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Normal form theory, Hamiltonian PDEs, Perturbation theory
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\begin{document}



\title{Perturbation theory for PDEs}


\date{\today} \maketitle


%\begin{abstract}

%\end{abstract}

%\tableofcontents

\setcounter{section}{0}

\noindent {\large Dario Bambusi$^{a}$}
\vskip20pt
\noindent
$^{a}$ Dipartimento di Matematica, Universit\`a
  degli Studi di Milano, Milano, Italia

\tableofcontents


\section{Glossary}

\noindent{\bf Perturbation theory.} The study of a dynamical systems
which is a perturbation of a system whose dynamics is known. Typically
the unperturbed system is linear or integrable.

\noindent{\bf Normal Form.} The method of normal form consists in
constructing a coordinate transformation changing the equations of a
dynamical system into new ones which are as simple as possible. In
Hamiltonian systems the theory is particularly effective and typically
leads to a very precise description of the dynamics.

\noindent{\bf Hamiltonian PDE.} A Hamiltonian PDE is a partial
differential equations (abbreviated, PDE) which is equivalent to the
Hamilton equations of a suitable Hamiltonian function. Classical
examples are the nonlinear wave equation, the Nonlinear Schr\"odinger
equation, the Kortweg de Vries equation.

\noindent{\bf Resonance-non resonance.} A frequency vector
$\{\omega_k\}_{k=1}^n$ is said to be non resonant if its components
are independent over the relative integers. If on the contrary there
exists a non vanishing $K\in\Z^n$ such that $\omega\cdot K=0$ the
frequency vector is said to be resonant. Such a property plays a
fundamental role in normal form theory. Nonresonance typically implies
stability.

\noindent{\bf Actions.} The action of a harmonic oscillator is its
energy divided by its frequency. It is usually denoted by $I$. The
typical issue of normal form theory is that in nonresonant systems the
actions remain approximatively unchanged for very long times. In
resonant systems there are linear combinations of the actions with
such properties.

\noindent{\bf Sobolev space.} Space of functions which have weak
derivatives enjoying suitable integrability properties. Here we will
use the spaces $H^s$, $s\in \Nn$ of the functions which are square
integrable together with their first $s$ weak derivatives.



\section{Definition of the subject and its importance.}

Perturbation theory for PDEs is a part of qualitative theory of
differential equations. One of the most effective methods of
perturbation theory is normal form theory which consists in using
coordinate transformations in order to come to the description of the
qualitative features of a given or generic equation. Classical normal
form theory for ordinary differential equations has been used all
along the last century in many different domains leading to important
results in pure mathematics, celestial mechanics, plasma physics,
biology, solid state physics, chemistry and many other fields.

The development of effective methods to understand the dynamics of
partial differential equations is relevant in pure mathematics as well
as in all the fields in which partial differential equations play an
important role.  Fluidodynamics, oceanography, meteorology, quantum
mechanics, electromagnetic theory are just example of potential
applications. More precisely the normal form theory allows to
understand whether a small nonlinearity can change the dynamics of a
linear PDE or not. Moreover, it allows to understand how the changes
can be avoided or forced. Finally, when the changes are possible it
allows to predict the behavior of the perturbed system.


\section{Introduction}\label{intro}

The method of normal form has been developed by Poincar\'e and
Birkhoff between the end of the 19$^{th}$ century and the beginning of
the 20$^{th}$ century. During the last 20 years the method has been
successfully generalized to a suitable class of partial differential
equations (PDEs) in finite volume ({in the case of infinite volume
dispersive effects appear and the theory is very different. See
e.g. \cite{SofW99}}). In this article we will give an introduction to
this recent field. We will almost only deal with Hamiltonian PDEs
since, on the one hand the theory for non Hamiltonian systems is a small
variant of the one we will present here, and on the other hand most
models are Hamiltonian.

We will start by a generalization of the Hamiltonian formalism to
PDEs, followed by a review of the classical theory and by the actual
generalization of normal form theory to PDEs.

In the next section we give a generalization the Hamiltonian formalism
to PDEs. The main new fact is that in PDEs the Hamiltonian is usually
a smooth function, but the corresponding vector field is nonsmooth (it
is an operator extracting derivatives). So the standard formalism has
to be slightly modified \cite{CheMa, Mar72, Wei69, K1, BG93,
KaP}. Here we will present a version of the Hamiltonian formalism
which is enough to cover the models of interest for local perturbation
theory. To clearly illustrate the situation we will start the article
with an introduction to the Lagrangian and Hamiltonian formalism for
the wave equation. This will lead to the introduction of the paradigm
Hamiltonian which is usually studied in this context. This will be
followed by a few results on the Hamiltonian formalism that are needed
for perturbation theory.

Subsequently we shortly present the standard Birkhoff normal form
theory for finite dimensional systems. This is useful since all the
formal aspects are equal in the classical and in the PDEs case.

Then we come to the generalization of normal form theory to PDEs.  In
the present paper we will concentrate almost only on the case of
1-dimensional semilinear equations. This is due to the fact that the
theory of higher dimensional and quasilinear equations is still quite
unsatisfactory.


In PDEs one meets essentially two kinds of difficulties. The first one
is related to the existence of non smooth vector fields. The second
difficulty is due to the fact that in the infinite dimensional case
there are small denominators which are much worse than in the finite
dimensional one.

%A further relevant property is
%the smoothness {\it of the nonlinear part of the equation}. If the
%nonlinearity has a smooth vector field then one speaks of semilinear
%equations, while if the nonlinearity is nonsmooth one speaks of
%quasilinear or fully nonlinear equations. The theory we will present
%pertains to semilinear equations: the results for quasilinear systems
%are at present quite unsatisf

We first present the theory for completely resonant systems
\cite{BG93,BN98} in which the difficulties related to small denominators do
not appear. It turns out that it is quite easy to obtain a normal form
theorem for resonant PDEs, but the kind of normal form one gets is
usually quite poor. In order to extract dynamical informations from
the normal form one can only compute and study it explicitly. Usually
this is very difficult. Nevertheless in some cases it is possible and
leads to quite strong results.  We will illustrate such a situation by
studying a nonlinear Schr\"odinger equation \cite{Bam99,Bo00}.

For the general case there is a theorem ensuring that a generic system
admits at least one family of ``periodic like trajectories'' which
are stable over exponentially long times \cite{BN02}. We will give its
statement and an application to the nonlinear wave equation
\begin{equation}
\label{wave}
u_{tt}-u_{xx}+\mu^2u+f(u)=0\ ,
\end{equation}
with $\mu=0$ and Dirichlet boundary conditions on a
segment\cite{PBC}. 

Then we turn to the case of nonresonant PDEs.  The main difficulty is
that small denominators accumulate to zero already at order 3. Such a
problem has been overcome in \cite{Bam03,BG06,BDGS06,Gre06,Bam07} by taking
advantage of the fact that the nonlinearities appearing in PDEs
typically have a special form. In this case one can deduce a very
precise description of the dynamics and also some interesting results
of the kind of almost global existence of smooth solutions
\cite{Kla83}. To illustrate the theory we will make reference to the
nonlinear wave equation \eqref{wave} with almost any $\mu$ and to the
nonlinear Schr\"odinger equation.

Another aspect of the theory of close to integrable Hamiltonian PDEs
concerns the extension of KAM theory to PDEs. We will not present it
here. We just recall the most celebrated results which are those due
to Kuksin \cite{K87}, Wayne \cite{W90}, Craig--Wayne \cite{CW92},
Bourgain \cite{Bo98,Bou03}, Kuksin-P\"oschel \cite{KP96},
Eliasson--Kuksin \cite{EK06}, Yuan \cite{Yua07}. All these results
ensure the existence of families of quasiperiodic solutions,
i.e. solutions lying on finite dimensional manifolds. We also mention
the papers \cite{Bo96,P02} where some Cantor families of full
dimension tori are constructed. We point out that in the dynamics on
such $\infty$--dimensional tori the amplitude of oscillation of the
linear modes decreases super exponentially with their index. A
remarkable exception is provided by the paper \cite{Bou05} where the
tori constructed are more ``thick'' (even if of course they lie on
Cantor families).

On the contrary the results of normal form theory describe solutions
starting on opens subsets of the phase space, and do not have
particularly strong localizations properties with respect to the
index. The price one has to pay is that the description one gets turns
out to be valid only over long but finite times.

Finally we point out a related research stream that has been carried on
by Bourgain \cite{Bo96,Bo96a,Bo97a,Bo00} who studied intensively the
behavior of high Sobolev norms in close to integrable Hamiltonian
PDEs (see also \cite{BouKal}).


