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\title{Quasi periodic breathers in Hamiltonian lattices with symmetries}
\author{Dario Bambusi\footnote{E-mail: bambusi@mat.unimi.it}, Davide Vella \
\\ {\it
Dipartimento di Matematica, Universit\'a di Milano} \\ {\it via
Saldini 50, 20133 Milano (Italy)} }
\date{\giorno}
\maketitle
\noindent
{\bf Summary.} {We prove existence of quasiperiodic breathers in
Hamiltonian lattices of weakly coupled oscillators having some
integrals of motion independent of the Hamiltonian. The proof is
obtained by constructing quasiperiodic breathers in the anticontinuoum
limit and using a recent theorem by N.N. Nekhoroshev \cite{N} as
extended in \cite{BG} to continue them to the coupled
case. Applications to several models are given.
\bigskip\bigskip\bigskip\bigskip
\section{Introduction.}
Existence of breathers (i.e. time periodic space localized solutions)
in infinite Hamiltonian lattices of weakly coupled oscillators has
been proved by Mac Kay and Aubry \cite{MA} (see also
\cite{B96,SM}) starting from the anticontinuoum limit.
Existence of (time) quasiperiodic breathers has also been
investigated, but such objects are expected not to exist in generic
models due to the presence of linear combinations of the frequencies
which fall in the continuous spectrum. As pointed out in \cite{KW1,KW2}
such ``resonances'' are expected to lead to a decay in time of
``quasiperiodic breather like solution''.
Nevertheless existence of quasiperiodic breathers has been proved in
some special models: in particular (1) quasiperiodic breathers with
two frequencies have been constructed in two models which have a non
trivial symmetry, namely the discrete nonlinear Schr\"odinger equation
\cite{JA} and the adiabatic Holstein model \cite{A}; (2) KAM
theory has been used to construct breathers with an arbitrary number
of independent frequencies in the case where the oscillators are
coupled via a first neighborhood potential which is at least cubic
\cite{Y}.
In the present paper we start from the anticontinuoum limit and
construct quasi periodic breathers in Hamiltonian systems having some
integrals of motion independent of the Hamiltonian. Our approach is
based on a theorem by N.N. Nekhoroshev \cite{N} that was recently
reformulated in a form suitable for our applications in \cite{BG}. This
theorem is an extension of the Poincar\'e--Lyapunov theorem of
continuation of periodic orbits to the case where some integrals of
motion independent of the Hamiltonian exist. It allows to continue
families of invariant tori to perturbations of the original system,
and to describe precisely the dynamics on them.
Applying this theorem
one can reproduce the known results on the DNLS and on the Holstein
model, and more generally obtain quasiperiodic breathers in any model
with symmetries. Here we will explicitly reconstruct the
quasiperiodic breathers of \cite{JA} for the DNLS, and we will construct a
new type of quasiperiodic breathers in the Holstein model. As an
example possessing 3--frequencies quasiperiodic breathers we also
study the vector discrete nonlinear Schr\"odinger equation.
We think that similar results could be obtained also by the approach
of \cite{M,WR}.
We point out that the present approach allows also to give more
details on the breather's dynamics. In particular breathers will
appear as invariant tori, which form smooth families with as many
parameters and as many dimensions as the number of independent
integrals of motion of the system. In many cases the frequencies can
be used to parametrize the family (in particular this is true for the
models studied here), so, in particular the set of the tori on which
the motion is dense has full measure.
We close this section by mentioning the very recent works \cite{AKK,J}
in which the authors prove existence of breathers in FPU type systems
by new approaches. We did not investigate the possibility of finding
quasiperiodic breathers in systems with symmetry using such
approaches.
\noindent {\it Acknowledgement.} We thank Andrea Carati for suggesting
the coordinate transformation \ref{cambio1}, \ref{cambio2}.
\section{The abstract theorem}
\subsection{Statement}
We state here Nekhoroshev's theorem \cite{N} in the improved form
of \cite{BG}. For the proof see \cite{BG}.
Let $(\B,\Omega )$ be a (weakly) symplectic Banach space (with
symplectic form $\Omega$). Let $\Fb^{\epsilon} := \{ H_1^{\epsilon} ,
... , H_s^{\epsilon} \}$ be $s$ real functions on $\B$, defined for
$\epsilon$ in a neighbourhood $E$ of zero.
Consider the Hamiltonian vector field $X_i^\epsilon$ of $H^\epsilon_i$
(defined as the unique vector field such that $dH^\epsilon_i
X=\Omega(X^\epsilon_i,X)$ $\forall X\in\B$).
