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\TITLE Nonlinear Waves and the KAM Theorem:
Nonlinear Degeneracies
\footnote*{\small To appear in the {\smallbold Proceedings
of the Conference on Nonlinear Waves}; Villefranche France }
\AUTHOR Walter Craig
\FROM
Department of Mathematics
Brown University
Providence, RI~ 02912
\AUTHOR C. E. Wayne
\FROM
Department of Mathematics
Pennsylvania State University
University Park, PA~ 16802
\ENDTITLE
\SECTION Introduction
This paper is concerned with solutions of nonlinear wave equations,
and other partial differential equations that model conservative
phenomena in
physics and applied mathematics. As the initial value problem is
increasingly well understood, the focus of our attention
is on the more detailed structure of the phase space in which the
evolution equations are posed. The nonlinear wave equation can be
viewed as an infinite dimensional Hamiltonian system, thus it is natural
to study important classes of periodic and quasiperiodic solutions
in the neighborhood of equilibrium. The paper (Craig \& Wayne [CW])
constructs periodic solutions for nonlinear wave equations, using a
version of the Nash-Moser technique to overcome the inherent small divisor
problem. In that reference, certain generic requirements of
nonresonance and genuine nonlinearity are needed in the existence
proof. This present paper addresses problems in which the hypotheses of
genuine nonlinearity are not satisfied, where nonetheless the
existence of families of periodic solutions near equilibrium are
obtained. Other recent work on the subject of perturbation theory
for Hamiltonian systems with infinitely many degrees of freedom
include Kuksin [K], Wayne [W], P\"oschel [P] and Albanese,
Fr\"ohlich and Spencer [AFS].
Some of the more interesting aspects of our approach
to these problems are the ties between partial differential equations,
Hamiltonian mechanics, and localization theory of mathematical
physics. Indeed, the central estimates in this work were pioneered
by Fr\"ohlich and Spencer [FS] in the study of the Green's function
for random Schr\"odinger operators. The nonlinear wave equation
is not the only equation of interest which has Hamiltonian structure,
for which results on periodic and quasiperiodic solutions are of
interest. We expect the techniques of [CW] and of this paper
to extend to the nonlinear Schr\"odinger equation, versions of
the KdV equation
and other problems with infinitely many degrees of freedom, for which
the equilibrium solution is an elliptic stationary point.
Moreover we expect the analysis of quasiperiodic solutions
to be similar to the analysis of periodic solutions in these
resonant cases, and plan a further publication on this subject.
This paper describes the construction of periodic solutions of
the nonlinear wave equation
$$
\partial_t^2 u = \partial_x^2 u - g(x,u)~~, \qquad
0 \leq x \leq \pi ~~,
\EQ(NLWaves)
$$
where the solution $u(x,t)$ satisfies either periodic or
Dirichlet boundary conditions at $x=0,\pi$. The nonlinear
term $g(x,u)$ is taken analytic, with the Taylor expansion in
the variable $u$ given by
$$
g(x,u) = g_1(x)u + g_2(x)u^2 + g_3(x)u^3 + \cdots.
\EQ(NLTerm)
$$
Well known examples are the sine-Gordon equation
$$
\partial_t^2 u = \partial_x^2 u - b^2\sin(u),
\EQ(sineGordon)
$$
and the $\phi^d$-nonlinear Klein-Gordon equation,
$$
\partial_t^2 u = \partial_x^2 u - b^2 u + u^{d-1}.
\EQ(KleinGordon)
$$
All of the above partial differential equations can be
considered as Hamiltonian systems with infinitely many
degrees of freedom. Indeed, we may define the Hamiltonian
$$
H(p,u) = \int {1 \over 2} p^2 + {1 \over 2} (\partial_x u)^2
+R(x,u) \, dx~~,
\EQ(Hamiltonian)
$$
with $\partial_u R(x,u) = g(x,u)$. Denoting $z = (u,p)^T$,
Hamilton's canonical equations read
$$
{\dot z} = J \nabla H(z),
\EQ(hameqns)
$$
where $J$ denotes the standard symplectic matrix.
The methods of this paper are
perturbative -- we construct solutions near the equilibrium point
$u=0$. For the wave equation \equ(NLWaves) $z = 0$ is
elliptic, thus by analogy with finite dimensional problems
one expects that the construction of quasiperiodic
solutions encounters small divisor problems, and a form
of the KAM theorem would be used. In fact the small divisor
problem arises even in the construction of periodic solutions,
as the presence of infinitely many degrees of freedom
introduces a dense set of resonances.
In the reference [CW] the existence theory for periodic solutions
is discussed, under hypotheses of nonresonance and genuine
nonlinearity. The results are essentially that there is an open
dense set $\GG$ of nonlinearities such that for
$g(x,\cdot) \in \GG$, there exist families of periodic
solutions of \equ(NLWaves). The character of these families is
typically that of a Cantor set foliated by invariant circles ---
a situation reminiscent of the conclusion of the KAM theorem
for quasiperiodic solutions in finite dimensional Hamiltonian
perturbation theory. In the results of [CW], the good set $\GG$
depends only upon $g_1(x), g_2(x)$, and $g_3(x)$ , the
{\sl 3--jet} of the nonlinearity $g(x,\cdot)$.
In the present paper we extend the results of [CW] to
cases which are equally nonresonant, but which
are nonlinearly degenerate. These problems fail to satisfy
the `twist condition' of the previous results, thus
the present work enlarges the class $\GG$ of nonlinearities
for which an existence theorem holds. For example, consider
the nonlinear term
$$
g(x,u) = g_1(x)u + g_M(x)u^M + \cdots~~.
