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%% A.Komech, H.Spohn
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\begin{document}
\newtheorem{theorem}{Theorem}[section]
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\begin{titlepage}
\begin{center}
{\Large\bf Soliton-Like Asymptotics for a Classical Particle
\medskip\\
Interacting with a Scalar Wave Field }\\
\vspace{2cm}
{\large Alexander Komech}\medskip
\footnote{Supported partly by
French-Russian A.M.Liapunov Center of Moscow State University, and
by research grants of RFBR and of
Volkswagen-Stiftung.}\\
Department of Mechanics and Mathematics of\\
Moscow State University, Moscow 119899, Russia\\
email: komech@sci.lpi.ac.ru\bigskip\\
{\large Herbert Spohn}\medskip\\
Theoretische Physik, Ludwig-Maximilians-Universit\"at\\
Theresienstrasse 37, D-80333 M\"unchen, Germany\\
email: spohn@stat.physik.uni-muenchen.de\bigskip\\
\end{center}
\date{April 11, 1996}
\vspace{4cm}
{\bf Abstract} We consider a scalar wave field translation invariantly
coupled to
a single particle. This Hamiltonian system admits soliton-like solutions,
where the particle and
comoving field travel with constant velocity.
We prove that a solution of
finite energy converges, in suitable local energy seminorms, to some
soliton-like solutions
in
the long time limit $t\to\pm\infty$.
\end{titlepage}
\section{Introduction and Main Results}
Let us consider a mechanical particle
coupled to a scalar wave field in a translation invariant manner.
The equations of motion read, for $x\in\R^3$ and $t\in\R$,
\beqn\la{1}
\ba{lll}
\dot \phi(x,t)=\pi(x,t)~,&\dot\pi(x,t)=\Delta\phi(x,t)-\rho(x-q(t))~,\\
~\\
\dot q(t)= p(t)/(1+p^2(t))^{1/2}~~,&\dot p(t)=
\int\limits d^3x \phi(x,t)\nabla\rho(x-q(t))~.
\ea
\eeqn
Here $\phi(x,t)$ is the real scalar field,
$q(t)$ the position of the particle, and
$\rho$
a form factor
of compact support providing a cut-off in the interaction at
small distances.
All derivatives in (\re{1}) are understood in the sense of
distributions. If we introduce the momentum $p$ as
canonically conjugate to $q$ and the field $\pi(x)$
as canonically conjugate to $\phi(x)$, then (\re{1})
is a Hamiltonian system with
Hamiltonian functional
\be\la{2}
h(\phi,q,\pi,p)=
(1+p^2)^{1/2}+
\fr 12\int\limits d^3x\,(|\pi(x)|^2+|\nabla\phi(x)|^2)+
\int\limits d^3x\,\phi(x)\rho(x-q)~.
\ee
Note that the kinetic energy of the particle is relativistic
and therefore $|\dot q|<1$.
Since the interaction is translation
invariant,
one expects soliton-like solutions to (\re{1}) of the form
\be\la{3}
\phi(x,t)=\phi_v(x-vt-q)~,~~~q(t)=vt+q
\ee
with $v\in V=\{v\in\R^3:~|v|<1\}$. Indeed they are easily determined. For
$v\in V$ there is a unique function $\phi_v$ which makes (\re{3})
a solution to (\re{1}). It is given by
\be\la{3'}
\phi_v(x) =
-\int d^3y(4\pi
|(y-x)_\|+\lambda(y-x)_\bot|)^{-1}\rho(y)~, ~~~\lambda=\sqrt{1-v^2}
\ee
as derived in Appendix 1.
We have set $z=z_\Vert+z_\bot$,
$z_\Vert~ \Vert v$ and $z_\bot~\bot v$
for $z\in\R^3$.
We are interested in the long time asymptotics
of a solution to (\re{1}), which turns out to be governed by the
following basic mechanism:
Whenever
$\ddot q(t)\not= 0$, energy is transferred from the particle
to infinity
via the wave field. Since the energy is bounded from below, this must
mean that
$\ddot q(t)\to 0$ as $t\to\infty$.
Indeed, such a result is proved in \ci{KSK}
using the boundedness of a certain energy dissipation
functional. One would expect then that also
the velocity has a limit,
$\dot q(t)\to v$ for some $v\in V$, and consequently
\be\la{4}
\phi(q(t)+x,t)\to \phi_v(x){\rm ~~~as~~}t\to\infty~.