We assume that there exists an $s$ dimensional compact submanifold
$\Lambda\subset\B$, {\it invariant under the flows of the fields
$X_i^0\equiv X_i^\epsilon\big\vert_{\epsilon=0}$ for all $i=1,...,s$.}
We also assume that
\begin{itemize}
\item[i] the functions $H_i^\epsilon$ and the vector fields
$X^\epsilon_i$ are of class $C^\infty$ in $\U\times E$, where $\U$
is a neighborhood of $\Lambda$,
\item[ii] the vector fields $X^\epsilon_i$ are
independent on $\Lambda$
\item[iii] the functions $\Fb^{\epsilon}$ are in involution in $\U$,
namely
$$
\left\{H_i^\epsilon,H^{\epsilon}_j\right\}\equiv
dH_i^{\epsilon}X_j^{\epsilon}=0\ ,\quad \forall i,j=1,...,s
$$
\end{itemize}
Remark that by our assumptions the manifold $\Lambda$ is diffeomorphic
to an $s$ dimensional torus.
Then we have to add the nonresonance assumption which extends that by
the Poincar\'e Lyapunov theorem to the quasiperiodic case. To this end
it is useful to restrict to the case where there exists a system of
canonical coordinates $(J,\psi,p,q)$ with $\psi\in\toro^s$ and
$(J,p,q)$ in an open set of a suitable Banach space, in which the
manifold $\Lambda$ has equation $p=q=J=0$, and the functions
$H^0_i\equiv H^\epsilon_i\big\vert_{\epsilon=0}$ take the form
\begin{equation}
\label{tilde}
H^0_i=\sum_{j=1}^{s}\omega^{(i)}_j\, J_j
+\sum_{j\geq 1}\nu^{(i)}_j\frac{p_j^2+q_j^2}2 + \tilde H^0_i
\end{equation}
where $\tilde H^0_i$ denotes a function with the property that the
coefficients of its Taylor expansion in $(J,p,q)$ are at least
quadratic in $J,p,q$ if they depend on $J$, while coefficients
independent of $J$ are at least cubic in $p,q$.
As proved by Kuksin \cite{Kuk}, such coordinates exist in quite
general situations.
Consider the matrix
$A\equiv\left\{\omega_j^{(i)}\right\}_{j=1,...,s}^{i=1,...,s}$
constituted by the frequencies of motion in the invariant torus
$\Lambda$, and the matrix $B\equiv\left\{\nu_j^{(i)}\right\}_
{j=1,...,\infty}^{i=1,...,s}$ constituted by the frequencies of small
oscillations in the transversal directions to the invariant torus.
We will denote by $A^{(k;j)}$ the matrix obtained from $A$
by substituting its $k$-th column with the $j$-th column
of $B$. In the forthcoming theorem we will denote by $\left|A\right|$
the determinant of the matrix $A$.
\begin{theorem}
\label{nek}
In the above situation, assume that there exists
$n\equiv(n_1,n_2,...,n_s)\in {\bf Z}^s$ and
$\gamma>0$ such that
\begin{equation}
\label{simple}
\left|\sum_{k=1}^s \ n_k \, | A^{(k;j)} | \ -\ N \ |A|\right|\geq \gamma \ \
\forall N \in {\bf Z} \ ,\quad \forall j=1,...,\infty\ .
\end{equation}
Then there exists $\epsilon_*>0$, such that, for all $\epsilon\in
E_0$, $E_0:=(-\epsilon_*,\epsilon_*)$, the following holds true:
\noindent {\rm (1)} there exists a family of symplectic submanifolds
$N^\eps $, of dimension $2s$, which is the union of $s$--dimensional
tori $\toro^\epsilon_{\beta}$, with $\beta\in\Re^s$ small. For each
$\eps \in E_0$, the tori $\toro_\b^\eps$ are invariant under the flow
of $X^\epsilon_i$; the tori $\toro_\b^\eps$ and the manifold $N^\eps$
depend in a $C^\infty$ way on $\eps \in E_0$; one has
$\Lambda=\toro^0_0$.
\noindent {\rm (2)} There exist symplectic action angle coordinates
$(I_1^\eps,...,I_s^\eps ; \phi_1^\eps , ... , \phi_s^\eps )$ in
$N^\epsilon$, such that the functions $H_i^\eps \big\vert_{N^\eps} $
depend only on the actions $I_j^\epsilon$, namely
$$
H_i^\eps
\big\vert_{N^\epsilon}\equiv H_i^\eps
\big\vert_{N^\epsilon} (I_1^\eps ,..., I_s^\eps )\ ,
$$
with
$i=1,...,s$. The coordinates $(I^\eps ; \phi^\eps )$ depend in a
$C^\infty$ way on $\eps \in E_0$, and so do the functions $H_i^\eps
\big\vert_{N^\eps}(I_1^\epsilon,...,I_s^\epsilon)$.
