This file has been bundled using the "shar" program.
Instructions for unbundling it follow.  Once unbundled
it will produce the files:
defs.tex
driver.tex
fig1.tex
fig2.tex
head.tex
intro.tex
refs.tex
sec3.tex
sec4.tex
sec5.tex
title.tex
The completed preprint can then be recovered by typing
"tex driver".  The preprint was prepared with plain
TeX, version 3.1.  The figures in this manuscript were prepared
with the "pictex" program.  If that is not available
on your machine, remove the lines 
\input fig1.tex
\input fig2.tex
from the file driver.tex, and you can produce the preprint
(less the figures) as above.  The figures can be obtained
by sending e-mail to "wayne@math.psu.edu".
########################################################
#! /bin/sh
# This is a shell archive.  Remove anything before this line, then unpack
# it by saving it into a file and typing "sh file".  To overwrite existing
# files, type "sh file -c".  You can also feed this as standard input via
# unshar, or by typing "sh <file", e.g..  If this archive is complete, you
# will see the following message at the end:
#		"End of shell archive."
# Contents:  defs.tex driver.tex fig1.tex fig2.tex head.tex intro.tex
#   refs.tex sec2.tex sec3.tex sec4.tex sec5.tex sec6.tex title.tex
# Wrapped by wayne@poincare on Mon Jan 13 16:01:20 1992
PATH=/bin:/usr/bin:/usr/ucb ; export PATH
if test -f 'defs.tex' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'defs.tex'\"
else
echo shar: Extracting \"'defs.tex'\" \(1000 characters\)
sed "s/^X//" >'defs.tex' <<'END_OF_FILE'
X\def\BGO{{\Omega}}
X\def\bgo{{\Omega}}
X\def\pp{{\bf p}}
X\def\bgnone{\overline G_{B_{n+1}}}
X\def\11{{\bf 1}}
X\def\go{{\omega}}
X\def\ga{\gamma}
X\def\nth{${\rm n}^{th}$\ }
X\def\rn{r$_n$}
X\def\sn{s$_n$}
X\def\X{{\bf XXX}}
X\def\QQ{{\cal Q}}
X\def\var{\pp,\bgo;z}
X\def\unone{u_{n-1}}
X\def\un{u_n}
X\def\l2{\ell^2}
X\def\Bnone{B_{n-1}}
X\def\Bn{B_n}
X\def\Bnp{B_{n+1}}
X\def\gre{\epsilon}
X\def\gs{\sigma}
X\def\gd{\delta}
X\def\Eta{{\cal N}}
X\def\11{{\bf 1}}
X\def\bGn{\overline{G_n}}
X\def\dist{{\rm dist}}
X\def\meas{{\rm meas}}
X\def\spec{{\rm spec}}
X\def\BP{{\bf P}}
X\def\BQ{{\bf Q}}
X\def\spec{{\rm spec}}
X\def\CC{{\cal C}}
X\def\HH{{\cal H}}
X\def\NN{{\cal N}}
X\def\LL{{\cal L}}
X\def\Cn1{C_{\ell_{n+1}}}
X\def\ZZ{{\bf Z}}
X\def\xx{{\bf x}}
X\def\zz{{\bf z}}
X\def\DD{{\cal D}}
X\def\Kappa{{\cal K}}
X\def\zsquared{\ZZ^+ \times \ZZ}
X\def\half{{{1}\over{2}}}
X\def\endproof{\vbox{\hfill{\vbox{\hrule height.5pt
X                 \hbox{\vrule width.5pt height5pt \kern5pt
X                 \vrule width.5pt}\hrule height.5pt}}}}
X
END_OF_FILE
if test 1000 -ne `wc -c <'defs.tex'`; then
    echo shar: \"'defs.tex'\" unpacked with wrong size!
fi
# end of 'defs.tex'
fi
if test -f 'driver.tex' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'driver.tex'\"
else
echo shar: Extracting \"'driver.tex'\" \(168 characters\)
sed "s/^X//" >'driver.tex' <<'END_OF_FILE'
X\input head.tex
X\input defs.tex
X\input title.tex
X\input intro.tex
X\input sec2.tex
X\input sec3.tex
X\input sec4.tex
X\input sec5.tex
X\input sec6.tex
X\input refs.tex
X
X\bye
END_OF_FILE
if test 168 -ne `wc -c <'driver.tex'`; then
    echo shar: \"'driver.tex'\" unpacked with wrong size!
fi
# end of 'driver.tex'
fi
if test -f 'fig1.tex' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'fig1.tex'\"
else
echo shar: Extracting \"'fig1.tex'\" \(2160 characters\)
sed "s/^X//" >'fig1.tex' <<'END_OF_FILE'
X\input pictex
X
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X%
X% arrow head
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X%
X% Fig POLYLINE object
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X% arrow head
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X% Fig POLYLINE object
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X%
X% Fig POLYLINE object
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X% arrow head
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X% Fig POLYLINE object
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X% Fig POLYLINE object
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X% Fig POLYLINE object
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X% Fig POLYLINE object
X%
X\plot  3.487  6.513  3.487  6.513 /
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X% Fig POLYLINE object
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X% Fig POLYLINE object
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X%
X% Fig POLYLINE object
X%
X\plot  3.487  4.513  3.487  4.513 /
X%
X% Fig POLYLINE object
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X\plot  1.488  6.513  3.487  4.513 /
X%
X% Fig POLYLINE object
X%
X\plot  3.487  8.512  3.487  8.512 /
X%
X% Fig POLYLINE object
X%
X\plot  3.487  8.512  3.487  8.512 /
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fi
# end of 'fig1.tex'
fi
if test -f 'fig2.tex' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'fig2.tex'\"
else
echo shar: Extracting \"'fig2.tex'\" \(2029 characters\)
sed "s/^X//" >'fig2.tex' <<'END_OF_FILE'
X\input pictex
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X% Fig POLYLINE object
X%
X\plot  2.362  7.638  3.362  6.888 /
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X% arrow head
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X% Fig POLYLINE object
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X% arrow head
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X\plot  3.700  7.612  3.800  7.638  3.700  7.663 /
X%
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X% Fig POLYLINE object
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X\setdots 
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X%
X% Fig POLYLINE object
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X\plot  3.612  6.825  4.112  8.137 /
X%
X% Fig POLYLINE object
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X% arrow head
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X\plot  4.939  5.756  4.862  5.825  4.897  5.728 /
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X%
X% Fig POLYLINE object
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X%
X% Fig POLYLINE object
X%
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X% Fig POLYLINE object
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X% Fig POLYLINE object
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X%
X% Fig POLYLINE object
X%
X\plot  3.487  4.513  5.487  6.513 /
X%
X% Fig POLYLINE object
X%
X\plot  3.487  4.513  3.487  4.513 /
X%
X% Fig POLYLINE object
X%
X\plot  1.488  6.513  3.487  4.513 /
X%
X% Fig POLYLINE object
X%
X\plot  3.487  8.512  3.487  8.512 /
X%
X% Fig POLYLINE object
X%
X\plot  3.487  8.512  3.487  8.512 /
X%
X% Fig POLYLINE object
X%
X\plot  3.487  8.512  5.487  6.513 /
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X% Fig TEXT object
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X%
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X% Fig TEXT object
X%
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X
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X%
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if test 2029 -ne `wc -c <'fig2.tex'`; then
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fi
# end of 'fig2.tex'
fi
if test -f 'head.tex' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'head.tex'\"
else
echo shar: Extracting \"'head.tex'\" \(9659 characters\)
sed "s/^X//" >'head.tex' <<'END_OF_FILE'
X
X\magnification \magstep1
X\vsize=22 truecm
X\hsize=16 truecm
X\hoffset=0.8 truecm
X\normalbaselineskip=5.25mm
X\baselineskip=5.25mm
X\parskip=10pt
X\hfuzz=2pt
X\font\titlefont=cmbx10 scaled\magstep1
X\font\sectionfont=cmbx10 scaled\magstep1
X\font\subsectionfont=cmbx10
X\font\small=cmr7
X%%%%%constant subscript positions%%%%%
X\fontdimen16\tensy=2.7pt
X\fontdimen17\tensy=2.7pt
X\fontdimen14\tensy=2.7pt
X%%%%%%%%%%%%%%%%%%%%%%%
X%%%  real math %%%%%%%%
X%%%%%%%%%%%%%%%%%%%%%%%
X\def\HB {\hfill\break}
X\def\AA{{\cal A}}
X\def\BB{{\cal B}}
X\def\CC{{\cal C}}
X\def\EE{{\cal E}}
X\def\HH{{\cal H}}
X\def\LL{{\cal L}}
X\def\MM{{\cal M}}
X\def\NN{{\cal N}}
X\def\OO{{\cal O}}
X\def\RR{{\cal R}}
X\def\TT{{\cal T}}
X\def\VV{{\cal V}}
X\def\HALF{{\textstyle{1\over 2}}}
X%%%%%%%%%%%%%%%%%%%%%%
X%%% macros  %%%%%%%%%%
X%%%%%%%%%%%%%%%%%%%%%%
X
X\headline={\ifnum\pageno>1 {\hss\tenrm-\ \folio\ -\hss} \else {\hfill}\fi}
X\newcount\EQNcount \EQNcount=1
X\newcount\SECTIONcount \SECTIONcount=0
X\newcount\CLAIMcount \CLAIMcount=1
X\newcount\SUBSECTIONcount \SUBSECTIONcount=1
X\def\undertext#1{$\underline{\smash{\hbox{#1}}}$}
X\def\QED{\hfill\smallskip
X         \line{\hfill\vrule height 1.8ex width 2ex depth +.2ex
X               \ \ \ \ \ \ }
X         \bigskip}
X\def\real{{\bf R}}
X\def\natural{{\bf N}}
X\def\complex{{\bf C}}
X\def\integer{{\bf Z}}
X\def\Re{{\rm Re\,}}
X\def\Im{{\rm Im\,}}
X\def\PROOF{\medskip\noindent{\bf Proof.\ }}
X\def\REMARK{\medskip\noindent{\bf Remark.\ }}
X\def\ifundefined#1{\expandafter\ifx\csname#1\endcsname\relax}
X\def\equ(#1){\ifundefined{e#1}$\spadesuit$#1\else\csname e#1\endcsname\fi}
X\def\clm(#1){\ifundefined{c#1}$\spadesuit$#1\else\csname c#1\endcsname\fi}
X\def\EQ(#1){\eqno\tag(#1)}
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X \def\beginAUTHOR{\start
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X \def\getAUTHOR{\getNORMAL}
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X \def\getFROM{\getNORMAL}
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X}
X%%%%%%%%%%%%%%%%BIBLIOGRAPHY%%%%%%%%%%%%%%%%%%%%
X%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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X%
X% ...invisible stuff
X%
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X\newbox\byboxJPE
X\newbox\paperboxJPE
X\newbox\secondpaperboxJPE
X\newbox\yrboxJPE
X\newbox\secondyrboxJPE
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X   \setbox\byboxJPE=\null             \gdef\isbyJPE{F}
X   \setbox\paperboxJPE=\null          \gdef\ispaperJPE{F}
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X   \setbox\bookboxJPE=\null           \gdef\isbookJPE{F}  \gdef\isinbookJPE{F}
X
X   \setbox\bybookboxJPE=\null         \gdef\isbybookJPE{F}
X   \setbox\publisherboxJPE=\null      \gdef\ispublisherJPE{F}
X
X}
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X\def\by   {\egroup\gdef\isbyJPE{T}\setbox\byboxJPE=\hbox\bgroup}
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X\def\secondpaper{\egroup\gdef\issecondpaperJPE{T}
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X\def\yr{\egroup\gdef\isyrJPE{T}\setbox\yrboxJPE=\hbox\bgroup}
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X\def\jour{\egroup\gdef\isjourJPE{T}\setbox\jourboxJPE=\hbox\bgroup}
X\def\secondjour{\egroup\gdef\issecondjourJPE{T}\setbox\secondjourboxJPE=\hbox\bgroup}
X\def\pages{\egroup\gdef\ispagesJPE{T}\setbox\pagesboxJPE=\hbox\bgroup}
X\def\secondpages{\egroup\gdef\issecondpagesJPE{T}\setbox\secondpagesboxJPE=\hbox\bgroup}
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X
X
X
X
X
X\SECTION Introduction:
X
XIn this paper we prove the existence of periodic
Xsolutions of nonlinear wave equations of the form
X$$
X   \partial_t^2 u = \partial_x^2 u - g(x,u)~,
X\EQ(origeqn)
X$$
Xfor nonlinearities $g(x,u)$ which satisfy certain
Xconditions of nonresonance and genuine nonlinearity.
XWe assume that one has either periodic or Dirichlet
Xboundary conditions at the ends of the interval
X$[0,\pi]$.  This problem is called the free vibration
Xproblem for the nonlinear string and has been extensively studied.
XThe review [B], contains over 60 references, and in the
Xeight years since it was written the number has increased
Xmuch further.  In this introduction we will try to describe
Xin very general terms the methods we use to study this
Xproblem and how these methods relate to, and differ
Xfrom, previous approaches to this problem.
X
XThe first real breakthrough on this problem was due to
XRabinowitz [R].  He rephrased the problem as a variational
Xproblem and was able to prove that under appropriate
Xassumptions on the non-linearity $g(x,u)$ one had
Xperiodic solutions whenever the time period  was
Xa rational multiple of $\pi$.  Many authors have
Xused Rabinowitz's variational methods to obtain related
Xresults.  None, however,
Xwere able to circumvent the restriction on the period.
XIn addition, these variational techniques have not yet
Xshed light on the existence of quasi-periodic solutions
Xof \equ(origeqn).  On the other hand, the variational
Xtechniques are global, and they place few
Xrestrictions on the strength of the non-linear term.
X
XMore recently, a quite different approach which uses the
XKolmogorov, Arnold, Moser (KAM) theory has been developed
Xby Kuksin [K] and Wayne [W].  This approach uses the fact
Xthat \equ(origeqn) is a hamiltonian system and modifies
Xthe classical KAM ideas to work in this infinite dimensional
Xcontext.  This has two advantages--first, it allows one to
Xconstruct solutions whose periods are irrational multiples of
X$\pi$ and second, it easily extends to give quasi-periodic
Xas well as periodic solutions.  A disadvantage is
Xthat since the KAM theory has an essentially
Xperturbative character, it is restricted to equations with
Xweak non-linearity, or equivalently, to solutions of
Xsmall norm.
X
XIn this paper we propose yet a third approach.  Our method
Xis reminiscent of the Lyapunov-Schmidt method of classical
Xbifurcation theory in that we split the problem into
Xtwo pieces, one of which is finite dimensional and corresponds
Xto the null space of the linearized operator, and the other
Xpiece infinite dimensional.  In contrast to the
XLyapunov-Schmidt method in our problem the linearization of the
Xinfinite dimensional piece does not have bounded inverse.
XIndeed, it's failure to have bounded inverse is related to
Xthe restriction in the work of Rabinowitz and others to
Xsolutions whose period is a rational multiple of $\pi$.
X
XOur method is perturbative and begins by expanding
Xthe nonlinear term in \equ(origeqn) as $g(x,u) =
Xg_1(x) u + g_2(x) u^2 + \dots$.  If one ignores terms of 
X$\OO(u^2)$ or higher, \equ(origeqn) becomes a linear equation
Xwhose solutions can be explicitly computed in terms of the eigenvectors,
X$\{ \psi_j(x) \}$, and eigenvalues, $\{ \omega_j^2 \}$, of the 
XSturm-Liouville operator $L = (-{{d^2}\over{dx^2}} + g_1(x) )$.
XWe then make the {\it Ansatz} that a periodic solution of \equ(origeqn)
Xwith angular frequency $\Omega$ exists and write it as
X$$
Xu(x,t) = \sum_{j,k} \hat{u}(j,k) \psi_j(x) e^{i k \Omega t}~~.
X\EQ(ex)
X$$
XSubstituting \equ(ex) into \equ(origeqn) gives an infinite system
Xof nonlinear algebraic equations which the coefficients
X$\{ \hat{u} (j,k) \}$ must solve.  We construct solutions of this
Xsystem of equations using Newton's method taking as our initial
Xapproximation to the solution linear combinations
Xof Kronecker $\delta$-functions at the lattice sites
X$(j,k) = (1, \pm 1)$.  This initial guess corresponds to a periodic
Xsolution of the linearized equation with angular frequency
X$\omega_1$.  We are able to prove that our interative scheme
Xconverges to a solution of the equations for $\{ \hat{u} (j,k) \}$,
Xwhich,
Xwhen substituted into \equ(ex) gives a periodic solution of \equ(origeqn).
XThis construction yields a family of periodic orbits with frequencies
Xin a Cantor set of positive measure.  One interesting point
Xis that the linear operator which must be inverted in order
Xto apply Newton's method is closely related to the lattice
XSchr\"odinger operators studied by Fr\"ohlich and Spencer [FS]
Xin their work on
Xlocalization theory.  From a technical point
Xof view, this connection between localization theory and
Xdynamical systems strikes us as one of the more interesting
Xaspects of the present approach.  Similar ideas, with
Xapplications to partial differential-difference equations have
Xbeen previously developed in the work of Albanese, Fr\"ohlich
Xand Spencer [AFS].
X
XOur method, like the KAM method is perturbative, and 
Xis restricted to the study of
Xsolutions of small norm.  However, it differs from the
XKAM method in several ways.  
XFirst of all,
Xthe present theory is not a transformation theory--we do
Xnot proceed by making a sequence of canonical transformations,
Xnor do we transform the system to some normal form.
XSecondly, the present
Xmethod makes no direct use of the hamiltonian nature of the
Xproblem. Thus, we hope it will be applicable
Xto non-hamiltonian problems.
X
XThe existence theorem does not apply to all choices of 
Xnonlinearity $g$, it requires that certain conditions of
Xlinear nonresonance and genuine nonlinearity are satisfied. 
XThese are analogs of well known difficulties in the theory
Xof dynamical systems. These conditions depend only upon the 
X{\it 3-jet} of $g$, and are finite in number. The set of
Xnonlinearities which satisfy them is generic, indeed it 
Xis open and dense, and is described more precisely in section 6.
XThese conditions can in principle be checked in explicit 
Xcases. For the classical examples of the nonlinear 
XKlein Gordon and the sine Gordon equations, the 
Xnonlinearity depends upon one parameter, and we 
Xshow that, for an open set of full measure of this 
Xparameter, the conditions are satisfied and the 
Xexistence theorem applies.
X
X
XThinking of the analogy with the Lyapunov center theorem
Xfor finite dimensional hamiltonian systems in the neighborhood
Xof an elliptic equilibrium point, one expects to obtain
Xa family of periodic orbits bifurcating from the orbit of the
Xlinearized equations whenever no integer multiple of the frequency
Xof the solution being perturbed coincides with the frequency
Xof any other normal mode.
XThere is a significant
Xdifference when the problem has infinitely
Xmany degrees of freedom.  In the finite dimensional
Xcase there are smooth families of periodic solutions bifurcating
Xfrom the solution of the linear problem.  In the case of the
Xwave equation we will construct a smooth curve bifurcating from the
Xsolution of the linear equation.  However we cannot prove
Xthat all points on this curve give rise to solutions of the
Xwave equation--only that there is a (Cantor) set of frequencies
Xof positive measure such that for any point on the curve whose
Xfrequency lies in this Cantor set one has a solution.
XThis difference arises from the fact that if there are
Xonly finitely many frequencies $\omega_1, \omega_2,
X\omega_3, \dots, \omega_N$, and if $m\omega_1 \ne
X\omega_j$, for $j=2,3,\dots,N$, then $m\Omega
X\ne \omega_j$, for all $\Omega$ in some interval surrounding
X$\omega_1$.  Thus, one typically obtains solutions for the
Xnonlinear problem for an
Xinterval of frequencies.  For infinitely many frequencies,
Xhowever, even if $m \omega_1 \ne \omega_j$, for
X$j = 2, 3, \dots$, there will in general be a dense
Xset of $\Omega$'s for which $m \Omega = \omega_j$, for some
X$j$.  It is the process of excising these resonant
Xfrequencies which gives rise to the Cantor set on which the
Xconstruction of solutions is successful.