\EQ(NLTerm)
$$
For $M > 3$, $g$ is not in the set $\GG$,
for the curvature of any
approximate solution branch will vanish. Among other
situations this appears for the nonlinear $\phi^d$
Klein-Gordon equation with $d > 4$. We show in this
paper that under more subtle conditions of
nondegeneracy, again families of periodic solutions
of the wave equation \equ(NLWaves) can be constructed.
These conditions depend upon the coefficient $g_1(x)$
of course, and if $M$ is odd, upon the first
nonzero coefficient $g_M(x)$ of the nonlinear term.
If the first nonzero coefficient $g_M(x)$ has even
order, then the existence criterion depends upon
a certain subset of the $(2M-1)$--{\sl jet} of
$g(x,u)$, that is, upon certain of the coefficients
$\{ g_1(x), \cdots , g_{2M-1}(x) \}$.
We feel that all these results are quite general,
and will extend to other equations and to the
construction of quasiperiodic solutions as well.
We point out that in the study of quasiperiodic
solutions, the analysis of higher order nonlinear
degeneracies has not been carried out, even in the
case of finite dimensional Hamiltonian systems in
the neighborhood of an elliptic stationary point.
\noindent{\bf Acknowledgements:} The authors would
like to thank the Universit\'e de Gen\`eve, the
Universit\'e de Paris 6, MSRI--Berkeley and Oxford
University for their hospitality, and the National
Science Foundation and the Alfred P. Sloan Foundation
for their support of our research.
\SECTION Results
It is instructive to solve the equation linearized
about $u = 0$,
$$
\partial_t^2 v = \partial_x^2 v - g_1(x)v~~~.
\EQ(LWaves)
$$
This is done by the elementary method of separation of
variables. Let $\{ (\psi_j(x),\omega^2_j) \}_{j=1}^\infty$
be the complete set of eigenfunction---eigenvalue pairs
for the linear Sturm-Liouville operator
$$
L(g_1) \psi = \bigl( -{d^2 \over dx^2} + g_1(x) \bigr) \psi
= \omega^2 \psi,
$$
imposing the proper boundary conditions,
($\psi(0) = \psi(\pi) = 0$ in the Dirichlet case, and
$\psi(x) = \psi(x + \pi)$ in the periodic case.) We will
assume that all $\omega^2$ are positive with little loss of
generality. Then a periodic solution to \equ(LWaves)
is given by
$$\eqalign{
v(x,t) = & r\cos(\Omega t + \xi) \psi_j(x) \cr
\Omega = & \omega_j~~~~~. \cr}
$$
The general solution of \equ(LWaves) is given by sums
of these solutions
$$
v(x,t) = \sum_{j=1}^\infty
r_j \cos(\omega_j t + \xi_j) \psi_j(x),
$$
parametrized by angles $\{ \xi_j \}_{j=1}^\infty$, and
amplitudes $\{ r_j \}_{j=1}^\infty$, (action variables
$\{ r_j^2 \}_{j=1}^\infty$.) These are not usually
periodic, but typically quasiperiodic if at most
finitely many amplitudes $r_j$ are nonzero, and in general
they are almost periodic, unless a full set of
rational relations (infinitely many)
exist among the frequencies
$\{ \omega_j \}$. Thus it is a natural question to pose whether
some of these periodic (or quasiperiodic, or almost periodic)
solutions persist for the nonlinear problem.
\noindent{\bf Hypothesis:}
(i) Let $g(x,u)$ be $\pi$ periodic in $x$, and analytic
in the strip
$\{ |{\rm Im} \ x | < {\overline \sigma} \}$ and in $u$
in some neighborhood of the origin.
In the case of Dirichlet boundary conditions we also ask
that $g$ be odd in the $(x,u)$--plane.
\noindent
(ii) We assume that in \equ(NLTerm), $M > 3$.
The cases $ M = 2,3$ were discussed in reference [1].
Define $m = M-1$ if $M$ is odd, and $m = ({\rm min} \
\{ R; M < R \leq 2M-1, \ R \ {\rm odd}, \
{\rm and} \ g_R(x) \not= 0 \} - 1)$
if $M$ is even. If no such $R$ exists, set $m=2M-2$.
\CLAIM Theorem(NLThm) There exists a generic set $\GG_M$
such that if $g \in \GG_M$ then there are uncountably
many small periodic solutions to the nonlinear equation
\equ(NLWaves). Furthermore
\item{(i)} The solutions are analytic in a smaller strip
$\{ |{\rm Im} \ x | < {\overline \sigma}/2 \}$.
\item{(ii)} The solutions are close to the linear
periodic solutions, and form a Cantor set foliated
by circles. More precisely, there is a small $r_0$
and a Cantor set $\CC \in (-r_0,r_0)$ such that
if $r \in \CC$ then there is an angle $\xi$ such that
$$\eqalign{
|u(x,t;r) - r \cos(\Omega(r) t + & \xi) \psi_j(x)|
\leq Cr^M , \cr
|\Omega(r) - \omega_j| & \leq C r^m. \cr}
\EQ(difference)
$$
\item{(iii)} The good set $\GG_M$ is open and dense.
If $M$ is odd, $\GG_M$
depends only upon the coefficients $g_1(x)$ and
$g_M(x)$. If $M$ is even, it depends upon $g_1(x)$
and $g_R(x)$, for the minimum $R$ odd, $M < R < 2M-1$,
$g_R(x) \not= 0$. If there is no such $R$, then
$\GG_M$ depends upon $g_1(x), g_M(x)$, and $g_{2M-1}(x)$.
