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\begin{opening}
\title{Wavelet Approach to Mechanical Problems.\protect\\
Symplectic Group, Symplectic Topology and \protect\\
Symplectic Scales}
\author{A.N.~Fedorova}
\author{M.G.~Zeitlin}
\institute{Institute of Problems of Mechanical Engineering,\\
Russian Academy of Sciences, Russia, 199178, St.~Petersburg,\\
V.O., Bolshoj pr., 61, e-mail: zeitlin@math.ipme.ru}
\end{opening}
\runningtitle{SYMPLECTIC WAVELETS }
\begin{document}
\begin{abstract}
We present the applications of methods from
wavelet analysis to polynomial approximations for
a number of nonlinear problems.
According to the orbit method and by using approach
from the geometric quantization theory we construct the symplectic
and Poisson structures associated with generalized wavelets by
using metaplectic structure. We consider wavelet approach to
the calculations of Melnikov functions in the theory of homoclinic
chaos in perturbed Hamiltonian systems, for parametrization of
Arnold--Weinstein curves in Floer variational approach and
characterization of symplectic Hilbert scales of spaces.
\end{abstract}
{\it 1.~Introduction}.
In this paper we consider the application of powerful methods of wavelet
analysis to polynomial approximations of nonlinear physical and mechanical
pro\-blems. In the related paper in this volume we considered
the general approach
for constructing wavelet representation for
solutions of nonlinear polynomial dynamical problems.
But now we are interested in underlying Hamiltonian
structures.
Therefore, we need to consider generalized wavelets, which
allow us to take into account the corresponding hidden symplectic, Poissonian,
quasicomplex structures, instead of simple
compactly supported wavelet representation.
By using the orbit method and constructions
from the geometric quantization theory we consider the
symplectic and Poisson structures associated with Weyl--\-
Heisenberg wavelets by using metaplectic structure and
the corresponding polarization in \S 2--\S 6. In \S 7 we consider applications
to construction of Melnikov functions in the theory of
homoclinic chaos in perturbed Hamiltonian systems.
In \S 8 we consider the generalization of our variational
wavelet approach [1]--[9] to the symplectic invariant calculation
of Arnold-Weinstein curves (closed loops) in Hamiltonian systems.
In \S 9 we consider wavelet characterization of symplectic Hilbert
scales of spaces, which is important in problems related with
KAM perturbations.
{\it 2.~Metaplectic Group and Representations.}
In wavelet analysis the following three concepts are used now:
1).\ a square integrable representation $U$ of a group $G$,
2).\ coherent states over G,
3).\ the wavelet transform associated to U.
We have three important particular cases:
a) the affine $(ax+b)$ group, which yields the usual wavelet
analysis:
$
[\pi(b,a)f](x)=1/{\sqrt{a}}f\left((x-b)/a)\right)
$,
b). the Weyl-Heisenberg group which leads to the Gabor
functions, i.e. coherent states associated with windowed Fourier
transform:
$
[\pi(q,p,\varphi)f](x)$ $=\exp(i\mu(\varphi-p(x-q))f(x-q)
$,
in both cases time-frequency plane corresponds to the phase
space of group representation.
c). also, we have
the case of bigger group, containing
both affine and We\-yl-\-Hei\-sen\-berg group, which interpolate between
affine wavelet analysis and windowed Fourier analysis: affine
Weyl--Heisenberg group [10]. But usual representation of it is not
square--integrable and must be modified: restriction of the
representation to a suitable quotient space of the group (the
associated phase space in that case) restores square --
integrability:
$G_{aWH\longrightarrow}$ homogeneous space.
Also, we have more general approach which allows to consider wavelets
corresponding to more general groups and representations [11], [12].
Our goal is applications of these results to problems of
Hamiltonian dynamics and as consequence we need to take into account
symplectic nature of our dynamical problem.
Also, the symplectic and wavelet structures
must be consistent (this must
be resemble the symplectic or Lie-Poisson integrator theory).
We use the
point of view of geometric quantization theory (orbit method)
instead of harmonic analysis. Because of this we can consider
(a) -- (c) analogously.