\section{The Hamiltonian formalism for PDEs}\label{hamilton}


\subsection{The gradient of a functional}

\begin{definition}
\label{grad}
Consider a function $f\in C^{\infty}(\U_s,\R)$, $\U_s\subset H^s(\T)$
open, $s\geq 0$ a fixed parameter and $\T:=\R/2\pi\Z$ is the 1
dimensional torus.  We will denote by $\nabla f(u)$ the gradient of
$f$ with respect to the $L^2$ metric, namely the unique function such
that
\begin{equation}
\label{h.1}
\left\langle \nabla f(u), h\right\rangle_{L^2}=\de f(u)h\ ,\quad
\forall h\in H^s
\end{equation}
where 
\begin{equation}
\label{l2}
\left\langle u,
v\right\rangle_{L^2}:=\int_{-\pi}^{\pi}u(x)v(x)\de x
\end{equation}
is the $L^2$ scalar product and $\de f(u)$ is the differential of $f$
at $u$. The gradient is a smooth map from $H^s$
to $H^{-s}$ (see e.g. \cite{Bamdarb}).
\end{definition}

\begin{example}
\label{ex.1}
Consider the function 
\begin{equation}
\label{e.1}
f(u):=\int_{-\pi}^\pi\frac{u_x^2}{2}\de x\ ,
\end{equation}
which is differentiable as a function from $H^s\to\R$ for any
$s\geq 1$. One has
\begin{equation}
\label{e.2}
\de f(u)h=\int_{-\pi}^\pi u_xh_x \de x =\int_{-\pi}^\pi -u_{xx}h \de x=
\langle -u_{xx}, h\rangle_{L^2}
\end{equation}
and therefore in this case one has $\nabla f(u)=-u_{xx}$.
\end{example}
\begin{example}
\label{ex.2}
Let $\F:\R^2\to\R$ be a smooth function and define
\begin{equation}
\label{e.4}
f(u)=\int_{-\pi}^\pi \F(u,u_x)\de x
\end{equation}
then the gradient of $f$ coincides with the so called functional
derivative of $\F$:
\begin{equation}
\label{e.5}
\nabla f\equiv \frac{\delta \F}{\delta u}:=\frac{\partial \F}{\partial
u} -\frac{\partial}{\partial x}\frac{\partial \F}{\partial
u_x} \ .
\end{equation}
\end{example}

\subsection{Lagrangian and Hamiltonian formalism for the wave
  equation}\label{lag} 

Until subsection \ref{rig} we will work at a formal level, without
specifying the function spaces and the domains. 

\begin{definition}
\label{d.2}
Let $L(u,\dot u)$ be a Lagrangian function, then the corresponding
Lagrange equations are given
\begin{equation}
\label{e.8}
\nabla_u L-\frac{\de}{\de t}\nabla _{\dot u}\ L=0
\end{equation}
where $\nabla_u L$ is the gradient with respect to $u$ only, and
similarly $\nabla_{\dot u}$ is the gradient with respect to $\dot u$. 
\end{definition}
 
\begin{example}
\label{ex.3}
Consider the Lagrangian
\begin{equation}
\label{e.7}
L(u,\dot u):=\intpi \left(\frac{\dot
  u^2}{2}-\frac{u_x^2}{2}-\mu^2\frac{
  u^2}{2}-F(u)\right) \de x\ .
\end{equation}
then the corresponding Lagrange equations are given by \eqref{wave}
with $f=-F'$.
\end{example}

Given a Lagrangian system with Lagrangian function $L$ one defines the
corresponding Hamiltonian system as follows. 

\begin{definition}
\label{d.3}
Consider the momentum $v:=\nabla_{\dot u}L$  conjugated to $u$;
  assume that $L$ is convex with respect to $\dot u$, then the
  Hamiltonian function associated to $L$ is defined by
\begin{equation}
\label{e.11}
H(v,u):=\left[\left\langle v;\dot u\right\rangle_{L^2}-L(u,\dot
  u)\right] _{\dot u=\dot u(u,v)}\ .
\end{equation}
\end{definition}
\begin{definition}
\label{d.4}
Let $H(v,u)$ be a Hamiltonian function, then the corresponding
Hamilton equations are given by
\begin{equation}
\label{e.12}
\dot v=-\nabla_u H\ ,\quad \dot u=\nabla_v H\ . 
\end{equation}
\end{definition}

As in the finite dimensional case one has that the Lagrange equations
are equivalent to the Hamilton equation of $H$.

An elementary computation shows that for the wave equation one has 
$v=\dot u$ and 
\begin{equation}
\label{e.13}
H(v,u)=\intpi\left(\frac{v^2+u_x^2+\mu^2 u^2}{2}+F(u)
\right)\de x
\end{equation}

\subsection{Canonical coordinates}\label{can}

Consider a Lagrangian system and 
let $\be k$ be an orthonormal basis of $L^2$, write  $u=\sum_k q_k\be
k$ and $\dot u=\sum_k \dot q_k\be k$, then one has the following
proposition
\begin{proposition}
\label{p.1}
The Lagrange equations \eqref{e.8} are equivalent to 
\begin{equation}
\label{e.14}
\frac{\partial L}{\partial q_k}-\frac{\de }{\de t}\frac{\partial
  L}{\partial \dot q_k}=0 
\end{equation}
\end{proposition}
\proof Taking the scalar product of \eqref{e.8} with $\be k$ one gets 
$$
\langle \be k;\nabla _uL\rangle_{L^2}-\frac{\de }{\de t}\langle \be
k;\nabla _{\dot u}L\rangle_{L^2}=0
$$
but one has $\langle\be k; \nabla_u L\rangle= \frac{\partial
  L}{\partial q_k}$ and similarly for the other term. Thus the thesis
follows. \qed

This proposition shows that, once a basis has been introduced, the Lagrange
equations have the same form as in the finite dimensional case. 

In the Hamiltonian case exactly the same result holds. Precisely,
denoting $v:=\sum_k p_k\be k$ one has 
\begin{proposition}
\label{p.2}
The Hamilton equations of a Hamiltonian function $H$, are equivalent
to
\begin{equation}
\label{e.16}
\dot p_k=-\frac{\partial
  H}{\partial q_k}\ ,\quad \dot q_k=\frac{\partial
  H}{\partial p_k}\ .
\end{equation}
\end{proposition}

In the case of the nonlinear wave equation, in order to get a
convenient form of the equations, one can choose the Fourier
basis. Such a basis is defined by 
\begin{equation}
\label{he.300}
\hat e_k:=\left\{
\begin{matrix}
\frac{1}{\sqrt\pi}\cos kx & k>0
\\
\frac{1}{\sqrt{2\pi}} & k=0
\\
\frac{1}{\sqrt\pi}\sin -kx& k<0
\end{matrix}
\right.
\end{equation}
Thus the Hamiltonian \eqref{e.13} takes the form
\begin{equation}
\label{e.15}
H(p,q)=\sum_{k\in\Z} \frac{p_k^2+\omega_k^2 q_k^2}{2}+ \intpi
F\left(\sum_k q_k\hat e_k(x )\right)\de x\ ,
\end{equation}
where $\omega_k^2:=k^2+\mu^2$.
For the forthcoming developments it is worth to rescale the variables by
defining 
\begin{equation}
\label{e.16a}
p'_k:=\frac{p_k}{\sqrt{\omega_k}}\ ,\quad q'_k:=\sqrt{\omega_k}q_k\ ,
\end{equation}
so that, omitting primes, the Hamiltonian takes the form 
\begin{equation}
\label{e.17}
H(p,q)=\sum_{k}\omega_k \frac{p_k^2+q_k^2}{2}+H_P(p,q)
\end{equation}
where $H_P$ has a zero of order higher than 2.  {\it In the
following we will always study systems of the form
\eqref{e.17}}. Moreover, possibly by relabeling the variables and the
frequencies it is possible to reduce to the case where $k$ varies in
$\Nn\equiv \{1,2,3...\}$. This is what we will assume in developing the
abstract theory.