\ee
Our main progress is to establish these limits.
Before entering into a more precise and technical discussion,
it may be useful to give a general idea of our strategy.
One first notices, that because of translation invariance the total
momentum
\be\la{5}
P(\phi,q,\pi,p)=
p-\int\limits d^3x\,\pi(x)~\nabla\phi(x)
\ee
is conserved. Through the canonical transformation
$(\Phi(x),Q,\Pi(x),P)=(\phi(q+x),q,\pi(q+x),P(\phi,q,\pi,p))$
one obtains the new Hamiltonian
\begin{eqnarray*}
&&H_P(\Phi,\Pi)=h(\phi,q,\pi,p)
\\
&&=\int~d^3x~\Big(\fr 12 |\Pi(x)|^2+\fr 12|\nabla\Phi(x)|^2+
\Phi(x)\rho(x)\Big)
+\Bigg(1+\Big( P+\int~d^3x~\Pi(x)~\nabla\Phi(x)\Big)^2\Bigg)^{1/2}~~.
\nonumber
\end{eqnarray*}
Since $Q$ is cyclic, we may regard $P$ as a parameter and consider
the reduced system
$(\Phi,\Pi)$ only. Let us define
\be\la{6'}
\pi_v(x)=-v\cdot\nabla\phi_v(x)~,
~~~P(v)=p_v+\int~d^3x~v\cdot\nabla\phi_v(x)~\nabla\phi_v(x)~,
~~p_v=v/(1-v^2)^{1/2}~~.
\ee
We will prove that
$(\phi_v,\pi_v)$ is the unique
critical point and global
minimum of $H_{P(v)}$~. Thus initial data
close to $(\phi_v,\pi_v)$ must remain close forever by
conservation of energy, which translates into the orbital stability
of soliton-like solutions. For a general class of nonlinear wave
equations with symmetries such orbital stability of
soliton-like solutions is proved in the well known work
\ci{GSS}. Our argument here follows in essence Bambusi and
Galgani \ci{BG} who discuss the coupled Lorentz-Maxwell equations.
Orbital stability by itself is not enough. It only ensures that
initial data close to a soliton remain so and does not
yield the convergence of $\dot q(t)$, even less the
convergence (\re{4}). Thus we need an additional,
not quite obvious argument which combines
the limit $\ddot q(t)\to 0$ as
$t\to\infty$ with the orbital stability in order
to establish the long time asymptotics.
As one essential input we will use the strong
Huygens principle for the wave equation.
We recall some definitions and assumptions from \ci{KSK}.
For the form factor $\rho$ we assume that
$$
~~~~~~~~~~~~~~~~~\rho\in C_0^\infty(\R^3)~,
~~~~~\rho(x)=0{\rm ~~for~~}|x|\geq R_\rho~,~~~~\rho(x)=\rho_r(|x|)~.
~~~~~~~~~~~~~~~~~~~(C)
$$
We require that all ``modes'' of the wave field
couple to the particle, which is formalized by the Wiener
condition
$$
~~~~~~~~~~~~~~~~~~~~~
\hat\rho(k)= \int\limits d^3x\, e^{ik x}\rho(x)\not=0
\mbox{ ~~~~~~for~~all~~ }k\in\R^3~.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(W)
$$
In \ci{KSK} generic examples of form factors
satisfying both $(C)$ and $(W)$ are constructed.
Next we
introduce the phase space for (\re{1}).
A point
in phase space is referred to as state.
Let $L^2$ be the real
Hilbert space $L^2(\R^3)$
with norm $\br\cdot\br$ and scalar product $(\cdot,\cdot)$~, and
$D^{1,2}$ be the completion of the real space $C_0^\infty(\R^3)$
with norm $\Vert\phi(x)\Vert=\br\nabla\phi(x)\br$~.
Equivalently, using Sobolev's embedding theorem,
$D^{1,2}=\{\phi(x)\in L^6(\R^3):~~|\nabla\phi(x)|\in L^2(\R^3)\}$
(see \ci{Li})~.
Let $\br\phi\br_R$ denote the norm in $L^2(B_R)$ for $R>0$~,
where $B_R=\{x\in\R^3: |x|\le R\}$. Then the seminorms
$\Vert\phi\Vert_R=\br\nabla\phi\br_R+\br\phi\br_R$ are
continuous on $D^{1,2}$~.