For an exact description of the topology in which
$\GG_M$ is dense, see [CW] section 6.
An immediate corollary applies to a specific choice
of nonlinearity. Consider the $\phi^d$ Klein-Gordon
equation \equ(KleinGordon) on the interval $[0,\pi]$.
For $d=4$ this is addressed in [CW], however for $d>5$
it fails to satisfy the hypothesis of genuine nonlinearity
of that paper. When periodic boundary conditions are
imposed, the problem can be reduced to an analysis of
the phase plane for a solution $u(x-ct)$. When Dirichlet
conditions are imposed this is not the case. For $d$ even,
\clm(NLThm) applies, giving the following result.
\CLAIM Corollary(NLKGThm) For an open set of parameters
$b^2$ of full measure, \equ(KleinGordon) has nonlinearity
within the good set $\GG_M$, and therefore there exist
families of periodic solutions, as described in \clm(NLThm).
This particular dependence of the condition of genuine
nonlinearity, and the power $m$ on the coefficients, is
natural in terms of the Birkhoff normal form for a dynamical
system in the neighborhood of an elliptic stationary point.
That is, odd terms in the Hamiltonian (even terms of the
nonlinearity) are generically nonresonant, and do not enter
the normal form at highest order. Even terms
of the Hamiltonian (odd terms of the nonlinearity) are
generically resonant, affecting the normal form and
the frequency of the solution at highest order. Furthermore,
the next to highest order corrections appear at order $2M-1$.
We remark here that for any $g_2(x), g_3(x), \dots$,
if $g_1(x)=0$ then the conditions of nonresonance of [CW] are
violated. Indeed both the Dirichlet problem and the
periodic problem are infinitely resonant,
as the equation linearized about $u=0$ is
$$
\partial_t^2 v = \partial_x^2 v,
$$
which has an infinite dimensional null space, spanned
respectively by
the functions $\{ \sin(\ell x) e^{\pm i \ell t} \}$,
$\{ \cos(2\ell x) e^{\pm 2i \ell t}, \sin(2\ell x)
e^{\pm 2i \ell t} \}$.
Problems which violate the nonresonance condition, with a
finite but possibly large null space will be addressed
in a subsequent paper. Other than solutions with rational
period that are obtained by global variational
methods [B,R], the infinitely resonant case has
not been addressed, so far as we know.
\SECTION A nonlinear lattice system
We will take the point of view of embedding
a circle into phase space, in such a manner that it
is invariant with respect to the flow determined by
the wave equation \equ(NLWaves). Denoting an
embedded circle by
$$\eqalign{
S(x,\xi) & = \sum_{j=1}^\infty s_j(\xi) \psi_j(x) \cr
s_j(\xi) & = s_j(\xi + 2\pi), \cr}
\EQ(embed)
$$
it will be invariant under flow by the wave equation,
and traversed with frequency $\Omega$, if $S(x,\xi)$
satisfies
$$
\Omega^2 \partial_\xi^2 S - \partial_x^2 S
+ g(x,S) = 0.
\EQ(Sembed)
$$
To treat the spatial and temporal variables on an
equal footing, one expands $s_j$
in Fourier series
$$
S(x,\xi) = \sum_{j=1 \atop k=-\infty}^\infty \
{\widetilde s}(j,k) \ e^{i k \xi} \psi_j(x).
$$
If $S(x,\xi)$ satisfies \equ(Sembed),
the coefficients of this
eigenfunction expansion of $S$ satisfy an equation
over the lattice, $(j,k) \in \ZZ^+ \times \ZZ$,
$$\eqalign{
0 = & (\omega_j^2 - \Omega^2 k^2){\widetilde s}(j,k)
+ W({\widetilde s})(j,k) \cr
= & V(\Omega){\widetilde s}(j,k)
+ W({\widetilde s})(j,k)~~~. \cr}
\EQ(modeinter)
$$
We call this the `mode interaction equation' of
the nonlinear problem \equ(Sembed).
The term $V(\Omega)$ is diagonal in the given basis,
while $W({\widetilde s})$ is nonlinear, and at least of
order $M$ for small ${\widetilde s}$. Linearizing
about ${\widetilde s} = 0$, we have
$$
V(\Omega)\phi = 0,
$$
with solutions $(\phi,\Omega) =
(\delta(j_0,\pm k_0),\omega_{j_0}/k_0)$
corresponding to a periodic solution of
\equ(LWaves). The point spectrum of $V(\Omega)$
is typically dense in the real line, in
particular it accumulates at zero;
this is often called the phenomenon of small divisors.
The fact that point spectra of the linearized problem
approach zero is the fundamental difficulty
of the problem. The technique that is presented in
[CW] and this paper shows that the
geometry of the lattice sites associated with
the small divisors also plays an important role
in the existence theory.
This lattice equation has
certain symmetries which are relevant to
the problem. Let $x = (j,k) \in
\ZZ^+ \times \ZZ$, and write
${\overline x} = (j,-k)$. Then $S$ is real
if and only if ${\widetilde s}({\overline x})
= {\overline {\widetilde s}(x)}$. The equation
respects this condition, for
${\overline {V(\Omega)(x)}} = V(\Omega)({\overline x})$,
and ${\overline {W({\widetilde s}(x))}}
= W({\overline {{\widetilde s}(x)}})$.