\begin{example}
\label{ex.NLS}
An example of a different nature in which the Hamiltonian takes the
form \eqref{e.17} is the nonlinear Schr\"odinger equation
\begin{equation}
\label{NLS}
-\im \dot \psi=\psi_{xx}+f(|\psi|^2)\psi\ ,
\end{equation}
where $f$ is a smooth function. Eq. \eqref{NLS} has the conserved
energy functional
\begin{equation}
\label{e.19}
H(\psi,\bar \psi):=\intpi\left( |\psi|^2+F(|\psi|^2)\right)\de x\ ,
\end{equation}
where $F$ is such that $F'=f$.
Introduce canonical coordinates $(p_k,q_k)$ by 
\begin{equation}
\label{e.20}
\psi=\sum_{k\in \Z}\frac{p_k+\im q_k}{\sqrt 2}\hat e_k\ ,
\end{equation} 
then the energy takes the form \eqref{e.17} with $\omega_k=k^2$ and
the NLS is equivalent to the corresponding Hamilton equations.
\end{example}
\begin{example}
\label{ex.KdV}
Consider the Kortweg de Vries equation 
\begin{equation}
\label{KdV}
u_t+u_{xxx}+uu_x=0\ ,
\end{equation}
in the space of functions with zero mean value. The conserved energy
is given by
\begin{equation}
\label{e.21}
H(u)=\intpi \left( \frac{u_x^2}{2}+\frac{u^{3}}{6}\right)\de x\ ,
\end{equation}
which again is also the Hamiltonian of the system. Canonical
coordinates are here introduced by 
\begin{equation}
\label{e.22}
u(x)=\sum_{k>0}\sqrt k 
(p_k\hat e_k+q_k\hat e_{-k})\ ,
\end{equation}
in which the Hamiltonian takes the form \eqref{e.17} with
$\omega_k=k^3$. 
\end{example}

\begin{remark}
\label{DBC}
It is also interesting to study some of these equations with Dirichlet
boundary conditions (DBC) typically on $[0,\pi]$. This will always be
done by identifying the space of the functions fulfilling DBC with the
space of the function fulfilling periodic boundary conditions on
$[-\pi,\pi]$ which are skew symmetric. Similarly, Neumann boundary
conditions will be treated by identifying the corresponding functions
with periodic even functions. In some cases (e.g. in equation
\eqref{wave} with DBC and an $f$ which does not have particular
symmetries) the equations do not extend naturally to the space of skew
symmetric and this has some interesting consequences (see
\cite{BMP07,BCP07}).  
\end{remark}

\subsection{Basic elements of Hamiltonian formalism for
  PDEs}\label{rig} 

A suitable topology in the phase space is given by a Sobolev like
topology. 

For any $s\in\R$, define the Hilbert space $\ell^{2}_s$ of the
  sequences $x\equiv \{x_k\}_{k\geq 1}$ with $x_k\in\R$ such that
\begin{equation}
\label{ells}
\norma{x}_{s}^2:=\sum_{k}|k|^{2s}|x_k|^2<\infty
\end{equation}
and the phase spaces $\Ph_s:=\ell^2_s\oplus \ell^2_s\equiv
z\ni(p,q)\equiv \left(\{p_k\},\{q_k\}\right)$. In $\Ph_s$ we will
sometimes use the scalar product
\begin{equation}
\label{e.29aa}
\langle(p,q),(p^1,q^1)\rangle_s:=\langle
p,p^1\rangle_{\ell^2_s}+\langle q,q^1\rangle_{\ell^2_s} \ .
\end{equation}

In the following we will always assume that 
\begin{equation}
\label{e.29a}
|\omega_k|\leq C|k|^d
\end{equation}
for some $d$.

\begin{remark}
\label{a0}
Defining the operator $A_0:D(A_0)\to \Ph_s$ by $A_0(p,q)=(\omega_k
p_k,\omega_kq_k)$ one can write $H_0=\frac{1}{2}\langle
A_0z;z\rangle_0$, $D(A_0)\supset\Ph_{s+d}$.  
\end{remark}

Given a smooth Hamiltonian function $\chi:\Ph_s\supset \U_s\to \R$, $\U_s$
being an open neighborhood of the origin, we define the corresponding
Hamiltonian vector field $X_{\chi}:\U_s\mapsto\Ph_{-s}$ by 
\begin{equation}
\label{xchi}
X_{\chi}\equiv\left(-\frac{\partial \chi}{\partial q_k},
\frac{\partial \chi}{\partial p_k}\right)\ .
\end{equation}

\begin{remark}
\label{a1}
Corresponding to a function $\chi$ as above we will denote by $\nabla
\chi$ its gradient with respect to the $\ell^2\equiv \ell^2_0$
metric. Defining the operator $J$ by $J(p,q):=(-q,p)$ one has
$X_\chi=J\nabla \chi$.
\end{remark}

\begin{definition}
\label{def.3}
The Poisson Bracket of two smooth functions $\chi_1$, $\chi_2$ is
formally defined by
\begin{equation}
\label{poi}
\left\{ \chi_1;\chi_2 \right\}:= d\chi_1 X_{\chi_2}\equiv
\langle\nabla \chi_1;J\nabla \chi_2 \rangle_{0}
\end{equation}
\end{definition}

\begin{remark}
\label{poi.1}
As the example $\chi_1=\sum_k kq_k$, $\chi_2:=\sum_kk p_k$ shows, there
are cases where the Poisson Bracket of two functions is not
defined. 
\end{remark}

For this reason a crucial role is played by the functions whose vector
field is smooth. 

\begin{definition}
\label{gen}
A function $\chi\in C^\infty(\U_s,\Ph_s)$, $\U_s\subset\Ph_s$ open, is said
to be of class $\Sev_s$, if the corresponding Hamiltonian vector field
$X_\chi$ is a smooth map from $\U_s \to \Ph_s$. In this case we will
write $\chi\in\Sev_s$
\end{definition}


\begin{proposition}
\label{p.ba}
Let $\chi_1\in\Sev_s$. If $\chi_2\in C^\infty(\U_s,\R) $ then
$\left\{\chi_1,\chi_2 \right\}\in C^\infty(\U_s,\R) $. If
$\chi_2\in\Sev_s$ then $\left\{\chi_1,\chi_2\right\}\in\Sev_s$.
\end{proposition}


\begin{definition}
\label{cano}
A smooth coordinate transformation $\Tr:\Ph_s\supset \U_s\to\Ph_s$ is
said to be canonical if for any Hamiltonian function $H$ one has
$X_{H\circ\Tr}=\Tr^*X_H\equiv \de \Tr^{-1}X_H\circ \Tr$, i.e. it
transforms the Hamilton equations of $H$ into the Hamilton equations
of $H\circ \Tr$.
\end{definition}

\begin{proposition}
\label{p.8}
Let $\chi_1\in\Sev_s$, and let $\Phi^t_{\chi_1}$ be the corresponding
time $t$ flow (which exists by standard theory). Then for
$\Phi^t_{\chi_1}$ is a canonical transformation.
\end{proposition}


\section{Normal form for finite dimensional Hamiltonian systems}

Consider a system of the form \eqref{e.17}, but with finitely many
degrees of freedom, namely a system with Hamiltonian of the form
\eqref{e.17} with
\begin{equation}
\label{h0}
H_0(p,q)=\sum_{k=1}^{n}\omega_k\frac{p_k^2+q_k^2}{2}\ ,\quad
\omega_k\in\R
\end{equation}
and $H_P\relax $ which is a smooth function having a zero of order at
least 3 at the origin.
\begin{definition}
\label{d.6}
A polynomial $Z$ will be said to be in normal form if $\{H_0;
Z\}\equiv 0$.
\end{definition}

\begin{theorem}
\label{birkhoff} 
(Birkhoff) For any positive integer $r\geq0$,  there exist a
neighborhood $\U^{(r)}$ of the origin and a canonical transformation
$\Tr_r:\R^{2n}\supset \U^{(r)}\to\R^{2n}$ which puts the system
(\ref{e.17}) in Birkhoff Normal Form up to order $r$, namely s.t. 
\begin{equation}
\label{eq:bir}
H^{(r)}:=H\circ \Tr_r=H_0+Z^{(r)}+\resto^{(r)}
\end{equation}
where $Z^{(r)}$ is a polynomial of degree $r+2$ which is in normal
form, $\resto^{(r)}$ is small, i.e.
\begin{equation}
\label{eq:resto}
|{{\resto^{(r)}}(z)}|\leq C_r\norma{z}^{r+3}\ ,\quad \forall z\in\U^{(r)}\ ;
\end{equation}
moreover, one has
\begin{equation}
\label{def}
\norma{z-\Tr_r(z)} \leq C'_r\norma{z}^2\ ,\quad \forall z\in\U^{(r)}\ .
\end{equation}
An inequality identical to \eqref{def} is fulfilled by the inverse
transformation $\Tr_r^{-1}$.