X
XAs remarked above, the KAM approach to this problem also
Xyields the existence of quasi-periodic solutions for equations
Xlike \equ(origeqn).  We believe that an extension of the
Xpresent method will also yield quasi-periodic solutions, and plan
Xfurther work on this problem.  We remark that
Xsuch results would be of interest even for finite dimensional
Xsystems since they would give a proof of the existence
Xof invariant tori for hamiltonian systems near an
Xelliptic equilibrium point whose dimension is less than
Xor equal to
Xthe number of degrees of freedom of the systems, different
Xfrom that of [E], or [P].
X
XWe conclude with an outline of the remainder of the paper.
XIn the next section we state our principle results,
Xtransform the wave equation to a problem on a two-dimensional
Xlattice, and apply our results to discuss
Xtwo well known examples, the Klein-Gordon and sine-Gordon
Xequations.  In Section 3 we state the induction hypotheses
Xwhich allow us to derive the results in Section 2.  Section
X4 contains the verification of these
Xinduction hypotheses, while Section 5 explains
Xhow to control the inverse of the linearized operator which
Xarises in Newton's method. This analysis is connected
Xwith the theory
Xof localization in Schr\"odinger operators.  Finally, in
XSection 6 we derive some estimates we need which link the wave
Xequation to the lattice problem, and discuss the issues of 
Xgenericity of the nonlinearity.
X
X
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X
X
X
X\SECTIONNONR References
X
X
X
X\ref
X \no AFS
X \by Albanese, C. and Fr\"ohlich, J. and
X     Albanese, C. Fr\"ohlich, J. and Spencer, T.
X \paper Periodic solutions of some infinite-dimensional
X        hamiltonian systems associated with non-linear
X        partial difference equations: Parts I and II
X \jour Commun. Math. Phys.
X \vol 116
X \pages 475-502
X \secondvol 119
X \secondpages 677-699
X \secondyr 1988
X\endref
X
X\ref
X \no B
X \by Brezis, H.
X \paper Periodic solutions of nonlinear vibrating
X        strings and duality principles
X \jour Bull. AMS
X \vol 8
X \pages 409-426
X \yr 1983
X\endref
X
X\ref
X \no C
X \by Chierchia, L.
X \paper A direct method for constructing solutions of
X            the Hamiltonian-Jacobi equation
X \preprint
X \yr November 1989
X\endref
X
X\ref
X \no CW
X \paper Nonlinear Waves and the KAM Theorem: Nonlinear
X           Degeneracies
X \jour To appear in the {\bf Proceedings of the Conference
X          on Nonlinear Waves}, Villefranche France
X \yr January 1991
X\endref
X
X
X
X\ref
X \no E
X \by Eliasson, H.
X \paper Perturbations of stable invariant tori
X \jour Ann. Sc. Super. Pisa, Cl. Sci.
X \vol IV Ser. 15
X \pages 115-147
X \yr 1988
X\endref
X
X\ref
X \no FS
X \by Fr\"ohlich, J. and Spencer, T.
X \paper Absence of diffusion in the Anderson tight binding
X      model for large disorder or low energy
X \jour Commun. Math. Phys.
X \vol 88
X \pages 151-184
X \yr 1983
X\endref
X
X\ref
X \no Ka
X \by Kato,  T.
X \book Perturbation Theory for Linear Operators; 2nd ed.
X \publisher Springer Verlag; Berlin
X \yr 1976
X\endref
X
X\ref
X \no KT
X \by Keller, J. and Ting, L.
X \paper Periodic vibrations of systems governed by
X        non-linear partial differential equations
X \jour Commun. Pure Appl. Math.
X \vol 19
X \pages 371-420
X \yr 1966
X\endref
X
X\ref
X \no K
X \by Kuksin, S.
X \paper Perturbation of quasiperiodic solutions of
X        infinite-dimensional linear systems with an
X        imaginary spectrum
X \jour Funct. Anal. Appl.
X \vol 21
X \pages 192-205
X \yr 1987
X\secondpaper Perturbation theroy for quasiperiodic solutions
X         of infinite-dimensional hamiltonian systems; Parts I-III
X \secondjour Preprint of Max-Plank-Institut, Bonn
X \secondyr 1990
X\endref
X
X
X\ref
X \no P
X \by P\"oschel, J.
X \paper On elliptic lower dimensional tori in
X        hamiltonian systems
X \jour Math. Z.
X \vol 202
X \pages 559-608
X \yr 1989
X\endref
X
X\ref
X \no P2
X \by P\"oschel, J.
X \paper On Fr\"ohlich Spencer estimates of Green's
X        function
X \jour Manuscripta Math.
X \vol 70
X \pages 27-37
X \yr 1990
X\endref
X
X\ref
X \no PT
X \by P\"oschel, J. and Trubowitz, E.
X \book Inverse Spectral Theory
X \publisher Academic Press; Boston, MA
X \yr 1987
X\endref
X
X\ref
X \no R
X \by Rabinowitz, P.
X \paper Free vibrations for a semilinear wave equation
X \jour Commun. Pure Appl. Math.
X \vol 30
X \pages 31-68
X \yr 1977
X\endref
X
X\ref
X \no Re
X \by Rellich, F.
X \book Perturation Theory for Eigenvalue Problems
X \publisher Gordon and Breach; New York
X \yr 1969
X\endref
X
X\ref
X \no W
X \by Wayne, C. E.
X \paper Periodic and quasi-periodic solutions
X        of nonlinear wave equations via KAM theory
X \jour Commun. Math. Phys.
X \vol 127
X \pages 479-528
X \yr 1990
X\endref
X
X
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X
X
X
X\SECTION Results
X
XWe present results for nonlinear wave equations, which
Xare obtained by a reduction of the problem to one for nonlinear
Xequations on lattices. The reduction to the lattice equations
Xand the corresponding existence results are also given in this section.
XSolutions are obtained by an induction procedure based on the
XNash Moser method which is described in Sections 3 and 4.
X
X\SUBSECTION Nonlinear wave equations
X
XThe equation we study in this paper is the nonlinear
Xwave equation on a bounded interval
X$0 \leq x \leq \pi$ in one space dimension.
X$$
X  \partial^2_t u = \partial^2_x u - g(x,u)
X\EQ(NLW)
X$$
XWe will assume that the nonlinear term is 
Xanalytic in both variables in the region
X$\{ (x,u); |{\rm Im} \ x| < \overline \gs \}$,
Xperiodic in $x$ with period $\pi$, and with Taylor
Xexpansion in $u$,
X$$
X   g(x,u) = g_1(x)u + g_2(x)u^2 + g_3(x)u^3 + \cdots
X\EQ(NL term)
X$$
XWe are seeking solutions which are periodic in time, with period
X$2\pi / \Omega$,
Xwhich satisfy certain self adjoint boundary
Xconditions on a spatial interval.
XIf we linearize about the solution $u \equiv 0$ we obtain
X$$
X   \partial^2_t v = \partial^2_x v -g_1(x) v.
X\EQ(LW)
X$$
XTwo examples of boundary conditions that we will address are
XDirichlet conditions, $u(0,t) = 0 = u(\pi,t)$, and periodic
Xconditions $u(x+\pi,t) = u(x,t)$.
XSolutions of the linear equation \equ(LW) are given by
Xseparation of variables and an eigenfunction expansion.
XLet $\{ \psi_j(x) \}_{j=1}^\infty $ be normalized eigenfunctions
Xfor the the linear differential operator
X$$
X  L(g_1) \psi = (-{d^2 \over dx^2} + g_1(x)) \psi
X\EQ(LOp)
X$$
Xwith the proper boundary conditions, with eigenvalues
X$\{ \omega^2_j \}_{j=1}^\infty$.
X(For periodic boundary conditions, it is more convenient
Xto begin labeling the eigenvalues and eigenfunctions
Xwith $j=0$--this causes no essential difference.)
XSolutions of \equ(LW) are given by
X$$
X   v(x,t) = \sum_{j=1}^\infty r_j
X               \cos(\omega_j t + \xi_j) \psi_j(x)
X\EQ(LSoln)
X$$
Xwhich are parametrized by the amplitudes $r_j$ and the phases
X$\xi_j$.
XFor real $\omega_j$ each function
X$\cos(\omega_j t + \xi) \psi_j(x)$
Xis time periodic, with frequency $\omega_j$.
XA more general solution to the linear equation \equ(LW) is time
Xperiodic only if for all nonzero amplitudes
X$r_j$ there exist a full set of rational
Xrelations between the associated frequencies $\omega_j$;
Xthat is, there exists $\omega$ and integers $k_j$ such that
Xfor all $j$ with $r_j \not= 0$, $\omega_j = k_j \omega$ .
XUnless a full set of resonance conditions are satisfied the
Xgeneral solution is quasiperiodic or almost periodic.
XWe seek periodic solutions to the full nonlinear problem
X\equ(NLW) near the linear solutions
X$r  \cos(\omega_j t + \xi) \psi_j(x)$, with frequency near the real
Xlinear frequency $\omega_j$. In this process any coincidence or
Xnear coincidence of linear frequencies causes resonance and other
Xphenomena related to small divisors in the full nonlinear
Xproblem.
XHowever the following results demonstrate that for most
Xnonlinearities $g(x,u)$ an iterative construction can overcome
Xthese difficulties, to prove the existence of periodic solutions
Xto \equ(NLW) of small amplitude.
X
X
X\CLAIM Theorem (NLWperiodic) Consider equation \equ(NLW) with
Xperiodic boundary conditions on the interval $0 \leq x \leq \pi$.
XFor an open dense set of nonlinear terms $g(x,u)$ there exist
Xtime periodic real analytic solutions. More precisely there exist
Xparameters $(\Omega, r, \xi)$ and solutions $u(x,t;r)$ such that
X$$\eqalign{
X   |u(x,t;r) - &r \cos(\Omega t + \xi)
X                            \psi_j(x)| < Cr^2 \cr
X      &|\Omega - \omega_j| < Cr^2    \cr }
X\EQ(2.3)
X$$
Xfor a set of $r$ of positive measure.
X
XIn section 6 the topology of the class of nonlinear
Xequations will be discussed, and a more precise statement
Xregarding the open dense set of nonresonant,
Xgenuinely nonlinear terms will be described.
XA similar statement holds for solutions of
Xthe nonlinear wave equation satisfying
XDirichlet boundary conditions.
X
XWe note that not only will the set of amplitudes $r$
Xof the solutions we construct have positive measure,
Xbut the set of frequencies of the periodic solutions
Xwill also have positive measure.  Thus, we are assured
Xof having solutions of irrational period.
X
X
X\CLAIM Theorem (NLWDir) Consider Dirichlet boundary
Xconditions for equation \equ(NLW).
XAdditionally ask that $g(x,u) = -g(-x,-u)$.
XAmong this class of nonlinearities there is
Xan open dense set
Xsuch that there exist time periodic solutions
Xof \equ(NLW). That is, there exist parameters $(\Omega,r,\xi)$
Xand solutions $u(x,t;r)$ satisfying
X$$\eqalign{
X  |u(x,t;r) - &r \cos(\Omega t + \xi)  \psi_j(x)| < Cr^2 \cr
X      &|\Omega - \omega_j| < Cr^2    \cr }
X\EQ(2.4)
X$$
Xfor a set of $r$ of positive measure.
X
X\noindent
X{\bf Remarks:} The conditions on the nonlinear
Xterm $g(x,u)$ in
X\clm(NLWperiodic) and \clm(NLWDir) are quite explicit.
XThey depend only
Xupon the coefficients $g_1(x), \, g_2(x), \, g_3(x)$,
Xin other words only upon the $3-jet$ of $g$.
XRoughly, there is a condition on $g_1$ in order
Xto avoid certain primary resonances in the
Xlinear equation, and a condition of genuine nonlinearity
Xplaced upon $g_1, \ g_2$ and $g_3$.
XBoth are open conditions, excluding sets
Xwhich are essentially of codimension $1$. The
Xprecise nature of the good set
Xwill be described in more detail below.
XUnfortunately the
Xcase $g_1(x) = 0$ is too resonant for the present methods to
Xhandle, and is in the excluded set.
XOn the other hand for an open set of constants
X$m^2$ of full Lebesgue measure the case
X$g_1(x) = m^2$ is included
Xin the conditions of the theorems, thus
Xthe nonlinear Klein Gordon-equation
Xand the sine-Gordon equations, and
Xnonlinear perturbations of them,
Xare covered by our results.
XIt is possible to prove similar existence
Xtheorems for other boundary
Xconditions as well; these should be self adjoint as well as
Xsatisfying other conditions on the
Xeigenfunction expansion and
Xthe description of the nonlinearity.
XThe precise conditions that are required
Xare discussed in Section 6.
X
X
X
XThe inverse of the linearized operator in \equ(LW) plays
Xa role in the existence results.
XWhen applied to time-periodic
Xfunctions with frequency $\BGO$ the point spectrum of the
Xoperator is
X$\{ \omega_j^2 - \BGO^2 k^2; 1 \leq j < \infty, \
X   -\infty < k < \infty \}$.
XFor most choices of $\BGO$ and coefficient $g_1(x)$
Xthis is a dense set in ${\bf R}$. In particular, spectrum
Xwill accumulate at zero, a phenomenon which is
Xoften called the small divisor problem.
XIt is in this case that the results \clm(NLWperiodic) and
X\clm(NLWDir) are most interesting.
XBecause of the small divisors,
Xthe method of solution is of the Nash Moser type,
Xalternating a Newton iteration with approximate
Xinversion of the linearized operator.
XBoth theorems above follow from a more general
Xresult in the form
Xof a Nash-Moser type theorem for nonlinear equations
Xposed on the lattice $\zsquared$.
XTwo dimensional lattices are not special,
Xand the theorem is easily generalized;
Xin our situation two lattice directions suffice to
Xindex the temporal and spatial eigenfunctions
Xused to describe the above
Xproblems in nonlinear waves.
X
XThe lattice problem arises by expanding solutions
X$u(x,t)$ of \equ(NLW) in eigenfunction-Fourier
Xseries.  The coefficients $U(i,j)$ in this expansion must
Xsatisfy nonlinear equations on the lattice of the form
X$$
X     W(U) + V(\Omega)U =0~~.
X\EQ(NLlattice)
X$$
XDenoting lattice sites $ x = (j,k) \in \zsquared $
Xand $\Omega \in \real$ a frequency parameter, the form of
X$V(\Omega)$ is a diagonal linear operator on sequences $U(x)$,
X$$
X   V(\Omega)(x,y) = (\omega_j^2 - \Omega^2 k^2) \delta(x,y).
X\EQ(VOp)
X$$
Xwhere $\delta(x,y)$ is the Kronnecker delta.
XThe sequence of frequencies
X$\{ \omega_j \}_{j=1}^\infty$ satisfies
X$$
X  \qquad |\omega_j - j| < C_g,
X$$
Xwhich is the case for the eigenvalues
Xof the linear operator \equ(LOp).
XLet $\HH_\gs = \{ U(x) \in \l2(\zsquared) ; \sum_{x \in \zsquared}
Xe^{2\gs|x|}|U(x)|^2 < \infty \}$,
Xa Hilbert space of sequences.
XFor $\gs < \overline \gs$ the nonlinear term
X$W(U)$ is a real analytic mapping from
X$\HH_\gs \rightarrow \HH_{\gs-\ga}$,
Xfor any $0 < \ga \le \gs$.
XWe ask that $W(0)=0, \, D_U W(0)=0$,
Xand furthermore that
X$W(U)$ satisfy certain natural conditions of genuine
Xnonlinearity, best explained below.
XFor the nonlinear wave equation the
Xconstant $\overline \gs$ is determined
Xby the analyticity properties of the term $g(x,u)$.
X
XAgain we are led to linearize the nonlinear problem
X\equ(NLlattice) about the solution $U(x)=0$,  obtaining
X$$
X    V(\Omega) \varphi = 0.
X$$
XSolutions of this are simply $\varphi(x) = \gd_y(x)$,
Xwith $y = (j,k)$, and $\Omega = (\omega_j / k)$.
XEach nonzero eigenspace of $V(\Omega)$
Xis at least two dimensional,
Xfrom the form of $V(\Omega) = (\omega_j^2-\Omega^2 k^2)$,
Xwhich is spanned by the vectors in
X$\HH_{\gs}$ supported on the lattice sites
X$y = (j,k)$ and $\overline y = (j,-k)$.
X
XThe nonlinear existence theorem focuses on
Xsolutions near the linearized solution space,
Xwith frequency near the value
X$\Omega = \omega_j$ of the linearized problem.
XIn fact with little loss
Xof generality we will assume that $\omega_1$
Xis real and focus on a neighborhood of
X$ \Omega = \omega_1$,
Xwith solutions supported near $y=(1,1)$ and
X$\overline y =  (1,-1)$.
X
X
X
X
X%%%%%%%%%%%%%
XCentral to the construction is the
Xinversion of the linearized
Xoperator $H(U) = V(\Omega) + DW(U)$ of
X, about an approximate solution $U$.
XThis involves an analysis of the small
Xeigenvalues of $H(U)$.
XThese are connected with the geometry
Xof the lattice points $x$ at
Xwhich  is close to zero.  
X%%%%%%%%%%%%%%%%%%%%%
X
XA major part of this paper is the analysis of the
Xlinearized operator $H(U) = V(\Omega) + DW(U)$ 
Xof \equ(NLlattice).
XWe assume that $DW(u)$ is selfadjoint, which will 
Xbe the case for the lattice problems which come from the
Xnonlinear wave equation. By an analogy with quantum mechanics
Xwe call $H(U)$ a {\bf Hamiltonian operator}, and the
Xmatrix of the inverse operator $G(U)(z) = (H(U) - z\11)^{-1}$ 
Xthe {\bf Green's function}. Central to the construction
Xis the approximate inversion of $H(U)$ about an approximate
Xsolution $U$. This involves an analysis of the small 
Xeigenvalues of $H(U)$, and the geometry of the lattice 
Xsites $x$ at which $V(\Omega)(x,x)$ is close to zero.
XWe define a {\bf singular site} 
Xto be a lattice point $x=(j,k)$ at which 
X$|V(\Omega)(x,x)| = |\omega_j^2 - \Omega^2 k^2| < d_s$, 
Xwhere $d_s$ is a small parameter which is specified
Xin the next section.
XAny connected set of singular sites will be
Xcalled a singular region.
XWe will show that by restricting the
Xfrequency $\BGO$ appropriately, singular
Xregions for the Dirichlet problem
Xconsist only of isolated sites,
Xwhile for the periodic problem singular regions will
Xconsist of no more than pairs of adjacent sites.
X
XIn order to make the first step in an existence
Xtheorem for \equ(NLW) with periodic or
XDirichlet boundary conditions we ask
Xfor certain conditions of
Xnonresonance among the linear frequencies
X$\{ \omega_j \}_{j=1}^\infty$.
XThis does not have to be a condition among
Xinfinitely many of them, but at least a large
Xenough number of the initial frequencies.
X
X\CLAIM Definition(L-nonresonance)
XA sequence $\{ \omega_j \}_{j=1}^\infty$ is
X$(L_0,d_0)$-nonresonant with $\omega_1$
Xif there exists some $\tau > 5$ such that
Xfor all $|j|+|k|~\leq~L_0$
Xthe following conditions hold:
X$$
X   |\omega_1^2 k^2 - \omega_j^2| >   d_0~~
X{\rm if}~~(j,k)~\ne~(1,\pm 1).
X\EQ(2.65)
X$$
Xand 
X$$
X   |k\omega_1 - j| > 
X      {d_0 \over (|j| + |k|)^\tau},~~{\rm for}~~
X(j,k)~\ne~(0,0).
X\EQ(2.6)
X$$
X
X
X\CLAIM Proposition(dense-L-nonres) An open dense set of
Xfrequency sequences $ \{ \omega_j \}_{j=1}^\infty $
Xare $(L_0,d_0)$-nonresonant with $\omega_1$ for
Xsome $L_0,d_0$ with $d_0  = o(L_0^{-1/2})$.
X
X
X
XThis condition is on the equation linearized about
X$U = 0$, thus depends only upon the coefficient
X$g_1(x)$. We defer to Section 6 the discussion of the
Xprecise topology in which the above set of
Xfrequency sequences
X$ \{ \omega_j \}_{j=1}^\infty $ is dense, and the
Xproof of this proposition; however the set of coefficients
X$g_1$ satisfying \equ(2.6) is open and dense
Xin $L^2(0,\pi)$,
Xand has open intersection with the $\pi$ periodic,
Xanalytic potentials.