Additionally there is the symmetry of an autonomous
system; for $ x = (j,k)$, define $T_\xi {\widetilde s}(x)
= e^{i k\xi} {\widetilde s}(x)$. This is a
unitary operator on $\ell^2(\ZZ^+ \times \ZZ)$.
The lattice equation is covariant with respect
to $T_\xi$, indeed $T_\xi$ commutes with $V(\Omega)$, and
$$
T_\xi W({\widetilde s})(x) = W(T_\xi {\widetilde s})(x).
$$
Other group actions may also respect the equation
\equ(NLWaves), however these will not be addressed
in this paper.
The existence theory is started by solving an
approximate problem, given by projection of
\equ(modeinter) onto a finite subregion of the
lattice; $B_0 = \{ x \in \ZZ^+ \times \ZZ;
|x| \leq L_0 \}$. The approximate problem
is solved under conditions of linear
nonresonance. Fix a constant $\tau > m + 3$.
\CLAIM Definition(nonres)
Define $\omega \equiv \omega_{j_0}/k_0$.
The frequency sequence $\{ \omega_j \}_{j=1}^\infty$
is $(d_0,L_0)$--nonresonant with
$\omega$ if
$L_0 >> |j_0|+|k_0|$, and
\item{(i)} for all $0 < |j|+|k| \leq L_0$,
$$
|k - \omega j| \geq {d_0 \over (|j|+|k|)^\tau}.
$$
\item{(ii)} For all $(j,k) \not= (j_0,\pm k_0)$,
with $|j|+|k| \leq L_0$,
$$
|\omega_j^2 - \omega^2 k^2| \geq d_0.
$$
\noindent{\bf Note:} If a sequence of $L_0 \to \infty$,
with $d_0 = o(L_0^{-1/2})$, then an open dense set of
coefficients $g_1(x)$ are $(d_0,L_0)$--nonresonant
with $\omega$ for some $L_0$. This is a result from [CW].
Writing $\Pi_0 V(\Omega) = V_0(\Omega)$, and
$\Pi_0 W(\Pi_0 {\widetilde s}) = W_0({\widetilde s})$,
the approximate equations on $\ell^2(B_0)$
are written
$$
V_0(\Omega){\widetilde s} + W_0({\widetilde s}) = 0.
\EQ(b0eqn)
$$
Then the linearized equation about ${\widetilde s} = 0$
is simply
$$
V_0(\Omega)(\delta {\widetilde s}) = 0.
\EQ(b0leqn)
$$
This linear operator has a nontrivial null space for
$\Omega = \omega = \omega_{j_0}/k_0$. Since the
problem is $(d_0,L_0)$--nonresonant, the null space is
two dimensional, spanned by
$\phi(p) = p\delta(j_0,k_0) +
{\overline p} \delta(j_0,-k_0)$, with $p \in \complex$.
Let $N = \{(j_0,k_0), (j_0,-k_0)\}$, the support
of the null vectors, and define orthogonal projections
$Q$ onto $\ell^2(N)$, and $P = (\11 - Q)$.
Equation \equ(b0eqn)
is solved via a Lyapounov-Schmidt decomposition.
$$
P\bigl( V_0(\Omega)u_0
+ W_0(\phi(p) + u_0) \bigr) = 0
\EQ(decompPb0)
$$
$$\eqalign{
Q\bigl( V_0(\Omega)\phi(p) &
+ W_0(\phi(p) + u_0) \bigr) = 0 \cr
u_0 = Pu_0 & \cr}
\EQ(decompQb0)
$$
Define spaces that account for exponential decay
of sequences;
$\HH_\sigma = \{ u \in \ell^2(\ZZ^+ \times \ZZ);
\|u\|_\sigma^2 \equiv \sum_{x \in \ZZ^+ \times \ZZ}
e^{2\sigma|x|} |u(x)|^2 < \infty \}$.
These form a scale of Hilbert spaces,
$\HH_\sigma \subseteq \HH_{\sigma-\gamma}$ for all
$0 \leq \gamma \leq \sigma$. We ask of the nonlinear
term that $W \in C^\omega (\HH_\sigma:\HH_{\sigma-\gamma})$
for all $0 < \gamma \leq \sigma < {\overline \sigma}$,
with norms
$$\eqalign{
\|W(u)\|_{\sigma-\gamma} & \leq {C_W \over \gamma^{M+1}}
\|u\|_\sigma^M \cr
\|D_u W(u) v \|_{\sigma-\gamma} &
\leq {C_W \over \gamma^{M+1}}
\|u\|_\sigma^{M-1}\|v\|_{\sigma-\gamma} \cr
\|D_u^2 W(u)[w,v]\|_{\sigma-\gamma} &
\leq {C_W \over \gamma^{M+1}}
\|u\|_\sigma^{M-2} \|w\|_\sigma
\|v\|_{\sigma-\gamma}~~~. \cr}
$$
The Taylor expansion of $W$ takes the form
$W(u) = W^{(M)}(u) + W^{(M+1)}(u) + \cdots$, where the
term $W^{(J)}$ is $J$--multilinear in $u$. We will
assume that $W^{(J)}$ is $J$--multilinear and symmetric
in $u$, although the symmetry is not essential for the
existence theorem.
The lattice nonlinearity that comes from the nonlinear
wave equation satisfies the above conditions.