\end{theorem}
We remark that while statement (1) ensures the existence of the
invariant tori and describes their shape, statement
(2) describes completely the motion on the invariant tori. Indeed, in
the coordinates $I,\phi$ (where we drop $\epsilon$) the equations of
motion of $H_i^\epsilon$, in $N^\epsilon$, have the form
$$
\dot I_k=0\ ,\quad \dot \phi_k=\frac{\partial H_i^\epsilon}{\partial
I_k}(I)\ ,
$$
which shows that on each of the tori the motion is
quasiperiodic. Moreover, the fact that the invariant tori, the
coordinates, and the Hamiltonians depend in a smooth way on $\epsilon$
ensures that the frequencies of the perturbed system are close to
those of the unperturbed one. Moreover, if for example one has (for a
fixed $i$)
$$
\left|\frac{\partial^2 H_i^0}{\partial I_l\partial I_k}\right|\not =0
$$
then the same holds for the perturbed system. In particular in
this case one has that the action to frequency map is one to one, and
therefore the tori can be parametrized by the frequencies.
\subsection{Idea of the proof}
First we recall the scheme of the proof of the Poincar\'e--Lyapunov theorem
(corresponding to $s=1$). In this case the manifold $\Lambda$ reduces
to a periodic orbit. Consider a Poincar\'e section and the
corresponding Poincar\'e map $P^\epsilon$; then periodic orbits of
$X_1^{\epsilon}$ are found as solutions of $P^\epsilon(x)=x$; such
solutions are constructed by using the implicit function theorem. The
standard condition that the Floquet multiplier 1 have multiplicity 2
ensures that the implicit function theorem applies.
In the case $s\geq2$ one tries to mimic the above proof. To define
the Poincar\'e map remark that, by the first step of the proof of
Arnold's theorem, there exists a function
$K^\epsilon(p,q):=\sum_{j}\alpha_{j}H_j^\epsilon(p,q)$ with the
property that all solution of the Hamilton equations of $K^0$ with
initial data in $\Lambda$ are periodic with a definite period
(actually there exist $s$ different functions with this property).
Then define a Poincar\'e section of one of these periodic orbits and
the corresponding Poincar\'e map $P^\epsilon$. Due to the symmetries
of the problem there exists a local foliation which is invariant under
the flow of the $X_i^\epsilon$, and moreover it turns out that $
P^\epsilon$ defines a natural map $\tilde P^\epsilon$ from one leaf of
the foliation to an other. It turns out that fixed points of $\tilde
P^\epsilon$ give rise to invariant tori. To find such fixed points one
uses again the implicit function theorem: the corresponding
invertibility condition is ensured by assuming that 1 is an isolated
eigenvalue of multiplicity $2s$ of the Floquet operator. Finally it
can be proved that such a condition is equivalent to \ref{simple}.
\section{Applications}
All the applications will deal with systems of the form
\begin{equation}
\label{gen}
H_1^{\epsilon}:=
\sum_{k\in\ra}H_{os}(P_k,Q_k)+\epsilon\sum_{k\in\ra}F(P_k-P_{k-1}
,Q_{k}-Q_{k-1})\
,
\end{equation}
where $(P_k,Q_k)\in\Re^{2j}$ are canonically conjugated variables, and
$H_{os}$ is the Hamiltonian of the on site system that we will assume
to have $j$ degrees of freedom. In our cases we will have $j=1$ or 2.
The phase space is defined formally as follows: Fix $\beta>0$ and
define the Banach space $\ell_\beta$ of the sequences $P=\{P_k\}$,
$P_k\in\Re^j$, such that
\begin{equation}
\label{ell}
\norma{P}_\beta:=\sup_{k\in\ra}\norma{P_k}e^{\beta|k|}<\infty
\end{equation}
where $\norma{P_k}$ denotes the euclidean norm.
The phase space is $(P,Q)\in\ell_\beta\times\ell_\beta$.
The system \ref{gen} will also have $s$ independent integral of motion
in involution given by $H^\epsilon_1$ and by $s-1$ more functions
$H_2,...,H_s$, that will turn out to be independent of $\epsilon$. In
our cases we will have $s=2$ or 3.