X
X\SUBSECTION Symmetries of the equation
X
XThe wave equation \equ(NLW) has certain elementary
Xproperties of
Xsymmetry, relevant
Xto this paper, that are reflected in the nonlinear
Xlattice systems \equ(NLlattice).
XThe sequences $u \in \HH_\gs$ among
Xwhich we construct solutions
Xare complex, however they will correspond to
Xreal solutions of
Xthe wave equation.
XDenote the involution on the lattice
X$$
X x = (j,k) \rightarrow \overline x = (j,-k)
X\EQ(invol)
X$$
Xand the complex conjugate of $U$ by $\overline U$, then the
Xreality condition on sequences is that
X$\overline{ U(x)} = U(\overline x)$.
XWe will require that the lattice equation
X\equ(NLlattice) is covariant
Xwith respect to this symmetry,
X$$\eqalign{
X  \overline {V(\Omega) U(x)}  &=
X    V(\Omega) \overline {U(x)} =
X          V(\Omega) U(\overline x ) \cr
X  \overline {W(U(x))} &=
X    W(\overline {U(x)}) = W(U(\overline x)). \cr }
X\EQ(sym1)
X$$
X
XThe wave equation respects an additional
Xtranslational symmetry;
X$t \rightarrow t+T, \, T \in \real$.
XThat is, time translation leaves the
Xequation and the boundary
Xconditions invariant.
XWe will consider lattice systems which also
Xpossess a continuous
Xsymmetry of this form.
XThe translation operator on sequences in
X$\l2(\zsquared)$ is the diagonal operator
X$$
X   T_{\xi} U(x) = e^{ik\xi} U(x),
X$$
Xwhere $x = (j,k)$.
X$T_{\xi}$ is a unitary operator on $\l2(\ZZ)$ and on all the
X$\HH_\gs$ spaces, and it preserves the reality 
Xcondition.  From the nature of the diagonal operator notice that
X$T_{\xi} V(\Omega) U = V(\Omega) T_{\xi} U$.
XWe will further require that the nonlinearity $W$ satisfy
X$$
X   T_{\xi} W(U) = W(T_{\xi} U).
X\EQ(sym2)
X$$
XThe nonlinear terms in lattice problems arising from the
Xnonlinear wave equation satisfy \equ(sym2),
Xbecause the system is autonomous.
XOur construction will be of families of
Xsolutions invariant with
Xrespect to this translation.
XThe interpretation is that this is the
Xconstruction of embedded
Xinvariant circles of solutions of \equ(NLW)
Xin the space $\HH_\gs$.
X
XThe final requirement on the lattice problem is that
Xthe linearized operator $H(U) = V(\Omega) + DW(U)$
Xis selfadjoint. Since $V(\Omega)$ is real and
Xdiagonal, this is the condition on the nonlinear 
Xterm $W(U)$ that 
X$$
X   DW(U)(x,y) = {\overline {DW(U)}}(y,x)~~.
X\EQ(selfadjDW)
X$$
XAgain, this condition holds for problems stemming
Xfrom the nonlinear wave equation.
X
X
XWe expect that similar constructions can be
Xobtained for invariant tori of
Xhigher dimension, giving rise to
Xsolutions of the nonlinear wave
Xequation \equ(NLW) which are quasiperiodic in time.
XIn a second publication we plan to address this and 
Xother problems from our point of view.
X
X
X
X
X\SUBSECTION Results for lattice problems
X
XThe main existence theorem for periodic solutions for nonlinear
Xlattice problems \equ(NLlattice) can now be stated. We fix
Xthe exponent $1/2 < \eta < 1$.
X
X\CLAIM Theorem (NLLat) Consider equations \equ(NLlattice) which
Xsatisfy the reality and translation invariance conditions
X\equ(sym1), \equ(sym2), \equ(selfadjDW).  
XSuppose further that the nonlinear
Xterm in \equ(NLlattice) satisfies hypotheses {\bf H1}-{\bf H3}
Xof Section 6. There is a constant $L_*$ such that
Xif $\{ \omega_j \}_{j=1}^\infty$ is
X$(L_0,d_0)$-nonresonant with $\omega_1$ for  $d_0 > L_0^{-\eta}$,
Xfor some $L_0 > L_*$,  then
Xthere is an open set of nonlinearities
X$W$ such that there exist uncountably many solutions
X$U(x) \in \HH_{{\overline \gs}/2}$ of \equ(NLlattice).
XMore precisely there exist $(r,\theta,\Omega)$
Xsuch that these solutions satisfy
X$$\eqalign{
X     \Vert U &- rT_{\theta}(\gd_y
X                      + \gd_{\overline y})\Vert_{\overline{\sigma}/2}
X                            \leq Cr^2  \cr
X     |\Omega &- \omega_1| \leq Cr^2  \cr}
X\EQ(2.8)
X$$
XThese sequences remain solutions when acted upon by the
Xtranslation $T_\xi$; they are embedded circles in
Xthe space $\HH_{{\overline \gs}/2}$.
X
XBoth  \clm(NLWperiodic) and \clm(NLWDir)
Xfollow from this theorem.
XIndeed, consider solutions to the nonlinear problem
X\equ(NLW) described in terms of the eigenfunctions,
Xor normal modes, of the linearized equations.
XOne expands a function $u(x,t)$ which is
X$2\pi / \Omega$ periodic in time,
Xsatisfying the correct spatial
Xboundary conditions (Dirichlet on $[0,\pi]$, or periodic),
Xin terms of the eigenfunction expansion.
XLet $\xi = \Omega t$, then
X$$
X   u(x,\xi) = \sum_{(j,k) \in \zsquared}
X                     U(j,k) e^{ik \xi} \psi_j(x).
X\EQ(e_funct)
X$$
XSquare integrable time periodic solutions
X$u$ to the wave equation \equ(NLW)
Xcorrespond to sequences $U(j,k) \in \l2(\zsquared)$
Xwhich solve the lattice equation \equ(NLlattice).
XThe nonlinear term in the wave equation is
X$g(x,u) - g_1(x)u$.
XThis corresponds to the nonlinearity
Xfor the lattice system
X$$
X    W(U)(j,k) = \int_0^{2\pi} \int_0^\pi
X        \psi_j(x) e^{-ik \xi} (g(x,u) - g_1(x)u) dx d\xi.
X\EQ(Wdef)
X$$
X
XWe call equation \equ(NLlattice) the
X{\bf mode interaction equation} for the
Xnonlinear wave equation \equ(NLW).
XIf $U(x) \in \HH_\gs$ then the solution $u(x,\xi) $
Xgiven by \equ(e_funct) is analytic.
XIf $U({\overline x}) = \overline {U(x)}$, then
X$u(x,\xi)$ is real, and vice versa.
XFurthermore, with $g$ analytic in the region
X$\{ (x,u); |{\rm Im} \  x| < \overline \gs \}$,
Xand with our choices of boundary
Xconditions, the lattice nonlinearity
X$W \in C^{\omega}(\HH_\gs;\HH_{\gs-\ga})$
Xfor all $0 < \ga \leq \gs < \overline \gs$.
XOther boundary conditions will also result in analytic
Xnonlinearities on the spaces $\HH_\gs$,
Xthis is discussed in detail in section 6.
XFor these, theorems similar to \clm(NLWDir)
Xand \clm(NLWperiodic) also hold.
X
XConversely, starting from a solution
X$U(x) \in \HH_\gs$ of the
Xlattice equation \equ(NLlattice),
Xconsider the function
X$u(x,\xi) = \sum_{(j,k) \in \zsquared}
XU(j,k) \psi_j(x) e^{ik \xi} $
Xand its translates by $T_\theta$.
XSince $U \in \HH_\gs$
Xthey form an analytic family, in fact an embedded circle.
XSetting $\xi = \Omega t$,
Xa real analytic solution of the nonlinear wave
Xequation \equ(NLW) is obtained.
X
X
XWe can now give a more precise
Xdescription of the existence result.
XThe proof of \clm(NLLat)
Xis by a Nash Moser iteration scheme,
Xproving a result which is in spirit
Xvery close to the theorem of
XKolmogorov, Arnold and Moser.
XIn fact the conclusions are reminiscent of these results;
Xwe construct families of solutions of
X\equ(NLlattice) invariant
Xunder translation $T_\xi$,
Xparametrized by $r \in C$ a Cantor set.
XThat is, the periodic solutions
Xthat we find occur not in smooth
Xcurves but in totally disconnected families,
XCantor sets foliated by invariant circles.
XThere is a set of positive measure of amplitudes
X$r$ for which
Xthere are solutions.
XThis feature of totally disconnected
Xfamilies of solutions is
Xfamiliar in the study of
Xinvariant tori of quasiperiodic orbits
Xfor Hamiltonian systems near
Xelliptic stationary points.
XThe fundamental reason behind this phenomenon
Xis that Hamiltonian systems
Xpossessing infinitely many degrees of freedom
Xhave the possibility for the
Xgeneration of a dense set of
Xlinear resonances even for periodic orbits.
X
XIn order to describe the detailed
Xexistence result we need to
Xdefine the condition of genuine nonlinearity.
XLet $B_0 = \{ x \in \zsquared : |j| + |k| \leq L_0 \} $
Xbe a bounded subdomain of the lattice $\zsquared$,
Xand define $\Pi_0$
Xto be the orthogonal projection onto
X$\l2(B_0)$.
XThis projection commutes with $T_\xi$,
Xand furthermore $\Pi_0$
Xcommutes with the lattice involution
X\equ(invol) and thus
Xpreserves the reality condition.
XConsider an approximate problem to
X\equ(NLlattice) for a sequence
X$U_0 \in \l2(B_0)$
X$$
X    \Pi_0 \bigl( W(U_0) + V(\Omega)U_0 \bigr) = 0.
X\EQ(P0NLLat)
X$$
XIf the sequence $\{ \omega_j \}_{j=1}^\infty$
Xis $(L_0, d_0)$ nonresonant then
X$\Pi_0 V(\Omega)$ restricted to $\ell^2(B_0)$
Xhas only a double eigenvalue at
X$\Omega = \omega_1$,
Xwith eigenvectors supported on the lattice sites
X$N = \{ (1,\pm 1) \}$.
XWe denote the orthogonal projection onto
X$\l2(N)$ by $Q$, and set
X$P= (\11 - Q)$.
XParametrize a neighborhood of zero in the
Xnull space of $\Pi_0 V(\go_1)$ by
X$\pp = (p_1,p_2) \rightarrow \varphi(\pp),
X\ \|\pp\| < r_0$,
Xwith $\varphi (\pp) =
X   (p_1 + i p_2)\delta_{(1,1)}(x) +
X   (p_1 - ip_2)\delta_{(1,-1)}(x)$.
XEquation \equ(P0NLLat) possesses a branch of nontrivial
Xsolutions $(U(\pp),\Omega_0(\pp))$
Xbifurcating from $(\pp,\BGO) = (0,\go_1)$.
XDenoting rotations by angle $\xi$ in the plane
X$\pp \in \real^2$ also by $T_\xi$, then
X$T_\xi \varphi(\pp) = \varphi(T_\xi \pp)$,
Xand the above branch of
Xsolutions is $T_\xi$ invariant.
XThe condition of genuine nonlinearity
Xis that for this branch of
Xapproximate solutions the frequency parameter
X$\Omega_0$ is nondegenerate in $\pp$.
X
X\CLAIM Definition (twist)
XThe problem \equ(NLlattice) is said to
Xsatisfy a twist condition
Xif the bifurcation surface of the
Xapproximate problem
X$ \CC_0 = (\pp,\Omega_0(\pp))$ satisfies
X$$\eqalign{
X     \Omega_0(0) &= \omega_1 \cr
X     \det \partial^2_{\pp} &\Omega_0 (0) \not= 0  \cr}
X\EQ(twistcond)
X$$
XSince the branch of solutions is
Xinvariant under $T_\xi$ then
X$\partial_{\pp} \Omega_0 (0) = 0$ automatically .
X
X
XRecall that $\eta$ is a small positive constant fixed above.
X\CLAIM Theorem (NLLat2)
XLet the sequence $\{ \go_j \}_{j=1}^\infty$
Xbe $(L_0,d_0)$ nonresonant with $\omega_1$
Xfor some $L_0 > L_*$  with $d_0 > L_0^{-\eta}$.
XIf the nonlinear term in \equ(NLlattice) satisfies
Xhypotheses {\bf H1}-{\bf H3}, there is a neighborhood
X$\Eta_0 = \{ (\pp,\BGO); \|\pp\| < r_0,
X               |\BGO - \go_1| < r_0^2 \}$
Xin the parameter space and a function
X$u(x;\pp,\BGO) \in \HH_{\gs/2}, \, Qu = 0$
Xwhich is $C^\infty$ in the parameters
X$(\pp,\BGO) \in \Eta_0$ satisfying:
X(i) there is a Cantor set $\Eta \subseteq \Eta_0$
Xin parameter space, invariant under
X$T_\xi$, such that for
X$(\pp,\BGO) \in \Eta, \, u(x;\pp,\BGO)$
Xis a solution of the first
Xbifurcation equation
X$$
X   P \bigl( W(\varphi(\pp) + u) + V(\BGO)u \bigr) = 0.
X\EQ(bifeq1)
X$$
X(ii) Set the exponent $0 < \nu < 1 - \eta$.
XIf additionally the approximate problem \equ(P0NLLat) on
X$\l2(B_0)$ satisfies a twist condition
X$$
X   \det\partial^2_{\pp} \BGO_0(0) \geq {\Kappa_0^2} > 0
X\EQ(twist2)
X$$
Xwith ${\Kappa_0^2} > L_0^{-\nu}$,
Xthen there exists a $C^\infty$ surface
X$ \CC = (\pp,\BGO(\pp))$ invariant
Xunder $T_\xi$ satisfying the second bifurcation equation
X$$
X  Q \bigl( W(\varphi(\pp) + u(\pp)) + V(\BGO(\pp))u(\pp) \bigr) = 0.
X\EQ(bifeq2)
X$$
Xwhere the intersection
X$C = \{ \pp ; \CC (\pp) \cap \Eta \not= \emptyset \}$
Xhas positive measure.
X
XThe intersection of $\CC$ with $\Eta$
Xis of course the solution
Xset for equation \equ(NLlattice),
Xconsisting typically of a
XCantor set foliated by invariant circles.
XThe measure of the set
X$C = \{ \pp ; \CC(\pp) \cap \Eta \not= \emptyset \}$
Xis relatively large,
Xon the order of $\pi r_0^2$.
X
XWe note that even if the non-degeneracy condition
Xfails to hold, one may have periodic solutions.
XSuch situations are explored in [CW].
X
XIt is possible that the actual bifurcation point
X$(0,\go_1) \notin \Eta$ due to an exact or
Xnear resonance of $\go_1$ and
X$\go_j$, with $j$ such that $(j,k) \notin B_0$.
XHowever if none of these
Xoccur then there is a result on the density of
Xthe periodic orbits within radii $0 < \|\pp\| < r_0$.
XFix exponents $0 < {\overline \tau}$ and
X$ 0 < {\overline \alpha}$ such that
X${\overline \alpha} + 1 > {\overline \tau}$.
X%%%%%%%%%%%%%%%%%%
X
X\CLAIM Theorem(density)
XAn $(L_0,d_0)$-nonresonant sequence $\{\omega_j \}_{j=1}^\infty $
Xis fully nonresonant
Xwith $\omega_1$ if there exist positive constants
X$c_1$ and $c_2$ such that for all
X$(j,k) \in \ZZ^+ \times \ZZ$, $(j,k) \ne (0,0)$,
X$$
X   |k \omega_1 - j| > \ { c_1 \over
X     (|j| + |k|)^{\overline \tau} },~~
X$$
Xand if for  $(j,k) \not= (1,\pm 1)$,
X$$
X   |k^2 \omega_1^2 - \omega_j^2| > \
X      { c_2 \over (|j|+|k|)^{\overline \alpha}} ~~.
X$$
XIn this case the Cantor set of \clm(NLLat2) has $\pp = 0$
Xas an accumulation point. Furthermore there is an
Xestimate of the density of periodic orbits near $\pp = 0$.
XThere are constants $\mu > 0, \ C_g$ such that for all
X$0 < r_1 < r_0$,
X$$
X     {\rm meas} \{ r \in (0,r_1); \|\pp\| = r, \
X	    (\pp,\Omega(\pp)) \in \Eta \}
X		 \geq r_1(1-C_g r_1^\mu).
X$$
X
X
X
XBecause it depends upon the details of the induction
Xprocess, the proof of this density
Xresult is deferred to section 6. In the proof
Xestimates of the size of the exponent $\mu$ will be given.
X
X
X
X
X%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%.
X\SUBSECTION The Nonlinear Klein-Gordon and sine-Gordon equations
X
XPrincipal examples of problems of the form \equ(NLW)
Xare the nonlinear Klein Gordon equation
X$$
X \partial_t^2 u = \partial_x^2 u - m^2u + (m^2/3) u^3
X\EQ(NLKG)
X$$
Xand the related sine-Gordon equation
X$$
X   \partial_t^2 u = \partial_x^2 u - m^2 \sin (u).
X\EQ(NLsG)
X$$
XBoth equations \equ(NLKG) and \equ(NLsG)
Xhave frequency sequences 
X$$\{ \omega_j \}_{j=1}^\infty =
X\{ \sqrt{j^2 + m^2} \}_{j=1}^\infty$$ for the
XDirichlet problem, and
X$$\{ \sqrt{4[(j+1)/2]^2 + m^2 } \}_{j=0}^\infty$$
Xfor the periodic problem. (The term
X$[(j+1)/2]$ inside the square root means the integer
Xpart of $(j+1)/2$.)
X
XWhen posed with periodic boundary conditions on the interval
X$[0,2\pi]$, \equ(NLsG) is a
Xcompletely integrable Hamiltonian system.
XThis is not the case for \equ(NLKG), or for
X\equ(NLsG) with Dirichlet conditions posed at
X$x=0,\pi$.  Traveling wave solutions for \equ(NLKG) and
X\equ(NLsG) satisfying
Xperiodic boundary conditions on the interval
X$[0,\pi]$ are easily
Xdescribed using phase plane analysis for
Xfunctions $u(x-ct)$.  Thus, we will concentrate on the
Xcase of Dirichlet boundary conditions.
XOn a formal level this problem was
Xdiscussed by J.B. Keller \& L. Ting [KT],
Xand the curvature of the solution surfaces
X$\partial^2_\pp \BGO(0)$ was derived.
XThese solutions are related to a class called
X`breather solutions' in the literature,
Xwhich are spatially localized, time
Xperiodic solutions to nonlinear wave
Xequations posed on all of
X$x \in \real$.
X\clm(NLWDir) implies the existence
Xof small amplitude time periodic solutions of both
X\equ(NLKG) and \equ(NLsG) for a
Xset of values of the parameter $m^2$ of full measure.
XWe have only to check that the hypotheses of \clm(NLLat2)
Xare satisfied for the associated lattice problem. Either 
Xof these equations could be perturbed
Xby an additional nonlinear term, $h(x,u)$, and as long
Xas the total nonlinearity satisfied the twist condition,
Xthe results discussed below would still hold.
X
X
X
X
X
X
X%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
X
X\CLAIM Theorem (nlkgfreqs)
XConsider a sequence of constants $d_0,L_0$
Xsuch that $d_0 \log (L_0) \rightarrow 0$.
XThere is an open set ${\cal M}$ of full
XLebesgue measure such that if $m^2 \in {\cal M}$,
Xthen the frequency sequences
X$\{ \omega_j \}_{j=1}^\infty$ for the equations
X\equ(NLKG) and \equ(NLsG) are
X$(d_0,L_0)$ nonresonant with $\omega_1$ for
Xsome $(d_0,L_0)$.
X
X
X
X\PROOF  Fix $d_0,L_0$ and consider an
Xarbitrary interval $[a,b]$  of parameters $m^2$.