Let $Sp(n)$ be
symplectic group, $Mp(n)$ be its unique two- fold covering --
metaplectic group. Let V be a symplectic vector space
with symplectic form ( , ), then $R\oplus V$ is nilpotent Lie
algebra - Heisenberg algebra:
$$[R,V]=0, \quad [v,w]=(v,w)\in
R,\quad [V,V]=R.$$
$Sp(V)$ is a group of automorphisms of
Heisenberg algebra.
Let N be a group with Lie algebra $R\oplus
V$, i.e. Heisenberg group. By Stone-- von Neumann theorem
Heisenberg group has unique irreducible unitary representation
in which $1\mapsto i$. This representation is projective:
$U_{g_1}U_{g_2}=c(g_1,g_2)\cdot U_{g_1g_2}$, where c is a map:
$Sp(V)\times Sp(V)\rightarrow S^1$, i.e. c is $S^1$-cocycle.
But this representation is unitary representation of universal
covering, i.e. metaplectic group $Mp(V)$. We give this
representation without Stone-von Neumann theorem.\
Consider a new group $F=N'\bowtie Mp(V),\quad \bowtie$ is semidirect
product (we consider instead of $ N=R\oplus V$ the $
N'=S^1\times V, \quad S^1=(R/2\pi Z)$). Let $V^*$ be dual to V,
$G(V^*)$ be automorphism group of $V^*$.Then F is subgroup of $
G(V^*)$, which consists of elements, which acts on $V^*$ by affine
transformations. \\
This is the key point!
Let $q_1,...,q_n;p_1,...,p_n$ be symplectic basis in V,
$\alpha=pdq=\sum p_{i}dq_i $ and $d\alpha$ be symplectic form on
$V^*$. Let M be fixed affine polarization, then for $a\in F$ the
map $a\mapsto \Theta_a$ gives unitary representation of G:
$ \Theta_a: H(M) \rightarrow H(M) $
Explicitly we have for representation of N on H(M):
$$
(\Theta_qf)^*(x)=e^{-iqx}f(x), \quad
\Theta_{p}f(x)=f(x-p)
$$
The representation of N on H(M) is irreducible. Let $A_q,A_p$
be infinitesimal operators of this representation
$$
A_q=\lim_{t\rightarrow 0} \frac{1}{t}[\Theta_{-tq}-I], \quad
A_p=\lim_{t\rightarrow 0} \frac{1}{t}[\Theta_{-tp}-I],
$$
$$\mbox{then}\qquad
A_q f(x)=i(qx)f(x),\quad A_p f(x)=\sum p_j\frac{\partial
f}{\partial x_j}(x)
$$
Now we give the representation of infinitesimal ba\-sic
elements. Lie algebra of the group F is the algebra of all
(non\-ho\-mo\-ge\-ne\-ous) quadratic po\-ly\-no\-mi\-als of (p,q) relatively
Poisson bracket (PB). The basis of this algebra consists of
elements
$1,q_1,...,q_n$,\ $p_1,...,p_n$,\ $ q_i q_j, q_i p_j$,\ $p_i p_j,
\quad i,j=1,...,n,\quad i\leq j$,
\begin{eqnarray*}
& &PB \ is
\quad \{ f,g\}=\sum\frac{\partial f}{\partial p_j}
\frac{\partial g}{\partial q_i}-\frac{\partial f}{\partial q_i}
\frac{\partial g}{\partial p_i} \quad
\mbox{and} \quad
\{1,g \}= 0 \quad for \mbox{ all} \ g,\\
& &\{ p_i,q_j\}= \delta_{ij},\quad \{p_i
q_j,q_k\}=\delta_{ik}q_j,\quad
\{p_i q_j,p_k\}=-\delta_{jk}p_i,\\
& & \{p_ip_j,p_k\}=0,\quad
\{p_i p_j,q_k \}=
\delta_{ik}p_j+\delta_{jk}p_i,\quad
\{ q_i q_j,q_k\}=0,\\
& &\{q_i q_j,p_k\}=-\delta_{ik}q_j-\delta_{jk}q_i
\end{eqnarray*}
so, we have the representation of basic elements
$ f\mapsto A_f : 1\mapsto i, q_k\mapsto ix_k $,
\begin{eqnarray*}
p_l\mapsto\frac{\delta}{\delta x^l}, p_i q_j\mapsto
x^i\frac{\partial}{\partial x^j}+\frac{1}{2}\delta_{ij},\qquad
p_k p_l\mapsto \frac{1}{i}\frac{\partial^k}{\partial x^k\partial
x^l}, q_k q_l\mapsto ix^k x^l
\end{eqnarray*}
This gives the structure of the Poisson mani\-folds to
representation of any (nilpotent) algebra or in other words to
continuous wavelet trans\-form.
