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\titlefont
\noindent
Solutions of Nonlinear Wave Equations and
Localization Theory
\footnote*{\small To appear in the {\smallbold Proceedings
of the $X^{\rm th}$ International Congress on Mathematical
Physics} }
\vskip.4in
\rm
\vbox{\settabs 2 \columns
\+\vbox{\noindent
Walter Craig \br
{\sl Department of Mathematics \br
Brown University \br
Providence, RI~ 02912 \br}}
&\vbox{
\noindent
C. Eugene Wayne \br
{\sl Department of Mathematics \br
Pennsylvania State University \br
University Park, PA~ 16802 \br}} \cr}
\vskip.2in
In this note we describe a new method for constructing time periodic
solutions of nonlinear wave equations of the form
$$
u_{tt} = u_{xx} - g(x,u) \qquad 0 < x < \pi,\;
\EQ(1)
$$
with either periodic or Dirichlet boundary conditions. For a
complete description of the results and methods with detailed proofs,
see [2]. This article will attempt only to explain the general ideas
behind this method and some of its possible extensions.
Our main result is the following:\br
\noindent
{\bf Theorem} {\it Consider equation (1) with periodic boundary conditions.
There is a set of nonlinearities $g(x,u)$, which is
dense in $L^2_{per}(0,\pi)$ and has open intersection with
the set of analytic functions, for which the following statements
are true: \br
There exists a Cantor set ${\cal C}\subset {\bf R}$, of positive measure,
and a $C^{\infty}$ curve $\rho(\Omega)$ such that for
every $\Omega \in {\cal C}$, equation (1) has a non-trivial,
analytic, time periodic solution $u(x,t;\Omega)$,
with angular frequency $\Omega$,
with $\|u(\cdot,\cdot;\Omega)\|_{L^2} = \rho(\Omega)$.
\rm
\medskip
Similar results hold if periodic boundary conditions
are replaced with Dirichlet boundary conditions.
The conditions which determine whether or not the theorem
applies to a given nonlinear term are finite in number, involve
only the first three terms in the Taylor series for $g(x,u)$, and
can, at least in principle, be explicitly verified for a given
$g(x,u)$. As a particular example of the sort of equation to
which the theorem applies, consider the Klein-Gordon equation,
$$
u_{tt} = u_{xx} - m^2 u + u^3~~.
$$
Then in the case of Dirichlet boundary conditions,
we are able to show that the theorem applies to this equation
for an open set of values of $m$ of full Lesbesgue measure.
(In this example, and in any cases in which the nonlinear
term $g(x,u)=g(u)$, a simple analysis of
the phase plane gives the existence result
when periodic boundary conditions are imposed.)
Our method is perturbative in nature and thus starts from an analysis
of the linearized problem. We assume that $g(x,u)$
is analytic and has a Taylor series at $u=0$
of the form
$g(x,u)= g_1(x)u + g_2(x)u^2 + \dots$ so that to first order (1) becomes
$$
v_{tt} = (\partial^2_x - g_u(x,0)) v\;\; , \quad 0< x < \pi,\;
\EQ(2)
$$
This equation is easily solved. If the eigenvectors and eigenvalues
of the Sturm-Liouville operator $\left(-{{d^2}\over{dx^2}} +
g_u(x,0)\right)$ with the appropriate boundary conditions are
$\{\psi_j(x)\}^\infty_{j=1}$ and $\{\omega^2_j\}^\infty_{j=1}$
(assume for simplicity that
all the eigenvalues are positive, though this is not necessary) then
$v^{(j)}(x,t) = \psi_j(x)\, \sin (\omega_jt)$ is a periodic solution of
(2), for every $j = 1,2\ldots\;$. We wish to show that there are
solutions of the full non-linear equation (1), which are ``close'' to
these solutions of the linearized equations. With only a slight loss
of generality we will focus on the case $j = 1$ -- i.e. we look for
solutions of (1), with frequency close to $\omega_1$. If such a solution
exists, we can write it in an eigenfunction-Fourier expansion as
$$
u(x,t) = \sum\limits^\infty_{j=1}\; \sum\limits_{k\in \ZZ}\; \widehat u(j,k) \psi_j (x) e^{i\Omega kt}
\EQ(3)
$$
where $\Omega \approx \omega_1$. If we substitute (3) into (1), we
obtain an infinite set of coupled equations for the
coefficients $\{\widehat u (j,k)\}$, which we write schematically as
$$
F(\widehat u) (j,k) = W(\widehat u) (j,k) + V(\Omega)
\widehat u(j,k) = 0, \quad j = 1,2\ldots, ~~;~ k \in \ZZ~~.
