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\begin{document}

\title{Solitons on pseudo-Riemannian manifolds: stability
and motion}

%    Information for first author
\author{David M. A. Stuart}
\address{Centre for
Mathematical Sciences, University of Cambridge, 
Wilberforce Road, Cambridge CB3 OWA, UK}
\email{D.M.A.Stuart@damtp.cam.ac.uk}
\thanks{The  author acknowledges support from   EPSRC Grant AF/98/2492.}
\subjclass[2000]{Primary 58J45, 37K45; Secondary 35Q75, 83C10, 37K40}

\date{April 30, 2000}
\revdate{August 7, 2000}

\keywords{Wave equations on manifolds, nontopological solitons, 
stability, solitary waves.}

\commby{Michael Taylor}

\begin{abstract}
This is an announcement of results concerning a class of solitary
wave solutions to semilinear wave equations. The solitary
waves studied are solutions of the form $\phi(t,x)=e^{i\omega 
t}f_\omega(x)$
to semilinear wave equations such as
$\Box\phi+m^2\phi=\beta(|\phi|)\phi$ on $\mathbb{R}^{1+n}$ 
and are called nontopological solitons.
The first preprint provides a new
modulational approach to proving
the stability of nontopological
solitons.
This technique, which makes strong use of the
inherent symplectic structure, provides explicit information
on the time evolution of the various parameters of the soliton.
In the second preprint a pseudo-Riemannian structure $\underline{g}$
is introduced onto $\mathbb{R}^{1+n}$ and the corresponding wave
equation is studied. It is shown that under the rescaling
$\underline{g}\to\epsilon^{-2} \ulg$, with $\epsilon\to 0$, it is 
possible to
construct solutions representing nontopological solitons
concentrated along a time-like geodesic. 
\end{abstract}

\maketitle

\section{Introduction}
\label{intro}
\subsection*{Solitons and classical field theory} 
The subject of classical field 
theory involves the study of partial
differential equations which describe the evolution and
interaction of a collection of fields, i.e. functions on the space-time
manifold $\Mink$ or other geometric objects such as sections, connections
and metrics defined on vector bundles over $\Mink$. The gravitational 
field
itself is described by a pseudo-Riemannian metric 
$\mg$ on $\Mink$. The situation
of a flat metric corresponds to the absence of a gravitational field, 
in which
case one is led to the study of equations on the Minkowski space-time
$(\Mink,\mg)=(\Real^{1+3},\eta)$ where $\eta$ is the 
metric \[\eta=-dt^2+\sum_{i=1}^3 (dx^i)^2\] in standard coordinates 
$(x^0=t,x^1,x^2,x^3)$. The requirement of invariance under the Lorentz
group, i.e. the set of (pseudo-orthogonal) linear transformations 
which preserve $\eta$, leads
to PDE's of hyperbolic type such as the wave equation
\[\Box\phi\equiv\partial_t^2\phi-\Delta\phi=0.\] This equation is an 
appropriate
prototype for understanding essentially linear effects such as
radiation. On the other hand important systems such as Einstein's
equation and the Yang-Mills-Higgs equations
are nonlinear and lead to different phenomena such as singularity 
formation (e.g. black holes) and energy concentration 
or localisation (e.g. monopoles
and other solitons).
Following \cite{lee88}
a soliton in classical field theory is, for present purposes, 
defined to be a 
spatially localised
solution to the equations which does not decay or disperse
as time progresses; the term is not intended to imply any properties
of complete integrability of the PDE. As in \cite{lee88}
it is helpful to distinguish between {\it topological} solitons
(such as BPS monopoles and Ginzburg-Landau vortices) which are 
stabilised by a nonzero winding number
and {\it nontopological solitons}
in which the stabilising mechanism is
dynamical.\footnote{Nontopological solitons correspond to
what are called relative equilibria in Hamiltonian mechanics;
see e.g. \cite[Chapter 4]{mars92}.} 
It is this latter case which is considered here: to be
precise, nontopological solitons on $\Mink$
are here defined as solutions to the equation
for $\phi:\Mink \to \Complex$,
\begin{equation}
\Box\phi =
\mcF(\phi),
\label{one} 
\end{equation}
of the form
\begin{equation}
\phi(t,x)  =e^{i\omega t}f_\omega(x).
\end{equation}
The basic properties of these solutions are explained below in 
\S\ref{notop}.

So far the discussion has been limited to the case when there is
no gravitational field present and the metric is flat. In the presence
of a gravitational field it is necessary to study the corresponding
PDE's with a background metric $\mg$ which itself evolves 
according to Einstein's
equation. As an intermediate problem I consider here the case of a 
fixed background metric: mathematically this means the
study of the semilinear wave equation on a pseudo-Riemannian manifold
$(\Mink,\mg)$. 