Then we will proceed by first defining the manifold $\Lambda$ for the
different models and then introducing the coordinates such that the
system takes the form \ref{tilde}; we will denote by
$\omega_1,...,\omega_s$ the frequencies of motion of the unperturbed
breather. We point out that in our cases the unperturbed quasiperiodic
breather will be concentrated at the sites $1,...,s$ of the
lattice. Then we will write down explicitly the nonresonance condition
\ref{simple}. Actually in all the cases that we will consider it takes
a quite simple form very similar to the nonresonance condition of the
Poincar\'e--Lyapunov theorem. To obtain such a simple form we will
compute explicitly the determinants involved in condition
\ref{simple} and simplify as much as possible the so obtained
expression.
\begin{proposition}
\label{app}
Assume that the system satisfy a suitable nonresonance condition that
depends on the model (see \ref{dnls1}, \ref{hol} and \ref{nr.3}
below); then there exists $\epsilon_*$ and a function
$\delta_*=\delta_*(\epsilon)$ defined for $|\epsilon|<\epsilon_*$,
such that for $|\epsilon|<\epsilon_*$ there exists a 2$s$--dimensional
manifold $N_\epsilon$ invariant under the flows of
$X^\epsilon_1,...,X_s$; moreover one has
$$
N_{\epsilon}=\bigcup_{|\delta_i|<\delta_*,i=1,...,s}
\toro^\epsilon_{\delta_1,...,\delta_s}
$$
with $\toro^\epsilon_{\delta_1,...,\delta_s}$ an $s$--dimensional torus
invariant under flow of $X^\epsilon_1$,..., $X_s$. On
$\toro^\epsilon_{\delta_1,...,\delta_s}$ the dynamics of $X_1^\epsilon$
(i.e. of the model we are interested in) is quasiperiodic with the
frequencies
$$
(\omega_{1}+\delta_1,...,\omega_{s}+\delta_s)\ ,
$$
The tori $\toro^\epsilon_{\delta_1,...,\delta_s}$ depend smoothly on
$\delta_i$ and on $\epsilon$. In particular
there exists a constant $C$ such that, for all points in
$N_\epsilon$ one has
\begin{equation}
\label{sti}
\norma{P_k}+\norma{Q_k}< C\epsilon e^{-\beta |k|} \ ,\quad\forall
k\not=1,...,s
\end{equation}
\end{proposition}
We point out that equation \ref{sti} is a consequence of smooth
dependence of the manifold $N_\epsilon$ on $\epsilon$ in the topology
of the phase space. The fact that it is possible to parameterize the
tori by the frequencies is a consequence of the fact that in the case
$\epsilon=0$ the application from the actions to the frequencies will
turn out to be one to one in all the models we are interested in.
\subsection{Discrete nonlinear Schr\"odinger equation}
Consider the discrete nonlinear Schr\"odinger equation
\begin{equation}
\label{dnls}
i\dot\psi_k=\psi_k\left(\frac{|\psi_k|^{2}}2\right)^{n-1}+\epsilon
\left[\left(\psi_{k+1}-\psi_k\right)+ \left( \psi_{k-1}-\psi_k
\right)\right]\ ,\quad k\in{\bf Z}
\end{equation}
where $n\geq2$ is an arbitrary integer. The Hamiltonian is
\begin{equation}
\label{dnls2}
H_1^\epsilon=\sum_{k\in{\bf Z}}\frac1n \left(\frac{p_k^2+q_k^2}2\right)^n
+\epsilon
\sum_{k\in{\bf Z}} \frac{(q_{k+1}-q_k)^2+(p_{k+1}-p_k)^2}2\ ,
\end{equation}
with $p_k+iq_k=\psi_k$. The second integral of motion is given by
$$
H_2(p,q):=\sum_{k\in{\bf Z}}\frac{p_k^2+q_k^2}2\ .
$$
In analogy with the standard Schr\"odinger such a quantity can be
called electron probability.
Fix positive $\omega_1,\omega_2$, then the manifold $\Lambda$ is
defined by
$$
\Lambda:=\left\{ \frac{p_1^2+q_1^2}2=\omega_1^{1/(n-1)} \ ,\quad
\frac{p_2^2+q_2^2}2=\omega_2^{1/(n-1)} \ ,\quad p_k=q_k=0\
,\quad\forall k\not=1,2 \right\}
$$
and
introduce action variables at the sites 1,2 by
$I_1=(p_1^2+q_1^2)/2,I_2=(p_2^2+q_2^2)/2$, and the corresponding angles.
To introduce the coordinates we need in order to apply theorem
\ref{nek} define
$J_1:=I_1-\omega_1^{1/(n-1)}$, and $J_2:=I_2-\omega_2^{1/(n-1)}$ so
that the Hamiltonians take the form
$$
H_1^0=\omega_{1}J_1+\omega_{2}J_2+\tilde H_1\ ,\quad
H_2=J_1+J_2+\sum_{k\not=1,2}\frac{p_k^2+q_k^2}2
$$
with $\tilde H_1$ having the same meaning as in \ref{tilde}.