\CLAIM Lemma(solnb0)
For $r_0^m < (d_0/3L_0^2), \rho_0=r_0$, the equation
\equ(decompPb0) has a solution $u_0(x;p,\Omega)$ which
is analytic in a complex $\rho_0$--neighborhood of the
set $\Eta_0 \equiv \{(p,\Omega); \|p\| < r_0, \
|\Omega - \omega| < r_0^m \}$. Furthermore, for
${\overline \sigma}/2 < \sigma_0 <
{\overline \sigma} - 1/L_0$ there is an estimate
$$
\|u_0(x;p,\Omega)\|_{\sigma_0}
\leq \|p\|^M {3C_W L_0 \over d_0 }.
$$
This solution is covariant with respect to the
translations $T_\xi$,
$$
T_\xi u_0(x;p,\Omega) = u_0(x;T_\xi p,\Omega)
$$
(where by notational abuse we denote rotations
in the $p$--plane also by $T_\xi$.)
These sequences form a family of embedded circles,
parametrized by $(\|p\|, \xi, \Omega)$,
which are solutions
of the approximate problem \equ(decompPb0).
To finish the approximate bifurcation problem, equation
\equ(decompQb0) is also solved. This is in the form
of a mapping, taking $(p,\Omega) \in \Eta_0 \to \real^2$.
The zero set of the mapping consists locally of the
$\Omega$ axis $\{ p=0\}$, and a surface
$(p,\Omega_0(p))$ given as a graph over a
neighborhood of zero in $\ell^2(N)$. A simple
analysis of the Taylor expansion of this mapping
determines that
$$
\Omega_0(p) = \omega
+ \lambda^{(m)}_0 \|p\|^m (1 + o(\|p\|)).
\EQ(Omega0)
$$
A straightforward perturbation expansion, which is
left to the reader, will determine the constant
$\lambda^{(m)}_0$. If $M$ is odd, then $m = M-1$ and
$$
\lambda^{(m)}_0 = {1 \over 2 k_0^2 \omega}
{\langle \phi(p) | W_0^{(M)}[(\phi(p))^M]
\rangle \over \|\phi(p)\|^{M+1} }~~~.
\EQ(lambda0odd)
$$
When $M$ is even, take $R$ to be the least odd index,
$M < R \leq 2M-1$, such that $W_0^{(R)} \not\equiv 0$.
If $R < 2M-1$, then $m = R - 1$, and
$$
\lambda_0^{(m)} = { 1 \over 2 k_0^2 \omega}
{ \langle \phi(p) | W_0^{(R)}[(\phi(p))^R] \rangle
\over \|\phi(p)\|^{R+1} }~~~.
\EQ(lambda0even1)
$$
If $R = 2M-1$, or there is no such $R$, then
$m = 2M-2$, and the
perturbation theory determines first that
$$
u_0^{(M)}(x;p,\Omega)
= -\bigl( PV_0(\Omega)P \bigr)^{-1}
P( W_0^{(M)}[(\phi(p))^M] )~~~,
$$
and then
$$\eqalign{
\Omega_0(p) = & \omega + { 1 \over 2 k_0^2 \omega}
{ \langle \phi(p) | W_0^{(2M-1)}[(\phi(p))^{2M-1}] \rangle
\over \|\phi(p)\|^2 } \cr
& + { M \over 2 k_0^2 \omega}
{ \langle \phi(p) |
W_0^{(M)}[(\phi(p))^{M-1},u_0^{(M)}] \rangle
\over \|\phi(p)\|^2 } + o(\|p\|^{2M-2})~~~. \cr}
\EQ(lambda2M)
$$
This perturbation analysis generalizes the formal results
of [KT], regarding solutions of the nonlinear Klein-Gordon
equation.
The analog of the `twist condition' of [CW] is a
condition on the nonvanishing of the coefficients
$\lambda_0^{(m)}$. This will ensure that the
dependence of the frequency of the solution upon
the amplitude is sufficiently nondegenerate.
The full nonlinear equations \equ(modeinter) are also
considered in a Lyapounov-Schmidt decomposition
$$
P \bigl( V(\Omega)u + W(\phi(p) + u) \bigr) = 0,
\EQ(decompP)
$$
$$
Q \bigl( V(\Omega)\phi(p) + W(\phi(p) + u) \bigr) = 0.
\EQ(decompQ)
$$
The approximate solution $u_0$ of \equ(decompPb0) is
a close approximation to the full equation \equ(decompP),
for it satisfies the estimate
$$
\|P\bigl( V(\Omega)u_0
+ W(\phi(p) + u_0) \bigr)\|_{\sigma_0-\gamma_0}
\leq {C_W \|p\|^M \over \gamma_0^{M+1} }
e^{-\gamma_0 L_0}.
\EQ(b0est)
$$
However, to adjust this approximate solution to a
full solution involves the small divisor problem.
The exact solution is obtained not over all of the
parameter region $\Eta_0$, but on a closed Cantor
subset $\Eta \subseteq \Eta_0$, on which the
resonances of the problem are under better control.
The solutions are obtained using Newton iteration
steps in conjunction with approximations of the
lattice $\ZZ^+ \times \ZZ$ by an increasing family
of finite subdomains $B_n = \{ x \in \ZZ^+ \times \ZZ;
|x| \leq L_02^n \}$. To state the existence result,
fix $ 1/2 < \eta < 1$.
\CLAIM Theorem(NLPsoln)
Assume that the sequence $\{\omega_j\}_{j=1}^\infty$ is
$(d_0,L_0)$--nonresonant with $\omega$ for
$d_0 \geq L_0^{-\eta}$, for $L_0$ sufficiently large.