If the frequencies are nonresonant at order $r+2$, namely if
\begin{equation}
\label{eq:nonr}
\omega\cdot K\not=0\ ,\quad \forall K\in\Z^n\ ,\quad 0<|K|\leq r+2
\end{equation}
the function $Z^{(r)}$ depends on the actions
$$
I_j:=\frac{p_j^2+q_j^2}{2}
$$
only.
\end{theorem}


\begin{remark}
\label{r.11}
If the nonlinearity is analytic and the frequencies are Diophantine,
i.e. there exist $\gamma>0$ and $\tau$ such that
\begin{equation}
\label{e.25}
\left|\omega\cdot K\right|\geq \frac{\gamma}{|K|^\tau}\ ,\quad \forall
K\in\Z^n-\{0\}\ .
\end{equation}
then one can compute the dependence of the constant $C_r$
(cf. eq.\eqref{eq:resto}) on $r$ and optimize the value of $r$ as a
function of $\norma z$. This allows to improve \eqref{eq:resto} and to
show that there exists and $r_{opt}$ such that (see e.g. \cite{Fas90})
\begin{equation}
\label{eq:resto1}
|{{\resto^{(r_{opt})}}(z)}|\leq C
 \exp\left(-\frac{c}{\norma{z}^{1/(\tau+1)}}\right)\ .
\end{equation}
In turn such an estimate is the starting point for the proof of the
celebrated Nekhoroshev's theorem \cite{Nek77}.
\end{remark}

The idea of the proof is to construct a canonical transformation
putting the system in a form which is as simple as possible, namely
the normal form. More precisely one constructs a canonical
transformation $\Tr^{(1)}$ pushing the non normalized part of the
Hamiltonian to order four followed by a transformation $\Tr^{(2)}$
pushing it to order five and so on, thus getting $\Tr_r=\Tr^{(1)}\circ
\Tr^{(2)}\circ...\circ \Tr^{(r)}$. Each of the transformations
$\Tr^{(j)}$ is constructed as the time one flow of a suitable
auxiliary Hamiltonian function say $\chi_j$ (Lie transform method). It
turns out that $\chi_j$ is determined as the solution of the
Homological equation
\begin{equation}
\label{homo}
Z_j:=\left\{\chi_j,H_0\right\}+H^{(j)}
\end{equation}
where $H^{(j)}$ is constructed recursively and $Z_j$ has to be
determined together with $\chi_j$ in such a way that $\{Z_j;H_0 \}=0$
and \eqref{homo} holds. In particular $H^{(1)}$ coincides with the
first non vanishing term in the Taylor expansion of $H_P$.

The algorithm of solution of the Homological equation \eqref{homo}
involves division by the so called small denominators $\im \omega\cdot
K$, where $K\in\Z^n-\{ 0\}$, fulfills $|K|\leq j+2$ and $\omega\cdot
K\not=0$.


The above construction is more or less explicit: provided one has at
disposal enough time, he can explicitly compute $Z^{(r)}$ up to any
given order. In the case of nonresonant frequencies this is not needed
if one wonts to understand the dynamics over long times. Indeed its features
are an easy consequence of the fact that $Z^{(r)}$ depends on the
actions only. A precise statement will be given in the case of
PDEs. It has to be noticed that the normal form can be used also as a
starting point for the construction of invariant tori through KAM
theory. To this end however one has to verify a nondegeneracy
condition and this requires the explicit computation of the normal
form. 

In the resonant case the situation is more complicated, however, it is
often enough to compute the first non vanishing term of $Z^{(r)}$ in order to
get relevant informations on the dynamics. This usually requires only
to be able to compute the function $Z_1$ defined by \eqref{homo}
with $H^{(1)}$ coinciding with the first non vanishing term of the
Taylor expansion of $H_P$. For a detailed analysis we referr to other
sections of the Encyclopedia.

A particular case where one can use a coordinate independent formula
for the computation of $Z_j$ and $\chi_j$ is the one in which the
frequencies are completely resonant.

Assume that there exists $\nu>0$ and integer numbers
$\ell_1,..,\ell_n$ such that 
\begin{equation}
\label{e.28}
\omega_k=\nu\ell_k\quad\forall k=1,...,n\ .
\end{equation}
\begin{remark}
\label{r.10}
Denote by $\Psi^t$ the flow of the linear system with Hamiltonian
$H_0$, then one has 
\begin{equation}
\label{e.29}
\Psi^{t+T}=\Psi^t\ ,\quad T:=\frac{2\pi}{\nu}\ ,\quad t\in\R\ .
\end{equation}
Moreover in this case one has $\omega\cdot K\not=0\Longrightarrow $
$\left|\omega\cdot K\right|\geq \nu>0$, so there are no small
denominators.
\end{remark}
In this case one has an interesting coordinate independent formula for
the solution of the homological equation \eqref{homo}.

\begin{lemma}
\label{l.34}
Let $f$ be smooth function, defined in neighborhood of the
origin. Define 
\begin{equation}
\label{Z}
Z(z)\equiv\left\langle f \right\rangle(z):= \frac{1}{T}\int_0^T
f(\Psi^t(z))\de t\ ,\quad 
\chi(z):= \frac{1}{T}\int_0^Tt\left[
f(\Psi^t(z))-Z(\Psi^t(z))  \right]\de t\ , 
\end{equation}
then such quantities fulfill the equation $\{ H_0,\chi\}+f=Z$.
\end{lemma}

\section{Normal form for Hamiltonian PDEs: general comments}
\label{PDE}

As anticipated in the introduction there are two problems in order to
generalize Birkhoff's theorem to PDEs: (1) the existence of nonsmooth
vector fields and (2) the appearance of small denominators
accumulating at zero already at order 3.

There are two reasons why (1) is a problem. The first one is that if
the vector field of $\chi_1$ were not smooth then it would be
nontrivial to ensure that it generates a flow, and thus that the
normalizing transformation exists. The second related problem is that,
if a transformation could be generated, then the Taylor expansion of
the transformed Hamiltonian would contain a term of the form $\{H_1;
\chi_1\}=\de H_1 X_{\chi_1}$, which is typically not smooth if
$X_{\chi}$ is not smooth. Thus one has to show that the construction
involves only functions which are of class $\Sev_s$ for some $s$ (see
def. \ref{gen}).

The difficulty related to small denominators is the following: In the
finite dimensional case $\{\omega\cdot K\not =0\ ,\ |K|\leq r+2\}$
implies $|\omega\cdot K|\geq \gamma>0$. Thus division by $\omega\cdot
K$ is a harmless operation in the finite dimensional case.  In the
infinite dimensional case this is no more true. For example when
$\omega_k=\sqrt{k^2+\mu^2}$ one has already
$$
\inf_{0\not=|K|\leq 3}|\omega\cdot K|=0\ .
$$ In order to solve such a problem one has to take advantage of a
property of the nonlinearity which typically holds in PDEs and is
called of having \emph{localized coefficients}. Assuming also a
suitable nonresonance property for the frequency vector one can deduce
a normal form theorem identical to \ref{birkhoff}. The main difficulty
consists in verifying the assumptions of the theorem. We will show how
to verify such assumptions by the application to some typical
examples.



\section{Normal form for resonant Hamiltonian PDEs and its
consequences}\label{res2sec} 

In the case of resonant frequencies and smooth vector field it is
possible to obtain a normal form up to an exponentially small remainder. 

Consider the system \eqref{e.17} in the phase space $\Ph_s$ with
some fixed $s$. Assume that the frequencies are completely resonant,
namely that \eqref{e.28} holds (with $k\in\Nn$); assume that
$H_P\in\Sev_s$ and that its vector field extends to a complex
analytic function in a neighborhood of the origin. Finally assume
that $H_P$ has a zero of order $n\geq 3$ at the origin. Then we have
the following theorem

\begin{theorem}
\label{resonant} \cite{BG93,BN98} There exists a neighborhood of the
origin $\U_s\subset\Ph_s$ and an analytic canonical transformation
$\Tr:\U_s\to \Ph_s$ with the following properties: $\Tr$ is close to
identity, namely it satisfies
\begin{equation}
\label{def11}
\norma{z-\Tr(z)}_s\leq C\norma{z}_{s}^{n-1}\ .
\end{equation}
$\Tr$ puts the Hamiltonian in resonant normal form up to an
exponentially small remainder, namely one has
\begin{equation}
\label{h.n}
H\circ\Tr=H_0+\langle H_P\rangle+Z_2+\resto 
\end{equation}
where $\langle H_P\rangle$ is the average (defined by \eqref{Z}) of
$H_P$ with respect to the unperturbed flow; $Z_2$ is in normal form,
namely $\left\{Z_2;H_0\right\}\equiv 0$, and has a zero of order
$2n-2$ at the origin; $\resto$ is an exponentially small remainder
whose vector field is estimated by
$$
\norma{X_\resto(z)}_s\leq C\norma{z}^{n-1}_s
\exp\left(-\frac C{\norma{z}^{n-2}_s}\right)\ .
$$
\end{theorem}

\begin{example}
\label{ex.11}
The nonlinear Schr\"odinger equation \eqref{NLS}. Here one has
$\ell_k=k^2$ and $\nu=1$. The Sobolev embedding theorems ensure that
the vector field of the nonlinearity is analytic if $f$ is analytic in
a neighborhood of the origin.  Thus theorem \ref{resonant} applies to
the NLS. To deduce dynamical consequences it is convenient to compute
explicitly $\langle H_P\rangle$. Assuming $f(0)=0$ and $f'(0)=1$ this
was done in \cite{Bam99} using formula \eqref{Z} which gives
\begin{equation}
\label{media}
\langle H_P\rangle(z)=\frac1{2}\left(\sum_k I_k  \right)^2-
\frac1{8}\sum_{k}|I_k|^2
\end{equation}
where $I_k=(p_k^2+q_k^2)/2$ are the linear actions. Thus one has that
$H_0+\langle H_P\rangle$ is a function of the actions only, and thus
it is an integrable system. It is thus natural to study the system
\eqref{h.n} as a perturbation of such an integrable system. This was
done in \cite{Bam99} and \cite{Bo00} obtaining the results we are
going to state. For simplicity we will concentrate here on the case of
Dirichlet boundary conditions, thus the function $\psi$ will always be
assumed to be skew symmetric with respect to the origin. Define
\begin{equation}
\label{n.1}
\epsilon_s:=\left(\frac12\int_{-\pi}^\pi|\partial_x^s
\psi^{(0)}(x)|^2\right)^{1/2} 
\end{equation}
i.e. the $H^s$ norm of the initial datum $\psi^{(0)}$, $s\geq 0$, and
denote by $I_k(0)$ the initial value of the linear actions.