XFor the Dirichlet problem, the first
X$(d_0,L_0)$-nonresonance
Xcondition is violated for those $m^2$ such that
X$| k^2(1+m^2) - (j^2+m^2)| \leq d_0$. That is
X$$
X   \bigl| \bigl( { j^2-k^2 \over k^2-1 } \bigr) - m^2 \bigr|
X	    \leq { d_0 \over |k^2-1|},
X$$
Xhence by excising a closed interval of length $2d_0/|k^2-1|$
Xabout every point
X$(j^2-k^2)/(k^2-1),\ |j|+|k| \leq L_0, \
X  (j,k) \not= (1, \pm 1)$
Xwhich falls within the interval $[a,b]$,
Xthe remaining values of $m^2$ satisfy the first condition
Xof \clm(L-nonresonance). Note that this imposes
Xno condition for $k= \pm 1$, and that
X$m^2 = 0$ is excised. The diophantine condition
Xof \clm(L-nonresonance) is violated for those
Xvalues of $m^2$ such that
X$|k \sqrt{m^2 + 1} - j| \leq d_0/(|j|+|k|)^\tau$.
XExcising  a closed interval of length
X$2d_0/(|j|+|k|)^\tau$ about every point
X$m^2 = (j/k)^2 - 1$ as well, the remaining
Xparameters satisfy both conditions in \clm(L-nonresonance).
XCall this open set ${\cal M}(d_0,L_0)$.
XThe only $(j,k)$ that need to
Xbe considered are those for which
X$(1+a)k^2 - C_0 \leq j^2 \leq (1+b)k^2 + C_0$.
XThe total measure of the excised intervals is estimated by
X$$
X   \sum_{{|j|+|k| \leq L_0 \atop k\not= \pm 1} \atop
X       (j^2/k^2) \in (a-C_1,b+C_1) }
X       {2d_0 \over |k^2-1|} \ + \
X       {2d_0 \over (|j|+|k|)^\tau}
X          \leq  C d_0 \log (L_0),
X$$
Xas long as $\tau \geq 2$.
XThe set ${\cal M} \cap [a,b] =
X   \cup_{d_0,L_0} {\cal M}(d_0,L_0)$,
Xthus if $d_0\log (L_0) \rightarrow 0$, the set ${\cal M}$ is
Xof full measure. The proof in the case of periodic
Xboundary conditions  is similar. 
X\endproof
X
XComputing the curvature of the approximate bifurcation branches
Xfor the equation \equ(2.6)
Xis a straightforward exercise, and it is non-zero and
Xindependent of the choice of approximate domain
X$B_0$ as long as $L_0 \geq 6$.
X\clm(NLLat) implies the following
Xexistence result
X
X\CLAIM Theorem (NLKG+sG)
XConsider the equations \equ(NLKG) and \equ(NLsG),
Xsatisfying Dirichlet boundary conditions
Xon the interval $[0,\pi]$.
XThe curvature of the branch bifurcating from the
Xfrequency $\go_1$ is
X$$
X   \partial_{\pp}^2 \BGO(0) =
X       {-3 m^2 \over 16 \sqrt{1+m^2}}{\bf 1}.
X\EQ(KGcurv)
X$$
XThus the nonlinear problem satisfies the
Xtwist condition, and for $m^2 \in {\cal M}$
Xand $d_0 > L_0^{-\eta}$, there exist small
Xperiodic solutions with frequency close to $\go_1$.
X
XIf the sign of the nonlinear term in the equation
X\equ(NLKG) is reversed, that is, if
X$g(x,u) = m^2 (u + (1/3)u^3)$ then the curvature
X\equ(KGcurv) reverses sign, but still retains a
Xtwist and the existence theorem holds.
X
X
X
X
X
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X
X
X\SECTION The Induction Argument
X
X\SUBSECTION Notation
X
XIn this subsection we introduce notation 
Xrelating to the function spaces in which we solve
X\equ(NLlattice) .
XWe wish to show that the solutions of these lattice problems
Xdecay exponentially;  to measure this
Xdecay we introduce a family of Hilbert spaces.
XWhen specifying points in $\zsquared$ by a single
Xsymbol, we will use $x$, $y$, etc.
XWhen we wish to specify their components we will
Xuse $(j,k)$, $(l,m)$, etc.
XFor points $x=(j,k) \in \zsquared$,
Xwe always use the $\ell^1$ norm,
X$|x|=|j|+|k|$.
X
XDefine a family of Hilbert spaces $\HH_{\sigma}
X\subset \l2 (\zsquared)$,
Xwhich consist of elements of $\l2 (\zsquared)$ for
Xwhich the norm
X$$
X   \|u\|^2_{\sigma} = \sum_{x \in \zsquared} |u(x)|^2
X         e^{2\sigma |x|}
X$$
Xis finite.  We denote the inner product in these Hilbert spaces
Xby $\langle \cdot ,\cdot \rangle_{\sigma}$.
XIf $\sigma = 0$,
X$\HH_{\sigma} = \l2$.  We will denote the inner product in $\l2$
Xby either  $\langle \cdot ,\cdot \rangle_{0}$ or
X$\langle \cdot ,\cdot \rangle$, depending on the circumstances.
XSimilarly, $\| \cdot \|$ will mean $\| \cdot \|_0$--$i.e.$
Xthe $\ell^2$ norm.
XThis is similar to sequence spaces used in [P2].
X
XIf $S : \HH_{\sigma} \to \HH_{\sigma} $ , let $\| S \|_{\sigma}$
Xbe the usual Hilbert space operator norm.  The following
Xproposition shows that these norms respect the exponential
Xweights.
X
X\CLAIM Proposition(opnorm)  The norm is a Banach
Xalgebra norm;
X$$\parallel ST\parallel_{\sigma} \leq \parallel
XS\parallel_{\sigma}\parallel T\parallel_{\sigma}.$$
XIf $\parallel S\parallel_{\sigma} \leq C_{s}$
Xthere is a sup norm estimate of the matrix elements
Xof $S$;
X$$|S(x,y)|\leq C_{s} e^{-\sigma |x-y|}.$$
XIf for some $\sigma$ the matrix elements of an
Xoperator $S$ are bounded by
X$$|S(x,y)|\leq C e^{-\sigma |x-y|},$$
Xthen for all $0\leq \gamma <\sigma$
X$$\parallel S\parallel_{\sigma-\gamma}\leq C/\gamma^{2}.$$
X
XAn additional simple but important property of these norms
Xis embodied in the following lemma.
X
X\CLAIM Lemma(cutoff) Let $B_{L} = \{(j,k) \in
X\ZZ^+ \times \ZZ ; |j|+|k| \le L \}$.  Let
X$\Pi_{L}$ be orthogonal projection onto
X$\ell^2(B_{L})$, and let ${\bf 1} - \Pi_{L}$
Xbe its orthogonal complement.  If
X$0 < \gamma \le \sigma$ then
X$$\eqalign{
X \| \Pi_L f \|_\gs &\leq
X          e^{\gamma L} \| f \|_{\gs-\gamma}     \cr
X  \| ({\bf 1} - \Pi_{L}) f \|_{\sigma - \gamma} &\leq
X          e^{-\gamma L} \| f \|_{\sigma}.       \cr}
X$$
X\PROOF  From the definition of the norm,
X$$\eqalign{
X \| \Pi_L f \|_\gs \leq&
X          \sup_{|j|+|k| \leq L}
X       e^{\gamma (|j|+|k|)} \| f \|_{\gs-\gamma}    \cr
X     \leq& e^{\gamma L} \|f\|_{\gs-\gamma}          \cr
X \|({\bf 1} - \Pi_{L}) f\|_{\sigma-\gamma} \le &
X           \sup_{|j|+|k| > L} e^{-\gamma(|j|+|k|)}
X              \| f \|_{\sigma} \cr
X   \le &  e^{-\gamma L} \|f\|_{\sigma}.             \cr}
X$$
X\endproof
X
XWe will also use the analyticity of these solutions
Xas functions of parameters.  For this purpose,
Xif $\NN \subset \real^2 \times \real$, we define complex domains
X$D(\NN;\rho)
X = \{(\zz,\BGO) \in \complex^2 \times \complex ;
X\sqrt{ \| \zz - \pp \|^2 + |\Omega_1 - \Omega|^2 }
X<  \rho \ {\rm for \ some} \ (\pp,\Omega_1) \in \NN \}$.
X
X
X
X\SUBSECTION The bifurcation problem on $B_0$
X
XThe induction procedure of this paper is started
Xby solving an approximate bifurcation problem
X\equ(P0NLLat), consisting of the full
Xequation \equ(NLlattice), restricted to the lattice
Xsubdomain $B_0$.  For $u \in \ell^2(B_0)$, we write
X$V_0(\Omega) u = \Pi_0 V(\Omega) u$ and $W_0(u) = \Pi_0 W(u)$.
XLinearizing \equ(P0NLLat) around $u \equiv 0$, we obtain
X$V_0(\Omega) \phi = 0$.  If the sequence
X$\{\omega_j \}_{j=1}^{\infty}$ is $(d_0,L_0)$-nonresonant
Xwith $\omega_1$, then $V_0(\omega_1)$ has a two-dimensional
Xnull space $\ell^2(N)$ parameterized by $\phi(\pp)$,
X$\pp \in \real^2$.  As we have indicated above,
Xthese will be solved using a Lyapunov-Schmidt decomposition;
X$$
X    P(W_0(\phi(\pp) +u) + V_0(\Omega) u ) = 0~~,
X\EQ(firstbifur)
X$$
X$$
X   Q(W_0(\phi(\pp) +u) + V_0(\Omega) u ) = 0~~.
X\EQ(secondbifur)
X$$
XThis pair of equations is equivalent to \equ(P0NLLat).
X
X
XWith these definitions in hand, we
Xcan construct the first approximation to the
Xsolution of \equ(NLlattice).
XFor any subset $B \subset \ZZ^2$, we write
X$\overline{B} = B \backslash N$, where $N$ is the set
Xof lattice points supporting the null space of $V_0(\omega_1)$.
XDefine $\NN_0 = \{ (\pp, \Omega)
X\in \real^2 \times \real ~|~
X|| \pp || < r_0, |\Omega - \omega_1| < r_0^2 \}$, a neigborhood
Xof the point $(0,\omega_1)$ at which bifurcation 
Xbranches are to be constructed.
X
X\CLAIM Lemma(firstbifonB0)
XSuppose that the sequence $\{ \omega_j \}_{j=1}^\infty$
Xis $(L_0,d_0)$ nonresonant with $\omega_1$. For any
X$0 < r_0 < (1/6C_W) d_0/L_0^2$ and
X$ \rho_0 < d_0/(3CL_0^2)$
Xthere exists a solution $u_0(x;\pp,\bgo) \in \l2(B_0)$
Xof (3.1) which is analytic in $D(\Eta_0,\rho_0)$.
XFurthermore,
X$$T_\xi u_0(x;\pp,\bgo) =
X  u_0(x;T_\xi \pp,\bgo),
X$$
Xand $u_0$ satisfies the estimate
X$$
X    \| u_0 \|_\gs < { 3C_WL_0 \over d_0} \|\pp\|^2
X\EQ (u_0est)
X$$
Xfor $\gs \leq \gs_{*}-(1/L_0)$.
X
XFor $(\pp,\BGO)$ in the complex subdomain
X$D(\Eta_0,\rho_0/2)$ the Cauchy estimates implies that
X$$\eqalign{
X   \| \partial_\BGO^\beta u_0 \|_\gs
X      \leq& {3C_W L_0 \over d_0}
X         {\|\pp\|^2 \over (\rho_0/2)^\beta }         \cr
X   \| \partial_{\pp} \partial_\BGO^\beta u_0 \|_\gs
X     \leq& {3C_W L_0 \over d_0}
X       \left( {\|\pp\| \over (\rho_0/2)^\beta } \right)
X         \left( 1 + {r_0 \over (\rho_0/2) } \right),  \cr}
X\EQ(u0derivs)
X$$
Xand in general for $|{\bf \alpha}| \geq 2$,
X$$
X  \| \partial_{\pp}^\alpha \partial_\BGO^\beta u_0 \|_\gs
X    \leq {3C_W L_0 \over d_0}
X          {1 \over (\rho_0/2)^{|\alpha|+\beta-2} }
X            \bigl( 1 + {r_0^2 \over (\rho_0/2)^2 } \bigr).
X$$
X
X\PROOF  Linearizing the  function on the left hand
Xside of (3.1) about $u_0=0$ and setting 
X$(\pp,\Omega) = (0,\omega_1)$ gives
X$$
X  PV_0(\omega_1) = H_{\overline {B_0}} (\omega_1).
X$$
XBy hypothesis this is invertible, with inverse
X$G_{\overline {B_0}}(\omega_1)$ bounded by $C/d_0$.
XEquation (3.1) is finite dimensional, solvable by
Xthe implicit function theorem to give a solution
Xanalytic in a complex neighborhood of
X$(\pp,\bgo) = (0,\omega_1)$.
XThe point of this lemma is to estimate the size
Xof this neighborhood.
XSolutions of (3.1) are fixed points of the mapping
X$$
X   u_1 = -G_{\overline {B_0}}(\omega_1)
X         \bigl( W_0(\phi(\pp) + u_0)
X            + (H_{\overline {B_0}}(\bgo)
X             - H_{\overline {B_0}}(\omega_1))u_0 \bigr).
X\EQ (mapping)
X$$
XThe symmetry of the equation with respect to $T_\xi$
Xassures the covariance property of the solution.
X
XUnique analytic solutions are assured for parameter
Xvalues such that the mapping in \equ(mapping) preserves a
Xneighborhood of the origin, on which it is a contraction.
XTo find a neighborhood that is mapped to itself,
Xwe use hypothesis {\bf H1} to estimate
X$$\eqalign{
X  \|u_1\|_\gs \leq& {C \over d_0}\|W_0(\phi(\pp) + u_0)\|_\gs
X        + \| G_{\overline {B_0}}(\omega_1)
X         \bigl( H_{\overline{B_0}}(\bgo)
X          - H_{\overline {B_0}}(\omega_1)u_0 \bigr) \|_\gs \cr
X              \leq& {C_W \over d_0 \gamma}
X                        (\|\pp\|^2 + {1 \over \gamma^2}
X                               \|u_0\|^2_{\gs+\gamma})
X             + {{\rm sup} \atop (j,k) \in {\overline {B_0}}}
X            \biggl| { \omega_j^2 - \bgo^2 k^2 \over
X        \omega_j^2 - \omega_1^2 k^2} - 1 \biggr| \|u_0\|_\gs \cr}
X$$
XUsing \clm(cutoff),
X$$\eqalign{
X   \|u_1\|_\gs \leq& {C_W \over d_0 \gamma}
X                      (\|\pp\|^2 +
X         {e^{2\gamma L_0} \over \gamma^2} \|u_0\|_\gs^2) \cr
X           &+ {C L_0^2 \over d_0} |\bgo-\omega_1|
X                       \|u_0\|_\gs,                      \cr}
X$$
Xand by optimizing over
X$\gamma$ such that $\gs + \gamma \leq \gs_{*}$
Xwe find that $\gamma = 1/L_0$ and
X$$
X  \|u_1\|_\gs \leq C_W { L_0 \over d_0}
X          (\|\pp\|^2 + L_0^2\|u_0\|_\gs^2)
X        + C {L_0^2 \over d_0} |\bgo-\omega_1| \|u_0\|_\gs~~.
X\EQ (mapest)
X$$
XWe require $r_0 = \rho_0 < d_0 / (3(C_W + C) L_0^2)$,
Xand define $C_u = 3 C_W L_0/ d_0$.
XFor $(\pp, \Omega) \in D(\NN_0 ; \rho_0)$, the complex
Xneighborhood of $\NN_0$, if $\Vert u_0 \Vert_{\sigma}
X\le C_u \Vert \pp \Vert^2 $
Xthen
X$\Vert u_1\Vert_\gs < C_u \Vert \pp \Vert^2$ also.
XThus the fixed point of the map will lie in this neighborhood.
X
XTo obtain a contraction estimate, we use hypothesis {\bf H2} of
Xsection 6.
X$$\eqalign{
X   \|G_{\overline {B_0}}(\omega_1)&(W_0(\phi(\pp) + u_2) -
X                   W_0(\phi(\pp) + u_1)\|_\gs \cr
X     \leq& {C \over d_0} \| \int_0^1 DW_0((\phi(\pp) +
X             tu_2+(1-t)u_1) \ dt \ (u_2-u_1)\|_\gs \cr
X     \leq& { C_W L_0^2 \over d_0} (\|\pp\| + L_0\|u_1\|_\gs
X               + L_0\|u_2\|_\gs) \|u_2-u_1\|_\gs \cr}
X$$
XIn order that this gives a contraction mapping when
X$\|\pp\| < r_0 + \rho_0
X$ and $\|u_1\|_\gs, \|u_2\|_\gs < C_u \|\pp\|^2$,
Xwe ask that $\rho_0 = r_0 < (1/9C_W)(d_0/L_0^2)$.
X
XThe second term is easier,
X$$
X   \|G_{\overline {B_0}}(\omega_1)(H_{\overline {B_0}}(\bgo) -
X        H_{\overline {B_0}}(\omega_1))(u_2-u_1)\|_\gs
X    \leq {CL_0^2 \over d_0} |\bgo-\omega_1| \|u_2-u_1\|_\gs
X$$
Xwhich gives a contracting estimate if
X$|\Omega - \omega_1| < r_0^2 + \rho_0$,
X and $2C(r_0^2+\rho_0) < d_0/L_0^2$.
X
XEstimate \equ(mapest) leads to an a priori estimate on the
Xfixed point.
X$$
X   \|u_0\|_\gs \leq C_u \|\pp\|^2
X          = {{3C_W L_0}\over{ d_0} }\|\pp\|^2.
X$$
XThis is the upper bound stated in \equ(u_0est).
X
X\endproof
X
X
XOver the neighborhood $\NN_0$ the problem \equ(P0NLLat)
Xis reduced to finding the zero set of the equation
X\equ(secondbifur).  A trivial solution branch is
X$\{(\pp,\Omega) : \pp =0, -r_0^2 < (\omega_1 -
X\Omega) < r_0^2 \}$, and we seek an additional family of
Xsolutions parametrized by $\{ \pp : \| \pp \| < r_0 \} $.
XAs the equations \equ(firstbifur) and \equ(secondbifur)
Xand the solution $u_0(x;\pp,\Omega)$ behave covariantly
Xwith respect to the translation $T_{\xi}$, the zero
Xset of \equ(secondbifur) is  invariant
Xunder $T_{\xi}$.  Properties of this set are given
Xin the next result.
X
X\CLAIM Lemma(secondbifonB0) The mapping $(\pp,\Omega) \to
XG_0(\pp,\Omega) = Q\left( W_0 (\phi(\pp) +
Xu_0(\pp,\Omega) ) + V_0(\Omega) \phi(\pp) \right)$
Xis analytic on $\NN_0$, zero for
X$\{(\pp,\Omega) : \pp = 0, -r_0^2 < (\omega_1 -
X\Omega) < r_0^2 \}$, and has a Taylor expansion at $\pp = 0$,
X$\Omega = \omega_1$ with the following Taylor coefficients:
X$$\eqalign{
X  \partial_{\pp} G_0(0,\omega_1) \ =& \ 0        \cr
X    \partial_{\Omega} G_0(0,\omega_1) \ =& \ 0,   \cr
X}
X\EQ(firstderiv)
X$$
Xand
X$$\eqalign{
X  \partial_{\pp}^2 G_0(0,\omega_1) =& \ 0         \cr
X     \partial_{\Omega}^2 G_0(0,\omega_1) =& \ 0   \cr
X     \partial_{\pp} \partial_{\Omega}
X      G_0(0,\omega_1) =&
X    \ Q\left( \partial_{\Omega} V_0(\omega_1))
X        \partial_{\pp} \phi(0) \right)             \cr
X   =& - 2\omega_1 \partial_{\pp} \phi(0)~~.        \cr
X}
X\EQ(secondderiv)
X$$
XIf $\omega_1 \ne 0$, the last term is nonvanishing.
X
XThus the zero set of $G_0(\pp,\Omega)$ in a neighborhood
Xof $(\pp,\Omega) = (0,\omega_1)$ consists of the
X$\Omega$ axis union a surface $(\pp,\Omega_0(\pp))$,
Xgiven as a graph over a neighborhood of zero in the
X$\pp$-plane.