Then there is a constant
$r_0$, a sequence
$u(x;p,\Omega) \in \HH_{{\overline \sigma}/2}$
which is $C^\infty$ on $\Eta_0 = \Eta_0(r_0)$,
and a Cantor subset $\Eta \subseteq \Eta_0$
such that for $(p,\Omega) \in \Eta$, $u$ is a
solution of \equ(decompP). Furthermore
$$
\|u - u_0\|_{{\overline \sigma}/2}
\leq C \|p\|^M e^{-\gamma_0 L_0 /2}.
\EQ(estdiff)
$$
The second bifurcation equation \equ(decompQ) can
also be solved, giving a $C^\infty$,
$T_\xi$--invariant solution surface $(p,\Omega(p))$
in addition to the trivial branch of solutions
$p = 0$. This solution surface is close to the
approximate surface $(p,\Omega_0(p))$, however
unless we specify further conditions it will
not necessarily intersect the remaining set
$\Eta$, and \equ(decompP) and \equ(decompQ)
will not be simultaneously satisfied.
We ask that in addition to being $(d_0,L_0)$--
nonresonant, the approximate nonlinear problem
satisfies a {\bf twist condition}. Then the surface
$(p,\Omega(p))$ will intersect $\Eta$, giving
rise to solutions of the full problem \equ(modeinter).
For the following we fix $0 < \nu < (1 - \eta)$.
\CLAIM Theorem(NLQsoln)
If the approximate problem \equ(b0eqn) also satisfies
the quantitative twist condition
$|\lambda^{(m)}_0| \geq L_0^{-\nu}$, then the
solution surface $(p,\Omega(p))$ of \equ(decompQ) intersects
$\Eta$. Define $\CC = \{ 0 < r < r_0; \|p\| = r, \
(p,\Omega(p)) \in \Eta \}$, the set for which
a solution of the full problem is obtained.
Then ${\rm meas} \, (\CC) > 0$, and is in fact of
order $r_0$.
The intersection points correspond to analytic
solutions of the nonlinear wave equation \equ(NLWaves),
through their eigenfunction expansion. This proves
\clm(NLThm) of the previous section.
Through exact or near resonance, the Cantor set
$\CC$ may not have $r=0$ as an accumulation point,
for $(p,\Omega) = (0, \omega)$ may be too resonant,
and not in $\Eta$. However if the frequency sequence
$\{ \omega_j \}_{j=1}^\infty$ is fully nonresonant
with $\omega$, then $\CC$ does accumulate at zero,
and in addition there is an estimate of its density
nearby. Let ${\overline \tau} > m + 3$ and
${\overline \alpha} > {\overline \tau} + 1$ be fixed.
\CLAIM Theorem(density)
Suppose that a $(d_0,L_0)$--nonresonant sequence
$\{ \omega_j \}_{j=1}^\infty$ satisfies the
conditions of full nonresonance.
\item{(i)} For all $0 < |(j,k)| < \infty$,
$$
|k - \omega j| \geq
{d_0 \over (|j| + |k|)^{\overline \tau} }.
$$
\item{(ii)} For all $(j,k) \not= (j_0,\pm k_0)$,
$$
|\omega^2 k^2 - \omega_j^2| \geq
{d_0 \over (|j| + |k|)^{\overline \alpha} }.
$$
Define $\CC(r_1) = \CC \cap [0,r_1]$. Then there
is an exponent ${\overline \mu}$ such that
$$
{\rm meas} \, (\CC(r_1)) \geq
r_1(1 - C r_1^{\overline \mu})
$$
for all $0 < r_1 < r_0$.
One can additionally make an estimate of the size of
${\overline \mu}$, there are similar estimates in [CW].
\SECTION Proof of \clm(NLPsoln)
The proof is via a modified Newton iteration scheme,
similar to the Nash-Moser method. The major difference
is the presence of a null space, and the
sensitive parametric dependence of the
approximate solutions and the linearized operator.
Thus during the iteration, an acceptable set of
parameters must be chosen as well, resulting
ultimately in the Cantor set $\Eta$ on which
the first bifurcation equation \equ(decompP) is
solved. The second bifurcation equation is a
finite dimensional mapping. The zero set
corresponding to a nontrivial
solution is given by a graph $(p,\Omega(p))$,
which gives a relationship between the action
and the frequency of a solution, called
the {\bf frequency map}. This exhibits one of
the differences of the present technique from
the more classical versions of the KAM theorem,
in which the problem is assumed
nonlinearly nondegenerate, the frequency map
is performed first, and only then does the
analysis of the invariant sets take place.
An outline of the iteration is as follows. We
choose:
\item{(1)} A sequence of length scales $L_n = L_02^n$
which define the approximating domains
$B_n = \{ |x| \leq L_n \}$ which exhaust
$\ZZ^+ \times \ZZ$.
\item{(2)} A sequence of tolerances for small divisors
(small eigenvalues) $\delta_n = L_n^{-\alpha}$, for a
suitable $\alpha > 0$.
\item{(3)} A sequence of lengths $\ell_n = L_n^\beta$
over which linear resonances are decoupled.
\item{(4)} A sequence $\gamma_n = c_0/(n+1)^2$
which governs loss of exponential decay of the
approximate solutions throughout the the iteration.
\item{(5)} And a rapidly convergent sequence
$\epsilon_n = \epsilon_0^{\kappa^n}$, for
$1 < \kappa < 2$, which will bound the error
terms during the iteration.
The size of the error is dominated by a rapidly
convergent sequence as the iteration scheme has
quadratic errors; this is the usual phenomenon
with the Nash-Moser technique.
The major issue to contend with is the
invertibility of relevant linearized operators.