\subsection*{The geodesic hypothesis} In general relativity
it is a standard assumption
(\cite{wein72}) that a ``test-particle'' introduced into
the space-time
will follow a geodesic with respect to the background metric
(to highest order): this will be referred to as the {\it geodesic 
hypothesis}.
In this work it is my aim to give a rigorous mathematical derivation of
this hypothesis for the case of a fixed background metric.
Of course there are
many possible ways to formulate a corresponding 
precise mathematical 
question as it is necessary to translate the notion of ``test-particle''
into definite mathematical terms. Here I do this by taking particle  
to mean nontopological soliton; this is reasonable in view of the strong
spatial localisation which these solitons 
exhibit\footnote{The choice of nontopological solitons to model test 
particles is not intended
to be physically realistic, but rather to provide an interesting and 
relatively clean analytical problem
in which to develop appropriate techniques.}. 
In order to consider
a {\it test}-particle it is necessary to also consider a 
rescaling of the metric
\[\mg\to\epsilon^{-2}\mg\qquad\hbox{ with }\quad\epsilon<<1\]
to ensure that the size of the particle is small
compared to scales over which the metric varies. All in all this leads
to the problem
of constructing
solutions to the equation\footnote{Here $\Box_{\mg}$ is the
wave operator on a pseudo-Riemannian manifold, an explicit formula for 
which 
is given in \eqref{gen} below.}
\begin{equation}
\Box_{\mg}\phi =
\frac{1}{\epsilon^2}\mcF(\phi)
\label{gone} 
\end{equation}
which are close to a nontopological soliton
concentrated along a geodesic.
This problem is given a 
solution in Theorem
\ref{stheorem}. The proof is an outgrowth of a new explicit approach to
the stability of the solitary wave solutions on flat Minkowski space,
as detailed in \S\ref{stab}. Before explaining these results in
detail a review of the basic properties of nontopological solitons 
will be given in the
next section.
%
%
\section{Nontopological solitons}
\label{notop}
In this section a general discussion of known existence and stability
results for the class of solitons under investigation will be given. The
domain will be 
$\Mink=\Real ^{1+n}$ with
Minkowski metric $\eta=-dt^2+\sum_{i=1}^n(dx^i)^2$ in standard 
coordinates.

\subsection*{Existence} 
Consider the semilinear wave equation
for $\phi:\Mink \to \Complex$ with the following type of nonlinearity:
\begin{equation}
\partial_t^2\phi-\Delta\phi+m^2\phi  =
\beta(|\phi|)\phi,\quad\hbox{where $\beta:\Real\to\Real$ and $m^2>0$.}
\label{main} 
\end{equation}
Observe that this form of nonlinearity ensures that the 
equation is invariant under the action of $S^1$ by
phase rotation $\phi\to\phi e^{i\chi}$, for any $\chi\in\Real$.
Nontopological solitons are solutions to \eqref{main} of the form
\begin{equation}
\phi(t,x)  =e^{i\omega t}f_\omega(x),
\label{bs}
\end{equation}
where (for appropriate $\beta$) the function $f_\omega(x)$ is the 
unique positive radial solution on
$\Real^n$ of the {\it elliptic} equation 
\begin{equation}
-\Delta f_\omega+(m^2-\omega^2)f_\omega= \beta(f_\omega)f_{\omega}
\qquad\quad\hbox{with \ $\omega^2< m^2$.}
\label{soli}
\end{equation}
Notice that although the solution is not independent of time it has a 
very simple time dependence: it maps out an orbit of the symmetry group
$S^1$ as a function of $t$. This concept can of course be generalised 
to solutions
which map out the orbit of a one-parameter subgroup of the symmetry 
group; see
\eqref{releq}.
Solutions of this form have been much studied in 
Hamiltonian mechanics and are called relative equilibria in that context
(\cite{mars92}); many examples of this type of solution for PDE's are 
given
in \cite[Chapter 7]{stra89} and \cite{GSS,GSS90}.

Existence and uniqueness of $f_\omega>0$ verifying \eqref{soli}
has been proved under very general conditions on $\beta$
(see \cite{beli83,blp83,mcl93,stra77,stra89} and references therein).
A useful model case to keep in mind
is the {\it pure power
nonlinearity} 
\[ \beta(|\phi|)\phi = |\phi|^{p-1}\phi,\]
in which case
a unique positive radial solution to \eqref{soli} exists if $10\right\}. 
\end{equation}
This is called the {\it stability interval}.
To be precise consider the Cauchy problem for \eqref{main} with
radial initial data
\[
\phi(0,x)=f_\omega(x)+F(|x|),\qquad
\phi_t(0,x)=i\omega f_\omega(x)+G(|x|).
\]
Then if $\|(F,G)\|_{H^1\times L^2}$ is sufficiently small
the quantity
\begin{equation}
\inf_{\chi\in\Real}\Bigl(\|\phi(t,x)
-e^{i\chi}e^{i\omega t}f_\omega(x)\|_{H^1}
+\|\phi_t(t,x)-i\omega e^{i\chi}e^{i\omega t}f_\omega(x)\|_{L^2}\Bigr)
\label{dist}\end{equation}
remains uniformly bounded by a multiple of $\|(F,G)\|_{H^1\times L^2}$
if $\omega$
lies in the stability interval. This provides a useful criterion for 
stability and
in fact  if
$\omega$ does not satisfy this condition, instability was proved.
Furthermore
for the case of a pure power nonlinearity $\beta(|\phi|)=|\phi|^{p-1}$
the 
condition can be made completely explicit:
a scaling argument gives
\[{\bf I} =
\left\{\omega\in\Real:\frac{1}{1+\frac{4}{p-1}-n}< 
\frac{\omega^2}{m^2}<1\right\},\]
which is nonempty when
$10$ such that if
\[\epsilon=
\|\phi(0,\cdot)-\phi_S(\cdot;\lambda_0)\|_{H^1}
+\|\psi(0,\cdot)-\psi_S(\cdot;\lambda_0)\|_{L^2}<\epsilon_*(\lambda_0)
\]
then there exist $\lambda\in C^1(\Real;O_{stab})$, a pair
$(\phi,\psi)\in C(\Real;H^1\oplus L^2)$ and $c_1>0$ such that
\begin{equation}
\sup_{t\in\Real}\Bigl(
\|\phi(t,\cdot)-\phi_S(\cdot;\lambda(t))\|_{H^1}
+\|\psi(t,\cdot)-\psi_S(\cdot;\lambda(t))\|_{L^2}
\Bigr)
0$ such that 
\begin{equation}
\label{bbtl1}
|\partial_t\lambda-V(\lambda)|\le c_2\epsilon.
\end{equation}
In this situation the exact solution  
$\bigl(\phi_S(x;\lambda^{(0)}(t) ),\, \psi_S(x;\lambda^{(0)}(t))\bigr)$ 
of \eqref{il}
is said to be {\em modulationally stable.}
\end{theorem}
Notice that since 
$\lambda_0=\bigl(\omega_0 ,\theta_0 ,\xi_0 ,u_0\bigr)$ 
can be anywhere in $O_{stab}$ this theorem asserts
that stability holds for all velocities $u_0$ of the soliton.
The fact that the stability
criterion is independent of velocity is certainly to be
expected in view of Lorentz invariance, although it is {\it not}
possible to reduce study of the Cauchy problem for different velocities
to that at zero because the Lorentz transform does not preserve
the $t=\operatorname{const.}$ hyperplanes. The assertion of stability for all velocities
and without radial symmetry does not appear to have been stated
explicitly in the literature before, although it should be derivable
from the general theorems in \cite{GSS90}. As already remarked 
the main point of the new
approach taken here is to give explicit information on the
curve $t\mapsto\lambda(t)$.  Indeed the system of equations satisfied by
$\lambda$ is precisely given in
\cite{stab99}. It is the corresponding set of equations in the case when
curvature is present that leads to the geodesic equation 
in the proof of Theorem \ref{stheorem}.