{\it 3.~ The Segal-Bargman Representation.}
Let $ z=1/\sqrt{2}\cdot(p-iq),\quad
\bar{z}=1/\sqrt{2}\cdot(p+iq),\quad
$
$ p=(p_1,...,p_n)
,\quad
F_n $ is the
space of holomorphic functions of n complex variables with
$(f,f)< \infty$, where $$ (f,g)=(2\pi)^{-n}\int
f(z)\overline{g(z)}e^{-|z|^2}dpdq $$
Consider a map $U:
H\rightarrow F_n$ , where H is with real polarization, $F_n
$ is with complex polarization, then we have $$(U\Psi)(a)=\int
A(a,q)\Psi(q)dq,\qquad \mbox{where}\quad
A(a,q)=\displaystyle\pi^{-n/4}e^{-1/2(a^2+q^2)+\sqrt{2}aq}
$$
i.e. the Bargmann formula produce wavelets.We also have
the representation of Heisenberg algebra on $F_n$ :
\begin{eqnarray*}
U\frac{\partial}{\partial q_j} U^{-1}=\frac{1}{\sqrt{2}}\left
(z_j- \frac{\partial}{\partial z_j}\right),\qquad
Uq_j
U^{-1}=-\frac{i}{\sqrt{2}}\left(z_j+\frac{\partial }{\partial
z_{j}} \right)
\end{eqnarray*}
and also : $ \omega=d\beta=dp\wedge dq,$
where
$\beta=i\bar{z}dz $.
{\it 4.~Orbital Theory for Wavelets.}
Let coadjoint action be
$=,$
where $<,>$ is pairing
$ g\in G,\quad f\in g^*,\quad Y\in{\cal G}$.
The orbit is
${\cal O}_f=G\cdot f\equiv G/G(f)$.
Also, let A=A(M) be algebra of functions,
V(M) is A-module of vector fields,
$A^p$ is A-module of p-forms.
Vector fields on orbit is
$$
\sigma({\cal O},X)_f(\phi)=\frac{d}{dt}(\phi(\exp tXf))\Big |_{t=0}
$$
where $\phi\in A({\cal O}),\quad f\in{\cal O}$. Then ${\cal O}_f$
are homogeneous symplectic manifolds with 2-form
$
\Omega(\sigma({\cal O},X)_f,\sigma({\cal O},Y)_f)=,
$
and $d\Omega=0$. PB on ${\cal O}$ have the next form
$
\{ \Psi_1,\Psi_2\}=p(\Psi_1)\Psi_2
$
where p is $ A^1({\cal O})\rightarrow V({\cal O})$ with
definition
$\Omega (p(\alpha),X)$ $=$ $i(X)\alpha$. Here $\Psi_1,\Psi_2\in
A(\cal {O})$ and $A({\cal O}) $ is Lie algebra with bracket
\{,\}.
Now let N be a Heisenberg group. Consider adjoint and
coadjoint representations in some particular case.
$N=(z,t)\in C\times R,
z=p+iq$; compositions in N are $(z,t)\cdot(z',t')=
(z+z',t+t'+B(z,z')) $, where $B(z,z')=pq'-qp'$. Inverse
element is $(-t,-z)$. Lie algebra n of N is $(\zeta,\tau)
\in C\times R$ with bracket $[(\zeta,\tau),(\zeta',\tau')]=
(0,B(\zeta,\zeta'))$. Centre is $\tilde{z}\in n $ and
generated by (0,1);
Z is a subgroup $\exp\tilde{z}$.