\begin{theorem}
\label{t.n.1}
\cite{Bam99} Fix $N\geq1$, then there exists a constant 
$\epsilon_*,$ with the property that, if the initial datum
$\psi^{(0)}$ is such that
\begin{equation}
\label{e.n.1}
\epsilon_1<\epsilon_*\ ,\quad \sum_{k\geq N+1}I_k(0)^2\leq
C \epsilon_1^4\epsilon^{2-1/N}_1\ ,
\end{equation}
then along the corresponding solution of \eqref{NLS} one has
\begin{equation}
\label{e.n.2}
\sum_{k\geq1}|I_k(t)-I_k(0)|^2\leq C' \epsilon^{4+1/N}_1
\end{equation}
for the times $t$ fulfilling
$$
|t|\leq C''\exp\left(\frac{\epsilon_*}{\epsilon_1}\right)^{1/N}\ .
$$
\end{theorem}
This result in particular allows to control the distance in energy
norm of the solution from the torus given by the intersection of the
level surfaces of the actions taken at the initial time. 

\begin{theorem}
\label{n.t.3}
\cite{Bo00} Fix an arbitrarily large $r$, then there
exists $s_r$ such that for any $s\geq s_r$ there exists
$\epsilon_{*s}$, such that the following holds true: most of the
initial data with $\epsilon_s<\epsilon_{*s}$ give rise to solutions
with
\begin{equation}
\label{n.e.5}
\norma{\psi(t)}_s\leq C\epsilon_s\ ,\quad \forall |t|\leq
\frac{C}{\epsilon_s^r}
\end{equation}
\end{theorem}
For the precise meaning of ``most of the initial data'' we refer to
the original paper. The result is based on the proof that the
considered solutions remain close in the $H^s$ topology to an infinite
dimensional torus. In particular the uniform estimate of the Sobolev
norm is relevant for applications to numerical analysis \cite{CHL1,CHL3}.
\end{example}

\begin{example}
\label{ex.35}
Consider the nonlinear wave equation \eqref{wave} with $\mu=0$. Here
the frequencies are given by $|k|$ and thus they are completely
resonant. Again the smoothness of the nonlinearity is ensured by
Sobolev embedding theorem. In the case of DBC in order to ensure
smoothness one has also to assume that the nonlinearity is odd,
namely that $f(-u)=-f(u)$, then theorem \ref{resonant} applies. However
in this case the computation of $\langle H_P\rangle$ is nontrivial. It
has been done (see \cite{PBC}) in the case of $f(u)=\pm u^3+O(u^4)$
and Dirichlet boundary conditions. The result however is that the
function $\langle H_P\rangle$ does not have a particularly simple
structure, and thus it is not easy to extract informations on the
dynamics.
\end{example}

In order to extract informations on the dynamics consider the
simplified system in which the remainder is neglected, namely the
system with Hamiltonian 
\begin{equation}
\label{Hs}
H_S:=H_0+\langle H_P\rangle+Z_2\ .
\end{equation}
Such a system has two integrals of motion, namely $H_0$ and $\langle
H_P\rangle+Z_2$. Let $\gamma_\epsilon$ be the set of the $z$'s at
which $\langle H_P\rangle+Z_2 $ restricted to the surface
$\Se_\epsilon:= \{z:H_0(z)=\epsilon^2\}$ has an extremum, say a
maximum.  Then $\gamma_\epsilon$ is an invariant set for the dynamics
of $H_S$. By the invariance under the flow of $H_0$, one has that
$\gamma_\epsilon$ is the union of one dimensional closed curves, but
generically it is just a single closed curve. In such a situation it
is also a stable periodic orbit of \eqref{Hs} (see \cite{Del89}). Actually
it is very difficult to compute $(\langle
H_P\rangle+Z_2)\vert_{\Se_\epsilon} $, but a maximum of such a
function can be easily constructed by applying the implicit function
theorem to a non degenerate maximum of $\langle
H_P\rangle\vert_{\Se_\epsilon} $. The addition of the remainder then
modifies the dynamics only after an exponentially long time.  We are
now going to state the corresponding theorem.

First remark that a critical point of $\ristr$ is a solution $z_a$ of the
system
\begin{equation}
\label{n.e.51}
\lambda_aA_0z_a+\nabla \langle H_P\rangle(z_a)=0\ ,\quad H_0(z_a)=1
\end{equation}
where we used the notations of remarks \ref{a0} and \ref{a1}. Here
$\lambda_a$ is clearly the Lagrange multiplier. The closed curve
$\gamma_a:=\bigcup_{t}\Psi^t(z_a)$ is (the trajectory of) a periodic
solution of $H_0+\langle H_P\rangle$. Consider now the linear operator
$B_a:=\de( \nabla \langle H_P\rangle)(z_a)$.
\begin{definition}
\label{nondeg}
The critical point $z_a$ will be said to be non degenerate if the
system
\begin{equation}
\label{e.n.9}
\lambda_a A_0h+B_a h=0\ ,\quad \langle A_0z_a;h\rangle_{0}=0
\end{equation}
has at most one solution. 
\end{definition}


Under the assumptions of theorem \ref{nekho} below it is easy to prove
that $\gamma_a$ is a smooth curve and that its tangent vector
$h_a:=\frac{\de}{\de t}\Psi^t(z_a)\big|_{t=0}$ is a solution of
\eqref{e.n.9}.

\begin{theorem}
\label{nekho}{(\cite{BN98, BN02}) Assume that $X_{H_P}\in\Sev$ for any
  $s$ large enough, assume also that there exists a non degenerate
  maximum $z_a$ of $\ristr$, then there exists a constant
  $\epsilon_*$, such that the following holds true: consider a
  solution $z(t)$ of the Hamilton equation of \eqref{e.17} with
  initial datum $z_0$; if there exists $\epsilon<\epsilon_*$, such
  that
\begin{equation}
\label{e.n.10}
d_{E}(\epsilon\gamma_a ,z_0)\leq C\epsilon^{n} \ , 
\end{equation}
then one has 
\begin{equation}
\label{e.n.11}
d_{E}(\epsilon\gamma_a ,z(t))\leq C' \epsilon^{n}\ ,
\end{equation}
for all times $t$ with $|t|\leq \frac C{\epsilon^{r-1}}\exp
\left(\frac{\epsilon_*}{\epsilon}\right)^{n-1}$}. Here $d_E$ is the
distance in the energy norm.  
\end{theorem}
Such a theorem does not ensure that there exist periodic orbits of the
complete system, but just a family of closed curves with the property
that starting close to it one remains close to it for exponentially
long times. Some results concerning the existence of true periodic
orbits close to such periodic like trajectories can also be proved
(see e.g. \cite{LS88,BP01,BB02,GMP05,BB06}).

\begin{example}
\label{ex.78}
In the paper \cite{PBC} it has been proved that the non degeneracy
assumption \eqref{e.n.9} of theorem \ref{nekho} holds for the equation
\eqref{wave} with $f(u)=\pm u^3+$higher order terms and Dirichlet
boundary conditions. In the case of such an equation an  extremum
of $\ristr$ is given by
$$
u(x)=V_m {\rm sn}({\tt w}x,{\tt m})\ ,\quad v(x)\equiv 0\ ,
$$ with $V_m$, ${\tt w}$ and ${\tt m}$ suitable constants, and sn the
Jacobi elliptic sine. Therefore the curve $\gamma_a$ is the phase
space trajectory of the solution of the linear wave equation with such
an initial datum. There are no other extrema of $\ristr$. Thus the
theorem \ref{nekho} ensures that solutions starting close to a
rescaling of such a curve remain close to it for very long times.
\end{example}

\section{Normal form for Hamiltonian nonresonant PDEs}\label{res2} 

\subsection{A statement}

We turn now to the nonresonant case. The theory we will present has
been developed in \cite{BDGS06,Gre06,Bam07}, and is closely related to
the one of \cite{Bam03,BG03,DS04a}. First we introduce the class
of equations to which the theory applies. To this end it is useful to
treat the $p$'s and the $q$'s exactly on an equal footing so we will
denote by $z\equiv (z_k)_{k\in\Zb}$, $\Zb:=\Z-\{0\}$ the set of all
the variables, where
$$
z_{-k}:=p_k\ ,\quad z_k:=q_k\quad k\geq 1\ .
$$ Given a polynomial function $f:\Ph_\infty\to\R$ of degree $r$ one
can decompose it as follows
\begin{equation}
\label{mod.2}
f(z)=\sum_{k_1,...,k_{r}}f_{k_1,...,k_{r}}z_{k_1}...
z_{k_r}
\end{equation}

