}{\sqrt{16\mu +\Vm^2}}F\left(\varphi\;\bigg\vert -\frac
{\Vm^2}{16\mu +\Vm^2}\right) \qquad \sin \varphi
=\frac{V}{\Vm} \ ,
$$
where $F$ is the incomplete elliptic integral of first
kind, and $E'=\Phi(\Vm)$. Using the definition of
elliptic sine, which is the inverse of such elliptic integral,
we get the solution of \ref{sdnl} (with $\kappa=0$) in the form
\begin{equation}
V(x)=\Vm \; {\rm sn}(\omega\; x\vert \mm) \ ,
\end{equation}
with
\begin{equation}
\label{omm}
\begin{split}
\omega=\frac{\sqrt{16\mu+\Vm^2}}{4m} \ ,
\\
\mm=-\frac{\Vm^2}{16\mu+\Vm^2} \ .
\end{split}
\end{equation}
Since the natural period of the elliptic sine is $4\KK$,
(\ref{condiz}) gives:
\begin{equation}
\omega\frac{2\pi}{n}=4\KK \quad
\Rightarrow \quad \omega_n=\frac{2\KK}{\pi}n \ .
\end{equation}
Now consider condition \ref{con.mu}: using properties of elliptic
functions we have
\begin{equation}
\mu=\frac38 \frac{\KK-\EE}{\mm \KK}\Vm^2 \ ;
\end{equation}
inserting in the second of \ref{omm}, we obtain \ref{EK}
which, as stated before, is the equation fixing the value of the
parameter $\mm$.
To complete the picture we give the value of the other
parameters involved. From \ref{con.en} we obtain a relation between
$E$ and $m$
\begin{equation}
m_n=\frac{9\pi^4}{128}\,\frac{-1}{(\mm^2+14\mm+1)\KK^5}
\;\,\frac{1}{n^4}\frac{E}{c^2} \ .
\end{equation}
>From this last equation and using the \ref{omm} we have
\begin{equation}
\left(\Vm^2\right)_n=\frac{9\pi^2}{2}\,\frac{\mm}{(\mm^2+14\mm+1)\KK^3}\;\,
\frac{1}{n^2}\frac{E}{c^2} \ ,
\end{equation}
and than also
\begin{equation}
\mu_n=\frac{9\pi^2}{32}\,\frac{-(1+\mm)}{(\mm^2+14\mm+1)\KK^3}\;\,
\frac{1}{n^2}\frac{E}{c^2} \ .
\end{equation}
\end{proof}
\finsection
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Extrema
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begsection{Extrema}
In this section we will show that, among all the critical point found,
there's only one extremum, while all the other points are saddle points.
%%%
\begin{proposition}
\fmed\ constrained to $\S_E$ has only one extremum, $V_1$ which is a
maximum; the other critical points $V_n,\ n\neq 1$ are saddle points.
\end{proposition}
%%%
%%%
\begin{proof}
We will show that all the critical points but
the first are saddle points. Then we will ensure the presence of a maximum
by variational methods.
First of all, we calculate the second differential of $\mfmed|_{\S_E}$
at the critical points: it is given by the following formula
$$
d^2\left(\mfmed\vert_{\S_E}\right)(V_n)(h,h)=\lambda d^2 h_\omega
(V_n)(h,h) + d^2 \mfmed (V_n)(h,h) \ ,
$$
where $\lambda$ is the lagrange multiplier at the critical point and
$h$ belongs to the tangent space to $\S_E$.
Thus we have
$$
d^2\left(\mfmed\vert_{\S_E}\right)(V_n)(h,h)=
c^2 \lambda\int_{-\pi}^{\pi}h_x^2(x)dx
+ \frac38\int_{-\pi}^{\pi}V_n^2(x)h^2(x)dx +
$$ $$
+\frac{3}{16\pi} \int_{-\pi}^{\pi}V_n^2(x)dx\int_{-\pi}^{\pi}h^2(x)dx +
\frac{3}{8 \pi}\left(\int_{-\pi}^{\pi}V_n(x)h(x)dx\right)^2,
$$
which can be expressed as an inner product $\langle h,A_nh\rangle
_{L^2}$, with
\begin{equation}
\label{AB}
\begin{split}
A_nh=&B_nh +\frac34\left[V_nh\right]V_n,
\\
B_nh=&m_nh_{xx}+\frac38\left(V_n^2+\left[V_n^2\right]\right)h.