X
X\PROOF  The analyticity follows immediately since
Xthe implicit function theorem guarantees that
X$u_0(\pp,\Omega)$ is analytic, $\phi(\pp)$ is
Xanalytic by construction, and $W$ and $V$ are analytic
Xby assumption.
XWe next compute the Taylor expansion of the function
X$G_0$ at $(\pp,\Omega) = (0,\omega_1)$.  Clearly
X$Q\left( W_0 (\phi(\pp) + u_0(\pp,\Omega) ) + V_0(\Omega)
X\phi(\pp) \right)|_{(\pp,\Omega) = (0,\omega_1)}
X= 0$, and the first derivatives are
X$$\eqalign{
X   \partial_{p_j} G_0(0,\omega_1) =&
X   Q\left( V_0(\omega_1) \partial_{p_j} \phi(0) \right)
X   = 0~~;~j=1,2~,                                    \cr
X   \partial_{\Omega} G_0(0,\omega_1) =&
X   Q\left( \partial_{\Omega} V_0(\omega_1) \phi(0) \right)
X   = 0 ~~,                                           \cr
X}
X\EQ(firstderivtwo)
X$$
Xwhere we use that $\phi(0) = 0$ and $\partial_{p_j}
X\phi(0)$ is in the null space of $V_0(\omega_1)$.
XTo compute the
Xsecond derivatives we use that $D_u W_0(0) = 0$,
X$\| u_0 \|_{\sigma} \le C_u \| \pp \|^2$, and the fact
Xthat \equ(secondbifur) is covariant with respect to
X$T_{\xi}$.  Indeed,
X$$\eqalign{
X   T_{\pi} Q& \left( W_0 (\phi(\pp) + u_0(\pp,\Omega) ) +
X     V_0(\Omega) \phi(\pp) \right)       \cr
X  =& \ Q\left( W_0 (\phi(-\pp) + u_0(-\pp,\Omega) ) +
X     V_0(\Omega) \phi(-\pp) \right)  \cr
X   =& - Q\left( W_0 (\phi(\pp) + u_0(\pp,\Omega) ) +
X     V_0(\Omega) \phi(\pp) \right)~~.\cr
X}
X$$
XThus, $G_0$ is odd in $\pp$, and both
X$\partial_{\pp}^2 G_0(0,\omega_1) = 0 $ and
X$\partial_{\Omega}^2 G_0(0,\omega_1) = 0 $.
XThe mixed partial is nonzero, however, as long as
X$\omega_1 \ne 0$.
X$$\eqalign{
X  \partial_{\Omega} \partial_{p_j} G_0(0,\omega_1) =& \
X    Q \big( D^2 W_0(0)(\partial_{p_j} \phi + \partial_{p_j}
X      u_0 ) (\partial_{\Omega} u_0)                     \cr
X     &+ D W_0(0)(\partial_{\Omega} \partial_{p_j} u_0) +
X      \partial_{\Omega} V_0(\Omega) \partial_{p_j}
X      \phi \big)|_{(\pp,\Omega)=(0,\omega_1)}           \cr
X    = & Q\left(\partial_{\Omega} V_0(\omega_1)
X          \partial_{p_j} \phi(0) \right)\cr
X    = & -2 \omega_1 \partial_{p_j} \phi(0)~~. \cr
X}
X\EQ(mixedpartial)
X$$
X\endproof
X
X
XThe $T_{\xi}$ invariance of the zero set of $G_0(\pp,\Omega)$
Xalso implies that the surface $(\pp,\Omega_0(\pp))$ obeys
X$\partial_{p_j} \Omega_0(0) = 0$, that is,
Xit is tangent to the  plane $\{ (\pp,\Omega) ;
X\Omega = \omega_1 \}$.   By a further Taylor expansion
Xwe will compute $\partial^2_{p_j p_k} \Omega_0(0)$,
Xwhich pertains to the twist condition of the nonlinear
Xproblem.
X
X
X\CLAIM Lemma(bifsurface)  If
X$
X\sqrt{r_0^2 + \rho_0^2}(3C_W L_0 / d_0) < C,
X$
Xand 
X$
X(r_0^2 + \rho_0^2) C_W L_0 / (d_0 \rho_0) << 1 $, 
Xthe solution surface $(\pp,\BGO_0(\pp))$ is defined
Xover the full neighborhood 
X$\{ \pp ; \| {\rm Re} \ \pp \| < r_0, 
X\| {\rm Im} \  \pp \| < \rho_0 \}$.
XThe surface satisfies
X$$\eqalign{
X    \BGO_0(0) &= \omega_1 \cr
X    \partial_{p_j} \BGO_0(0) &= 0 \cr
X    \partial^2_{p_j p_k} \BGO_0(0) &= \Kappa_0 \delta_{jk},
X  \cr}
X\EQ(OmegaTaylor)
X$$
Xwhere
X$$\eqalign{
X   \Kappa_0 =& { 1 \over 2\omega_1} \biggl(
X     {1 \over 6}\langle \partial_\pp \phi,
X       (D^3W_0(0)(\partial_\pp \phi)^3) \rangle \cr
X    &- {1 \over 2} \langle (D^2W_0(0)(\partial_\pp \phi)^2),
X      G_{\overline {B_0}}(\omega_1)
X              (D^2W_0(0)(\partial_\pp \phi)^2) \rangle
X     \biggr). \cr}
X\EQ(curvature)
X$$
X
X\PROOF   The expression for the curvature
Xis derived from the Taylor expansion of the mapping
X$G_0(\pp,\BGO)$ at $(0,\omega_1)$. Differentiating (3.1)
Xwith respect to $\pp$, we easily find that
X$$\eqalign{
X   u_0(0,\omega_1) &= 0 \cr
X    \partial_{p_j} u_0(0,\omega_1) &=
X       \partial_\bgo u_0(0,\omega_1) = 0 \cr}
X\EQ(taylor12)
X$$
Xand $\partial^2_{p_j p_k} u_0(0,\omega_1)$ satisfies
X$$
X  P(D^2W_0(0)[\partial_{p_j} \phi,\partial_{p_k} \phi]
X    + V_0(\omega_1)\partial^2_{p_j p_k} u_0 ) = 0.
X$$
XSince $PV_0(\omega_1)$ has inverse $G_{\overline {B_0}}$,
X$$
X  \partial^2_{p_j p_k} u_0(0,\omega_1)
X = -G_{\overline {B_0}}(\omega_1)(D^2W_0(0)
X        [\partial_{p_j} \phi,\partial_{p_k} \phi] ).
X$$
XIn order to compute the curvature of the  surface,
Xthe relevant third order term in the Taylor expansion
Xof the mapping $G_0(\pp,\bgo)$ is
X$\partial^3_\pp G_0(0,\omega_1)$.
X$$
X   \partial^3_\pp G_0(0,\omega_1) =
X       Q((D^3W_0(0)(\partial_\pp \phi)^3)
X     + 3(D^2W_0(0)[\partial_\pp \phi,
X         \partial_\pp^2 u_0]))~~.
X\EQ(taylor3)
X$$
X
X
XUsing the expression for $\partial^2_\pp u_0$
Xand \equ(taylor12)
Xthis gives \equ(curvature) for the curvature of the
Xnontrivial zero set $(\pp,\BGO_0(\pp))$
Xof the mapping $G_0$ at $\pp=0$.
XIncidentally we may easily deduce that
X$\partial_\BGO \partial^2_\pp G_0(0,\omega_1) = 0, \
X\partial^3_\bgo G_0(0,\omega_1) = 0$
Xsince the mapping is odd.
X
XWe also establish that the surface $(\pp,\BGO_0(\pp))$
Xis defined and analytic throughout the complex neighborhood
X$\{ \pp; \Vert \pp - \pp_1 \Vert < \rho_0 ~~{\rm for~~
Xsome}~~ \pp_1 \in \real~~{\rm  with}~~ \Vert \pp_1 \Vert
X< r_0 \}$.
XSince the branch is simple it suffices to show that
X$\partial_\pp \BGO_0(\pp)$ is bounded.
XUsing that $G_0(\pp,\BGO_0(\pp)) = 0$ and
Xdifferentiating with respect to $\pp$, we find
X$$
X   Q \bigl( V_0(\BGO) \partial_{p_j} \phi
X     + \partial_\bgo V_0(\BGO) \phi \partial_{p_j}\BGO
X      + DW_0(\phi(\pp) + u_0)(\partial_{p_j} \phi
X        +\partial_{p_j} u_0
X         + \partial_\BGO u_0 \partial_{p_j} \BGO)
X     \bigr) = 0.
X$$
XThus
X$$
X   Q \bigl( \partial_\BGO V_0(\BGO) \phi
X    + (DW_0) \partial_\BGO u_0) \bigr) \partial_{p_j} \BGO
X   = -Q \bigl( V_0(\BGO) \partial_{p_j} \phi
X       + DW_0 \partial_{p_j}(\phi + u_0) \bigr).
X$$
XQuite clearly $\| Q(V_0(\BGO) \partial_{p_j} \phi +
XDW_0 \partial_{p_j}( \phi + u_0) ) \|_0 < C$,
Xso that the only possible singularities occur when
X$Q (\partial_\BGO V_0(\BGO) \phi + DW_0 \partial_\bgo u_0)$
Xvanishes. The first term is explicit --
X$ Q \partial_\BGO V_0 \phi(\pp)  =   2\BGO \phi(\pp)$,
Xwhich is bounded below by $c \|\pp\|$ for
X$\BGO$ bounded away from zero.
XUsing \clm(firstbifonB0),
Xand a Cauchy estimate for the second term,
X$$\eqalign{
X   \| Q DW_0 \partial_\BGO u_0 \|_0 &\leq
X      C_W ( \|\pp\| + \|u_0\|_\gs)
X          \|\partial_\BGO u_0 \|_\gs \cr
X    &\leq C_W( \|\pp\| +
X     {3C_W L_0 \over d_0} \|\pp\|^2)
X       { 6C_W L_0 \over \rho_0 d_0 }\|\pp\|^2 \cr
X    &\leq {c \over 2} \|\pp\|~~,   \cr
X}
X$$
Xas long as constants are chosen as in the hypotheses of the 
Xlemma.
X
X
X\endproof
X
X
X
X\SUBSECTION The Induction Hypotheses
X
XStarting from the approximate solution of the previous
Xsubsection we will inductively construct
Xbetter approximate solutions in ever larger
Xregions of the lattice and prove that in the limit
Xthey converge to a solution of the original problem
X\equ(NLlattice).  In this subsection, we will state
Xthe inductive hypotheses, and in the next,
Xwe verify that they suffice to prove \clm(NLLat2).
X
XWe begin the statement of the induction scheme
Xwith the definition of the constants which appear in
Xthe induction and a description of their role which we
Xhope will help the readers orient themselves.
X
X{\bf Inductive Constants}:
X
X
X(1) At the ${n}^{\rm th}$ step in the iteration
Xwe solve a finite dimensional
Xproblem in a  lattice box
X$B_{n} = \{(j,k) : |j|+|k| \le L_n \}$,
Xwhose size is given by
X$$
XL_n =  2^n L_0\qquad n \ge 1~~.
X$$
X
X(2) The amount by which our ${n}^{\rm th}$
Xapproximate solution fails to be a
Xtrue solution is measured by
X$$
X\epsilon_n = \epsilon_0^{{\kappa}^n}~~,
X~{\rm with}~~\kappa > 1
X~{\rm and}~~ n\ge 1~~.
X$$
XThe constants $\epsilon_0$  and
X$\kappa$ are fixed in section 3.5.
X
X(3) As the iteration proceeds we encounter
Xworse and worse small denominators
Xwhose effects are estimated by
X$$
X\delta_n = {1 \over {L_n^{\alpha}}}~~,
X$$
Xwhere $\alpha$ is a fixed
Xconstant.
X
X
X(4) There is a length scale $\ell_n$ which estimates the distance
Xover which the effects of a resonance $\delta_n$ can be felt.  Set
X$$
X\ell_n = L_n^{\beta}~~,
X$$
Xwith $\beta$ is a small constant. We will need to estimate
Xthe spectra of the local Hamiltonian operators, 
X$H_{C_{\ell_n}(S)} = (V(\Omega) + D_u W)|_{C_{\ell_n}(S)}$,
Xwhich are defined on disks
X$C_{\ell_n}(S) \equiv
X\{ x\in \ZZ^+ \times \ZZ : \dist(x,S)  < \ell_n \}$,
XHence we require
Xthat $\ell_n << L_n$.
X
X(5)  The approximate solutions will decay exponentially
Xin the size of the indices $|(j,k)|$.
XThe exponential decay rate will change during
Xthe iteration.  This rate is determined by
X$$
X\sigma_{n+1} = \sigma_n -  6 \gamma_n~~,~\gamma_n
X= {{\sigma_0}\over{64 (n+2)^2}}~~, ~n\ge 0~~.
X$$
XHere, $\sigma_0 < \gs_* $ is defined in Section 6,
Xand is related to ${\overline \gs}$,
Xthe width of the strip in which the
Xcoefficients of (2.1) are analytic. Note that for all
X$n \geq 0, \ \gs_n > {\overline \gs}/2$.
X
X(6) The solutions will depend on parameters
X(such as the frequency $\Omega$ and $\pp$, the
Xamplitude and phase of the periodic solution).
XWe require them to be analytic in a
Xcomplex neighborhood, whose size is governed by
X$$
X\rho_n = {{\rho_0 \delta_n}\over{L_n^2}}~~,~n\ge 1~,
X$$
Xwhere $\rho_0$ is the size of our original analyticity
Xdomain.
X
XWe are now in a position to state the induction
Xhypotheses.
XThe equation \equ(NLlattice) is equivalent to two
Xequations obtained by the Lyapunov Schmidt decomposition
Xof the full lattice problem
X$$
X   F(\pp,\BGO,u) \equiv P
X         \bigl( W(\phi(\pp) + u) + V(\BGO)u \bigr) = 0
X\EQ(Peqn)
X$$
X$$
X   G(\pp,\BGO,u) \equiv Q
X        \bigl( W(\phi(\pp) + u) + V(\BGO)\phi(\pp)
X            \bigr) = 0~~~.
X\EQ(Qeqn)
X$$
XThere exists a constant $C_G$ and a positive 
Xexponent $\mu$ such that, provided 
Xthe conditions stated in section 3.5
Xare satisfied, the following induction statements
Xare true.
X
X
X\item{(n.1)} There is a function $u_n(\pp,\BGO ;x) =
Xu_{n-1}(\pp,\BGO ;x) + v_{n-1}(\pp,\BGO ;x) =
Xu_0(\pp,\BGO ;x) + \sum_{j=0}^{n-1} v_j(\pp,\BGO ;x)$,
Xdefined on $\Eta_0 \times \Bn$, and analytic on
X$D(\NN_n,\rho_n)$ satisfying:
X\itemitem{(i)} For any $(\pp,\Omega) \in D(\NN_j,\rho_j)$,
X$$
X  \|v_{j}(\pp,\BGO ;\cdot)\|_{\sigma_j-\gamma_j} \le
X   \|\pp\|^2 \gre_{j} C_G^{j+1}/\delta_{j+1} \gamma_{j}^{12}~~,
X~~ j= 0, \dots , n-1~~,
X$$
Xand on $\NN_0$, one has
X$$\eqalign{
X   \|\partial_\BGO^\beta
X       v_j(\pp,\BGO;\cdot ) \|_{\gs_j - \ga_j}   \leq&
X     {C_G^{j+1}\epsilon_j \over \gd_{j+1} \ga_j^{12} }
X          (\beta + 3)!
X                {\|\pp\|^2 \over (\rho_{j+1}/2)^\beta}     \cr
X   \|\partial_\pp \partial_\BGO^\beta v_j (\pp,\BGO;\cdot )
X         \|_{\gs_j-\ga_j}
X   \leq& {C_G^{j+1} \epsilon_j \over \gd_{j+1}\ga_j^{12}   }
X         (\beta + 4)!
X      {\|\pp\| \over (\rho_{j+1}/2)^\beta }
X         \bigl( 1 + {r_0 \over \rho_0} \bigr).
X\cr}
X$$
XFor higher $\pp$ derivatives there is a general estimate
X$$
X  \|\partial_{\pp}^{\bf \alpha} \partial_{\Omega}^{\beta}
X     v_{j}(\pp,\BGO ;\cdot)\|_{\sigma_j-\gamma_j}
X  \le {C_G^{j+1} \gre_j  \over \delta_{j+1} \gamma_{j}^{12}}
X     {(|{\bf \alpha}|+\beta +3)!
X       \over \rho_{j+1}^{|{\bf \alpha}| + \beta -2}}~~.
X$$
XThis estimate quantifies our control of the
X$C^k$  norms of $v_j$.
X\itemitem{(ii)} The function $u_n(x,\pp,\BGO)$ is an
Xapproximate solution of the first bifurcation equation
X\equ(Peqn).  For $(\pp,\Omega) \in D(\NN_n,\rho_n/2)$,
X$$
X   \|F(u_n)\|_{\sigma_n} \le \|\pp\|^2 \epsilon_n~~.
X$$
X
X\item{(n.2)} There exists a closed set of parameters,
X$(\pp, \BGO)$ $\in \Eta_{n+1} \subset \Eta_{n} \subset
X\dots \subset \Eta_0$, with the following properties:
X\itemitem{(i)} If $(\pp, \BGO)\in \Eta_{n+1}$,
Xand if
X$x_i$ and $x_j$ are any two singular sites
Xin $B_{n}^c$, which are not in the same singular region,
Xthen the distance between $x_i$ and $x_j$ is greater
Xthan $2 \ell_{n+1}$.
X\itemitem{(ii)} If $S$ is a singular region in $B_{n+1}\backslash
XB_{n}$,
Xthen for any $(\pp,\BGO) \in D(\Eta_{n+1}, \rho_{n+1})$,
X$$
X   {\rm dist}({\rm spec}
X     (H_{S}(\pp, \BGO; u_n)),0) > \delta_{n+1}
X$$
X$$
X{\rm dist}({\rm spec}
X(H_{C_{\ell_{n+1}}(S)}(\pp, \BGO;u_n)),0) > \delta_{n+1}
X$$
X\itemitem{(iii)} For any $C^\infty$ curve, $\BGO(\pp)
X= \Kappa \|\pp\|^2(1+C(\|\pp\|)) + \go_1$, with
X$|\Kappa| \geq L_0^{-\nu}$ and
X$|C(\cdot)|_{C^1} < 1/2$, then
X$$
X  {\rm meas} \left\{ \|\pp\| \in (-r_0,r_0);
X    (\pp,\BGO(\pp))\in \Eta_{n+1} \right\} \ge
X      r_0(1-Cr_0^\mu)~~.
X$$
X(Recall that the exponent $\nu$ was defined in
X\clm(NLLat).)
X
XThe induction hypotheses $(n.1)$ and $(n.2)$ suffice
Xto prove \clm(NLLat2).  We will verify this in the next
Xsubsection. The inductive estimates imply in particular
Xthat throughout the parameter domain $\Eta_0$ there is
Xa bound on the derivatives of $u_n$.
X$$
X   \| \partial_\pp^{\bf \alpha} \partial_\BGO^\beta u_n
X         \|_{\gs_n-\ga_n} \
X  \leq \
X     \| \partial_\pp^{\bf \alpha} \partial_\BGO^\beta u_0
X         \|_{\gs_n-\ga_n}
X     + \| \sum_{j=0}^n \  \partial_\pp^{\bf \alpha}
X            \partial_\BGO^\beta v_j \|_{\gs_n-\ga_n}.
X$$
X Thus for ${\bf \alpha} = 0$, we estimate
X$$
X    \| \partial_\BGO^\beta u_n \|_{\gs_n-\ga_n} \
X  \leq \ {CL_0 \over d_0}
X      \bigl( 1 + {C\epsilon_0^{1/ \kappa} \over \gd_0}
X          ( \beta !)^b \bigr)
X            {{\|\pp\|^2 }\over {(\rho_0/2)^\beta}}~~.