\EQ(4)
$$
Here, $V(\Omega)$ is the diagonal operator $V(\Omega) \widehat u(j,k) =
(\omega^2_j - k^2 \Omega^2)\widehat u(j,k)$, which arises from the
linearized equation (2), while $W(\widehat u)$ is a non-diagonal
operator coming from the non-linear part of the function $g(x,u)$,
whose form we will comment upon below.
We will construct solutions $\{\widehat u(j,k)\}$ of (4) by applying
Newton's method, taking as our starting point the function $\varphi(p)
= p\delta(1,1) + \overline p \delta(1,-1)$, where $p\in {\bf C}$. Note
that $\varphi$ is a solution of the linearized problem (i.e. when $W =
0)$ if $\Omega = \omega_1$. (Here, $\delta(j,k)$ is the
Kronecker $\delta$-function on the lattice $\ZZ^+ \times \ZZ$.)
The most difficult part of this process is to control the
inverse of the operator
$$
D_{\tilde u} F = D_{\tilde u} W + V(\Omega)
\EQ(5)
$$
which occurs when we linearize about an approximate solution
$\widetilde u$. We control this operator in two stages. The first
problem we encounter is that when $\Omega = \omega_1$, the linear
operator $V(\omega_1)$ has a two-dimensional null space
$\ell^2(N)$, where $N = \{(1,1), (1,-1)\}$.
This is a common problem in bifurcation theory and we
treat it using a standard method, the Lyapunov-Schmidt method. Define
$Q =$ orthogonal projection onto $\ell^2(N)$ and $P={\bf 1}-Q$. Then (4)
is equivalent to a pair of equations,
$$
QF(\widehat u) = QW(\widehat u) (j,k) + (\omega^2_1-\Omega^2)
\widehat u(j,k) = 0, \quad (j,k) \in N~~,
\EQ(6)
$$
and
$$
PF(\widehat u) = PW(\widehat u) (j,k) +
(\omega^2_j-k^2 \Omega^2) \widehat u(j,k) = 0, \quad (j,k) \not\in N.
\EQ(7)
$$
Note that the ``$Q$-equation'' ({\it i.e.} (6)) is a finite dimensional
problem (in fact two dimensional) and can be handled by ordinary
implicit function theorem arguments. Thus, we will concentrate in the
remainder of this note on how one can solve the ``$P$-equation'' (7).
Equation (7) is solved inductively by finding better and better
approximate solutions, on larger and larger subsets of the lattice,
${\ZZ}^+ \times {\ZZ}$. In the limit, the subset converges to the
whole lattice and the solution converges to a true solution of (7).
In order to start the induction we make certain assumptions about
finitely many of the frequencies $\{\omega_j\}$, which is equivalent to
making assumptions about the first order term in the Taylor expansion
of the non-linear term $g(x,u)$ in (1). We comment below on just how
restrictive these assumptions are.
The initial ``box'' on which we solve (7) is the set $B_0 = \{(j,k)
\in \ZZ^+ \times \ZZ\; |\; j + |k| \le L_0\}$, with $L_0$ a large
constant. We then assume that for $(j,k)$ in $B_0$ the frequencies of
the linearized problem satisfy (i) $|\omega^2_j - k^2
\omega_1^2 | \ge
d_0$ and (ii) $|k\omega_1 - j | \ge c_0 (j+|k|)^{-\tau}$, where $d_0,
c_0$ and $\tau$ are positive constants.