We will assume suitable localization properties for the coefficients
$f_{k_1,...,k_{r}} $ as functions of the indexes
$k_1,...,k_r$.

\begin{definition}
\label{d.4.1}
Given a multi-index $k\equiv(k_1,...,k_r)$, let $(k_{i_1},k_{i_2},
k_{i_3}...,k_{i_r})$ be a reordering of $k$ such that 
$$
|k_{i_1}|\geq |k_{i_2}|\geq |k_{i_3}|\geq...\geq |k_{i_r}|\ .
$$
We define
$\mu(k):=|k_{i_3}|$ and 
\begin{equation}
\label{e.4.1}
S(k):=\mu(k)+||k_{i_1}|-|k_{i_2}||\ . 
\end{equation} 
\end{definition} 

\begin{definition}
\label{d.4.2}
Let $f:\Ph_\infty\to\R$ be a polynomial of degree $r$. We will say
that $f$ has localized coefficients if there exists
$\nu\in[0,+\infty)$ such that $\forall N\geq 1$ there exists $C_N$
such that for any choice of the indexes $k_1,...,k_r$
following inequality holds
\begin{eqnarray}
\label{e.4.2}
\left|f_{k_1,...,k_{r}}\right|
\leq C_N
\frac{\mu(k)^{\nu+N}}{S(k)^{N}}\ .
\end{eqnarray} 
\end{definition}

\begin{definition}
\label{d.4.2a}
A function $f\in\Sev_s$ for any $s$ large enough, will be said to have
localized coefficients if all the terms of its Taylor expansion have
localized coefficients.
\end{definition}

Some important properties of functions with localized coefficients are
given by 

\begin{theorem}
\label{t4.1}
Let $f:\Ph_\infty \to\R$ be a polynomial of degree $r$ with localized
coefficients, then there exists $s_0$ such that for any $s\geq s_0$
the vector field $X_f$ extends to a smooth map from $\Ph_s$ to itself;
moreover the following estimate holds
\begin{equation}
\label{e.4.7}
\norma{X_f(z)}_s\leq C\norma{z}_s\norma z_{s_0}^{r-2}\ .
\end{equation}
\end{theorem}

In particular it follows that a function with localized coefficients
is of class $\Sev_s$ for any $s\geq s_0$.

\begin{theorem}
\label{c.4.4}
The Poisson Bracket of two functions with localized coefficients has
localized coefficients.
\end{theorem}

In order to develop perturbation theory we need also a quantitative
nonresonance condition.

\begin{definition}
\label{d.4.5}
Fix a positive integer $r$. The frequency vector $\omega$ is said to
fulfill the \emph{property ($r$--NR)} if there exist $\gamma>0,$
and $\alpha\in\R$ such that for any $N$ large enough one has
\begin{eqnarray}
\label{nr.1}
\left|\sum_{k\geq 1}\omega_kK_k
\right|\geq\frac{\gamma}{N^\alpha}\ ,
\end{eqnarray}
for any $K\in\Z^\infty$, fulfilling
$0\not=|K|:=\sum_k|K_k|\leq r+2$, $\sum_{k>N}|K_k|\leq2$.
\end{definition}


It is easy to see that under this condition one can solve the
homological equation and that, if the known term of the equation has
localized coefficients, then also the solution has localized
coefficients.


\begin{theorem}
\label{main}(\cite{BG06,BDGS06})
Fix $r\geq1$, assume that the frequencies fulfill the nonresonance
condition ($r$-NR); assume that $H_P$ has localized
coefficients. Then there exists a finite $s_r$ a neighborhood
$\U_{s_r}^{(r)}$ of the origin in $\Ph_{s_r}$ and a canonical
transformation $\Tr:\U_{s_r}^{(r)}\to\Ph_{s_r}$ which puts the system
in normal form up to order $r+3$, namely
\begin{equation}
\label{eq:bir1}
H^{(r)}:=H\circ \Tr=H_0+Z^{(r)}+\resto^{(r)}
\end{equation}
where $Z^{(r)}$ has localized coefficients and is a function of the
actions $I_k$ only; $\resto^{(r)}$ has a small vector field, i.e.
\begin{equation}
\label{resto1}
\norma{X_{\resto^{(r)}}(z)}_{s_r}\leq C\norma{z}_{s_r}^{r+2}\ ,\quad
\forall z\in\U_{s_r}^{(r)}\ ;
\end{equation}
one  has
\begin{equation}
\label{def1}
\norma{z-\Tr_r(z)}_{s_r} \leq C\norma{z}_{s_r}^2\ ,\quad \forall
z\in\U_{s_r}^{(r)}\ .
\end{equation}
An inequality identical to \eqref{def1} is fulfilled by the inverse
transformation $\Tr_r^{-1}$. Finally for any $s\geq s_r$ there exists
a subset $\U_s^{(r)}\subset\U_{s_r}^{(r)}$ open in $\Ph_s$ such that
the restriction of the canonical transformation to $\U^{(r)}_s$ is
analytic also as a map from $\Ph_s\to\Ph_s$ and the inequalities
\eqref{resto1} and \eqref{def1} hold with $s$ in place of $s_r$.
\end{theorem}

This theorem allows to give a very precise description of the
dynamics.

\begin{proposition}
\label{time}
Under the same assumptions of theorem \ref{main}, $\forall s\geq s_r$
there exists $\epsilon_{*s}$ such that, if the initial datum fulfills
$ \epsilon:=\norma{z_0}_s<\epsilon_{*s} $, then one has
\begin{equation}
\label{esti}
\norma{z(t)}_s\leq 4\epsilon\ ,\quad \sum_{k}k^{2s} 
 \left|I_{k}(t)-I_{k}(0)\right|\leq C\epsilon^3
\end{equation}
for all the times $t$ fulfilling $|t|\leq \epsilon^{-r}$. Moreover
there exists a smooth torus $\T_0$ such that, $\forall M\leq r$
\begin{equation}
\label{ds1}
d_s(z(t),\T_0)\leq C \epsilon^{(M+3)/2}\ ,\quad
\text{for}\quad |t|\leq\frac 1{\epsilon^{r-M}}
\end{equation}
where $d_s(.,.)$ is the distance in $\Ph_s$.  \end{proposition}

A generalization to the resonant or partially resonant case is easy to
be obtained and can be found e.g. in \cite{BG03}.


\subsection{Verification of the property of localization of
coefficients}\label{coef}

The property of localization of coefficients is quite abstract. We
illustrate through a few examples some ways to verify it.

\begin{example}
\label{ex.vlc1}
Consider the nonlinear wave equation \eqref{wave} with Neumann
boundary conditions on $[0,\pi]$. We recall that the corresponding
space of functions will be considered as a subset of the space of
periodic functions.

Consider the Taylor expansion of the nonlinearity, i.e. write
$F(u)=\sum_{r\geq 3} c_r\intpi u^r$. Then one has to prove that the
functions $f_r(u)\equiv \intpi u^r$ have localized coefficients. The
coefficients $f_{k_1,...,k_r}$ are given by
\begin{equation}
\label{vlc.1}
f_{k_1,...,k_r}=\intpi \cos(k_1x)\cos(k_2x)...\cos(k_rx) \de x\ .
\end{equation}
One could compute and estimate such a quantity directly, but it is
easier to proceed in a different way: to show that $f_3$ has localized
coefficients and then to use theorem \ref{c.4.4} to show that each
$f_r$ has localized coefficients for any $r$. This is the path we will
follow.  Consider
\begin{equation}
\label{vlc.2}
f_{k_1,k_2,k_3}=\intpi \cos(k_1x)\cos(k_2x)\cos(k_3x) \de x
\end{equation}
since the estimate \eqref{e.4.2} is symmetric with respect to the
indexes, we can assume that they are ordered, $k_1\geq k_2\geq k_3$, so
that $\eqref{vlc.2}=\pi\delta_{k_1}^{k_2+k_3}/2 $, $\mu(k)=k_3$,
$S(k)=k_3+k_1-k_2$ from which one immediately sees that \eqref{e.4.2}
holds with $\nu=0$. As a consequence one also gets that the function
$g_3:=\intpi v u^2$ has localized coefficients. Since $\{g_3;f_r \}=r
f_{r+1} $, by induction theorem \ref{c.4.4} ensures that $f_r$ has
localized coefficients for any $r$.
\end{example}

Often it is impossible to explicitly compute the coefficients
$f_{k_1,...,k_r}$, so one needs a different way to verify the
property. 
\begin{example}
\label{ex.vlc2}
Consider the nonlinear wave equation
\begin{equation}
\label{vlc.3}
u_{tt}-u_{xx}+V u= f(u)
\end{equation}
with Neumann boundary conditions.  Here $V$ is a smooth, even, periodic
potential. The Hamiltonian reduces to the form \eqref{e.17} by
introducing the variables $q_k$ by $u(x)=\sum_k q_k\varphi_k(x)$ where
$\varphi_k(x)$ are the eigenfunctions of the Sturm Liouville operator
$-\partial_{xx}+V$, and similarly for $v$. In such a case one has
\begin{equation}
\label{vlc.4}
f_{k_1,k_2,k_3}=\intpi
\varphi_{k_1}(x)\varphi_{k_2}(x)\varphi_{k_3}(x) \de x\ .
\end{equation}
Here the idea is to consider \eqref{vlc.4} as the matrix element
$L_{k_1,k_2}$ of
the operator $L$ of multiplication by $\varphi_{k_3}(x)$ on the basis of
the eigenfunctions of the operator $S:=-\partial_{xx}+V$. The key
idea is to proceed as follows.