\end{split}
\end{equation}
We must evaluate this second differential along directions lying on
the tangent space to the energy surface.
In the forthcoming computation we use the remark that if $i$ is an
even integer, and $j$ is an odd integer then $V_i$ is $L^2$ and $H^1$
orthogonal to $V_j$. So, having fixed $n$, consider a value of $i$ with
a different parity. One has
\begin{equation*}
\begin{split}
\langle V_i,A_nV_i\rangle_{L^2} = \langle V_i,
m_n V_{i\, xx} +\frac38\left(V_n^2+\frac{1}{2\pi}\int
V_n^2 \right) V_i +\frac{3}{8\pi}V_n\int V_nV_i \rangle_{L^2}
\\
= \left(\mu _n - \frac{m_n}{m_i}\mu _i \right)
\int V_i^2 +\frac38 \int V_n^2V_i^2 -\frac18\frac{m_n}{m_i}
\int V_i^4 +\frac{3}{8\pi}\left(\int V_nV_i\right)^2
\\
= \pi c_4^2 \frac{E^2}{c^4}\frac{1}{n^4}
\left[ 3(1+\mm)^2\frac{n^2-i^2}{i^2} + 4(1+5\mm+\mm^2) \right]
+\frac38 \int V_n^2V_i^2 \ ,
\end{split}
\end{equation*}
with
\begin{equation*}
c_4:=\frac{3\pi^2}{8}\,\frac{1}{(\mm^2+14\mm+1)\KK^3} \ .
\end{equation*}
When $i$ is sufficiently large the above expression is negative,
indeed $1+5\mm+\mm^2$ is easily seen to be negative (using the value
of $\mm$), and all the positive terms vanish as $i\to\infty$.
Concerning the non existence of maxima it is easy to see that for any
$n\geq2$ there are directions $h$ making the value of the second
differential positive. For even $n$ take $i=1$, and for odd $n\ne 1$
take $i=2$ using $3(1+\mm)^2+4(1+5\mm+\mm^2)>0$ one obtains that the
square bracket is positive.
\vskip15pt
Up to now we showed that, among all the critical points, only $V_1$
can be an extremum, in particular a maximum; by variational
techniques we now prove that a maximum actually exists, and therefore it
must coincide with $V_1$.
We exploit the fact that \fmed\ is positive and homogeneous, and can be
extended to a continuous functional on $L^\infty$.
Introduce the 0 degree homogeneous functional
$$
F(V):=E^\frac32 \frac{\mfmed(V)}{\left(\int V_x^2\right)^\frac32}
\; , \qquad\qquad V\in\Pt\setminus\{ 0\}.
$$
We have
$$
\sup_{V\in\Pt\setminus \{ 0\} } F = \sup \; F\vert_{_{\S_E}}
= \sup \; \mfmed\vert_{_{\S_E}} \; .
$$
>From Sobolev embedding theorem $H^1(\Omega )\subset C(\overline \Omega )$ it
follows $\sup F\equiv\sup \; f\vert_{_{\S_E}} <\infty$, while
from positivity of \fmed\ we have $\sup F>0$.
Let $\{ V_n\}\subset\Pt$ be a maximizing sequence, and $v_n\in\S_E$
be given by:
$$
v_n:=\sqrt{E} \frac{V_n}{\sqrt{\int V_{nx}^2}} \; ,
$$
which is also maximizing, since $F$ is homogeneous. $\{ v_n\}$ is
bounded in $H^1$, and thus there exists a subsequence (that we call
again $\{ v_n\}$) weakly
converging to a point $v$: $v_n \rightharpoonup v$.
By compact embedding we have $v_n
\buildrel{L^{\infty}}\over{\longrightarrow} v$.