Adjoint representation N on n is given by formula
$
Ad(z,t)(\zeta,\tau)=(\zeta,\tau+B(z,\zeta))
$
Coadjoint:
for $f\in n^*,\quad g=(z,t)$,
$(g \cdot f)(\zeta,\zeta)=f(\zeta,\tau)-B(z,\zeta)f(0,1)$ then
orbits for which $f|_{\tilde z}\neq 0$ are plane in $n^*$
given by equation $ f(0,1)=\mu$ . If $X=(\zeta,0),\quad
Y=(\zeta ',0),\quad X,Y\in n$ then symplectic structure
is
\begin{eqnarray*}
\Omega (\sigma({\cal O},X)_f,\sigma({\cal
O},Y)_f)==
f(0,B(\zeta,\zeta'))\mu B(\zeta,\zeta')
\end{eqnarray*}
Also we have for orbit ${\cal O}_\mu=N/Z$ and ${\cal O}_\mu $ is
Hamiltonian G-space.
{\it 5.~Kirillov Character Formula or
Analogy of Gabor Wavelets.}
Let U denote irreducible unitary representation of N with
condition $U(0,t)$ $=\exp(it\ell)\cdot 1$, where $ \ell\neq
0 $,then U is equivalent to representation $T_\ell$ which acts in
$L^2(R)$ according to
$$
T_\ell(z,t)\phi(x)=\exp\left(i\ell(t+px)\right)\phi(x-q)
$$
If instead of N we consider E(2)/R we have $S^1$ case and we
have Gabor functions on $S^1$.
{\it 6.~Oscillator Group.}
Let O be an oscillator group, i.e. semidirect product of R
and Heisenberg group N.
Let H,P,Q,I be standard basis in Lie algebra o of the group O
and $H^*,P^*,Q^*,I^*$ be dual basis in $o^*$. Let functional
f=(a,b,c,d) be
$
aI^*+bP^*+cQ^*+dH^* .
$
Let us consider complex polarizations
$
h=( H,I,P+iQ ), \quad
\bar{h}=(I,H,P-iQ)
$
Induced from $h$ representation, corresponding to functional f
(for $a>0$), unitary equivalent to the representation
$$
W(t,n)f(y)=\exp (it(h-1/2)) \cdot U_{a} (n)V(t),
$$
\begin{eqnarray*}
\mbox{where}\qquad V(t)=\exp[-it(P^2+Q^2)/2a] ,\quad
P=-{d}/{dx} ,\quad
Q=iax,
\end{eqnarray*}
and $U_a(n)$ is irreducible representation of N, which have the
form $U_a(z)=exp(iaz)$ on the center of N.
Here we have:
U(n=(x,y,z)) is Schr\"{o}\-din\-ger representation, $U_t(n)=U(t(n))$
is the representation obtained from previous by
automorphism (time translation)
$n\longrightarrow t(n);\quad U_t(n)=U(t(n))$
is also unitary irreducible representation of N.
$
V(t)=\exp(it(P^2+Q^2+h-1/2))
$
is an operator, which according to Stone--von Neumann theorem
has the property
$
U_t(n)=V(t)U(n)V(t)^{-1}.
$
This is our last private case, but according to our approach we
can construct by using methods of geometric quantization theory
many "symplectic wavelet constructions" with corresponding
symplectic or Poisson structure on it.
Very useful particular spline--wavelet basis with uniform
exponential control on stratified and nilpotent Lie groups
was considered in [12]. In particular case of Heisenberg group $N \ni
(p,q,t)$ we may use as bases in our space of representations $L^2(N)$ the
following orthonomal bases [12] $\psi_{i,j,k}$ generated from a finite set
of 15 regular, localized and oscillating functions
$\psi_i: \psi_{i,j,k}=4^j\psi_i(2^j p-k_1, 2^jq-k_2, 2^jt-k_3+2^jk_2p-2^jk_1q)$.
{\it 7.~Melnikov Functions Approach.}
We give now some points of applications
of wavelet methods from the preceding consideration to Melnikov approach
in the theory of
homoclinic chaos in perturbed Hamiltonian systems.
We consider some points of our program of
understanding routes to chaos in Hamiltonian
systems in the wavelet approach [1]-[9]. All points are:
1. A model.
2. A computer zoo. The understanding of
the computer zoo.