We remark that here the manifold $N^\epsilon\big\vert_{\epsilon=0}$ is
just the phase space of the first two oscillators, and the restriction
of $H_1^0$ to $N^0$ is simply given by
$$
\frac{I_1^n}n+\frac{I_2^n}n\ .
$$
\begin{lemma}
The nonresonance condition \ref{simple} takes here the form
\begin{equation}
\label{dnls1}
\frac{\omega_{1}}{\omega_{2}-\omega_{1}}\not\in\ra
\end{equation}
\end{lemma}
\sp {\bf Proof.}
The
matrices $A$ and $B$ are given by
$$
A:=\pmatrix{\omega_{1} &\omega_{2}\cr 1 &1 \cr}\ ,\quad B:=\pmatrix{
0 & 0 & 0 & ...
\cr
1 & 1& 1 & ...\cr}
$$
from which one has
$$
A^{(1,j)}=\pmatrix{0 & \omega_{2} \cr 1 &1 \cr}\ ,\quad
A^{(2,j)}=\pmatrix{\omega_{1} & 0 \cr 1 &1 \cr}\ ,\quad \forall j\geq 1
$$
and thus the nonresonance condition \ref{simple} takes
the form `there exist $(n_1,n_2)\in\ra^2$ such that
$$
-n_1\omega_{2}+n_2\omega_{1}\not= N(\omega_{1}-\omega_{2})\ ,
$$
for all $N\in\ra$'. This can be rewritten as
$$
(-n_1+n_2)\omega_2+n_2(\omega_1-\omega_2)\not= N(\omega_{1}-\omega_{2})\ ,
$$
from which it is evident that the second term at left hand side
(l.h.s.) does not affect the condition and can be dropped. So the
condition is equivalent to
`there exist $n\in\ra$ such that
\begin{equation}
\label{po}
n\frac{\omega_{2}}{\omega_{1}-\omega_{2}}\not= N\ ,
\end{equation}
for all $N\in\ra$', and in turn this is equivalent to
\ref{dnls1},
since, in case the fraction at l.h.s. of \ref{po} is an integer,
the l.h.s. of \ref{po} is an integer for any choice of $n$, while in
case the fraction is not an integer, just choose $n=1$. So equation
\ref{dnls1} is the wanted nonresonance condition under which the
theorem \ref{app} applies. \EOP
So we have that proposition \ref{app} holds for DNLS.
In this case the quasiperiodic breather is a solution in which the
electron probability is essentially concentrated at two lattice cites.
\subsection{Adiabatic Holstein model}
The equations of motion of the adiabatic Holstein model \cite{A} are given by
$$
-i\dot\psi_{i}= -q_i\psi_{i}-\epsilon[(\psi_i-\psi_{i-1})+(\psi_i-\psi_{i+1})]
$$
$$
\ddot q_i=-\omega_0^2q_i+|\psi_{i}|^2
\ .\qquad i\in\ra
$$
which are Hamiltonian with Hamiltonian function
$$
H^\epsilon_1\equiv H:=\sum_{i}H_{os}(p_i,q_i,x_i,y_i)+\frac12\epsilon
\sum_{i} \left[(x_{i}-x_{i-1})^2+ (y_{i}-y_{i-1})^2\right] \ ,
$$
where
$$
H_{os}(p_i,q_i,x_i,y_i):=\frac{p_i^2+\omega_0
^2q_i^2}2+q_i\frac{ x_{i}^2+y_{i}^2}2\ ,
$$
$(p_k,q_k)$, $(x_{i},y_{i})$ are canonically conjugated
variables, and one has $\psi_j=x_j+iy_j$.
The additional integral of motion is
$$
H_2:=\sum_{i}\frac{x_{i}^2+y_{i}^2}2\ .
$$
To construct the manifold $\Lambda$
we proceed as follows:
Following \cite{B99} we
introduce action angle variables for $H_{os}$ at the sites 1,2 by
first defining the variables
$(I_{i},\phi_{i})$ by
$$
x_{i}=\sqrt{I_{i}}\cos\phi_{i}\ ,\quad
y_{i}=\sqrt{I_{i}}\sin\phi_{i}
\ ;\quad i=1,2
$$
and then performing the canonical transformation
$$
\xi_i=q_i+\frac{I_i}{\omega_0^2}\ ,\quad \eta_i=p_i\ ,\quad I_i'=I_i,\quad
\phi_i'=\phi_i+\frac{\eta_i}{ \omega_0^2} \ ,\quad i=1,2
$$
which gives $H_{os}$ the form
$$
\frac{\eta_i^2+\omega_0
^2\xi_i^2}2-\frac1{2\omega_0^2}I_i^2
$$
where we omitted the prime from $I$.