Now, by construction, $\mfmed(v_n)=F(v_n)\rightarrow \sup F$; by
continuity of \fmed\ in $L^\infty$, $\mfmed(v_n)\rightarrow \mfmed(v)$, and so
$\mfmed(v)=\sup F>0$, which gives $v\ne 0$. It remains to prove that
$v$ satisfies the constraint.
>From weak lower semicontinuity of the $H^1$ norm it follows
$$
\left(\int v_x^2\right)^\frac32 \le \liminf
\left(\int v_{n\, x}^2\right)^\frac32=E^\frac32 \; .
$$
Since $v\not =0$ we also have
$$
\frac{E^{\frac{3}{2}}}{\left(\int v_x^2\right)^{\frac{3}{2}}} \ge
\limsup \frac{E^{\frac{3}{2}}}{\left(\int v_{x\,n }^2\right)
^{\frac{3}{2}}}=1 \; ,
$$
and remembering the definition of $F$
$$
F(v)=E^{\frac{3}{2}} \frac{f(v)}{\left(\int v_x^2\right)^{\frac{3}{2}}}
\ge f(v)=\sup F
$$
which implies $F(v)=\sup F$, and $\sqrt{\int v_x^2}=E$.
\end{proof}
%%%
\finsection
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Non degeneration
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begsection{Non degeneration}
Finally, to apply the theory of \cite{BN} we have to verify that the
second differential of $\mfmed|_{\S_E}$ is non degenerate, in the
directions transversal to the orbit of the linearized system, at
$V_1\equiv V$. We know that $V_1(x+s)$ is again a critical point at
the same level, so $V_{1x}$ is a null direction of the second
differential of the functional (remark that $A_1 V_{1x}=0$ because $V$
and $V_x$ are $L^2$ orthogonal and $B_1 V_{1x}=0$ being the derivative
of the equation defining $V$). We have to prove that this is the only
null direction. So we have to find all solutions $h\in T_{V_1}S_{E}$
of the equation
\begin{equation}
\label{nodeo}
0=d^2\left(\mfmed |_{\S_E}\right)(V)(h,h)\equiv
\left\langle
h,Ah\right\rangle \ ,
\end{equation}
with $A:=A_1$, and $A_1$ defined by \ref{AB} with $n=1$. It is easy to
see that \ref{nodeo} is equivalent to
\begin{equation}
\label{node1}
0=d^2\left(\mfmed\vert_{\S_E}\right)(V)(\tilde h,h)\equiv\left\langle
\tilde h,Ah\right\rangle \ , \quad \forall \tilde h\in T_{V_1}{\S_E}
\ ,
\end{equation}
which in turn is equivalent to the following equation for $h\in
T_{V_1}{\S_E}$ and $\alpha\in\Re$
\begin{equation}
\label{nodev}
Ah=-\alpha V_{xx}\ .
\end{equation}
We study equation \ref{nodev}
in the space $\Pt$ adding the constrain that the solutions must
belong to $T_{V_1}{\S_E}$, i.e. that its $H^1$ scalar product with $V$
must vanish or, more explicitly that
\begin{equation}
\label{orto}
[h_x V_{x}]=0\ .
\end{equation}
We will prove that the only solution of this problem is
$h\propto V_x$.
To find the solutions of \ref{nodev} we introduce a new variable $g$
defined by
$$
h=V_xg\ .
$$
Remark that $g$ could be discontinuous, but this will not cause any
problem. The equations we will write will be intended to hold at the
points $x$ such that $g$ is continuous and differentiable ($x\not
=\pm\pi/2$). In terms of $g$, \ref{nodev} takes the form
$$
m\partial_x\left(V_x^2g_x
\right)=\partial_x\left(-\frac38[VV_xg]V^2-\frac \alpha2V_x^2\right)\ ,
$$
from which
\begin{equation}
\label{n1}
m V_x^2g_x
=-\frac38[VV_xg]V^2-\frac \alpha2V_x^2+\gamma\ ,
\end{equation}
with a real $\gamma$.