Furthermore we assume that the nonlinearity
satisfies a ``twist'' condition. This condition
ensures that the bifurcation curve $\rho(\Omega)$
has non-zero curvature at $\Omega = \omega_1$,
and it can be explicitly
computed from a formula involving the second and
third order terms $g_2(x)$
and $g_3(x)$ of the Taylor series of
the nonlinearity and the eigenfunctions of the
linearized operator. When restricted to $B_0$, (7)
becomes a finite dimensional problem and
conditions (i) and (ii) guarantee that it can be
solved by the
ordinary implicit function theorem. One finds further that the
coefficients $\{u^{(0)} (j,k)\}$ of this solution decay exponentially
in $(|j| + |k|)$, which means that $u^{(0)}$ fails to be a true
solution of (7) (i.e. a solution on the whole lattice $\ZZ^+ \times
\ZZ$ rather than just on $B_0$) only by an amount of
${\cal O}(e^{-\sigma L_0})$, for some $\sigma > 0$. Thus, if $L_0$ is
large we have a very good approximate solution.
To improve this solution we use Newton's method. If we restrict (7)
to a new, larger lattice region $B_1 \supset B_0$,
where $B_1$ is defined analagously to $B_0$, but
with $L_0$ replaced by $L_1 > L_0$, and try to solve
$PF(u^{(0)} + v^{(0)})\; |_{B_1} \;= 0$ by Newton's method, we find
$$
v^{(0)} = - \left\{P(D_{u^{(0)}} W + V(\Omega))\; |_{B_1} P\right\}^{-1} PF|_{B_1} (u^{(0)}).
\EQ(8)
$$
From the comments above, we know that $PF|_{B_1}$ is small. Thus, the
only difficulty we encounter is controlling the inverse of the
operator $P(D_{u^{(0)}} W + V(\Omega)) |_{B_1} P$. We have an
explicit form for the diagonal term $V(\Omega)$, so this causes no
problems. To understand the off-diagonal term $D_{u^{(0)}} W$, we note
that for the example of the Klein-Gordon equation,
$u_{tt} = u_{xx} - m^2 u + u^3$, one
can compute $D_uW$ exactly if one takes $u$ to be the solution of the
linearized problem. One finds that $D_{u} W + V(\Omega) =
\gamma \Delta + V(\Omega)$, where $\gamma$ is a constant, and $\Delta$
is the lattice laplacian. Thus, in this example, the task of
inverting $D_{u^{(0)}} W + V(\Omega)$ is the same as controlling the
inverse of a lattice Schr\"odinger operator. This is a problem that
has been studied in depth in the physics literature. In particular,
Fr\"ohlich and Spencer [3] have shown that if one can insure that the
``resonances'' in $V(\Omega)$ are not too close together and not too
severe, then the matrix elements of $(\gamma\Delta + V(\Omega))^{-1}$
will decay exponentially fast as one moves away from the diagonal.
For more general non-linearities than that in the Klein-Gordon
equation, or if we evaluate the derivative at a point other than the
solution of the linearized equation, one will not obtain $D_{u^{(0)}} W
= \gamma\Delta$. However, $D_{u^{(0)}} W$ will still be a bounded
operator with matrix elements which decay exponentially fast as one
moves away from the diagonal, and the methods of Fr\"ohlich and
Spencer are still applicable.
Applying the methods of Fr\"ohlich and Spencer places restrictions on
the allowed frequencies of our periodic orbits. In the present
problem the ``resonances'' correspond to points $(j,k)$ at which the
potential $V(\Omega)(j,k)$ approaches zero. A simple calculation
using the fact that asymptotically the linear frequencies in this
problem satisfy $\omega_j \sim j$, shows that if we eliminate frequencies
$\Omega$ which fail to satisfy the diophantine inequality $|k\Omega -
j | \ge c (j+ |k|)^{-\tau}$, then for any remaining frequency,
$\Omega$, two lattice sites at which $V(\Omega)$ is small, must be
widely separated. One must also insure that at any remaining sites
where $V(\Omega)$ is small, the linearized operator $(D_u W +
V(\Omega))$ does not have points in its spectrum that are too close to
zero. This entails a further excision of frequencies $\Omega$, so
that at the conclusion of this induction argument we are left with a
family of periodic orbits, but the frequencies in this family form a
Cantor set of positive measure, not an interval as one typically finds
in ordinary finite dimensional bifurcation theory. Once one makes
these excisions of frequencies, the inverse of the linearized operator
in Newton's method is controllable, and the induction argument
converges to give a family of periodic orbits. We remark that
methods related to ours have been applied to construct periodic
solutions for some nonlinear versions of the Anderson localization
model by Albanese, Fr\"ohlich and Spencer [1].