Let $B \subseteq \ZZ^+ \times \ZZ$ be a subdomain
of the lattice. We define the {\bf Hamiltonian operator}
on $\ell^2(B)$ by
$$
H_B(p,\Omega;u) = \bigl(
V(\Omega) + D_u W(\phi(p) + u) \bigr)_B~~.
$$
The subscript $B$ denotes the restriction of the
operators to $\ell^2(B)$. Invertibility depends
crucially upon the small spectra of the operator
$V_B(\Omega)$, as the following result demonstrates.
\CLAIM Lemma(nonres)
Let $A \subseteq \ZZ^+ \times \ZZ$ be a domain
such that $|V(\Omega)(x,x)| > d_0$ for all
$x \in A$. Then for $r_0^{m-1}/d_0 << 1$ the
Green's function
$$
G_A(x,y) = \bigl(
V(\Omega) + D_u W(\phi(p) + u) \bigr)^{-1}_A(x,y)~~
$$
satisfies the estimate
$$
\|G_A\|_{\sigma_0} \leq {C \over d_0}~~~.
$$
We call a lattice site $s \in \ZZ^+ \times \ZZ$
{\bf singular} if $|V(\Omega)(s,s)| < d_0 $, and
regular otherwise.
Connected regions of singular sites are called
singular regions. The wave equation has singular
regions consisting of either isolated sites, or
else two adjacent sites, $S=\{s_1,s_2\}$, with
$s_1 = (j,k),\ s_2 = (j+1,k)$. We will consider
local Hamiltonians defined on neighborhoods of
singular regions. Let $S \subseteq B_{n+1}
\backslash B_n$ and $C_{\ell_{n+1}}(S) = \{ x;
{\rm dist} \, (x,S) \leq \ell_{n+1} \}$. We
will be concerned with the operators
$H_S(p,\Omega;u_n)$ and
$H_{C_{\ell_{n+1}}(S)}(p,\Omega;u_n)$.
The proof of \clm(NLPsoln) is by induction on the
following statements.
\noindent
$(n.1)$ There is a sequence $u_n(x;p,\Omega) =
u_0(x;p,\Omega) + \sum_{j=0}^n v_j(x;p,\Omega)$
in $\ell^2(B_{n+1})$, which is $C^\infty$ on $\Eta_0$,
analytic in a $\delta_{n+1}r_0/L_n^2$ complex
neighborhood of $\Eta_{n+1}$ such that
$$\eqalign{
\|P(V(\Omega)u_n + & W(\phi(p) + u_n))\|_{\sigma_n}
\leq \|p\|^M \epsilon_n \cr
\|v_n\|_{\sigma_n-\gamma_n} \leq &
{ C_G^n \epsilon_n \over \delta_{n+1} \gamma_n^s }
\|p\|^M \cr}
$$
for some fixed constant $s$.
\noindent
$(n.2)$ There exists a closed domain
$\Eta_{n+1} \subseteq \Eta_n \subseteq \cdots \Eta_0$
with the following properties.
\item{(i)} If $(p,\Omega) \in \Eta_{n+1}$, and
$S_1,S_2 \subseteq B_n^c$ are any two singular
regions, then
$$
{\rm dist} \, (S_1,S_2) > 2\ell_{n+1}~~~.
$$
\item{(ii)} If $(p,\Omega) \in \Eta_{n+1}$, and
$S$ is a singular region in
$B_{n+1} \backslash B_n$, then
$$\eqalign{
{\rm dist} \, ({\rm spec} \, &
(H_S(p,\Omega;u_n)),0) > \delta_{n+1} \cr
{\rm dist} \, ({\rm spec} \, &
(H_{C_{\ell{n+1}}(S)}(p,\Omega;u_n)),0)
> \delta_{n+1}~~~. \cr}
$$
\item{(iii)} Any $C^\infty$, $T_\xi$--invariant
surface
$\Omega(p) = \omega + \lambda\|p\|^m(1+o(\|p\|))$,
with $|\lambda| > L_0^{-\nu}$ intersects
$\Eta_{n+1}$ with nonzero measure;
$$
{\rm meas} \, (\{ r \in [0,r_0); \|p\| = r,
(p,\Omega(p)) \in \Eta_{n+1} \})
\geq r_0(1 - Cr_0^\mu)~~~.
$$
A consequence of $(n.2)(i)(ii)$ is that the
Green's function for any $E \subseteq B_n
\backslash N$ is controlled on the parameter
region $\Eta_{n+1}$.
\CLAIM Lemma(Gfunction)
Let $A$ be a nonsingular region, and
$E \subseteq (B_{n+1} \backslash N) \cup A$.
The Green's function satisfies
$$
\|G_E(p,\Omega;u_n)\|_{\sigma_n} \leq
{C_G^n \over \delta_{n+1} \gamma_n^s }~~,
$$
and under perturbations of $u_n$ of size
$\|u - u_n\|_{\sigma_n-\gamma_n} \leq
\|p\|^M \epsilon_n / \delta_{n+1} \gamma_n^s $,
$$
\|G_E(p,\Omega;u)\|_{\sigma_n-2\gamma_n} \leq
{2C_G^n \over \delta_{n+1} \gamma_n^s }~~.
$$
\PROOF The proof is the same as in [CW], Section 5.