X$$
XFor $|{\bf \alpha}|=1$,
X$$
X    \| \partial_\pp \partial_\BGO^\beta u_n
X         \|_{\gs_n-\ga_m} \
X  \leq \ {CL_0 \over d_0}
X      \bigl( 1 + {C\epsilon_0^{1/\kappa} \over \gd_0}
X          (( \beta+1 )!)^b \bigr)
X            {\|\pp\| \over (\rho_0/2)^\beta}
X               \bigl( 1 + {r_0 \over (\rho_0/2)} \bigr)~~.
X$$
Xand in general
X$$
X   \| \partial_\pp^{\bf \alpha} \partial_\BGO^\beta u_n
X         \|_{\gs_n-\ga_m} \
X  \leq \ {C L_0 \over d_0}{1 \over
X     (\rho_0/2)^{|{\bf \alpha}| + \beta - 2} } \bigl(
X       1 + {C\epsilon_0^{1/\kappa} \over \gd_0}
X          ((|{\bf \alpha}|+\beta)!)^b
X               (1+ {r_0^2 \over (\rho_0/2)^2}) \bigr)~~.
X\EQ(allderivsu)
X$$
XThese are obtained using elementary bounds and the
Xabove choices of inductive constants. The exponent
X$b = 1/\log \kappa + 1$. The approximate solution $u_n$
Xis analytic in $D(\NN_n,\rho_n/2)$, and an estimate similar
Xto \equ(allderivsu) holds over the complex neighborhood
X$D(\NN_n,\rho_n/4)$, where in fact the exponent $b= 1/\kappa$.
X
X
XThe induction hypotheses  also allow us to prove the following
Ximportant result about the inverse of the linearized
Xoperator which appears in Newton's method.
X
X\CLAIM Theorem(Greens) Suppose that the induction
Xhypotheses $(j.1)$ and
X$(j.2)$ hold for $j=0,1,\dots,n$.
XFor any non-singular lattice
Xregion  $A$ and
X$E_{n+1} \subset \overline{B_{n+1}} \cup A$, the
XGreen's function is analytic on
X$D(\NN_{n+1},\rho_{n+1})$, and satisfies
X$$
X  \|G_{E_{n+1}}(\pp,\Omega,u_n)\|_{\sigma_n-\gamma_n} \le
X  {{C_G^{n+1}}\over{ \delta_{n+1} \gamma_{n}^{12}}}~~.
X\EQ(redGreen1)
X$$
XUnder perturbations of $u_n$ a similar estimate holds.
XIf
X$$
X   \| u - u_n \|_{\sigma_n -\gamma_n} \le
X      \| \pp \|^2 \epsilon_n C_G^{n+1}
X                      /\delta_{n+1} \gamma_{n}^{12}~~,
X$$
Xthe estimate holds,
X$$
X\|G_{E_{n+1}}(\pp,\Omega,u)\|_{\sigma_n-2\gamma_n} \le
X{{2 C_G^{n+1}}\over{ \delta_{n+1} \gamma_{n}^{12}}}~~.
X\EQ(redGreen2)
X$$
X
X
XWe delay the proof of this theorem until section 5,
Xwhere a detailed analysis of the linearized operators
Xis presented.
X
X\SUBSECTION Proof of \clm(NLLat2).
X
XIn this section we demonstrate that the
Xinduction hypotheses lead
Xto a proof of \clm(NLLat2).
X
XSet
X$\NN = \cap_{n\ge0} \NN_n$, the parameter domain
Xfor which we obtain a solution of the   first
Xbifurcation equation \equ(Peqn) .
XThe parameters $\ga_n$ governing loss of exponential decay
Xin the induction process satisfy
X$\sigma_n \ge
X   {\overline \gs}/2$, for all $n$.
XThus the induction
Xhypothesis $(n.1)(i)$ implies that $u_n(\pp,\BGO;x)$
Xconverges to a sequence
X$u(\pp,\BGO;x) \in \HH_{{\overline \gs}/2}$,
X$$
X   u(\pp,\Omega;x) \equiv u_0(\pp,\Omega;x)
X     + \sum_{j=0}^{\infty} v_j(\pp,\Omega;x),
X$$
Xwhich is $C^\infty$ as a function of
X$(\pp,\Omega)$ on the parameter domain $\NN_0$.
XInduction hypothesis $(n.1)(ii)$ implies that for
X$(\pp,\Omega) \in \NN \subseteq \Eta_0$ the sequence
X$u$ satisfies the first bifurcation equation \equ(Peqn).
X
XTo obtain a solution to the full nonlinear lattice problem,
Xit remains to solve the second bifurcation equation
X\equ(bifeq2).  This is a finite
Xdimensional problem, recovering the zero set of the mapping
X$G$. Not surprisingly, we will show that the solution
Xis close to the approximate solution that was
Xdiscussed in Section 3.2.
X
X
X\CLAIM Lemma(map)  The mapping $(\pp,\Omega) \to
XG(\pp,\Omega)$ $= Q ( W(\phi(\pp) + u(\pp,\Omega) )
X$ $+ V(\Omega) \phi(\pp))$ is $C^{\infty}$ on $\NN_0$,
Xzero for $\{ (\pp,\Omega) ; \pp =0,$
X$ -r_0^2 < (\omega_1 - \Omega) < r_0^2 \}$ and
Xhas a Taylor expansion at $\pp = 0$, $\Omega = \omega_1$
Xwith the following Taylor coefficients:
X$$
X\partial_{\pp} G (0,\omega_1) = 0~~,~~
X\partial_{\Omega} G (0,\omega_1) = 0~~,
X$$
Xand
X$$\eqalign{
X\partial_{\pp}^2 G (0,\omega_1) = 0~~, &~~
X\partial_{\Omega}^2 G (0,\omega_1) = 0~~,\cr
X\partial_{\pp} \partial_{\Omega} G (0,\omega_1)
X= Q(\partial_{\Omega} V(\omega_1) )  & \partial_{\pp}\phi(0)
X= -2 \omega_1 \partial_{\pp} \phi(0)~~.\cr
X}
X$$
X
X\PROOF The fact that $G$ is $C^{\infty}$ follows
Xfrom the fact that $u$ is $C^{\infty}$ on $\NN_0$, which
Xfollows from the uniform bounds in \equ(allderivsu).
XThe Taylor coefficients are computed as in the proof
Xof \clm(secondbifonB0).
X
X\endproof
X
XThe zero set of the mapping $G(\pp,\BGO)$ consists of the
X$\BGO$-axis union a surface $(\pp,\BGO(\pp))$ which is
Xinvariant under the translations $T_\xi$, and is given as
Xa graph over the neighborhood
X$\{ \pp \in \real^2; \|\pp\| < r_0 \}$. This is the surface
Xof solutions of the second bifurcation equation
X\equ(bifeq2).
XIt is close to the approximate solution surface
X$(\pp,\BGO_0(\pp))$. Intersections of $(\pp,\BGO(\pp))$ with
X$\NN$ correspond to solutions of the first bifurcation
Xequation as well, thus are solutions of the full problem
X\equ(NLlattice). Such intersections are guaranteed if the
Xsurface $(\pp,\BGO(\pp))$ has nonzero curvature at zero.
X
X
X\CLAIM Lemma(bend) The solution surface $(\pp,\Omega(\pp))$
Xof \equ(bifeq2) is defined in
X$\{\pp \in \real^2; \| \pp \| < r_0\}$.
XThe surface satisfies
X$$
X   \Omega(0) = \omega_1~~,~\partial_{p_j} \Omega(0) = 0~~,
X      \partial_{p_j p_k}\Omega(0) = \Kappa \delta_{j k}~~,
X$$
Xwhere
X$$
X   \Kappa \ = \ {1 \over 6\omega_1}
X      \langle \partial_\pp \phi, (D_u^3 W(0)
X        (\partial_\pp \phi)^3) \rangle
X   + {1 \over 2\omega_1}
X     \langle \partial_\pp \phi, D_u^2 W(0)
X       [\partial_\pp \phi,(\partial_\pp^2u)] \rangle.
X$$
XThe difference $|\Kappa - \Kappa_0| <
X   C \bigl( e^{-({\overline \gs} L_0)/2}
X     + \epsilon_0^{1/\kappa} L_0/d_0 \bigr)
X       /(\gd_0 \rho_0^2)$.
X
X
X\PROOF The proof is essentially the same as that for
X\clm(bifsurface), using the solution $u$ in place of
Xthe approximate solution $u_0$. The estimates \equ(u0derivs)
Xon $u_0$ from the bifurcation theory on $B_0$ are
Xreplaced by \equ(allderivsu) from the induction
Xargument. The expression for the curvature can
Xbe rewritten as;
X$$\eqalign{
X   \Kappa \ =& \ \Kappa_0 + {1 \over 6\omega_1}
X     \langle \partial_\pp \phi, (\11-\Pi_0)
X       D_u^3 W(0)(\partial_\pp \phi)^3 \rangle    \cr
X   &+ \ {1 \over 2\omega_1} \langle \partial_\pp \phi,
X     (\11-\Pi_0) D_u^2 W(0) [\partial_\pp \phi,
X      \partial_\pp^2u] \rangle                    \cr
X   &+ {1 \over 2\omega_1} \langle \partial_\pp \phi,
X      D_u^2 W_0(0) [\partial_\pp \phi,
X       (\partial_\pp^2 u - \partial_\pp^2 u_0)] \rangle
X\cr}
X$$
XThe difference is estimated using \equ(allderivsu).
X$$\eqalign{
X   |\Kappa - \Kappa_0| \ \leq& \ C\bigl( \|(\11-\Pi_0)
X     D_u^3 W(0)(\partial_\pp \phi)^3 \|_0
X      + \| (\11-\Pi_0)D_u^2 W(0)[\partial_\pp \phi,
X       \partial_\pp^2 u ] \|_0           \cr
X   &+ \ \|D_u^2 W_0(0)[\partial_\pp \phi,
X     (\partial_\pp^2 u - \partial_\pp^2 u_0)] \|_0 \bigr)
X \cr
X   \leq& \ Ce^{-({\overline \gs}L_0)/2}
X      (1 + \|\partial_\pp^2 u \|_{{\overline \gs}/2})
X        + {C_W \over ({\overline \gs}/2)^2}
X         \|\partial_\pp^2(u - u_0)\|_0         \cr
X   \leq& \  Ce^{-({\overline \gs}L_0)/2}
X      \bigl( 1 + {1 \over \gd_0\rho_0^2} \bigr)
X    +  {C_W \over ({\overline \gs}/2)^2}
X       {L_0 \epsilon_0^{1/\kappa} \over d_0 \delta_0}
X       \left( 1 + {r_0^2 \over \rho_0^2} \right)
X.  \cr}
X\EQ(curvediff)
X$$
XAssuming that $|\Kappa_0| > L_0^{-\nu}$, if
X$$
X  \bigl( e^{-({\overline \gs} L_0 / 2)} +
X   \epsilon_0^{1/\kappa} L_0/d_0 \bigr)
X    /(\delta_0 \rho^2_0) << L_0^{-\nu},
X$$
Xthe curvature
Xof the surface $(\pp,\BGO(\pp))$ satisfies
X$|\Kappa| > cL_0^{-\nu}$, and thus by the induction
Xhypothesis $(n.2)(iii)$ the set
X$(\pp,\BGO(\pp)) \cap \Eta$ is nonempty.  This inequality will
Xbe verified in Section 3.5 which follows
X
X\endproof
X
XThe Klein-Gordon case $G(x,u) = m^2(u - u^3)$
Xis somewhat special in these considerations of
Xcurvature of the bifurcation branches. Firstly
X$D_u^2 W(0) = 0$, since the nonlinearity appears only at
Xcubic order. Additionally for
X$L_0 \geq 6$, $ (\11-\Pi_0) D_u^3
X   W(0)(\partial_\pp \phi)^3 = 0$, thus the
Xexpression \equ(curvediff) gives that
X$\Kappa = \Kappa_0$, and the twist condition is
Xsatisfied automatically.
X
X
X
X
X
X
X
X\SUBSECTION Final Reckoning
X
XThe proof of convergence of this induction is based on a
Xproper choice of constants that appear in the analysis
Xthroughout the paper. There are roughly three requirements
Xto be satisfied. First the inductive constants $\alpha,
X\beta, \tau$ and $\kappa$ must be chosen so that the Newton iteration
Xmethod is convergent. This involves estimates of the
Xtruncation in approximating the nonlinearity, the excision
Xprocess in the parameter domain, and the iterative
Xconstruction of the Green's function. Secondly, the initial
Xapproximate bifurcation problem and parameter neighborhood
Xmust be sufficiently large so that, even after the excisions
Xof the induction steps, the remaining solution set has
Xlarge measure. Finally, we must be able to carry out
Xthe analysis of convergence for an open dense set of
Xnonlinearities $g(x,u)$ for the wave equation, thus
Xrequirements of Proposition 2.4 and Theorems 2.8 and 2.9
Xmust be satisfied. In this subsection we show that all
Xthe requirements of the paper can be simultaneously satisfied.
XWe accordingly make choices of the
Xinductive parameters $\alpha,\beta, \tau$ and $\kappa$,
Xand show that the principal requirements on the parameters
X$r_0, \rho_0$ and $d,d_0$ and $d_s$  reduce to a condition
Xthat $L_0$, the initial radius of an approximating lattice domain,
Xbe sufficiently large. As technical aspects of the induction 
Xverification appear in  sections 4 and 5, this subsection 
Xoccasionally refers ahead to to formulae in these sections. 
X
XBefore the induction starts, a bifurcation analysis is
Xperformed on the initial domain $B_0$ of radius $L_0$.
XThe apriori estimates on the initial domain require that
Xthe induction starts with
X$u_0 \in \HH_{\gs_0}, \ \gs_0 + \ga_0 < \gs_* - 1/L_0$.
XSetting $d_0 = L_0^{-\eta}$ for some $1/2 < \eta < 1$,
Xand $r_0 = \rho_0 = c L_0^{-(2+\eta)}$, for $c$ some
Xsmall positive number, we are able to
Xsatisfy the hypotheses of \clm(firstbifonB0) and
X\clm(bifsurface),
X$$\eqalign{
X   r_0 < & C { d_0 \over L_0^2}   \cr
X   \rho_0 < & C{d_0 \over L_0^2}   \cr
X   r_0^2 {L_0 \over d_0 \rho_0} & << 1  \cr}
X\EQ(FR1)
X$$
X
X
XWe next  turn to the choice of the inductive constants
Xand the question of convergence. The sequence
X$u_0 \in \ell^2(B_0) \cap \HH_{\gs_0}$ is an
Xexact solution of the restricted equation $\Pi_0 F(u_0) = 0$,
Xthus there exists $c_0 > 0$ such that 
X$$\eqalign{
X   \| F(u_0(\pp,\BGO) ) \|_{\gs_0}
X    & = \| (\11 - \Pi_0) W(\phi(\pp) + u_0) \|_{\gs_0} \cr
X    & \leq {C_W \over \ga_0} e^{-\ga_0 L_0}
X       \bigl( \|\pp\|^2
X       + {1 \over \ga_0^2}\|u_0\|^2_{\gs_0+\ga_0}  \bigr) \cr
X   & \leq e^{-c_0L_0} \|\pp\|^2       \cr}
X$$
Xas long as $L_0 \geq L_*$, is chosen sufficiently large so that
X$\gs_*-1/L_0 - \ga_0 \geq \gs_0 > {\overline \gs}/2$. We have
Xused \equ(FR1) and {\bf H1} to estimate the nonlinear term.
XMake the choice
X$$
X   \epsilon_0 = e^{-c_0L_0}.
X$$
XFor $n>0$ the iteration will have a supergeometric rate
Xof convergence as long as the hypotheses of Proposition
X4.1 are satisfied;
X$$
X   \epsilon_{n-1} {C C_G^{n+1} \over \gd_n \ga_{n-1}^{14} }
X     \left( e^{-\ga_{n-1}L_n}
X       + {r_0^2 \epsilon_{n-1} C_G^{n+1}
X           \over \gd_n \ga_{n-1}^{13}  } \right)
X     \leq \epsilon_n.
X\EQ(FR2)
X$$
XUsing the definition of the inductive constants, \equ(FR2)
Xwill follow from the pair of inequalities
X$$
X   C_G^{n+1} L_0^\alpha e^{\alpha \log(2)n}
X     \bigl( {c \over |n|^2} \bigr)^{14}
X      e^{-L_0 c2^n/|n|^2} \leq
X   {1 \over 2} \epsilon_0^{\kappa^{n-1}(\kappa-1)}
X\EQ(FR3)
X$$
Xand
X$$
X   L_0^{-2(2+\eta)} C_G^{2(n+1)} L_0^{2\alpha}
X    e^{2\alpha \log (2) n}
X     \bigl( {c \over |n|^2} \bigr)^{27}
X    \leq {1 \over 2}
X       \epsilon_0^{\kappa^{n-1}(\kappa-2)}
X\EQ(FR4)
X$$
XThe governing factor of the left hand side of \equ(FR3) is
X$e^{-cL_0 2^n/|n|^2}$, while from the definition of
X$\epsilon_0$ that of the right hand side is
X$e^{-c_0L_0(1-(1/\kappa))\kappa^n}$.
XIf necessary decrease $c_0$ so that
X$c_0(1-(1/\kappa)) < 2c$. For $L_*$ sufficiently large
Xthe estimate \equ(FR3) will hold for all $n>0$, uniformly
Xin $L_0 > L_*$.
X
XA similar discussion verifies that \equ(FR4) will hold
Xfor all $n > 0$ if $L_0 > L_*$ is sufficiently large.
XThe right hand side is governed by the factor
X$e^{(\ga_0L_0(2-\kappa)\kappa^{n-1})}$, while the
Xleft hand exponent grows at most linearly in $n$ and
Xlogarithmically in $L_0$. For $L_0 > L_*$, $L_*$ chosen
Xsufficiently large, it too will hold for all $n > 0$.
X
XIn order to make the excision process work, relations (4.4)--(4.7)
Xof Proposition 4.6 will have to hold. Set
X$$
X   2d = 2d_s = d_0 = L_0^{-\eta}
X$$
XIf we use the definitions above, we see that if
X$(1-\tau\beta) > 0, \ \beta(1 +c_0 \log (2)) < 1$ and
X$L_0 > L_*$ is chosen large, then the first two relations
Xof (4.4)-(4.7) are satisfied. The following three relations
Xwill have already been satisfied in \equ(FR1). Without yet
Xdiscussing the measure of the remaining parameter
Xregion, we will verify the rest of the requirements
Xof section 4. In order to separate the localizing
Xneighborhoods $C_{\ell_n}(S)$, the hypothesis of
XLemma 4.8 is that
X$$
X   d > L_0^{\tau\beta}(L_0^{-1} + r_0^2L_0^\beta )
X    = L_0^{\tau\beta-1} + L_0^{\beta(\tau+1)-2(2+\eta)}~~,
X$$
Xwhich will hold for $\eta < (1 - \tau\beta)$
Xand $\beta(\tau+1) < \eta + 4$, which can be
Xsatisfied by taking $\beta$ small. The extension of
Xthese estimates to the complex domain will go through
Xif (4.17) holds. By definition
X$L_n \rho_n < \gd_n/2$, and one checks readily that
X$(1 + L_0/d_0(1+(r_0/\rho_0)))\rho_{n+1} =
X   (1 + 2L_0^{(1+\eta)}) L_0^{-(2+\eta)}
X    \gd_{n+1} /L_{n+1}^2 \leq \gd_{n+1}$.
X
XIn order to construct the Green's function using the
Xprocedure of section 5, define $2d_s = d_0 = L_0^{-\eta}$,
Xand ask that $L_0 > L_*$ is sufficiently large so that
X$C_W r_0/ ({\overline \gs}^2 d_s) \leq C\gs_*$.
XThis permits the construction of the Green's
Xfunction on non-singular domains.  If we choose$$
X   \gs_0 + 5 \gamma \equiv ({{35}\over{32}}) < \gs_* -
X      2\log (1+4\sqrt{C_W r_0/({\overline \gs}^2 d_s)})~~,
X$$
Xthen the restrictions on $\sigma_0$ in both Theorem 5.1 and
XProposition 5.3 can be satisfied.