3. A naive Melnikov function approach.
4. A naive wavelet description of (hetero) homoclinic orbits
(separatrix) and quasiperiodic oscillations.
5. Symplectic Melnikov function approach.
6. Splitting of separatrix... $\longrightarrow$stochastic web
with magic symmetry, Ar\-nold diffusion and all
that.
1. As a model we have two frequencies perturbations of two-mode Ga\-ler\-kin
approximations to beam oscillations in liquid [7]:
\begin{eqnarray*}
\dot x_1&=&x_2,\qquad \dot x_3=x_4,\qquad \dot x_5=1,\qquad \dot x_6=1,\nonumber\\
\dot x_2&=&-ax_1-b[\cos(rx_5)+\cos(sx_6)]x_1-dx^3_1-
mdx_1x^2_3-
px_2-\varphi(x_5) \nonumber\\
\dot x_4&=&ex_3-f[\cos(rx_5)+\cos(sx_6)]x_3- gx_3^3-
kx_1^2x_3-gx_4-
\psi(x_5) \nonumber
\end{eqnarray*}
or in Hamiltonian form
\begin{eqnarray*}
\dot{x}=J\cdot\nabla H(x)+\varepsilon g(x,\Theta), \
\dot{\Theta}&=&\omega,\
(x,\Theta)\in R^4\times T^2,\
T^2=S^1\times S^1,
\end{eqnarray*}
for $\varepsilon=0 $ we have:
\begin{equation}
\dot{x}=J\cdot\nabla H(x),\quad
\dot\Theta=\omega
\end{equation}
2. For pictures and details one can see [4], [7].
The key point is the
splitting of separatrix (homoclinic orbit) and transition to
fractal sets on the Poincare sections.\\
3. For $\varepsilon=0$ we
have homoclinic orbit $\bar{x}_{0}(t)$ to the hyperbolic fixed
point $x_0$. For $\varepsilon\neq 0$ we have normally hyperbolic
invariant torus $T_{\varepsilon}$ and condition on transversally
intersection of stable and unstable ma\-ni\-folds
$W^s(T_{\varepsilon})$ and $W^u(T_{\varepsilon})$ in terms of
Melnikov functions $M(\Theta)$ for $\bar{x}_{0}(t)$.
$
M(\Theta)=\int\limits_{-\infty}^{\infty}\nabla
H(\bar{x}_{0}(t)) \wedge g(\bar{x}_{0}(t),\omega t+\Theta)dt
$.
This condition has the next form:\\
$
M(\Theta_0)=0, \quad
\sum\limits_{j=1}^{2}\omega_j\frac{\partial}{\partial\Theta_j}
M(\Theta_0)\neq0
$.
According to the approach of Birkhoff-Smale-Wiggins we
determined the region in parameter space in which we observe the
chaotic behaviour [7].\\
4. If we cannot solve equations (1)
explicitly in time, then we use the wavelet approach from our other paper
for the computations of homoclinic (heteroclinic) loops as
the wavelet solutions of system (1).
For computations of quasiperiodic Melnikov functions:
$
M^{m/n}(t_0)=\int^{mT}_0 DH(x_\alpha(t))\wedge g(x_\alpha(t),t+t_0)dt
$,
we used periodization of wavelet solution [3], [4].\\
5. We also used symplectic Melnikov function approach:\\
$
M_i(z)=$ $\lim_{j\rightarrow\infty}\int\limits_{-T_j^*}
^{T_j}\{h_i,\hat{h}\}_{\Psi (t,z)}dt,
d_i(z,\varepsilon)=h_i(z^u_\varepsilon)-h_i(z^s_\varepsilon)=
\varepsilon M_i(z)+O(\varepsilon^2)
$,
where $\{,\}$ is the Poisson bracket,
$d_i(z,\varepsilon)$ is the Melnikov distance. So, we need symplectic
invariant wavelet expressions for Poisson brackets. The computations
are produced according to \S 2--\S 6.\\
6. Some hypothesis about
strange symmetry of stochastic web
in multi-degree-of freedom Hamiltonian systems [8].