Let $L$ be a linear operator which maps $D(S^{r})$ into itself for all
$r\geq 0$, and define the sequence of operators
\begin{equation}
\label{e.7.3}
L_N:=[S,L_{N-1}]\ ,\quad L_0:=L\ .
\end{equation}

\begin{lemma}
\label{l.7.2}
Let $S$ be as above, then, for any $N\geq 0$ one has
\begin{equation}
\label{e.7.8}
\left| L_{k_1,k_2}\right|=\left|\left\langle L
\varphi_{k_1};\varphi_{k_2}\right\rangle\right|\leq
\frac{1}{|\lambda_{k_1}-\lambda_{k_2}|^N} \left|\left\langle L_N
\varphi_{k_1};\varphi_{k_2}\right\rangle\right|
\end{equation}  
where $\lambda_{k_j}$ is the eigenvalue of $S$ corresponding to
$\varphi_{k_j}$. 
\end{lemma}
 
Then, in order to conclude the verification of the property of
localization of the coefficients one has just to compute $L_N$ and
to estimate the scalar product product at r.h.s. All the computations
can be found in \cite{Bam07}.
\end{example}

\begin{example}
\label{ex.vlc3}
A third example where the verification of the property of localization
of coefficients goes almost in the same way 
is the nonlinear Schr\"odinger equation
\begin{equation}
\label{vlc.10}
\im \dot \psi=-\psi_{xx}+V\psi+\frac{\partial F(\psi,\bar
  \psi)}{\partial \bar \psi} 
\end{equation}
with Dirichlet Boundary conditions on $[0,\pi]$. Here one has to
assume that $V$ is a smooth even potential and that $F$ is smooth and
fulfills $F(-\psi,-\bar \psi)=F(\psi,\bar \psi)$ (this is
required in order to leave invariant the space of skew symmetric
functions, see remark \ref{DBC}). Here the variables $(p,q)$ are
introduced by
\begin{equation}
\label{vlc.12}
\psi=\sum_{k\in \Z}\frac{p_k+\im q_k}{\sqrt 2}\varphi_k\ ,
\end{equation}
with $\varphi_k$ the eigenfunctions of $S$ with Dirichlet boundary
conditions. Here the Taylor expansion of the nonlinearity has only even
terms. Thus the building block for the proof of the property of
localization of coefficients is the operator $L$ of multiplication by $
\varphi_{k_3}\varphi_{k_4}$. Then, mutatis mutandis the proof goes as
in the previous case.  \end{example}



\subsection{Verification of the nonresonance property}\label{nonre}

Finally in order to apply theorem \ref{main} one has to verify the
nonresonance property ($r$--NR). As usual in dynamical system this is
done by tuning the frequencies using parameters. In the case of the
nonlinear wave equation \eqref{wave} one can use the mass
$\mu$. 

\begin{theorem}
\label{non.res}
\cite{Bam03, BG06, DS04} There exists a zero measure set $S\subset\R$
such that, if $\mu\in \R-S$, then the frequencies
$\omega_k=\sqrt{k^2+\mu^2}$, $k\geq1$ fulfill the condition ($r$-NR)
for any $r$.
\end{theorem}

Thus the theorem \ref{main} applies to the equation \eqref{wave} with
almost any mass.

A similar result holds for the equation \eqref{vlc.3}, where the role of the
mass is played by the average of the potential.

The situation of the nonlinear Schr\"odinger is more difficult. Here
one can use the Fourier coefficients of the potential as parameters. 

Fix $\sigma>0$ and, for any positive $R$ define the space of the
 potentials, by
\begin{equation}
\label{Vnls}
\V_R:=\left\{V(x)= \sum_{k\geq1} v_k\cos kx\mid 
v'_k:=v_k R^{-1}e^{\sigma k}
\in\left[-\frac{1}{2},\frac{1}{2} \right]
 \mbox{ for }k\geq 1 \right\}
\end{equation}
Endow such a space with the product normalized probability measure. 

\begin{theorem}
\label{NLSdir} (\cite{BG06}, see also \cite{Bo96})
For any $r$ there exists a positive $R$ and a set $\Se\subset \V_R$
such that property ($r$-NR) holds for any potential $V\in\Se$ and
$\left|\V_R-\Se \right|=0$.
\end{theorem}

So, provided the potential is chosen in the considered set theorem
\ref{main} applies also to the equation \eqref{vlc.10}.

We point out that the proof of theorem \ref{non.res} and of theorem
\ref{NLSdir} consists essentially of two steps. First one proves that
for most values of the parameters one has
\begin{eqnarray}
\label{nr.1a}
\left|\sum_{k\geq 1}\omega_kK_k
\right|\geq\frac{\gamma}{N^\alpha}\ ,
\\
\nonumber
\forall K\in\Z^\infty\ ,\quad \text{with}\quad 0\not=|K|:=\sum_k|K_k|\leq r+2
\end{eqnarray}
and then one uses the asymptotic of
the frequencies, namely $\omega_k\sim ak^{d}$ with $d\geq 1$ in order
to get the result.

\section{Non Hamiltonian PDEs}\label{honh}

In this section we will present some results for the non Hamiltonian
case. 

It is useful to complexify the phase space. So all along this
section we will denote by $\Ph_s$ the space of the complex sequences
$\{z_k\}$ whose norm (defined by \eqref{ells}) is finite.

In the space $\Ph_s$ consider a system of differential equations of
the form 
\begin{equation}
\label{non.1}
\dot z_k=\lambda_k z_k+P_k(z)\ ,\quad k\in \Z-\left\{0\right\}
\end{equation}
where $\lambda_k$ are complex numbers and $P(z)\equiv \{P_k(z)\}$ has
a zero of order at least 2 at the origin. Moreover we will assume $P$
to be a complex analytic map from a neighborhood of the origin of
$\Ph_s$ to $\Ph_s$. The quantities $\lambda_k$ are clearly the
eigenvalues of the linear operator describing the linear part of the
system, for this reason they will be called ``the eigenvalues''.

\begin{example}
\label{ex.non1}
A system of the form \eqref{e.17} with $H_P$ having a vector field
which is analytic. The corresponding Hamilton equations have the form
\eqref{non.1} with $\lambda_k=-\lambda_{-k}=\im \omega_k$, $k\geq1$.
\end{example}

\begin{example}
\label{ex.non2}
Consider the following nonlinear heat equation with periodic boundary
conditions on $[-\pi,\pi]$
\begin{eqnarray}
\label{he.1}
u_t=u_{xx}-V(x)u+f(u)\ .
\end{eqnarray}
If $f$ is analytic then it can be given the form \eqref{non.1} by
introducing the basis of the eigenfunctions $\varphi_k$ of the Sturm
Liouville operator $S:=-\partial_{xx}+V$, i.e. denoting $u=\sum_k
z_k\varphi_k$. In this case the the eigenvalues $\lambda_k$ are the
opposite of the periodic eigenvalues of $S$. Thus in particular one
has $\lambda_k\in\R$ and $\lambda_k\sim -k^2$.
\end{example}

In this context one has to introduce a suitable concept of
nonresonance:

\begin{definition}
\label{non.d1}
A sequence of eigenvalues is said to be resonant if there exists a
sequence of integer numbers $K_k\geq 0$ and an index $i$ such that 
\begin{equation}
\label{non.e2}
\sum_k\lambda_kK_k-\lambda_i=0\ .
\end{equation}
\end{definition}

In the finite dimensional case the most celebrated results concerning
systems of the form \eqref{non.1} are the Poincar\'e theorem, the
Poincar\'e--Dulac theorem and the Siegel theorem \cite{Arn}. The
Poincar\'e theorem is of the form of Birkhoff's theorem
\ref{birkhoff}, while Poincar\'e Dulac and Siegel theorem guarantee
(under suitable assumptions) that there exists an analytic coordinate
transformation reducing the system to its normal form or linear part
(no remainder!).