X
X
XThis is the second of two conditions to be satisfied
Xin defining $\gs_0$, the starting decay rate of the
Xinduction. The requirements of the extension lemmas
X5.3 and 5.4 are that $r_0 < Cd_0$ and $r_0 < \ga_0^4 d_s$,
Xwhich are both satisfied as in \equ(FR1) by an increase
Xin $L_*$ if necessary. The more intricate patching techniques
Xof Theorem 5.6 require additionally that
X$$
X  { Cr_0 e^{-\ga_{n-1}\ell_{n+1}/2}
X    \over \gd_{n+1}^2 \ga_{n-1}^{34} }
X   \leq 1.
X\EQ(FR5)
X$$
XUsing the definitions above, \equ(FR5) reads
X$$
X   CL_0^{-(2+\eta)} L_0^{2\alpha}
X     e^{\alpha \log (2) (n+1)}
X    \times C|n|^{68}
X    \times e^{-(c/2|n|^2) L_0^\beta}{
X        e^{\beta \log (2) (n+1)}    }   < 1~~~.
X$$
XAs before, for any choices of $\alpha, \beta$ there is an
X$L_*$ such that this will hold for all $n\geq 1$, uniformly
Xin $L_0 > L_*$. Using \equ(FR5), the hypotheses of Lemmas
X5.9 and 5.10 follow accordingly.
XThe second type of extension of the Green's function is
Xaddressed in Theorem 5.12, where it is asked that
X$$
X   r_0^2 \epsilon_n
X     \leq { \gd_{n+1}^2 \ga_n^{27} \over 2C_W C_G^{2n}}.
X\EQ(FR6)
X$$
XUsing the definitions, it is clearly tantamount to
X$$
X   L_0^{-2(2+\eta)} \epsilon_0^{\kappa^n}
X     \leq  L_0^{-2\alpha} e^{-2\alpha \log (2) (n+1)}
X      \bigl( {c \over |n+1|^2 } \bigr)^{27}
X        {1 \over 2C_W C_G^{2n} },
X$$
Xwhich can be uniformly satisfied for $L_0 > L_*$, for all
X$n \geq 0$ as long as $L_*$ is sufficiently large.
XCorollary 5.13 follows from Theorem 5.12 if the correction
Xterm $v_j$ is small in norm. That is, given \equ(FR2)
Xand \equ(FR3), we have
X$$
X   \|v_j\|_{\gs_j-\ga_j} \leq \|\pp\|^2
X      { \epsilon_j C_G^j \over \gd_{j+1} \ga_j^{12} }
X      \leq {1 \over 4^{j+1}}
X         { \ga_{j+1} \ga_0^4 d_s \over 2C_W C },
X$$
Xwhich is even easier to satisfy than \equ(FR6).
X
XThe remaining requirements that are placed on the
Xinductive constants come from Proposition 4.6,
Xwhere an estimate is made of the measure of the
Xparameter set after the ${n}^{\rm th}$ excision. This will
Xinvolve the choice of $\alpha$ and $\tau$. To satisfy
X(4.5),
X$$\eqalign{
X   c \sum_{j=0}^n  \ {d \over L_j^\tau} &
X                  ( 1 + 2r_0^2 L_{j+1})  \cr
X       = CL_0^{-(\tau+\eta)} &
X     \sum_{j=0}^n e^{-\tau \log (2) j}
X       + 4Cr_0^2 L_0^{-(\tau+\eta-1)}
X        \sum_{j=0}^n e^{-\tau \log (2) j}     \cr
X    \leq C {L_0^{-(\tau + \eta)} \over \tau}
X    &\leq  r_0^2           \cr}
X\EQ(FR7)
X$$
Xwhich will hold, uniformly in $n$, as long as
X$\tau > 4 + \eta$ and $L_*$ is large. We will construct
Xthe $\Eta_{n+1}$ by first constructing two intermediate
Xsets $\Eta_{n+1}^{1}$ and $\Eta_{n+1}^{2}$.
XThe estimate of \equ(FR7) bounds the measure of the excisions that
Xare made in constructing $\Eta_{n+1}^{(1)}$.
XTo control the size of excisions in constructing
X$\Eta_{n+1}^{(2)}$, we require
X$$\eqalign{
X    \sum_{j=0}^n & \ C L_j(1 + r_0^2 L_{j+1})
X       \ell_{j+1}^2 {\gd_{j+1} \over \Kappa L_j^2}    \cr
X    \leq & C L_0^{-(\alpha +1 -2\beta -\nu)}
X      \sum_{j=0}^n  \ e^{-(\alpha +1 - 2\beta) \log (2)j} \cr
X    \qquad &+ CL_0^{-(\alpha + 2(2+\eta) - 2\beta - \nu)}
X       \sum_{j=0}^n \ e^{-(\alpha - 2\beta) \log (2) j}  \cr
X    \leq &   r_0^2~~~~.
X\cr}
X\EQ(FR8)
X$$
XAs long as we satisfy the requirements that
X$$\eqalign{
X    \alpha  & > 2\beta + \nu  \cr
X    \alpha + 1 - 2\beta & > 2(2+\eta)  \cr
X    \alpha + 4 +2\eta - 2\beta & > 2(2+\eta)  \cr}
X$$
Xthen \equ(FR8) will hold uniformly in $L_0 > L_*$ and $n>0$.
X
X
XAddressing (4.6), we see that the inequalities will be
Xsatisfied if the curvature $\Kappa$ of the surface is
Xrestricted from being too small. Using the above choices,
X$$
X   C({1 \over d_s} +
X        {L_0 \over d_0}(1+{r_0 \over \rho_0}) )
X   =  C(L_0^\eta + 2L_0^{1+\eta})
X   \leq \Kappa L_0^2 2^{2n}.
X$$
XThis will hold for all $n \geq 0$, uniformly in $L_0 > L_*$
Xif the curvature satisfies
X$$
X   \Kappa > L_0^{-\nu}, \qquad 0 \leq \nu < 1-\eta
X\EQ(FR9)
X$$
XWe define $\nu$ in the twist condition of Theorem 2.7, for
Xa bifurcation surface must intersect the sets
X$\Eta_{n+1} = \Eta_{n+1}^{(1)} \cap \Eta_{n+1}^{(2)}$
Xto exhibit a solution of the equation. To estimate the
Xmeasure of this intersection, we show that for $\alpha$
Xand $\tau$ reasonably large, (4.7) is satisfied for some
Xexponent $\mu$. The first inequality is
X$$\eqalign{
X   \sum_{j=0}^n & \sqrt{ {C \over \Kappa} {d \over L_j^\tau}
X       (1 + r_0^2 L_0^{j+1}) }                         \cr
X   & \qquad \le \sum_{j=0}^n  \sqrt{C \over \Kappa} \bigl(
X      L_0^{-(\eta + \tau)/2} e^ {-\tau \log (2) j/2} \cr
X    & \qquad \qquad \qquad+ L_0^{-(\eta + \tau -1)/2} L_0^{-(2+\eta)}
X       e^{-(\tau - 1) \log (2) (j+1)/2}   \bigr)      \cr
X    & \qquad \le  C \sqrt{1 \over \Kappa}
X      \bigl( L_0^{-(\tau + \eta)/2} +
X                  L_0^{-(\tau + \eta +3)/2}  \bigr)     \cr}
X\EQ(FR10)
X$$
Xwhere the exponential sums are bounded uniformly
Xin $n$ as long as $\tau > 1$. This sum is dominated
Xby $r_0^{(1+\mu)} = L_0^{-(2+\eta)(1+\mu)}$ if
X$\Kappa > L_0^{-\nu}$
Xand $0 < \mu < (\tau - (\eta + \nu + 4))/2$.
XConsiderations of the second inequality are similar.
X$$\eqalign{
X   \sum_{j=0}^n & \sqrt{CL_j(1+r_0^2  L_0^{j+1})
X     \times \ell_{j+1}^2 \times {1 \over \Kappa }
X     \times {\gd_{j+1} \over L_j^2}  }            \cr
X   & \leq \sum_{j=0}^n   \sqrt{C \over \Kappa} \bigl(
X      L_0^{-(\alpha + 1 -2\beta)/2}
X         e^{-(\alpha + 1 - 2\beta) \log (2) j/2} \cr
X    & \qquad \qquad + L_0^{-(\alpha +2(2+\eta) - 2\beta)/2}
X         e^{-(\alpha -2\beta) \log (2) j/2}   \bigr)  \cr
X    &  \leq \sqrt{C \over \Kappa} 
X     \bigl( L_0^{-(\alpha+1-2\beta)/2}
X       + L_0^{-(\alpha + 2(2+\eta) - 2\beta)/2} \bigr), \cr}
X\EQ(FR11)
X$$
Xwhere we must require that $\alpha > 2\beta$. Allowing
X$\Kappa > L_0^{-\nu}$, this expression will be dominated
Xby $r_0^{(1+\mu)}$ for
X$$
X    0 < \mu < {\alpha - (\nu +2(\beta + \eta) + 3)
X                \over 2(2+\eta) }.
X$$
XClearly a choice of $\alpha$ and $\tau$ large,
Xand $\beta$ small is available so that these
Xinequalities will hold throughout the induction.
X
XThe last relations that should be verified pertain
Xto the the density of the nonlinearities to which
Xthe results apply. We require that $d_0 = o(L_0^{-1/2})$
Xin order to satisfy the hypotheses of Proposition 2.4.
X This gives a genericity result for the coefficients $g_1$.
XWe thus restrict $1/2 < \eta < 1$ in order to do this.
XSecondly, we ask for bounds on the curvature in the
Xtwist condition. Setting  $|\Kappa| \geq L_0^{-\nu}$ with
X$\nu > 0$ gives a bound which is decreasing in $L_0$, and again
Xthere is an open dense set of nonlinearities possessing a
Xsufficiently large twist for Theorem 2.7 to apply.
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
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X
X
X\SECTION Verification Of The Induction Hypotheses
X
XIn this section we begin the verification of the
Xinduction hypotheses.  There are two main subsections;
Xin the first, we show that if $(j.1)$
Xand $(j.2)$
Xhold for $j=0,1, \dots , n-1$ then we can prove
X$(n.1)$, and in the second we construct sets $\NN_{n+1}$
Xwhich satisfy $(n.2)$.
X
X\SUBSECTION Verification Of  $(n.1)$
X
XAssume that $(j.1)$ and $(j.2)$
Xhold  $j=0,1, \dots , n-1$.  Then $(n.1)$
Xis a consequence of the following estimate on the
XNewton iteration.
X
X\CLAIM Proposition(newton) 
XIf the inductive parameters satisfy
X$$
X\epsilon_{n-1}
X{{C C_G^{n+1} }\over{\delta_n \gamma_{n-1}^{14} }}
X\left[ e^{-\gamma_{n-1} L_n }+
X{{r_0^2 \epsilon_{n-1} C_G^{n+1} }\over{\delta_n
X\gamma_{n-1}^{13} }} \right] \le \epsilon_{n}~,
X\EQ(4.1)
X$$
Xthen  there is a function $u_{n}(\var)
X$ $= u_0(\var) + \sum_{j=0}^{n-1}
Xv_j(\var)$, defined on $\NN_0 \times \Bn$,
Xwhich obeys
Xthe following estimates:
X\item{(1)} For any $j=0,\dots ,n-1$, $v_{j}$ is analytic
Xon $D(\NN_{j};\rho_{j}/2)$ and for any $(\pp, \Omega)$
Xin this set it satisfies
X$$
X    \|v_{j}(\pp,\Omega;\cdot)\|_{\sigma_{j}
X      - \gamma_{j}} \le  \epsilon_{j}\|\pp\|^2 C_G^{j+1} /
X          \delta_{j+1} \gamma_{j}^{12}~~.
X$$
XFurthermore, there exists a positive constant $C_3$ such that
X$v_{j}$ is a $C^{\infty}$ function of
X$(\pp,\BGO)$ on all of $\NN_0$, satisfying the
Xfollowing estimates.
X$$\eqalign{
X  \| \partial_{\Omega}^{\beta} v_{j}(\pp,\Omega;\cdot)
X          \|_{\sigma_{j}-\gamma_{j}}
X  \le& C_3  \epsilon_{j} C_G^{j+1}
X      { (\beta + 3)! \over
X         (\delta_{j+1} \gamma_{j}^{12}
X            \rho_{j+1}^{\beta}) } \|\pp\|^2~~,      \cr
X  \|\partial_{\pp} \partial_{\Omega}^{\beta}
X     v_{j}(\pp,\Omega;\cdot)\|_{\sigma_{j}-\gamma_{j}}
X  \le& C_3  \epsilon_{j} C_G^{j+1}
X     { (\beta + 4)!
X       \over (\delta_{j+1} \gamma_{j}^{12}
X         \rho_{j+1}^{\beta}) }
X   \bigl( 1 + { r_0 \over \rho_0} \bigr) \|\pp\|~~. \cr}
X$$
Xand for $|{\bf \alpha}| \geq 2$,
X$$
X  \|\partial_{\pp}^{{\bf \alpha}} \partial_{\Omega}^{\beta}
X   v_{j}(\pp,\Omega;\cdot)\|_{\sigma_{j}-\gamma_{j}} \le
X    C_3  \epsilon_{j} C_G^{j+1}
X     { (|{\bf \alpha}| + \beta + 3)!
X       \over (\delta_{j+1} \gamma_{j}^{12}
X         \rho_{j+1}^{|{\bf \alpha}| + \beta -2}) }~~.
X$$
XThe usual multi-index notation for
Xderivatives is used in this estimate.
X\item{(2)} For every $(\pp,\Omega) \in D(\NN_n;\rho_n/2)$,
Xthe function $u_{n}$ satisfies
X$$
X   \| F(u_n(\pp,\Omega)) \|_{\sigma_{n}}
X \le \|\pp\|^2 \epsilon_n~~.
X$$
X
X\PROOF The idea of this proposition is to construct
X$v_{n-1}$ using Newton's method.
XWe expect the solution to be small
Xfar from the origin, so rather than trying
Xto solve $F(u_{n-1}+v_{n-1}) = 0$, we study
Xthe approximation
X$\Pi_{n}  F(u_{n-1}+v_{n-1}) = 0$, where
X$\Pi_n$ is the orthogonal projection onto $\l2(B_n)$.
XSince
X$$
X  \Pi_{n} F(u_{n-1}+v_{n-1}) \approx
X  \Pi_{n} \left[ F(u_{n-1}) + DF(u_{n-1}) v_{n-1} \right]~~,
X$$
Xa better approximation of a solution to the problem
Xis given by performing an iteration step based on
XNewton's method. This entails inverting the
Xlinearized operator
X$ \Pi_{n}(DF(u_{n-1}))^{-1} \Pi_{n}, \
X =  G_{\overline{B_{n}}}(u_{n-1}) $.  This Green's function
Xis not defined on all of $\NN_0$, but on the subset 
X$D(\NN_n,\rho_n)$ it is boundedly invertible.
XOn this set we define the correction to $u_{n-1}$,
X$$
X \tilde{v}_{n-1} = - \Pi_{n}
X    (DF(u_{n-1}))^{-1} \Pi_{n} F(u_{n-1}).
X$$
X From the estimates of
X\clm(Greens),  $G_{\overline{B_{n}}}$ is analytic
Xon $D(\NN_{n};\rho_{n})$, and satisfies
X\equ(redGreen2).
XFurthermore by $(n-1.1)(ii)$, we know that
X$\|P F(u_{n-1}( \pp,\Omega))\|_{\sigma_{n-1}}
X      \le  \|\pp\|^2 \epsilon_{n-1}$, 
Xon $D(\NN_{n-1},\rho_{n-1}/2)$.
XThus, on $D(\NN_{n};\rho_{n}/2)$,
X$G_{\overline{B_{n}}}$ and
X$\Pi_{n}F(u_{n-1})$ are analytic, and
X$$\eqalign{
X   \|\tilde{v}_{n-1}\|_{\sigma_{n-1}-\gamma_{n-1}} \le&
X   \| G_{\overline{B_{n}}} \|_{\sigma_{n-1}-\gamma_{n-1}}
X  \| \Pi_{n}F(u_{n-1}) \|_{\sigma_{n-1}-\gamma_{n-1}}   \cr
X  \le& \epsilon_{n-1}\|\pp\|^2 C_G^{n} /
X      (\delta_{n} \gamma_{n-1}^{12}) ~~,         \cr}
X$$
Xwhere we used \equ(redGreen2) to estimate
X$\| G_{\overline{B_{n}}}
X    \|_{\sigma_{n-1}-\gamma_{n-1}}$.
XOne has
X
X\CLAIM Lemma(vn) $\tilde{v}_{n-1}(\pp,\Omega;x)$
Xis analytic on
X$D(\NN_{n}; \rho_{n})$ and for any $(\pp, \BGO)$ in
Xthis domain it satisfies
X$$
X   \| \tilde{v}_{n-1}(\pp,\Omega;\cdot)\|_{
X          \sigma_{n-1}-\gamma_{n-1}}
X         \le \epsilon_{n-1}\|\pp\|^2
X    {{C_G^{n}}\over{ \delta_{n} \gamma_{n-1}^{12}}} ~~.
X\EQ(vntilde)
X$$
X
X
XWe now construct $v_{n-1}$ by smoothly
Xextending $\tilde{v}_{n-1}(\pp,\Omega;x)$
Xto all of $\NN_0$,
Xdoing this essentially by setting $\tilde{v}_{n-1}=0$ on
X$\NN_{0} \backslash \NN_n$,
Xhowever using some care in order to obtain in the
Xlimit a $C^{\infty}$ function. We use the
Xfollowing from the appendix of [C].
X
X\CLAIM Lemma(chierchia) For every $R>0$ and for every
Xcompact set $\Delta \subset \complex^d$, there exists
Xa $\chi \in C^0(\complex^d) \cap C^{\infty}(\real^d):
X\complex^d \to [0,1]$.  $\chi$ has support in
X$Y_R(\Delta) \equiv \cup_{\eta_0 \in \Delta}
X\{ \eta \in \complex^d ; \|\eta - \eta_0 \| \le R \}$
Xand $\chi(\eta) = 1$ for every $\eta \in Y_{R/2}(\Delta)$.
XFinally, for every positive integer $k$,
X$$
X\sup_{\real^d} \| \partial_{\eta}^k \chi \| \le
X{{|k| (|k|+2)!}\over{R^{|k|} }}~~.
X$$
X
XDefine $\chi_{n-1}$ to be a function satisfying
Xthe hypotheses of the lemma with $d=3$, 
X$\Delta = \overline{\Eta_n}$,
Xand $R= \rho_n/2$, and set $v_{n-1} = \chi_{n-1}
X\tilde{v}_{n-1}$.
X
X
X\CLAIM Lemma(vntwo)
XWithin the domain of analyticity $D(\Eta_n;\rho_n/2)$
Xthe Cauchy estimate applied to \equ(vntilde) controls
Xall derivatives
X$\partial_{\Omega}^{\beta} \partial_{\pp}^{{\bf \alpha}}
X         v_{n-1}( \pp,\Omega;\cdot) $.
XUsing \clm(chierchia), the estimate on all of $\Eta_0$ is,
X$$\eqalign{
X  \| \partial_{\Omega}^{\beta} v_{n-1}(\pp,\Omega;\cdot)
X          \|_{\sigma_{n-1}-\gamma_{n-1}}
X  \le& C_3 \epsilon_{n-1} C_G^{n}
X      { (\beta + 3)! \over
X         (\delta_{n} \gamma_{n-1}^{12}
X            \rho_{n+1}^{\beta}) } \|\pp\|^2~~,      \cr
X  \|\partial_{\pp} \partial_{\Omega}^{\beta}
X     v_{n-1}(\pp,\Omega;\cdot)\|_{\sigma_{n-1}-\gamma_{n-1}}
X  \le& C_3 \epsilon_{n-1} C_G^{n}
X     { (\beta + 4)!
X       \over (\delta_{n} \gamma_{n-1}^{12}
X         \rho_{n+1}^{\beta}) }
X   \bigl( 1 + { r_0 \over \rho_0} \bigr) \|\pp\|~~. \cr}
X$$
Xand for $|{\bf \alpha}| \geq 2$,
X$$
X  \|\partial_{\pp}^{{\bf \alpha}} \partial_{\Omega}^{\beta}
X   v_{n-1}(\pp,\Omega;\cdot)\|_{\sigma_{n-1}-\gamma_{n-1}} \le
X    C_3 \epsilon_{n-1} C_G^{n}
X     { (|{\bf \alpha}| + \beta + 3)!