The analysis is similar to that
in \cite{wein85}, where a linearised stability result
for \eqref{bs} in the nonlinear Schr\"{o}dinger
equation was obtained. Here there is the additional complication 
(not present in the Schr\"{o}dinger case) caused by the
dependence of the stability condition on $\omega$
(see \eqref{stabint}). Indeed this latter fact makes even
the understanding of stability at the linear level somewhat subtle 
due to the following rescaling argument.
In the pure power case
$ \beta(|\phi|)\phi = |\phi|^{p-1}\phi$ 
the equation \eqref{soli} becomes
\[
-\Delta f_\omega+(m^2-\omega^2)f_\omega=f_\omega^p
\label{soli2}\]
so that it is easy to check that the solutions are all
related by rescaling:
\begin{equation}
f_\omega(x)=(m^2-\omega^2)^{\frac{1}{p-1}}f(\sqrt{m^2-\omega^2}x),
\label{res}
\end{equation}
where $f$ is the corresponding solution with $m^2-\omega^2=1$.
Furthermore
the linear operators which appear on linearisation, namely
\begin{align*}
&L_+=-\Delta_Z+(m^2-\omega^2)-pf_\omega^{p-1}(Z),\\
&L_-=-\Delta_Z+(m^2-\omega^2)-f_\omega^{p-1}(Z)
\end{align*}
are also transformed into a universal form (independent of $\omega$)
under this rescaling.
So at first sight it is unclear how the stability properties can change 
as 
$\omega$ changes. This apparent paradox is settled by Lemma \ref{infstab}
below.

\subsection*{Remarks on the proof}
To start with write 
\[
\left(
\begin{array}{l}
\phi(t)\\
\psi(t)
\end{array}
\right) = \left(
\begin{array}{l}
\phi_S(x;\lambda(t))+ \tilde\phi(t)\\
\psi_S(x;\lambda(t))+ \tilde\psi(t)
\end{array}
\right).
\]
The purpose is to determine $\lambda(t)$ so that the error
terms $\tilde\phi, \ \tilde\psi$ can be estimated in $H^1\oplus  L^2$
uniformly in time. 
The crucial points for achieving this are: 
\begin{itemize}
\item[(i)] to identify, at the linear level, the origin of the
stability interval as a condition for positivity of the Hessian of an
appropriate functional called the 
{\it augmented Hamiltonian} on an appropriate subspace
which is {\it the symplectic normal subspace};
\item[(ii)] to show that there is a locally well-posed system of
ordinary differential equations for $t\mapsto \lambda(t)$ which forces
$ (\tilde{\phi}, \ \tilde{\psi})=
 (\phi-\phi_S, \ \psi-\psi_S)
$  to lie in the subspace referred to in (i).  
\end{itemize}

In order to fully explain these points, definitions of the italicised
phrases in (i) are needed.