{\it 8.~Symplectic Topology and Wavelets.}
Now we consider the generalization of our wavelet variational
approach to symplectic invariant calculation of closed loops in
Hamiltonian systems [13].
We also have the parametrization of our solution by some
reduced algebraical problem but in contrast to the general case where
the solution is parametrized by construction based on scalar
refinement equation, in symplectic case we have
parametrization of the solution
by matrix problems -- Quadratic Mirror Filters equations [14].
The action functional for loops in the phase space is [13]
$$
F(\gamma)=\displaystyle\int_\gamma pdq-\int_0^1H(t,\gamma(t))dt
$$
The critical points of $F$ are those loops $\gamma$, which solve
the Hamiltonian equations associated with the Hamiltonian $H$
and hence are periodic orbits. By the way, all critical points of $F$ are
the saddle points of infinite Morse index, but surprisingly this approach is
very effective. This will be demonstrated using several
variational techniques starting from minimax due to Rabinowitz
and ending with Floer homology. So, $(M,\omega)$ is symplectic
manifolds, $H: M \to R $, $H$ is Hamiltonian, $X_H$ is
unique Hamiltonian vector field defined by
$
\omega(X_H(x),\upsilon)=-dH(x)(\upsilon),\quad \upsilon\in T_xM,
\quad x\in M,
$
where $ \omega$ is the symplectic structure.
A T-periodic solution $x(t)$ of the Hamiltonian equations:
$
\dot x=X_H(x) \quad \mbox{ on $M$}
$
is a solution, satisfying the boundary conditions $x(T)$ $=x(0), T>0$.
Let us consider the loop space $\Omega=C^\infty(S^1, R^{2n})$,
where $S^1=R/{\bf Z}$, of smooth loops in $R^{2n}$.
Let us define a function $\Phi: \Omega\to R $ by setting
$$
\Phi(x)=\displaystyle\int_0^1\frac{1}{2}<-J\dot x, x>dt-
\int_0^1 H(x(t))dt, \quad x\in\Omega
$$
The critical points of $\Phi$ are the periodic solutions of $\dot x=X_H(x)$.
Computing the derivative at $x\in\Omega$ in the direction of $y\in\Omega$,
we find
\begin{eqnarray*}
\Phi'(x)(y)=\frac{d}{d\epsilon}\Phi(x+\epsilon y)\vert_{\epsilon=0}
=
\displaystyle\int_0^1<-J\dot x-\bigtriangledown H(x),y>dt
\end{eqnarray*}
Consequently, $\Phi'(x)(y)=0$ for all $y\in\Omega$ iff the loop $x$ satisfies
the equation
$$
-J\dot x(t)-\bigtriangledown H(x(t))=0,
$$
i.e. $x(t)$ is a solution of the Hamiltonian equations, which also satisfies
$x(0)=x(1)$, i.e. periodic of period 1. Periodic loops may be represented by
their Fourier series:
$
x(t)=\sum_{k\in{\bf Z}}e^{k2\pi Jt}x_k, \quad x_k\in R^{2k},
$
where $J$ is quasicomplex structure. We give relations between
quasicomplex structure and wavelets in \S 9.
But now we use the construction [14]
for loop parametrization. It is based on the theorem about
explicit bijection between the Quadratic Mirror Filters (QMF) and
the whole loop group: $LG: S^1\to G$.
In particular case we have relation between {\bf QMF}-systems and
measurable functions
$\chi: S^1 \to U(2)$ satisfying
\begin{displaymath}
\chi(\omega+\pi)=\chi(\omega)\left [ \begin{array}{ll}
0 & 1\\
1 & 0
\end{array}\right ],
\end{displaymath}
in the next explicit form
\begin{eqnarray*}
\left [ \begin{array}{ll}
\hat\Phi_0(\omega) & \hat\Phi_0(\omega+\pi)\\
\hat\Phi_1(\omega) & \hat\Phi_1(\omega+\pi)
\end{array}\right ]
&=& \chi(\omega)\left [\begin{array}{ll}
0 & 1\\
1 & 0
\end{array}\right ]
+\chi(\omega+\pi)\left [\begin{array}{ll}
0 & 0\\
0 & 1
\end{array}\right ],
\end{eqnarray*}
where
$
\left |\hat\Phi_i(\omega)\right |^2+\left |\hat\Phi_i(\omega+\pi)\right |^2=2,
\quad i=0,1.