X       \over (\delta_{n} \gamma_{n-1}^{12}
X         \rho_{n+1}^{|{\bf \alpha}| + \beta-2}) }~~.
X$$
X
X\clm(vn) and \clm(vntwo), together with the remark that
Xfor $\dist(( \pp,\Omega),\NN_{n}) \le  \half \rho_{n}$,
X$v_{n-1} = \tilde{v}_{n-1}$, imply the first assertion of
X\clm(newton).
X
XWe complete the proof of \clm(newton)
Xby verifying that $u_{n} \equiv u_{n-1} + v_{n-1}$
Xis an approximate solution of the first bifurcation equation
X$ F(u) = 0$.
XNote that
X$$\eqalign{
X  F(u_n) =& F(u_{n-1}+v_{n-1}) = \{ (\11 - \Pi_{n}) F(u_{n}) \cr
X    &+ \Pi_n \left( F(u_{n-1}+v_{n-1}) - [F(u_{n-1}) +
X              DF(u_{n-1}) v_{n-1}] \right) \}~~,          \cr}
X\EQ(newton)
X$$
Xwhere we used the fact that $v_{n-1}$ was constructed so that
X\hfill \break  $\Pi_{n}[F(u_{n-1})+ DF(u_{n-1}) v_{n-1}] = 0$,
X(on $D(\NN_n;\rho_n)$.)
X
XWe can bound the various terms in \equ(newton) by using
Xthe following observations:
X\item{(i)} From hypotheses ${\bf H2}$ and ${\bf H3}$ of Section 6,
Xthere exist positive
Xconstants, $C_W$ and $\sigma$, (with $\sigma < \sigma_0$)
Xsuch that the
Xoperator norm satisfies $\| D_u W(u)\|_{\sigma -\gamma}
X\le C_W/\gamma^2$ and
X$\| D^2_u  W(u)[v,w] \|_{\sigma -\gamma}
X   \le C_W/\gamma^3 \|v\|_\gs \|w\|_{\gs-\ga}$,
X  for all $u$ with $\| u \|_{\sigma} \le 1$.
X\item{(ii)} Using the explicit form of $V(\BGO)$, we see
Xthat $|V(\Omega)(j,k)| \le (\Omega_m^2 + 1)(j^2+k^2)$, thus
X$$
X\|V(\Omega) v\|_{\sigma - \gamma} \le
X(\Omega_m^2+1) \sup_{j,k} (j^2+k^2)e^{-\ga(|j|+|k|)}
X\|v\|_{\sigma}
X\le  {{4 (\Omega_m^2 + 1)}\over{\ga^2}} \|v\|_{\sigma}~.
X$$
Xwhere $\Omega_m = $ the maximum value of $\Omega =
X\omega_1 + \sqrt{ r_0^2 + \rho_0^2}$.
X\item{(iii)} By \clm(cutoff)
X$\|(\11 - \Pi_{n})w\|_{\sigma - \ga}
X   \le  e^{-\ga L_{n}} \|w\|_{\sigma}~~.$
X
XPoint $(iii)$ implies that
X$$
X\|(\11 - \Pi_{n})F(u_{n})\|_{\sigma_{n}} \le
Xe^{-\ga_{n-1} L_{n}} \|F(u_{n})\|_{
X\sigma_{n-1}-2\gamma_{n-1}}~~.
X$$
XThe fundamental theorem of calculus implies that
X$$\eqalign{
X   \|F(u_{n})\|_{\sigma_{n-1}-2\gamma_{n-1}}
X\le& ||F(u_{n-1})||_{\sigma_{n-1}-2\gamma_{n-1}}  \cr
X  &+ \| \int_0^1 (DF(u_{n-1}+tv_{n-1}) v_{n-1})dt
X          \|_{\sigma_{n-1}-2\gamma_{n-1}}        \cr}
X$$
XBy $(n-1.1)$ the first term is bounded
Xby $\epsilon_{n-1} \|\pp\|^2$.
XPoints $(i)$ and $(ii)$ imply that
X$$
X   \| \int_0^1 (DF(u_{n-1}+tv_{n-1}) v_{n-1})dt \
X          \|_{\sigma_{n-1}-2\gamma_{n-1}} \le
X  \left[ {{C_W +4(\Omega_m^2 +1)}
X                   \over{\gamma_{n-1}^2}} \right]
X         \|v_{n-1}\|_{\sigma_{n-1}-\gamma_{n-1}}~~,
X$$
Xcompleting the estimate of the truncation error.
X
XWe  now estimate $[F(u_{n-1}+v_{n-1})
X- F(u_{n-1}) - DF(u_{n-1})
Xv_{n-1}]$,
Xusing the fundamental theorem of calculus to rewrite it as
Xas \relax $\int_0^1 \int_0^t D^2F(u_{n-1}+sv_{n-1})
Xv_{n-1} v_{n-1} ds dt
X$. This is quadratic in $v_{n-1}$.
X{}From the first of the observations above,
X$$
X  \| \int_0^1 \int_0^t D^2F(u_{n-1}+sv_{n-1})
X               v_{n-1} v_{n-1} ds dt
X             \|_{\sigma_{n-1}-2\ga_{n-1}}  \le
X       {C_W \over \ga_{n-1}^3}
X         \|v_{n-1}\|_{\sigma_{n-1} - \ga_{n-1}}^2~~.
X$$
XIn deriving
Xthis estimate we used the fact that
X$D_u^2 F(u_{n-1}) = D_u^2 W(u_{n-1})$.
XCombining these three estimates, we obtain the lemma:
X
X\CLAIM Lemma(L0estimate) The exists a constant $C$,
Xdepending on $C_W$ and $\Omega_m$,
Xsuch that if
X$$
X\epsilon_{n-1}
X{{C C_G^{n+1} }\over{\delta_n \gamma_{n-1}^{14} }}
X\left[ e^{-\gamma_{n-1} L_n }+
X{{r_0^2 \epsilon_{n-1} C_G^{n+1} }\over{\delta_n
X\gamma_{n-1}^{13} }} \right] \le \epsilon_{n}
X$$
Xthen
X$$
X       \|F(u_{n})\|_{\sigma_{n}}
X             \le \|\pp\|^2 \epsilon_n~~,
X$$
Xfor $(\rho, \Omega) \in D(\NN_n, \rho_n/2)$.
X
X\PROOF The estimates above imply that the truncation
Xerror is
X$$\eqalign{
X     \|(\11 - \Pi_n) F(u_n) \|_{\sigma_n}
X           &\le e^{-\gamma_{n-1} L_n}
X            \| F(u_n) \|_{\sigma_{n-1}
X                      - 2\gamma_{n-1} }            \cr
X          &\le  \|\pp \|^2 \epsilon_{n-1}
X                   e^{-\gamma_{n-1} L_n}
X                  \left[1 + {{(C_W + 4(\Omega_m^2+1)
X               C_G^{n} }\over{\delta_n
X                   \gamma_{n-1}^{14} }} \right]    \cr
X          &\le  \|\pp \|^2 \epsilon_{n-1}
X               e^{-\gamma_{n-1} L_n}
X                    {{C C_G^{n} }\over
X                 {\delta_n \gamma_{n-1}^{14} }} ~~.\cr
X}
X$$
X
XThe contribution from the quadratic error is bounded by
X$$
X      \| \Pi_n \left\{ F(u_{n-1}+v_{n-1})-[F(u_{n-1})
X           +DF(u_{n-1}) v_{n-1}]\right\} \|_{\sigma_n} \le
X         {{ \epsilon_{n-1}^2 \|\pp \|^4 C_W C_G^{2n}}
X         \over{ \delta_n^2 \gamma_{n-1}^{27} }}~~.
X$$
X
XThe proposition follows if the sum of
Xthese two terms to be less than $\epsilon_n$.
XCombining these two estimates and using
Xthe fact that $\| \pp \|^2 \le r_0^2$, this follows from
Xthe hypothesis of \clm(L0estimate).
X
X
X
X\SUBSECTION Verification of $(n.2)$
X
XWe continue now by showing how one constructs the set
X$\NN_{n+1}$, described in the induction hypothesis
X$(n.2)$.  We will
Xconstruct $\NN_{n+1}$ using the properties of $u_n$ given
Xby $(n.1)$.
XRecall that we constructed
X$\NN_0$ in Section 2.3.  We denote by $C$ a constant
Xindependent of $n$ and of the inductive constants.
X The following proposition not only implies
X$(n.2)$, but includes some other useful information as well.
X
X
X
X\CLAIM Proposition(parameters) Assume that the following
Xrelationships hold between the inductive constants.
X$$\eqalign{
X  {Cd \over 2^{\tau + 1}} < L_n^{(1-\tau \beta)},
X      & \quad \tau \beta < 1                    \cr
X\beta(1+  c_0 \log(2) )  & < 1                        \cr
X   {C_W L_0 r_0^2 \over
X       {\overline \gs}^3 d_0 \rho_0} <& \ 1
X         < \BGO_{min} L_n^2                     \cr
X   {16 C_W \over {\overline \gs}^2} r_0
X        <& \ d_s                                \cr
X   (1 + {L_0 \over d_0} r_0) <& \ C             \cr}
X\EQ(indconstconds)
X$$
XThen there exists a closed set $\Eta_{n+1}
X\subseteq \Eta_n$ such that:
X\item{(a)} If $(\pp,\BGO) \in \Eta_{n+1}$, then
Xany two singular sites in $B_{n}^c$, which
Xare not in the same singular region, are separated by
Xa distance of at least $2 \ell_{n+1}$.
X\item{(b)} If $S$ is a singular region
Xin $B_{n+1} \backslash B_{n}$  and $(\pp, \BGO )
X\in D(\NN_{n+1},\rho_{n+1}) $ then
X$$\eqalign{
X     &{\rm dist}({\rm spec}(H_{C_{\ell_{n+1}}(S)}
X        (\pp, \BGO; u_n)),0) >  \delta_{n+1} \cr
X         {\rm and}~~~&                       \cr
X    &{\rm dist}({\rm spec}(H_{S}
X       (\pp, \BGO ; u_n)),0) >  \delta_{n+1} \cr}
X$$
XIf the inductive constants satisfy
X$$\eqalign{
X    \sum_{j=0}^n& \ C {d \over L_j^\tau }
X           (1 + 2r_0^2 L_{j+1}) < r_0^2     \cr
X    \sum_{j=0}^n& \ CL_j (1 + r_0^2 L_{j+1})
X       \times  \ell_{j+1}^2
X        \times {\gd_{j+1} \over L_j^2 }
X           < r_0^2                         \cr}
X\EQ(4.4)
X$$
Xthen $\Eta_{n+1}$ has positive measure. If
Xadditionally,
X$$
X    C\bigl( {1 \over d_s} + {L_0 \over d_0}
X        (1 + {r_0 \over \rho_0}) \bigr)
X      < {\Kappa L_n^2 \over 4}
X\EQ(4.5)
X$$
Xand for some $\mu > 0$,
X$$\eqalign{
X     \sum_{j=0}^n& \
X      \sqrt{ {Cd \over \Kappa L_j^\tau }
X        (1 + r_0^2 L_{j+1})    } < r_0^{1+\mu} \cr
X   \sum_{j=0}^n& \ \sqrt{ C L_j (1 + r_0^2L_{j+1})
X       \times \ell_{j+1}^2
X         \times {\gd_{j+1} \over \Kappa L_j^2} }
X           < r_0^{1+\mu}                   \cr}
X\EQ(4.6)
X$$
Xthen the set $\NN_{n+1}$ satisfies the
Xfollowing intersection condition:
X\item{(c)} Let $\BGO(\pp) =\omega_1 + \Kappa
X\| \pp \|^2(1+C(\|\pp \|)) $, be a surface with
Xnon-degenerate quadratic contact at $\pp =0$.
XIf $|\Kappa| > L_0^{-\nu}$, 
Xand $|C(\cdot )|_{C^{1}} \le 1/2$,
Xthen the set $ \{ 0 \leq r < r_0 ; \ r = \|\pp\|, \
X(\pp, \BGO(\pp) ) \in \NN_{n+1} \}$
Xhas measure greater than or equal to
X$r_0(1 - Cr_0^\mu)$.
X
X\REMARK In fact the set $\NN_{n+1}$ is invariant under
Xthe rotations $\pp \to T_{\xi} \pp$.  This invariance is a
Xconsequence of the invariance of the
Xspectrum of the linearized
Xoperator $H_{B_{n+1}}(u)$, under the rotations $u \to
XT_{\xi} u$.
X
X\CLAIM Lemma(invariance) The spectrum of the operator
X$H_{E_{n+1}}(T_{\xi} u)$ is independent of $\xi$.
X
X\PROOF The properties of the nonlinear function
Xin \equ(NLlattice) are that $T_{\xi} F(u) = F(T_{\xi} u)$,
Xand furthermore $T_{\xi}$ commutes with orthogonal
Xprojection onto $\ell^2(E)$ for any $E \subset
X\zsquared$.  Differentiating with respect to $u$ we find
X$T_{\xi} DF(u) v = DF(T_{\xi} u) T_{\xi}v$, thus
X$DF(u)$ and $DF(T_\xi u)$ are unitarily equivalent, and
Xthe result follows.    
X
X\endproof
X
X
XPart $(a)$ of \clm(parameters) can be proven
Xwithout restricting $\pp$--we
Xneed only place conditions on $\Omega$.
X{}From the diophantine condition on $\omega_1$
Xthere is an integer $N_0$ such that for the
Xinitial induction steps $0 \leq n \leq N_0$,
Xthe separation condition $(a)$ is satisfied
Xfor all $(\pp,\BGO) \in \Eta_0$. This is the
Xresult of the next lemma, whose hypotheses form
Xa subset of those of \clm(parameters).
X
X\CLAIM Lemma(separation) Assume that
X$\tau \beta < 1$, $\beta(1+c_0 \log(2) ) <1$,
Xand that $d > L_0^{\tau \beta}
X(1/L_0 + r_0^2 L_0^\beta)$. There exists a
Xconstant $c_0 = c_0(\tau,\beta)$ such that,
Xif $L_0$ is sufficiently large and
X$N_0 = c_0 \log (L_0)$, then for all $n \le N_0$
Xand $(\pp,\BGO) \in \Eta_n$, if
X$x = (k,j)$, $x' = (k',j')$ are a pair of singular
Xsites in $B_n^c$ which are not in the same singular
Xregion, they are well separated; $i.e.$
X$$
X\dist (x,x') \ge 2 \ell_{n+1}~~.
X$$
X
X
X
X\PROOF Assume that $|x - x'| < 2 \ell_{n+1}$.  We will derive a
Xcontradiction.  If $x$ is a singular site then
X$|j^2 \Omega^2 - \omega_k^2| < d_s$.  Factoring the left
Xhand side of this inequality and assuming that $j$ and
X$k$ are non-negative this implies
X$|j \Omega - \omega_k| < d_s/|j\Omega + \omega_k|
X\le C(\Omega_m) d_s /(|j|+|k|)$, for some constant $C(\Omega_m)$.
XSimilar estimates hold for $|j' \Omega - \omega_{k'}|$,
Xand for the cases when $j$ and $j'$ are negative.
XWe now use the asymptotics of the Sturm-Liouville operator.
XFor the case of Dirichlet boundary conditions there exists
Xa constant $C_{g_1}$ such that $|\omega_k - k| < C_{g_1}/k$, for all $k \ge 1$.
X (For a review of the properties of these
Xeigenvalues see, for example [PT].)  For periodic boundary
Xconditions, if $k = 2m$ or $k = 2m -1$, $m \ge 1$, then
X$|\omega_k - 2m| < C_{g_1}/m$.
XNote that since $x$ and $x'$ are
Xin $B_n^c$, there exists a constant $C(\Omega_m)$ such that
X$\min (|k|,|k'|) \ge C(\Omega_m) L_n$.  In the case of
XDirichlet boundary conditions we have
X$$\eqalign{
X   |(j-j') \Omega - (k-k')| \ =& \
X      |(j-j') \Omega - (\omega_k - \omega_{k'})
X              + (\omega_k - \omega_{k'}) - (k-k')| \cr
X   \le& \ {{2 C(\Omega_m) d_s}\over{L_n}} +
X        {{2 C(\Omega_m) C_{g_u}}\over{L_n}} \ \le \
X             {{C(g_1,\Omega)}\over{L_n}}~~.        \cr
X}
X\EQ(upperbnd)
X$$
XSince $x \in B_n^c$ we have $|j|+|k| \ge L_n$, with a
Xsimilar estimate for $x'$. We are assuming that
X$\dist (x,x') = |j-j'|+|k-k'| \le 2 \ell_{n+1}$ $=
X2 L_{n+1}^{\beta} = 2 (2^{n+1} L_0)^{\beta}$.  But
X$n \le N_0 = c_0 \log (L_0)$ so
X$\dist (x,x') \le 2(2^{c_0 \log(L_0) }L_0)^{\beta}
X= 2(L_0^{\beta(1+c_0 \log 2)}) < L_0$, if
X$ \beta(1+c_0 \log 2) < 1$ and $L_0$ is sufficiently
Xlarge. The principal frequency $\omega_1$ is
X$(d_0,L_0)$-nonresonant, thus it satisfies a
Xfinite diophantine condition over the lattice
Xpoints within $B_0$. Hence as long as
X$2 \ell_{n+1} < L_0$ and $|\BGO-\omega_1| < r_0^2$,
X$(x - x') \in B_0$ and the components satisfy
X$$\eqalign{
X       |(j-j') \Omega - (k-k')| \ \ge &
X            \ |(j-j') \omega_1 - (k-k')|
X               - |(j-j')(\Omega - \omega_1)|       \cr
X            \ge & \ {{d}\over{(2 \ell_{n+1})^{\tau}}} -
X               2 \ell_{n+1} r_0^2~~.        \cr
X}
X\EQ(lowerbnd)
X$$
XBy the hypotheses of the lemma, $4 \ell_{n+1} r_0^2
X<  \half {{d}\over{(2 \ell_{n+1})^{\tau}}}$, so we find
X$|(j-j') \Omega - (k-k')| \ge
X \half {{d}\over{(2 \ell_{n+1})^{\tau}}}$.  Combining this with
X\equ(upperbnd) gives
X$$
X    \half {{d}\over{(2 \ell_{n+1})^{\tau}}} \le
X      {{C(g_1,\Omega_m)}\over{L_n}}
X$$
Xor, using $\ell_{n+1} = L_{n+1}^{\beta}$,
X$L_{n+1}^{(1-\tau \beta)} \le 2^{\tau(1+\beta)} C(g_1,\Omega)/d$.
XIf $\tau \beta < 1$, and $L_0$ is sufficiently
Xlarge, this inequality
Xcannot be true and the lemma follows.
X
X\endproof
X
XIn the case of periodic boundary conditions, the argument
Xneeds to be modified only slightly to accound for the different
Xasymptotics of the eigenvalues.  The estimate in
X\equ(upperbnd) is replaced by
X$$
X     |(j-j') \Omega - 2([k/2]-[k'/2])| \le
X           {{C(g_1,\Omega_m)}\over{L_n}}~~,
X\EQ(pupperbnd)
X$$
Xwhere $[\cdot]$ denotes the integer part.  Combining
Xthis estimate with the lower bound coming from
Xthe fact that $\omega_1$ is $(d_0, L_0)$
Xnonresonant, we find that $\dist (x,x')$
Xcannot be less than $2 \ell_{n+1}$ except in one special
Xcase.  This special case occurs when $k=2m$
Xand $k' = 2m-1$ or vice-versa.
XThen  \equ(pupperbnd) becomes
X$|(j-j') \Omega| \le C(g_1,\Omega_m)/L_n$.
XBut if, in addition, $j=j'$ we cannot apply the diophantine
Xestimate to obtain the contradiction.  Thus, if
X$(j,k)$ and $(j',k')$ are any pair of singular sites with either
X$j \not= j'$ or $\{ k,k' \} \not= \{ 2m, 2m-1 \}$, then they are
Xseparated by a distance of at least $2 \ell_{n+1}$.
X
X\REMARK As a corollary of the proof of this
Xlemma, we see that in the case of Dirichlet
Xboundary conditions, the singular
Xregions consist of isolated singular sites, while in
Xthe case of periodic boundary conditions the singular
Xreg