(a) {\it The augmented Hamiltonian.} 
It is useful to note, at the formal level, that $(\phi_S,\psi_S)$ are
critical points of $H$ subject to the constraints
that $\Pi,Q$ be fixed, where
\begin{align}
&\Pi_i(\phi,\psi)=\int\Big\langle\psi,\frac{\partial\phi}{\partial
x^i}\Big\rangle dx,\qquad\qquad\qquad\hbox{(momentum)}\label{cons1}\\
&Q(\phi,\psi)=\int\langle i\psi,\phi\rangle dx
\qquad\qquad\qquad\qquad\hbox{(charge).}
\label{cons2}
\end{align}
These are the conserved quantities deriving from  translation and
phase ($S^1$) invariance on account of Noether's theorem.
In this setting
$u^i$ and ${\omega}/{\gamma}$ emerge as the
corresponding Lagrange multipliers and
thus $\phi_S,\psi_S$ are
critical points of the enlarged functional
\begin{equation}
F(\phi,\psi;\lambda)=H(\phi,\psi)
+u^i\bigl(\Pi_i(\phi,\psi)-p_i\bigr)
+\frac{\omega}{\gamma}\bigl(Q(\phi,\psi)-q\bigr),
\end{equation}
for appropriate $p,q$: this quantity is called the augmented
Hamiltonian.
Thus
the Hessian of $F$ at
$\bigl(\phi_S(\cdot,\lambda),\psi_S(\cdot,\lambda)\bigr)$
should be
an important quantity for the stability analysis.

(b) {\it The symplectic normal subspace.}\footnote{$\Upsilon_\lambda$ 
is the space of
vectors $\Omega$-orthogonal to the tangent space to $\cs$. The 
terminology is justified
in the stable case because $\cs$ is then symplectic.}
Introduce the subspace 
\begin{equation}
\Upsilon_{\lambda}\equiv
\left\{(\tilde\phi,\tilde\psi)\in H^1\oplus L^2:
\bfOmega\Bigl((\tilde\phi,\tilde\psi),\frac{\partial}{\partial\lambda_A}
\bigl(\phi_S,\psi_S\bigr)
\Bigr)=0\hbox{ for all $A$}\right\},
\label{ups}
\end{equation}
where $\bfOmega$ is the standard symplectic form 
for wave equations,
\begin{align}
\begin{split}
&\bfOmega: \Bigl(L^2(\Sigma;\Complex)\Bigr)^2\times 
\Bigl(L^2(\Sigma;\Complex)\Bigr)^2\to\Real\\
&\bfOmega\Bigl((\tilde\phi,\tilde\psi),(\phi',\psi')\Bigr)
=\int\Bigl(
\langle\tilde\phi,\psi'\rangle-\langle\tilde\psi,\phi'\rangle
\Bigr)dx.
\end{split}
\label{sympform}
\end{align}
$\Upsilon_{\lambda}$ is the symplectic normal subspace to 
${\mathcal S}=\bigl(\phi_S(\cdot,\lambda),\psi_S(\cdot,\lambda)\bigr)
\bigr|_{\lambda\in O}$.

Having made these definitions return now to a discussion of the two 
crucial points
of the proof referred to above.
Point (ii) is resolved by a lengthy but 
straightforward calculation relying on the following observation, 
which is the first
intimation of the significance of the condition on $\omega$ 
in \eqref{stabint}.

\begin{lemma}[\cite{stab99}]
\label{ssm}
The subset
${\mathcal 
S}_{stab}=\bigl(\phi_S(\cdot,\lambda),\psi_S(\cdot,\lambda)\bigr)
\bigr|_{\lambda\in O_{stab}}
\subset H^1\times L^2$
is a local $C^1$ symplectic submanifold, in particular 
$\bar{{\mbox{\boldmath$\Omega$}}}$, 
the restriction of $\bfOmega$ to ${\mathcal S}_{stab}$,
is nondegenerate. As $\lambda$ approaches the boundary of 
$O_{stab}$ in $O$, i.e., as $\frac{\omega^2}{m^2}$ tends to
$\frac{1}{1+\frac{4}{p-1}-n}$, 
$\bar{{\mbox{\boldmath$\Omega$}}}$ degenerates.
\end{lemma}

The stability condition
thus makes it possible to restrict the Hamiltonian flow
\eqref{il} to $\cs$ in a nondegenerate fashion: this is important
for the well-posedness of the system of ODE's for $\lambda$
referred to in (ii) above.
Notice also that Lemma \ref{ssm} justifies the terminology 
``symplectic normal subspace'' in (b) above.

Point (i) is dealt with by the
following {\it infinitesimal stability}  lemma.
For fixed $\lambda=(\omega,\theta,\xi,u)\in O$
consider the quadratic form on $H^1\times L^2$:
\begin{align*}
{\mathcal E}(\tilde\phi,\tilde\psi;\lambda)
& =\frac{1}{2}\int \Bigl( |\tilde\psi|^2
+|(\nabla_x)_i\tilde\phi|^2+
m^2|\tilde\phi|^2-\beta(|\phi_S|)|\tilde\phi|^2
\\
&\qquad\qquad -\beta'(|\phi_S|)\frac{\langle 
\phi_S,\tilde\phi\rangle^2}{|\phi_S|}
-\frac{2\omega}{\gamma}\langle\tilde\psi,i\tilde\phi\rangle
 +2{u}^i\langle(\nabla_x)_i\tilde\phi,\tilde\psi\rangle \Bigr) dx,
\end{align*}
where $\phi_S=\phi_S(x;\lambda)$ is as above in \eqref{e3}
($ {\mathcal E}$ is independent of $\psi_S$ because the
nonlinearity appears only in $\phi$). The significance of the
quantity $\ce$ is that it is the Hessian of the augmented Hamiltonian
$F$.
\begin{lemma}[\cite{stab99}]
\label{infstab}
The 
quadratic form $\ce(\tilde\phi,\tilde\psi;\lambda)$,
restricted to the subspace $\Upsilon_{\lambda}$,
is equivalent to the $H^1\oplus L^2$ norm, uniformly for $\lambda$ in 
compact subsets of
$O_{stab}$.
\end{lemma}
\noindent This information can be combined with the conservation laws for
$H,\Pi,Q$ to imply a uniform $H^1\times L^2$ bound for 
$(\tilde\phi,\tilde\psi)$ in the stable case and thus prove Theorem 
\ref{purepower}.