Fix now two positive frequencies $\omega_1$ and $\omega_2$, and define
$$
\Lambda:=\left\{ I_1=\omega_1\omega_0^2\ ,\quad
I_2=\omega_2\omega_0^2\ ,\quad \xi_i=\eta_i=p_k=q_k=0\ ,\quad \forall
i=1,2,\ k\not=1,2\right\}
$$
Defining $J_1:=I_1-\omega_1\omega_0^2$ and
$J_2:=I_2-\omega_2\omega_0^2$ one has that the Hamiltonians of the
system take the form we need, namely
$$
H_1^0=-\omega_{1}J_1-\omega_{2}J_2+\sum_{i=1,2}\frac{\eta_i^2+\omega_0
^2\xi_i^2}2+\sum_{i\not=1,2}\frac{p_i^2+\omega_0
^2q_i^2}2 +\tilde H_1
$$
$$
H_2=J_1+J_2+\sum_{i\not=1,2}\frac{x_{i}^2+y_{i}^2}2\ .
$$
\begin{lemma}
The nonresonance condition \ref{simple} takes here the form
\begin{equation}
\label{hol}
\frac{\omega_{1}}{\omega_{2}-\omega_{1}}\not\in\ra\ {\rm and }\
\frac{\omega_0}{\omega_{2}-\omega_{1}}\not\in\ra\
\end{equation}
\end{lemma}
\sp {\bf Proof.} One has
$$
A=\pmatrix{-\omega_{1} & -\omega_{2}\cr 1 &1 \cr}\ , \quad
B:=\pmatrix{ \omega_0 &\omega_0 &\omega_0&0& \omega_0&0&\omega_0&0&...
\cr
0&0&0&1&0&1&0&1\cr
}
$$
where the first two columns of the matrix $B$ refer to the variables
$\xi,\eta$ at the lattice sites 1,2. The $2i-3$ column refers to the
variables $(p_i,q_i)$ $(i\geq3)$, and the $2i-2$ columns refer to the
variables $(x_i,y_i)$ $(i\geq 3)$. We thus obtain the matrices
$A^{(k,j)}$, and from them the nonresonance conditions which have the
form `there exists $(n_1,n_2)\in\ra^2$ such that
$$
m(\omega_{2}-\omega_{1})\not= n_1\omega_0-n_2\omega_0\ ,\quad
m(\omega_{2}-\omega_{1})\not= n_1\omega_{2}-n_2\omega_{1}\ ,
$$
for all $m\in\ra$'. In turn this is equivalent to the nonresonance
conditions \ref{hol}.
\EOP
The kind of breathers we constructed here are new and consist of
solutions in which the electron probability is essentially
concentrated at two lattice sites, and the oscillators are at
rest. Notice that the rest position of the oscillators at the sites
where the breather is localized are translated with respect to the
unperturbed ones.
\subsection{Vector discrete nonlinear Schr\"odinger equation}
Consider the vector DNLS equation whose equations of motion are
$$
i\dot
u_k=u_k\left(\frac{|u_k|^2+|v_k|^2}2\right)^{n-1}+\epsilon
\left(u_{k-1}+u_{k+1}-2u_k
\right)
$$
$$
i\dot
v_k=v_k\left(\frac{|u_k|^2+|v_k|^2}2\right)^{n-1}+\epsilon
\left(v_{k-1}+v_{k+1}-2v_k
\right)
$$
with $k\in\ra$. Introducing the real and imaginary parts of $u_k,v_k$
as new variables, namely
$$
u_k=x_{k,1}+iy_{k,1}\ ,\quad v_k=x_{k,2}+iy_{k,2}
$$ one sees that the system is Hamiltonian with
Hamiltonian function
$$
H_1^\epsilon=\sum_{k\in\ra}\frac1n\left(\frac{x_{k,1}^2+y_{k,1}^2}2+
\frac{x_{k,2}^2+y_{k,2}^2}2\right)^{n}
$$
$$
+\epsilon\frac12\sum_{k\in\ra}\left[(x_{k,1}-x_{k-1,1})^2+(y_{k,1}-y_{k-1,1})^2
+(x_{k,2}-x_{k-1,2})^2+(y_{k,2}-y_{k-1,2})^2\right]\ .
$$
The additional integrals of motion are due to the phase shift
symmetries, and to the rotation invariance in the plane of $u$ and
$v$. They are given by
$$
F_{2}=\sum_{k\in\ra}\left(\frac{x_{k,1}^2+y_{k,1}^2}2\right)\ ,\quad
F_3:=\sum_{k\in\ra}\left(\frac{x_{k,2}^2+y_{k,2}^2}2\right)\ ,
$$
and by the angular
momentum
$$
F_4:=\sum_{k\in\ra}\left(x_{k,1}y_{k,2}-x_{k,2}y_{k,1}\right)\ ,
$$
which are independent but not in involution. As functions independent
and in involution on which we will base our construction we choose
$$
H_2:=F_2+F_3\ ,\quad H_3:=F_4\ .
$$
Obviously other choices are possible, and they would lead to different
kinds of breathers. For example one could choose
$$
H_2:=F_2\ ,\quad H_3:=F_3\ ,
$$
and obtain a kind of breathers that exist also in the anisotropic
vector DNLS. Our choice is motivated by the fact that a construction
very similar to ours can be performed also in other interesting
models, like an infinite lattice of three dimensional oscillators
interacting via a spherically symmetric potential.