The arguments involve the decoupling of the local
Hamiltonians at singular regions of
$B_{n+1} \backslash N$. As long as the spectra
of the local Hamiltonians are controlled, and
the singular regions are sufficiently separated,
resolvant expansions can be employed to recover
the full Green's function. \endproof
\smallskip
Induction step $(n.1)$ will follow from
$((n-1).2)(i)(ii)$ and \clm(Gfunction). Indeed
the Newton iteration step is
$$\eqalign{
v_{n-1} = & -G_{B_{n} \backslash N}(p,\Omega;u_{n-1})
\bigl( V(\Omega) u_{n-1} + W(\phi(p) + u_{n-1})
\bigr)_{B_{n} \backslash N}~~~, \cr
u_{n} & = u_{n-1} + v_{n-1}~~. \cr}
$$
With this definition of $u_{n}$ the Taylor
remainder theorem will exhibit a quadratic error,
and the error due to domain truncation will be
exponentially small if some decay is sacrificed.
We again refer to [CW] for details of the
convergence proof.
The remaining task is to realize a large set of
parameters $\Eta_{n+1} \subseteq \Eta_n$ such
that $(n.2)$ is satisfied. Conditions $(i)$ and $(ii)$
decrease the size of $\Eta_{n}$, while condition
$(iii)$ requires that it be sufficiently large, and
further satisfy certain geometrical properties
related to the order of contact of the nonlinear
degeneracy. Central to the verification of this
induction step is a lemma on eigenvalue perturbation
theory for the local Hamiltonians. For the case
$M$ even we introduce an additional hypothesis
on the lattice nonlinearity $W$, a restriction
on the self-interaction of the system within a
singular region. It will always be satisfied
for the nonlinear wave equation. The case $M$
odd has no such requirement.
\noindent
{\bf Hypothesis:} If $z,w \in S$ a singular region,
then for $M \leq J < R$,
$$
\langle \delta(z) | D^J_u W(0)
[(\partial_p \phi(p))^{J-1},\delta(w)] \rangle = 0~~.
$$
Consider a self adjoint operator $H(a)$ depending upon a
parameter $a$, and suppose an eigenvector--eigenvalue
pair $(\psi(a), e(a))$
of $H(a)$ is smooth. Then
$$
\partial_a e(a) = \langle \psi(a) | \partial_a H(a)|
\psi(a) \rangle~~~,
\EQ(Feynmanhellman)
$$
which is known as the Feynman--Hellman formula.
\CLAIM Lemma(evalue) Let $(\psi(p,\Omega),e(p,\Omega))$
be an eigenvector--eigenvalue pair for
$H_{C(S)}(p,\Omega)$. Then
$$
|\langle \psi(p,\Omega)|\partial_\Omega
H_{C(S)}(p,\Omega) | \psi(p,\Omega) \rangle|
\geq C_1L_n^2~~~.
\EQ(one)
$$
For $(p,\Omega)$ satisfying $(n+1.2)(i)$,
$$
|\langle \psi(p,\Omega)|\partial_p
H_{C(S)}(p,\Omega) | \psi(p,\Omega) \rangle|
\leq C_2 \|p\|^{m-1}.
\EQ(two)
$$
Let $e(p,\Omega)$ be an eigenvalue of a
local Hamiltonian (labeled by ordering), and
$Z$ be the set in $\Eta_0$ on which
$e(p,\Omega)$ vanishes. $Z$ is given by a graph
$(p,\Omega_Z(p))$, and if $(p_1,\Omega_Z(p_1)),
(p_2,\Omega_Z(p_2))$ are nearby points satisfying
$(n.2)(i)$, then
$$
|\Omega_Z(p_2) - \Omega_Z(p_1)| \leq
{C_3 \over L_n^2}
\bigl| \|p_2\|^m - \|p_1\|^m \bigr|
\EQ(three)
$$
The proof of this is similar to Lemma 4.14 of [CW].
%%%%%%%%%%%%%%%%%%
This result allows us to control the excisions of
parameters in order to satisfy $(n.2)(iii)$. Consider
a $T_\xi$ invariant surface $(p,\Omega(p))$, with
$\Omega(p) = \omega + \lambda\|p\|^m(1 + o(\|p\|))$,
and $|\lambda| > L_0^{-\nu}$. Let
$S \subseteq B_{n+1} \backslash B_n$ be a singular
region, and $e(p,\Omega)$ an eigenvalue of a local
Hamiltonian $H_{C(S)}$. Suppose that for some $p_1$,
$e(p_1,\Omega(p_1)) = 0$, and that $(p_1,\Omega(p_1))$
satisfies $(n.2)(i)$. We are concerned
with nearby points on the
surface $(p,\Omega(p))$. In order to inductively
construct the next set $\Eta_{n+1}$ a
$\delta_{n+1}/L_n^2$--neighborhood of $Z$ is excised
from $\Eta_n$. If the point $(p,\Omega(p))$ is excised
in this process, then
$$\eqalign{
{\delta_{n+1} \over L_n^2} & \geq
|\Omega(p) - \Omega_Z(p)| \cr
& \geq |\Omega(p) - \Omega(p_1)|
- |\Omega_Z(p_1) - \Omega_Z(p)| \cr
& \geq (|\lambda/2| - C_3/L_n^2)
\bigl| \|p_2\|^m - \|p_1\|^m \bigr|~~~. \cr}
$$
Hence any $p$ such that $\|p - p_1\|^m >
(4/|\lambda|) (\delta_{n+1}/L_n^2)$ is not excised,
and $|e(p,\Omega(p))| > \delta_{n+1}$.
\clm(evalue) provides the main result needed to
verify the induction statement $(n.2)(iii)$, and with
some patience the convergence proof will follow.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\SECTIONNONR References
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