\subsection*{More general nonlinearities}
The class of nonlinearities ${\mathcal F}(\phi)=\beta(|\phi|)\phi$ 
to which the present development
can be extended is delineated primarily by the requirement that
Lemma \ref{infstab} holds. This is essentially ensured 
by the conditions for uniqueness 
of positive $f_\omega$ (see
\cite{mcl93} and references therein)
which guarantee that the subspace on which $\ce$ is negative is 
one-dimensional. 
(It is of course also necessary that $f_\omega$ exist, and that the 
Cauchy problem is well posed, so that
it is always assumed that the nonlinearity is subcritical.) 

In the pure power case
Lemma \ref{infstab} can be deduced from the fact 
(\cite{wein83}) that $f_\omega$ are optimizers of the Gagliardo-Nirenberg
quotient:
\begin{equation}
J^{p,n}(u)
\equiv\frac{\|\nabla u\|_{L^2}^{\frac{(p-1)n}{2}}
\|u\|_{L^2}^{2+\frac{(p-1)(2-n)}{2}}}
{\|u\|_{L^{p+1}}^{p+1}}.
\label{gnq}
\end{equation}
This is related to the scaling and invariance properties of 
\eqref{soli} in the
pure power case (also see \eqref{res} and surrounding discussion) and 
allows an explicit determination of the stability interval 
\eqref{stabint}. For
more general nonlinearities Lemma \ref{infstab} can be proved under the 
assumptions
just mentioned using \eqref{stabint} as a characterisation of the 
stability interval 
${\bf I}$. However,
unlike the pure power case it may not be possible to delineate ${\bf 
I}$ any more
explicitly than this.
%
%
%
%
\section{Nontopological solitons on pseudo-Riemannian manifolds}
\label{pseud}
In this section I explain a special case of a result from 
\cite{pseud00} which
generalises the techniques described in \S\ref{stab} to the wave 
equation on a
manifold. So fix $n=3$ and introduce on $\Mink=\Real^{1+3}$ a 
pseudo-Riemannian structure
$\ulg$ so that 
$\bigl(\Mink, \ulg\bigr)$ is a pseudo-Riemannian manifold, and assume 
there
is a global system of coordinates $(x^0=t, x^1,x^2,x^3)$ in which the
metric is of the form\footnote{The summation convention will be used. 
Greek (space-time) indices
run from $0$ to $3$, and Latin (space) indices run from $1$ to $3$.}
\begin{equation}
ds^2=\ulg_{\mu\nu}dx^\mu dx^\nu=
\Bigl(-p^2dt^2+g_{ij}dx^idx^j\Bigr).
\label{metric}
\end{equation}
$(\Mink,\ulg)$
admits a $t$-foliation  into space-like hypersurfaces which are the 
level sets of
the time function $t:\Mink\to\Real$, i.e., 
\begin{equation}
\Mink\approx\Real\times\Sigma,\qquad\Sigma\approx\Real^n,\qquad
\Sigma_{t_0}\equiv\{t_0\}\times\Sigma=t^{-1}(t_0).
\label{fol}
\end{equation}
In \eqref{metric} $p:\Mink \to\Real$ is called
the {\it lapse} function and $g_{ij}(t,x)$ is the metric induced on
$\Sigma_t$ from $ds^2$.  It is assumed that $p,g$ are $C^2$ and there 
exists
a number $K>1$ such that\footnote{The exponents are chosen to make the 
assumptions
scale invariant.}
\begin{itemize}
\item
all derivatives up to second order of $p^2$ and $g_{ij}$ are bounded by 
$K$;
\item
 $p^6$ and ${\bf g}=\dt g_{ij}$ are bounded below by $\frac{1}{K^3}$.  
\end{itemize}
As usual the induced inner product on the cotangent
space is represented by the inverse matrix $g^{ij}(t,x)$ and the
notation ${\bf g}=\dt g_{ij}$ will be used throughout.  

\subsection*{A scaling limit}
The aim is to construct solutions to the wave equation on 
$(\Mink,\ulg)$ using
the flat Minkowski space solutions in Lemma \ref{exsol} as basic 
building
blocks. This is possible in a certain scaling limit in which the size 
of the soliton
is small compared to length scales over which the metric varies. It is 
worthwhile to
introduce this scaling first in the flat case: here it corresponds to 
rescaling the
metric
\[-dt^2+\sum_{i=1}^3 (dx^i)^2\Rightarrow 
\frac{1}{\epsilon^2}\Bigl(-dt^2+\sum_{i=1}^3 (dx^i)^2\Bigr).
\]
Under this scaling the equation \eqref{main} becomes
\begin{equation}
\partial_t^2\phi-\Delta\phi+m^2\phi  =
\frac{1}{\epsilon^2}\beta(|\phi|)\phi.
\end{equation}
The basic soliton solution is thus now
\begin{equation}
\phi(t,x)  =e^{i\omega t/\epsilon}f_\omega(x/\epsilon).
\label{bse}
\end{equation}
Recalling that $f$ has exponential decay at rate $m$ it follows that
the soliton is now exponentially localised in a region of size 
$\epsilon$. Now in the
presence of a nontrivial metric consider the rescaling:
\[
\ulg\Rightarrow 
\frac{1}{\epsilon^2}\ulg
=\frac{1}{\epsilon^2}\Bigl(-p^2dt^2+g_{ij} dx^i dx^j\Bigr),
\]
where $p,g$ are fixed (independent of $\epsilon$) as above.
In the limit $\epsilon\to 0$ the size of the soliton will be small 
compared to
distances over which $p,g$ vary.