In order to continue the analysis it is useful to
perform the following change of variables
\begin{equation}
\label{cambio1}
p_{1,k}:=\frac1{\sqrt2}\left({x_{k,1}}+y_{k,2}
\right)\ ,\quad
q_{k,1}=
\frac1{\sqrt2}\left({x_{k,2}}-y_{k,1}
\right)
\end{equation}
\begin{equation}
\label{cambio2}
p_{2,k}:=\frac1{\sqrt2}\left({x_{k,2}}+y_{k,1}
\right)\ ,\quad
q_{k,2}=
\frac1{\sqrt2}\left({x_{k,1}}-y_{k,2}\right)
\end{equation}
the functions $H_i^\epsilon$, for $\epsilon=0$ take the form
$$
H_1^0=\sum_{k\in\ra}\frac1n\left(\frac{p_{k,1}^2+q_{k,1}^2}2+
\frac{p_{k,2}^2+q_{k,2}^2}2\right)^{n}
$$
$$
H_2=\sum_{k\in\ra}\frac{p_{k,1}^2+q_{k,1}^2}2+
\frac{p_{k,2}^2+q_{k,2}^2}2
$$
$$
H_3=\sum_{k\in\ra}\frac{p_{k,1}^2+q_{k,1}^2}2-
\frac{p_{k,2}^2+q_{k,2}^2}2
$$
To define $\Lambda$ we fix arbitrary positive quantities $\omega_{1},\omega_{2},
\omega_{3}$ and put
$$
\Lambda:=\left\{ \frac{p_{1,1}^2+q_{1,1}^2}2=\omega_{1}^{1/(n-1)}\
,\quad
\frac{p_{2,1}^2+q_{2,1}^2}2=\omega_{2}^{1/(n-1)}\ ,\quad
\frac{p_{3,2}^2+q_{3,2}^2}2=\omega_{3}^{1/(n-1)}\ ,\right.
$$
$$
\left. p_{k,j}=q_{k,j}=0\ {\rm otherwise} \
\right\}
$$
Finally we introduce action angle variables by putting
$$
I_1:= \frac{p_{1,1}^2+q_{1,1}^2}2\ ,\quad I_2:=
\frac{p_{2,1}^2+q_{2,1}^2}2\ ,\quad I_3:= \frac{p_{3,2}^2+q_{3,2}^2}2\ ,
$$
and define
$$
J_i:=I_i-\omega_i^{1/(n-1)}\ ,\quad i=1,2,3\ ,
$$
so that the three independent integrals of motion take the form
$$
H_1^0=\omega_{1}J_1+\omega_{2}J_2+\omega_{3} J_3 +
\omega_{1}\frac{p_{1,2} ^2+q_{1,2}^2}2+\omega_{2}\frac{p_{2,2} ^2+
q_{2,2}^2}2 +\omega_{3}\frac{p_{3,1} ^2+q_{3,1}^2}2+\tilde H_1
$$
$$
H_2=J_1+J_2+J_3+\frac{p_{1,2} ^2+q_{1,2}^2}2+\frac{p_{2,2}
^2+q_{2,2}^2}2
+\frac{p_{3,1} ^2+q_{3,1}^2}2+
\sum_{k\not=1,2,3} \frac{p_{k,1} ^2+q_{k,1}^2}2+\frac{p_{k,2}
^2+q_{k,2}^2}2
$$
$$
H_3=J_1+J_2-J_3-\frac{p_{1,2} ^2+q_{1,2}^2}2-\frac{p_{2,2}
^2+q_{2,2}^2}2
+\frac{p_{3,1} ^2+q_{3,1}^2}2+
\sum_{k\not=1,2,3} \frac{p_{k,1} ^2+q_{k,1}^2}2-\frac{p_{k,2}
^2+q_{k,2}^2}2
$$
Then one can apply our theory and obtain a family of three dimensional
invariant tori continuing to the coupled case the manifold
$\Lambda$. In particular one has
\begin{lemma}
The nonresonance condition \ref{simple} takes here the form
\begin{equation}
\label{nr.3}
\frac{\omega_{1}-\omega_{3}}{\omega_{1}-\omega_{2}}\not\in\ra
\ ,\quad
\frac{\omega_{1}}{\omega_{1}-\omega_{2}}\not\in\ra
\ ,\quad
\frac{\omega_{3}}{\omega_{1}-\omega_{2}}\not\in\ra\ .