$
Also, we have symplectic structure
on $LG$
$$
\omega(\xi,\eta)=\frac{1}{2\pi}\int_0^{2\pi}<\xi(\theta),\eta'(\theta)>d\theta
$$
So, we have the parametrization of periodic orbits (Arnold--Weinstein cur\-ves)
by reduced QMF equations.
{\it 9.~Symplectic Hilbert Scales via Wavelets.}
We can solve many important dynamical problems such that KAM
perturbations, spread of energy to higher modes, week turbulence, growths of
solutions of Hamiltonian equations only if we consider scales of spaces instead
of one functional space. For Hamiltonian system and their perturbations
for which we need take into account underlying symplectic structure we
need to consider symplectic scales of spaces. So, if
$\dot{u}(t)=J\nabla K(u(t))$
is Hamiltonian equation we need wavelet description of symplectic or
quasicomplex structure on the level of functional spaces. It is very
important that according to [16] Hilbert basis is in the same time a
Darboux basis to corresponding symplectic structure.
We need to provide Hilbert scale $\{Z_s\}$ with symplectic structure [15], [17].
All what we need is the following.
$J$ is a linear operator, $J : Z_{\infty}\to Z_\infty$,
$J(Z_\infty)=Z_\infty$, where $Z_\infty =\cap Z_s$.
$J$ determines an isomorphism of scale $\{Z_s\}$ of order $d_J\geq 0$.
The operator $J$ with domain of definition $Z_\infty$ is
antisymmetric in $Z$:
$
_Z=-_Z, z_1,z_2 \in $ $ Z_\infty
$.
Then the triple $\{Z,\{Z_s|s\in R\},\quad
\alpha=<\bar J dz,dz>\}$ is symplectic Hilbert scale. So, we may consider
any dynamical Hamiltonian problem on functional level.
As an example, for KdV equation we have
$ Z_s=\{u(x)\in H^s(T^1)|\int^{2\pi}_0 u(x)dx=0\},\
s\in R,\quad J=\partial/\partial x,$
J is isomorphism of the scale of order one, $\bar J=-(J)^{-1}$ is
isomorphism of order $-1$.
According to [18] general functional spaces and scales of spaces such as
Holder--Zygmund, Triebel--Lizorkin and Sobolev can be characterized
through wavelet coefficients or wavelet transforms. As a rule, the faster
the wavelet coefficients decay, the more the analyzed function is
regular [18]. Most important for us example is the scale of Sobolev spaces.
Let $H_k(R^n)$ is the Hilbert space of all distributions with finite norm
$
\Vert s\Vert^2_{H_k(R^n)}=\int d\xi(1+\vert\xi\vert^2)^{k/2}\vert
\hat s(\xi)\vert^2.
$
Let us consider wavelet transform
$$
W_g f(b,a)=\int_{R^n}dx\frac{1}{a^n}\bar g\lgroup\frac{x-b}{a}\rgroup f(x),
\quad b\in R^n, \quad a>0
$$
w.r.t. analyzing wavelet $g$, which is strictly admissible, i.e.
$$
C_{g,g}=\int_0^\infty\frac{da}{a}\vert\bar{\hat g(ak)}\vert^2<\infty.
$$
Then there is a $c\geq 1$ such that
$$
c^{-1}\Vert s \Vert^2_{H_k(R^n)}\leq\int_{H^n}
\frac{dbda}{a}(1+a^{-2\gamma})\vert W_gs(b,a)\vert^2\leq
c\|s\|^2_{H_k(R^n)}
$$
This shows that localization of the wavelet coefficients at small
scale is linked to local regularity.
Extended version and related results may be found in [1]-[9].
We would like to thank Zohreh Parsa (BNL) for many discussions and
continued encouragement during and after workshop "New Ideas for
Particle Accelerators" and Institute for Theoretical Physics,
University of California, Santa Barbara for hospitality.
This research was supported in part under "New Ideas for Particle
Accelerators Program" NSF-
Grant No.~PHY94-07194.
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\end{document}