Under this rescaling the
corresponding Lebesgue and Sobolev spaces on $\Sigma\approx\Real^n$
defined with the metric $\frac{1}{\epsilon^2}g(t,\,\cdot\,)$ are denoted
by $L^{2}_{\epsilon}$ and $H^{1}_{\epsilon}$,
and their norms analogously:
\begin{align}
\begin{split}
\|f\|_{L^2_\epsilon}^2&=\epsilon^{-n}\int_\Sigma |f(x)|^2\sqrt{\bf 
g}\,dx,\\
\|f\|_{H^1_\epsilon}^2&=\int_\Sigma \bigl(\epsilon^{2-n}
g^{jk}\nabla_j f(x)\nabla_k f(x)+\epsilon^{-n} |f(x)|^2\bigr)
\sqrt{\bf g}\,dx.
\end{split}
\label{norm}
\end{align}
The assumptions made on the metric $g$ ensure that the norms so 
defined, which 
are different for different $t$, are in fact uniformly equivalent with 
constant depending
upon $K$ only, so there is no need to distinguish between
them for the purposes 
of this paper.

\subsection*{The wave equation on $(\Mink,\ulg)$}
With the metric $\epsilon^{-2}\ulg$ the wave equation \eqref{main} 
generalises to 
\begin{equation}
\partial_t\left(\frac{\sqrt{\bf g}}{p}\partial_t\phie\right)-\partial_i
(p\sqrt{\bf g}g^{ij}\partial_j\phie)
+\frac{1}{\epsilon^2}p\sqrt{\bf g}(m^2\phie-\beta(|\phie|))\phie=0.
\label{gen}
\end{equation}
From this it is again clear that $\epsilon$ controls the size of the 
soliton
compared to length scales over which the metric varies:
the size of the soliton is $O(\epsilon)$ as $\epsilon\to 0$,  while 
$p,g$ vary over
scales of $O(1)$. 

\subsection*{Geodesics on $(\Mink,\ulg)$}
To describe time-like geodesics on $(\Mink,\ulg)$ 
it is convenient to parametrise them with the time coordinate $t$.
So using the foliation structure in \eqref{fol} this gives a
curve $t\mapsto (t,\xi(t))\in M=\Real\times\Sigma$ and hence
a curve $t\mapsto\xi(t)\in \Sigma $. To write down the conditon for 
this curve to be
a geodesic introduce the velocity
\[u^j=\frac{d\xi^j}{dt}.\]
At fixed $t$ this is a vector in $T_{\xi(t)}\Sigma$. Introduce also the 
evaluations of the
metric coefficients along the curve:
\begin{align}
\begin{split}
q(t)&=p(t,\xi(t)),\qquad\;\;\, q_{,k}(t)=\frac{\partial p}{\partial
x^k}(t,\xi(t)),\\
h_{ij}(t)&=g_{ij}(t,\xi(t)),\qquad h^{ij}_{,k}(t)=\frac{\partial 
g^{ij}}{\partial x^k}(t,\xi(t)).\cr
\end{split}
\label{antiball}
\end{align}
Since  $u(t)\in T_{\xi(t)}\Sigma$, operations to raise/lower indices for 
$u$ and
compute the norm are carried out with $h_{ij}(t)$:
\[u_j(t)=h_{jk}(t)u^k(t), \qquad |u|_h^2=h_{jk}u^ju^k.\]
In terms of this  the Lorentz
contraction factor is 
\[\gamma=\gamma(t)=\bfga(q^{-1}|u(t)|_h)\] 
(where $\bfga(s)\equiv (1-s^2)^{-1/2}$ as above).
In terms of $u$ and $\gamma$ the geodesic equation
takes the form
\begin{equation}
\frac{d}{dt}\biggl(\frac{\gamma u_k}{q}\biggr)
+\gamma q_{,k}+\frac{\gamma u_i u_j}{2q}h^{ij}_{,k}=0.
\label{geo}
\end{equation}
In terms of $\xi$ this is a second order nonlinear differential equation.

\subsection*{Statement of theorem on geodesic hypothesis}
The objective is to construct solutions of  \eqref{gen} which 
``look like'' \eqref{bs} centred
at a point $\xi(t)$ at each time $t$. In view of the discussion of the 
geodesic hypothesis
in \S\ref{intro} it is to be anticipated that this will only be 
possible if $t\mapsto\xi(t)$ is
close to being a geodesic, i.e. a solution of \eqref{geo}.
To achieve and make precise this objective I 
will now introduce, given $\epsilon>0$, a function 
$\phi^\epsilon_{0}:\Mink\to\Complex$ which 
(by definition) represents a nontopological soliton centred on the
curve $\xi(t)$. The idea is to ``freeze'' the metric coefficient 
functions at their values
at the centre $\xi(t)$ as in \eqref{antiball} and regard these as 
defining a constant
coefficient metric. But in the constant coefficient case the formulae 
\eqref{e1}--\eqref{e4}
give exact solutions (after making a linear change of variables to put 
the metric in standard
Minkowski form). It is these formulae which motivate the function 
introduced in the
following definition. The fact that this process of ``freeezing the 
coefficients'' has any
validity of course has to be proved. It is only expected to be a good 
idea in the scaling
limit in which the soliton is small so that it only ``sees'' the values 
of the coefficients
in a small neighbourhood of the point $\xi(t)$. It 
is the purpose of Theorem \ref{stheorem} to justify this expectation.