There are several related problems which we think can be treated by
this method. First of all, there is the existence of quasi-periodic
(in time) orbits for (1). One can again make an eigenfunction-Fourier
expansion similar to (3), but with $n$-frequencies in time -- i.e.
the exponential becomes $\exp \{ i( \sum_{\ell=1}^n \,
\Omega_\ell k_\ell ) t \}$.
In this case the original partial differential equation is reduced to
a problem on the lattice $\ZZ^+ \times \ZZ^n$. One can then apply
Newton's method just as in the periodic case. While formally
the same, the problems one encounters when studying the resonances of
the linearized problem are more complicated than in the periodic case,
and we have not yet finished this analysis. However, we believe that
this construction will also quasi-periodic solutions of (1). We note
that if successful this method would also give a new construction of
quasi-periodic orbits of finite dimensional hamiltonian systems of the
form $H(p,q) = \sum_{j=1}^N \, {{1}\over{2}} p^2_j + V(q)$, in the
neighborhood of an elliptic equilibrium point which is
different from the usual construction using the KAM Theorem.
Another case which seems difficult to treat with KAM methods, but
which we believe the present methods can treat, is the case of
resonances in the linear problem. Suppose, for instance, that the
linear frequencies $\omega_1$ and $\omega_2$ satisfy
$\omega_2 = 2\omega_1$.
In this case when we convert the problem to one on the
lattice we find the null space of the diagonal operator $V(\Omega)$ is
four dimensional rather than two dimensional. We can still proceed
with the Lyapunov Schmidt procedure, however, and the ``$P$-equation''
(7) is solved just as in the non-resonant case. This leaves us with
the ``$Q$-equation'', (6). This is now a finite (in this example,
four) dimensional problem. Although the resonance makes the ordinary
implicit function theorem inapplicable, we believe other methods will
allow us to solve this problem --- either perturbative techniques like
those of Schmidt [6], or variational methods like those used by
Weinstein [8] and Moser [5] to extend Lyapunov's center theorem to the
case in which the linear problem contains resonances. The advantage
that the present method offers, in our opinion, is that it allows one
to treat quite independently the problems arising from the linear
resonance (``$Q$-equation'') and the infinite dimensionality of the
problem (``$P$-equation'').
We conclude with a short discussion of how these methods compare with
KAM approaches to the problem which have been used by Kuksin [4] and
Wayne [7]. The principle difference is in the sort of non-resonance
conditions that are imposed on the frequencies of the linear problem.
The conditions imposed by the present method require essentially that
no multiple of the frequency of the periodic orbit we are constructing
resonate with any of the other linearized frequencies. This condition
is an infinite dimensional generalization of the
hypothesis of the Lyapunov center theorem. In contrast, the KAM type
methods require in addition that certain non-resonance conditions hold
among those linear frequencies that we are not perturbing. These
additional conditions arise because in the KAM method one transforms
the hamiltonian of the problem to some normal form. The present
method is not a transformation theory in this sense, but rather a more
direct implicit function theorem. In contrast to previous approaches
like that of Zehnder [9],
which used the ``group structure'' given by the transformation theory
to invert the linearized operator in Newton's method explicitly, in
the present method we obtain bounds on this inverse without using a
transformation theory. Indeed, even the hamiltonian nature of the
problem plays little explicit role in this approach, so that we hope
it may be applicable to a wide variety of problems.
\noindent{\bf Acknowledgements:} The authors would
like to thank the D\'epartement de Physique
Th\`eorique, Universit\'e de Gen\`eve, and Oxford
University for their hospitality, and the National
Science Foundation (grants \#DMS-8858218 and
\#DMS-9002059) and the Alfred P. Sloan Foundation
for their support of our research.
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