\end{equation}
\end{lemma}
\sp {\bf Proof.}
A long but straightforward computation (just write down the
determinants and compute them) shows that the nonresonance
condition has here the form ``there exist $(n_1,n_2,n_3)\in\ra^3$ such
that
$$
n_1(\omega_{3}-\omega_{1})+n_2(\omega_{1}-\omega_{3})+
n_3(\omega_{2}-\omega_{1})\not= N(\omega_{2}-\omega_{1})
$$
$$
n_1(\omega_{3}-\omega_{2})+n_2(\omega_{2}-\omega_{3})+
n_3(\omega_{2}-\omega_{1})\not= N(\omega_{2}-\omega_{1})
$$
$$
n_1(\omega_{2}-\omega_{3})+n_2(\omega_{3}-\omega_{1})+
\not= N(\omega_{2}-\omega_{1})
$$
$$
n_1\omega_{3}-n_2\omega_{1} \not= N(\omega_{2}-\omega_{1})
$$
$$
n_1\omega_{3}-n_2\omega_{3} \not= N(\omega_{2}-\omega_{1})
$$
for all $N\in\ra$'. Introducing the variable
$\mu_1:=\omega_{2}-\omega_{1}$ in order to eliminate $\omega_{2}$,
remarking that the last terms at l.h.s. of the first two equations
are inessential and denoting $m:=n_1-n_2$
one sees that the above equations are equivalent to
$$
m(\omega_{3}-\omega_{1})\not = N\mu_1
$$
$$
m(\omega_{3}-\omega_{1}-\mu_1)\not = N\mu_1
$$
$$
-(n_2+m)(\omega_{3}-\omega_{1}-\mu_1)+n_2(\omega_{3}-\omega_{1})
\not = N\mu_1
$$
$$
(n_2+m)(\omega_{1}+\mu_1)+n_2\omega_{1}\not = N\mu_1
$$
$$
m\omega_{3}\not = N\mu_1
$$
then it is clear that the terms containing $\mu_1$ at l.h.s. do not
affect the nonresonance conditions, and therefore this is equivalent to
the stated conditions.\EOP
\section{Discussion}
We first discuss briefly the relation with the papers
\cite{KW1,KW2,Y}.
To fix ideas we consider a quasiperiodic breather of the DNLS, and consider
the motion of one observable, for example $q_1$, then one can consider
the Fourier expansion of $q_1(t)$, namely write
$$
q_1(t)=\sum_{(n_1,n_2)\in\ra^2}
c_{n_1n_2}e^{i(n_1\omega_1+n_2\omega_2)t}\ :
$$
generically all coefficients $c_{n_1n_2}$ are different from zero,
i.e. the motion contains all the frequencies
$n_1\omega_1+n_2\omega_2$. In particular some of these frequencies will
fall in the continuous spectrum, and as shown by \cite{KW1,KW2} in general
this phenomenon creates a coupling between the quasiperiodic motion
and the continuous spectrum, and as a consequence the breather begins
to radiate energy and therefore to decay.
So at first sight it is quite surprising that quasiperiodic motions exist in
the considered models. However, as it is clear from the
perturbative construction of quasiperiodic solutions (see
e.g. \cite{Kuk94}), in order to destroy a quasiperiodic motion two
ingredients are needed: the first one is a resonance, and the
second one is a coupling term in the nonlinearity.
{\it Generically} all possible coupling terms are present in the
nonlinearity, but {\it a system with symmetry is not generic!} The
symmetry actually prevents the existence of such coupling terms. For
this reason the mechanism of \cite{KW1,KW2} is not active in this
case.
Concerning the relation with the work by Yuan \cite{Y} we point out
that his situation is completely different from ours, indeed, while in
our case radiation is not possible due to the non generiticity of the
nonlinearity, in his case radiation is not possible due to the
nongeneriticity of the linear part of the system, indeed in his case
there is no continuous spectrum.
In conclusion we have shown that quasiperiodic breathers exist in
Hamiltonian lattices having some integrals of motion independent of
the Hamiltonian, and that Nekhoroshev's theorem is a powerful tool in
order to actually construct such breathers in concrete cases.
We also emphasize that systems with symmetry are exceptional, but at
the same time they are quite common and interesting: we think that the
same is true for quasiperiodic breathers.
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\end{document}