In terms of $g$, eq.~\ref{orto} takes the form $[V_x^2g_x]=0$. So,
averaging \ref{n1}, we get
\begin{equation}
\label{vvxg}
-\frac38[VV_xg]=\frac\alpha2\frac{[V_x^2]}{[V^2]}-\frac{\gamma}{[V^2]}\ ,
\end{equation}
from which, inserting in \ref{n1} we obtain
$$
m g_x=\frac{\alpha}2\left( \frac{[V_x^2]}{[V^2]}\frac{V^2}{V_{x}^2}-1
\right) +\gamma\left(\frac1{V_x^2}-\frac{V^2}{[V^2]V_x^2} \right)\ .
$$
Remembering that $V=\Vm\sn(\omega x|\mm) $ and using known expressions
for the integrals of elliptic functions one obtains the general
solution of the above equation. This is given by
\begin{equation}
m g(x)=\xi + \frac{\alpha}2
\left[\frac{\mm^2+14\mm+1}{\omega(3+\mm)(1-\mm)^2}
\Big(A-B\Big) -x\right] + \gamma \left[\frac{(1+7\mm)A-\mm(7+\mm)B}
{\Vm^2 \omega^3 (1+\mm)(1-\mm)^2}\right]
\end{equation}
with
\begin{equation*}
\begin{split}
A&\equiv-\EE (\omega x) + (1-\mm)\omega x + \frac{\sn(\omega x|\mm)
\dn(\omega x|\mm)}{\cn(\omega x|\mm)} \ ,
\\
B&\equiv\EE (\omega x) - \mm\frac{\sn(\omega x|\mm)
\cn(\omega x|\mm)}{\dn(\omega x|\mm)} \ ,
\end{split}
\end{equation*}
and $\xi$ is a real parameter.
Imposing now the conditions of periodicity of $g$ and that of
orthogonality (see \ref{orto} and \ref{vvxg}) we get a linear homogeneous
system in the variables $\alpha$ and $\gamma$; the determinant of such
system is given by
$$
\KK^4\frac{(1+\mm)(255-3444\mm-32661\mm^2-27448\mm^3-6147\mm^4
-180\mm^5-7\mm^6)}{216} \ ,
$$
and this quantity is different from zero (remember that $\mm$, and
hence $\KK$, are real numbers fixed by \ref{EK}).
The conclusion is than that $\alpha$ and $\gamma$ must be zero, and the
only null direction is $h=\xi V_x$.
\begin{rem}
It is possible to prove the non degeneracy of all the other critical
points $V_n$ exactly in the same way, obtaining for the determinant
precisely the same expression, which turns out to be independent from $n$.
\end{rem}
\finsection
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Appendix
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begsection{Appendix}
\begin{proof}[Proof of lemma \ref{dopo}]
The idea is to show that, when $\kappa\neq0$, due to the
broken symmetry of the potential $\Phi_\kappa$, the average of the
function $V$ is no more zero.
We can write
\begin{equation}
\begin{split}
\left[V\right] &= \frac n\pi \int_{V_{{\rm min}}}^{V_{{\rm max}}}
\frac {V dV} {\sqrt{\frac 2m \left(E'-\Phi_\kappa(V)\right)}}
\\
&\propto \int _{s_m}^{s_M} \frac{s ds}{\sqrt{p(s,E',c,\mu)}} \ ,
\end{split}
\end{equation}
where we changed the variable of integration from $x$ to $V$ and where
we
denoted $p(s,E',\kappa,\mu)=E'-\Phi(s,\kappa,\mu)$, and again $p(s_m)=p(s_M)=0$.
Call this last integral $I(\kappa)$, and
observe that $I(-\kappa)=-I(\kappa)$, so we restrict to the case $\kappa>0$. We
restrict also to the case $E'>0$, because otherwise the inversion
points are both positive and the proof is finished.