Corresponding to the projection operator in the direction of $u$ which 
appears in the
Lorentz transformation formulae \eqref{e1}--\eqref{e2} it is now 
necessary to introduce
the projection operator
$P_t:\Real^n\to\Real^n$ along $u(t)$ {\it defined with respect to the 
inner product $h(t)$}. The operator
$P_t$ and its orthogonal complement
$Q_t$ are given explicitly by:
\begin{align}
&(P_{t}V)^i=\frac{\langle u,V\rangle_{h}u^i}{|u|_h^2},\\
&(Q_{t}V)^i=V^i-\bigl(P_{t}V\bigr)^i\\
&\phantom{(Q_{t}V)^i}=\frac{|u|_h^2 V^i-\langle 
u,V\rangle_{h}u^i}{|u|_h^2},\notag
\end{align}
where the $t$ dependence of $h_{ij}$ and $u$ is suppressed.
\begin{definition}
\label{def}
Given
$(\omega,{\eta},\xi,u)\in C([0,t_1];\Real^2\times T\Sigma)$ and 
$\epsilon>0$,
define a nontopological soliton on 
$\bigl(\Mink,\epsilon^{-2}\ulg\bigr)$ centred on the
curve $\xi(t)$, with frequency $\omega(t)$ and phase $\eta(t)$, to be the
function $\phi^\epsilon_{0}:\Mink\to\Complex$ given by 
\[
\phi^{\epsilon}_{0} (t,x)=f_\omega\Bigl(\frac{1}{\epsilon}
|\gamma P_{t}(x-\xi)+Q_{t}(x-\xi)|_{h}\Bigr)
\exp\Bigl[
\frac{i}{\epsilon}
\Bigl(\int_0^t\frac{\omega q}{\gamma}dt'+{\eta}-
\frac{\gamma}{q}\langle x-\xi,u\rangle_{h}\Bigr)
\Bigr].
\]
(In this formula the $t$ dependence of
$\omega,\eta,\xi,u,h,q,\gamma$ is supressed. The notation 
$|\,\cdot\,|_h$ 
and $\langle\,\cdot\,\rangle_h$ is used for, respectively, the norm and 
inner product defined by $h_{ij}$ as above.) 
\end{definition}
\begin{remark} As remarked above, the origin of this formula is as a 
change
of variables from \eqref{e1}--\eqref{e4} obtained by freezing the metric 
coefficients.
Another helpful way to understand the formula is
to use the coordinate system
$(\hat t,\hat x)$ adapted to an observer moving along
the curve $t\mapsto\xi(t)$ (see \cite[\S 13.6]{mtw73}):
in this system of coordinates the function $\phi^\epsilon_{0}\sim 
e^{i\omega\hat t}
f_\omega(|\hat x|)$ as explained in \cite{pseud00}.
\end{remark}
Again it is important to emphasize that the function in Definition 
\ref{def} is not an exact
solution of \eqref{gen} in the nonflat case. Nevertheless
the following theorem indicates that there does exist a solution close 
to it
when $\xi(t)$ is close to a geodesic.
\begin{theorem}[\cite{pseud00}]
\label{stheorem}
Let $n=3$ and $\beta(|\phi|)=|\phi|^{p-1}$ with
$2\le p<1+4/n=7/3$, so that in particular the stability interval ${\bf 
I}$ defined
in \eqref{stabint} is nonempty.
Given $\omega(0)\in{\bf I}$ and a time-like geodesic 
\[t\mapsto\xi^{(0)}(t)\in\Sigma,\]
 there exist 
\begin{itemize}
\item[(i)] positive numbers
$c_*>0,\,t_*>0,\,\epsilon_*>0$,
\item[(ii)] a $C^1$ function
$(\omega,{\eta},\xi,u)\in C^1([0,t_*];\Real^2\times T\Sigma)$, and 
\item[(iii)] initial data 
$(\phi^\epsilon(0),\phi^\epsilon_t(0))$, for $0<\epsilon<\epsilon_*$,
\end{itemize} 
such that  for all $0\le t\le t_*$ and $0<\epsilon<\epsilon_*$,
\[
\epsilon^{-2}|\omega(t)-\omega(0)|+\epsilon^{-2}|{\eta}(t)-{\eta}(0)|+
\epsilon^{-1}|\xi(t)-\xi^{(0)}(t)|+\epsilon^{-1}\Big|u(t)-
\frac{d\xi^{(0)}}{dt}\Big|
\le c_*,
\]
while the solution $\phie$ to \eqref{gen}
satisfies, for all \ $0\le t\le t_*$ and \ $0<\epsilon<\epsilon_*$, the 
estimate
\begin{align}
\bigl\|\phi^\epsilon(t,x)-\phi_{0}^\epsilon(t,x)
\bigr\|_{H_\epsilon^1}+
\epsilon\bigl\|\partial_t\bigl(\phi^\epsilon(t,x)-\phi_{0}^%
\epsilon(t,x)\bigr)
\bigr\|_{L^2_\epsilon}
\le c_*\epsilon,\notag
\end{align}
where $H^1_\epsilon, L^2_\epsilon$ are the scaled
norms 
in \eqref{norm}
and $\phi^\epsilon_{0}$ is as above, determined by the curve
$(\omega,{\eta},\xi,u)\in C^1([0,t_*];\Real^2\times T\Sigma),$ whose 
existence
is asserted in (ii).
\end{theorem}
\begin{remark}
\label{previous}
The condition $\omega(0)\in {\bf I}$
is crucial because it determines the
stability of \eqref{bs} as discussed above. The upper limit $p<1+4/n$ 
ensures that
the interval ${\bf I}$ is nonempty, while the lower limit $p\ge 2$ is 
for regularity purposes,
and could possibly be relaxed. More significantly Theorem 
\ref{stheorem} is only a very
special case of a general theorem proved in \cite{pseud00} for a large 
class of nonlinearities
in arbitrary space dimension.
\end{remark}
\begin{remark}
In terms of the standard (unscaled) $L^2$ norm the estimate in Theorem
\ref{stheorem} can be written:
\begin{align}
\epsilon^{-\frac{n}{2}}\Bigl(
\bigl\|\nabla_x(\phi^\epsilon(t,x)-\phi_{0}^\epsilon(t,x))
\bigr\|_{L^2}&+
\bigl\|\partial_t\bigl(\phi^\epsilon(t,x)-\phi_{0}^\epsilon(t,x)\bigr)
\bigr\|_{L^2}\Bigr)\notag\\\qquad&+
\epsilon^{-1-\frac{n}{2}}\bigl\|\bigl(\phi^\epsilon(t,x)-\phi_{0}^%
\epsilon(t,x)\bigr)
\bigr\|_{L^2}
\le c_*.\notag
\end{align}
\end{remark}
%
%