At present there is not a satisfactory extension of Poincar\'e Dulac
theorem to PDEs (some partial results have been given in
\cite{FS87}). We are now going to state a known extension of Siegel
theorem to PDEs.

\begin{theorem}
\label{the.non.1} \cite{Nik86,Zhe78}
Assume that the eigenvalues fulfill the Diophantine type condition
\begin{equation}
\label{non.e3}
\left|\sum_k\lambda_kK_k-\lambda_i\right|\geq\frac{\gamma}{|K|^\tau}\
,\quad \forall i,K \ \text{with}\ 2\leq|K|\ , 
\end{equation}
where $\gamma>0$ and $\tau\in\R$ are suitable parameters;
then there exists an analytic coordinate transformation defined in a
neighborhood of the origin, such that the system \eqref{non.1} is
transformed into its linear part 
\begin{equation}
\label{non.e4}
\dot z_k=\lambda_kz_k
\end{equation}
\end{theorem}
The main remark concerning this theorem is that the condition
\eqref{non.e3} is only exceptionally satisfied. If $\lambda\in\C^n$
then condition \eqref{non.e3} is generically satisfied only if
$\tau>(n-2)/2$. Nevertheless some examples where such an equation is
satisfied are known \cite{Nik86}.

The formalism of sect. \ref{res2} can be
easily generalized to the non Hamiltonian case giving rise to a
generalization of Poincar\'e's theorem that we are going to state. 

Given a polynomial map $P:\Ph_{\infty}\to \Ph_{-\infty}$ one can
expand it on the canonical basis $\be k$ of $\Ph_0$ as follows
\begin{equation}
\label{non.e5}
P(z)=\sum_{k_1,..,k_r,i}P^i_{k_1,..,k_r}z_{k_1}...z_{k_r}\be i\ ,\quad
P^i_{k_1,..,k_r}\in\C
\end{equation}

\begin{definition}
\label{non.d2}
A polynomial map $P$ is said to have localized coefficients if
there exists $\nu\in[0,+\infty)$ such that $\forall N\geq 1$ there
exists $C_N$ such that for any choice of the indexes $k_1,...,k_r,i$
following the inequality holds
\begin{eqnarray}
\label{e.4.2a}
\left|P^i_{k_1,...,k_{r}^i}\right|
\leq C_N
\frac{\mu(k,i)^{\nu+N}}{S(k,i)^{N}}\ ,
\end{eqnarray} 
where $(k,i)=(k_1,..,k_r,i)$. A map is said to have localized
coefficients if for any $s$ large enough it is smooth as a map from
$\Ph_s$ to itself and if each term of its Taylor expansion has
localized coefficients. 
\end{definition}
\begin{definition}
\label{non.d.5}
The eigenvalues are said to be strongly nonresonant at order $r$ if
for any $N$ large enough , any $K=(K_{k_1},...,K_{k_r})$ and any 
index $i$ such that $|K|\leq r$ and there are at most two of the
indexes $k_1,..,k_r,i$ larger than $N$ the following inequality holds
\begin{equation}
\label{non.e7}
\left|\sum_k\lambda_kK_k-\lambda_i\right|\geq\frac{\gamma}{N^\alpha}\ .
\end{equation}
\end{definition}

\begin{theorem}
\label{th.34}
Assume that the nonlinearity has localized coefficients and that the
eigenvalues are strongly nonresonant at order $r$, then there exists
$s_r$ and an analytic
coordinate transformation $\Tr_r:\U_{s_r}\to\Ph_{s_r}$ which
transforms the system \eqref{non.1} to the form 
\begin{equation}
\label{non.e8}
\dot z_k=\lambda_kz_k+\resto_k(z)\ ,
\end{equation}  
where the following inequality holds
\begin{equation}
\label{non.e9}
\norma{\resto(z)}_{s_r}\leq C\norma{z}_{s_r}^{r}\ .
\end{equation}
\end{theorem}

\section{Extensions and related results}\label{disc}

The theory presented here applies to quite general semilinear
equations in one space dimension. At present a satisfactory theory
applying to quasilinear equations and/or equations in more than one
space dimensions is not available. The main difficulty for the
extension of the theory to semilinear equations in higher space
dimension is related to the nonresonance condition. The general theory
can be easily extended to the case where the differences
$\omega_k-\omega_l$ between frequencies accumulate only at a set
constituted by isolated points. 

\begin{example}
\label{ex.re}
Consider the nonlinear wave equation on the $d$-dimensional sphere
\begin{equation}
\label{wave.d}
u_{tt}-\Delta_gu+\mu^2 u=f(x,u)\ ,\quad x\in S^d
\end{equation}
with $\Delta_g$ the Laplace Beltrami operator; the frequencies are
given by $\omega_k=\sqrt{k(k+d-1)+\mu^2}$ and their differences
accumulate only at integers.  A version of theorem \ref{main}
applicable to \eqref{wave.d} was proved in \cite{BDGS06}. As a
consequence in particular one has a lower bound on the existence time $t$
of the small amplitude solutions of the form $|t|\geq\epsilon^{-r}$,
where $\epsilon$ is proportional to a high Sobolev norm of the initial
datum. An extension to $u_{tt}-\Delta_gu+V u=f(x,u)$ is also known.
\end{example}

\begin{example}
\label{ex.re1}
A similar result was proved in \cite{BG03} for the nonlinear
Schr\"dinger equation 
\begin{equation}
\label{NLS.d}
-\im \dot \psi=-\Delta\psi+V*\psi+f(|\psi|^2)\psi\ ,\quad x\in\T^d
\end{equation}
where the star denotes convolution. 
\end{example}

The only general result available at present for quasilinear system is
the following theorem. 

\begin{theorem}
\label{quasi}
\cite{Bam05} Fix $r\geq1$, assume that the frequency vector fulfills condition
\eqref{nr.1a} and that there exists $d_1$ such that the vector field
of $H_P$ is smooth as a map from $\Ph_{s+d_1}$ to $\Ph_s$ for any $s$
large enough. Then the same result of theorem \ref{main} holds, but
the functions do not necessarily have localized coefficients and, for
any $s$ large enough {\bf the remainder is estimated by}
\begin{equation}
\label{resto.q}
\norma{X_{\resto^{(r)}}(z)}_{s}\leq C\norma{z}_{s+d_r}^{r+2}\ ,\quad
\forall z\in\U_{s}^{(r)}\ ;
\end{equation}
where $d_r$ is a large positive number.
\end{theorem}

The estimate \eqref{resto.q} shows that the remainder is small only
when considered as an operator extracting a lot of derivatives. In
particular it is non trivial to use such a theorem in order to extract
informations on the dynamics. Following the approach of \cite{BCP} and
\cite{Bam05} this can be done using the normal form to construct
approximate solutions and suitable versions of the Gronwall lemma to
compare it with solutions of the true system. This however allows to
control the dynamics only over times of the order of $\epsilon^{-1}$,
$\epsilon$ being again a measure of the size of the initial
datum. Such a theory has been applied to quasilinear wave equations in
\cite{Bam05} and to the Fermi Pasta Ulam problem in \cite{BP06}. Among
the large number of papers containing related results we recall
\cite{FS87,Kro89,Ver87,Pal96}. A stronger result for the nonlinear
wave equation valid over times of order $\epsilon^{-2}$ can be found
in \cite{DS04}.





\section{Future Directions.}

Future directions of research include both purely theoretical aspects
and applications to other sciences. 

From a purely theoretical point of view the most important open
problems pertain the validity of normal form theory for equation in
which the nonlinearity involves derivatives, and for general equations
in more than one space dimensions.

These results would be particularly important since the kind of
equations appearing in most domains of physics are quasilinear and
higher dimensional. 

Concerning applications we would like to describe a few of them which
would be of interest.

\begin{itemize}
\item Water wave problem. The problem of description of the free
  surface of a fluid has been shown to fit in the scheme of
  Hamiltonian dynamics \cite{zak}. Normal form theory could allow to
  extract the relevant informations on the dynamics in different
  situations \cite{DZ94,Cra96}, ranging from the theory of
  Tsunamis \cite{Cra06} to the theory of fluid interface, which is
  relevant e.g. to the construction of oil platforms \cite{cra05}.

\item Quantum mechanics. A Bose condensate is known to be well
  described by the Gross Pitaevskii equation. When the potential is
  confining, such an equation, is of the form \eqref{e.17}. Normal
  form theory has already been used for some preliminary results
  \cite{BS06}, but a systematic investigation could lead to
  interesting new results.

\item Electromagnetic theory and magnetohydrodynamics. The equations
  have a Hamiltonian form; normal form theory could help to describe
  some instability arising in plasmas.

\item Elastodynamics. Here one of the main theoretical open problem is
  that of the stability of equilibria which are a minimum of the
  energy. The problem is that in higher dimensions the conservation of
  energy does not ensure enough smoothness of the solution to ensure 
  stability. Such a problem is of the same kind of the one solved in
  \cite{BDGS06} when dealing with the existence times of the nonlinear
  wave equation. 
\end{itemize}




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\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
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\providecommand{\href}[2]{#2}
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