Let's write the polynomial $p$ in the following way
\begin{equation}
\begin{split}
&E' + \kappa s - \frac{\mu^2}{2}s^2 - \frac{1}{32}s^4 =
\\
= & (s-s_m) (s-s_M) \left(\frac{E'}{s_m s_M} - \frac{s_m + s_M}{32}s
- \frac{1}{32}s^2 \right) \ ,
\end{split}
\end{equation}
and comparing the coefficient of $s$ and $s^2$ deduce the following relations:
\begin{equation}
\label{relc}
\kappa=-(s_m+s_M)\left(\frac{E'}{s_ms_M}+\frac{s_ms_M}{32}\right) \ ,
\end{equation}
\begin{equation}
\label{relmu}
-\frac{\mu^2}{2}-\frac{(s_m+s_M)^2}{32} = \left(\frac{E'}{s_ms_M}-
\frac{s_ms_M}{32}\right) \ .
\end{equation}
>From \ref{relc} you see that when $\kappa>0$, then $s_M>|s_m|$. Now look at
the other two roots of $p$
\begin{equation}
s_{3,4}=-\frac{s_m+s_M}{2} \pm \frac{\sqrt{(s_m+s_M)^2 +
\frac{128E'}{s_ms_M}}}{2} \ ,
\end{equation}
which are not real (use \ref{relmu}).
Now define
\begin{equation}
\begin{split}
\Omega &:= \frac{s_M+s_m}{2} \;>0 \qquad (\kappa>0)
\\
\omega &:= \frac{s_M-s_m}{2} \;>0
\\
\Delta &:= (s_m+s_M)^2 + \frac{128E'}{s_ms_M} \;<0
\end{split}
\end{equation}
and write $p$ in the following way
\begin{equation}
\begin{split}
p(s,E',\kappa,\mu)&=-(s-s_M)(s-s_m)
\left(s+\Omega + i\frac{\sqrt{-\Delta}}{2} \right)
\left(s+\Omega - i\frac{\sqrt{-\Delta}}{2} \right)
\\
p(\Omega\pm t,E',\kappa,\mu)&=(\omega^2-t^2)\,Q^\pm (t)
\\
Q^\pm (t):&=(2\Omega\pm t)^2-\frac\Delta 4
\end{split}
\end{equation}
Our integral takes now the form
\begin{equation}
\begin{split}
I(\kappa) &= \int _{s_m}^{s_M} \frac{s ds}{\sqrt{p(s,E',\kappa,\mu)}}
\\
&= \frac12\int _{-\omega}^{\omega} \left[
\frac{\Omega+t}{\sqrt{p(\Omega+t,E',\kappa,\mu)}} +
\frac{\Omega-t}{\sqrt{p(\Omega-t,E',\kappa,\mu)}} \right] \,dt
\\
&= \int _{0}^{\omega} \frac{dt}{\sqrt{\omega^2-t^2}\sqrt{Q^+Q^-}}
\left[t \left(\sqrt{Q^-}-\sqrt{Q^+}\right) +
\Omega \left(\sqrt{Q^-}+\sqrt{Q^+}\right)\right]
\\
&= \int _{0}^{\omega} \frac{\Omega \,w(t)\,dt}
{\sqrt{\omega^2-t^2}\sqrt{Q^+Q^-}\left(\sqrt{Q^-}+\sqrt{Q^+}\right)}
\end{split}
\end{equation}
with
\begin{equation}
\begin{split}
w(t):&= -2t^2 - \frac\Delta2 + 8\Omega^2 + 2\sqrt{Q^+Q^-}
\\
&= -2t^2 - \frac\Delta2 + 8\Omega^2 +
2\sqrt{\left(t^2-\frac\Delta4\right)^2+2\,\Omega^2\,s(t)}
\\
s(t):&=-4t^2-\Delta+8\Omega^2
\end{split}
\end{equation}
If we show that $w(t)>0$, then the proof is finished. One has
\begin{equation}
s(t)\ge s(\omega)= 4\left[s_Ms_m - \frac{32E'}{s_Ms_m}\right]>0 \ ,
\end{equation}
using again \ref{relmu}. Thus
\begin{equation}
\begin{split}
w(t) &> -2t^2 - \frac\Delta2 + 8\Omega^2 +
2\sqrt{\left(t^2-\frac\Delta4\right)^2}
\\
&\geq - \Delta + 8\Omega^2 > 0
\end{split}
\end{equation}
\end{proof}
\finsection
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Bibliography
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\vskip45pt
\end{document}