\section{Further generalisations}
In this final section I briefly mention some generalisations of Theorems
\ref{purepower} and \ref{stheorem}.

\subsection*{Modulational stability in general Hamiltonian systems}
The problem discussed in \cite{stab99} is one of a large class of 
stability
problems arising for Hamiltonian systems with symmetry for which a very 
general
theory was developed in \cite{GSS,GSS90}. It would be worthwhile to 
extend the
method in \cite{stab99} to prove general {\it modulational} stability 
results
(in the sense of Theorem \ref{purepower}) for
Hamiltonian systems, in particular Hamiltonian PDE's.
The abstract setting is this: there is a submanifold
${\mathcal S}$ of the phase space foliated by integral curves of a
vector field which are solutions   
of the original Hamiltonian equations.
In the case that ${\mathcal S}$ is a {\it symplectic} submanifold with a 
tubular neighbourhood given by the symplectic normal bundle it is 
possible to make a decomposition
of the solution (at each time) into a soliton, i.e. a point in 
${\mathcal S}$, and a deformation
which lies in the symplectic normal subspace. The stability condition of 
\cite{GSS,GSS90}
is then used 
to ensure the positivity of the Hessian of the augmented Hamiltonian
as a quadratic form {\it on the symplectic normal subspace}.  This can 
then be combined
with the conservation laws to derive
stability theorems which generalise Theorem \ref{purepower} to an 
abstract setting.
General results in this direction
will be presented in \cite{gen00}.  

\subsection*{Einstein's equation and the geodesic hypothesis}
In the work \cite{pseud00} summarised in \S\ref{pseud}
 I have considered the problem of motion of nontopological solitons
in the semilinear wave equation on a manifold $(\Mink,\ulg)$. 
Thus $\ulg$ has so far been a fixed background metric, corresponding to 
the notion
of an external, or applied, gravitational field.
In the general theory of relativity
the metric $\ulg$ becomes a dynamical variable itself and evolves 
according to Einstein's
equation:
\begin{equation}
{R}_{\mu\nu}(\ulg)-\frac{1}{2}{R}(\ulg) 
\ulg_{\mu\nu}=T_{\mu\nu}(\phi,\ulg).
\label{einstein}\end{equation}
Here ${R}_{\mu\nu}(\ulg)$ is the Ricci curvature of the metric $\ulg$ 
and its trace
${R}(\ulg)$ is the scalar curvature. The right hand side $T_{\mu\nu}$ 
is the
stress-energy tensor; its precise form need not be given here. 
Einsten's equation
is to be solved in conjunction with \eqref{gone}: together they form a 
quasilinear
system which is essentially hyperbolic (modulo the usual proviso 
regarding gauge invariance).
In this situation it is to be expected that the geodesic hypothesis is 
still valid as long as
the amplitude of the soliton is small enough that its effect on the 
metric $\ulg$ can be
treated perturbatively. (This assumption is in addition to the 
assumption 
already introduced that the size of the
soliton is small compared to other length scales in the problem.) 
It is possible to arrange for
the soliton to have small amplitude by taking advantage of the freedom
to choose the nonlinear term $\mcF$ in \eqref{gone}. It is then 
possible to
prove an analogue of Theorem \ref{stheorem} for the 
Einstein semilinear wave system
comprising \eqref{gone} and \eqref{einstein}: such a theorem will be 
presented in 
\cite{eins00}.

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\end{document}