We denote by ${\cal E}$ the Hilbert space
$D^{1,2}\oplus\R^3\oplus L^2\oplus\R^3$ with finite norm
$$
{\Vert\,Y\Vert}_{\cal E}=\Vert\phi\Vert + |q| + \br\pi\br +|p|
\quad\mbox{for}\quad Y=(\phi, q, \pi, p)\,.\nonumber
$$
For smooth $\phi(x)$ vanishing at infinity we have
\begin{eqnarray*}
-\fr 1 {8\pi}\int\limits\int\limits~d^3x~d^3y~\fr{\rho(x)\rho(y)}{|x-y|}
=
\fr 12(\rho,\Delta^{-1}\rho)
&\leq&
\fr {1}{2}\br\nabla\phi\br^2+
(\phi(x),\rho(x-q))\nonumber\\
&\leq&
~~\br\nabla\phi\br^2-
\fr 12(\rho,\Delta^{-1}\rho)~.\nonumber
\end{eqnarray*}
Therefore ${\cal E}$ is the space of finite energy states
and
in particular
for the soliton-like solutions
\be\la{7}
y_{v,q}(t)=(\phi_v(x-vt-q),~vt+q,~\pi_v(x-vt-q),~p_v)
\ee
we have $\Vert y_{v,q}(t)\Vert_{\cal E}<\infty$.
Note that $\Vert\phi_v\Vert<\infty$, but
$\br\phi_v\br=\infty$.
\begin{pro}\la{Ex}
For every $y^0=(\phi^0~,~ q^0~, ~\pi^0~,~ p^0)\in {\cal E}$
the Hamiltonian system (\re{1}) has a unique solution $y(t)
= (\phi(t),~ q(t), ~\pi(t),~ p(t))\in C(\R,{\cal E})$
with $y(0) = y^0$~. Energy and total momentum
are conserved.
\end{pro}
We refer to Section 2 where also the precise notion of solution
is explained.
On physical grounds one is tempted to conjecture that
every solution $y(t)$ of finite energy will converge to
some
soliton-like solution as $t\to\infty$. We do not achieve such a
global result in two respects:
The $t=0$ fields are required to
decay at
infinity so to have a
finite energy. But in addition some
smoothness is imposed. More severely, we do not control the asymptotics for the position
in the form $q(t)\sim vt+q$~. We only prove that $\dot q(t)$
has a limit.
\begin{theorem}\la{A}
Let $(C)$ and $(W)$ hold. The
initial state $y^0=(\phi^0~, q^0~, \pi^0~, p^0)\in {\cal E}$
is required to have the following
decay at infinity: For some $R_0>0$ the functions
$\phi^0(x)$,$\pi^0(x)$
are $C^2$,$C^1$-differentiable outside the ball
$B_{R_0}$, respectively, and
\be\la{8}
|\phi^0(x)|+
|x|(|\nabla\phi^0(x)|+|\pi^0(x)|)+
|x|^2(|\nabla\nabla\phi^0(x)|+|\nabla\pi^0(x)|)={\cal O}(|x|^{-\sigma})
{\rm ~~~as~~}|x|\to\infty
\ee
with some $\sigma>1/2$~. Let $y(t)\in C(\R,{\cal E})$ be the solution
to (\re{1}) with $y(0)=y^0$~. Then
there exists a velocity $v\in V$ such that for every $R>0$
\be\la{10}
\lim\limits_{t\to\infty}
\Big(\Vert \phi(q(t)+\cdot~,~t)-\phi_v(\cdot)\Vert_R+
\br \pi(q(t)+\cdot~,~t)-\pi_v(\cdot)\br_R+
|\dot q(t)-v|\Big)=0~.
\ee
\end{theorem}
{\bf Remarks}
(i) Since the Hamiltonian system (\re{1}) is invariant under
time-reversal, our results also hold for $t\to-\,\infty$.\\
(ii)
The assumption $(C)$ can be weakened to finite differentiability and to
some decay of $\rho(x)$ at infinity.
\medskip
In \ci{KSK} we consider the system (\re{1}) with an additional
external potential $V(q)$ which confines the particle.
In this case the system has stationary solutions of the form
$(\phi_{(q^*)}(x), q^*,0,0)$ where $\nabla V(q^*)=0$ and
$\phi_{(q^*)}(x)=\phi_0(x-q^*)$. If the set of critical points for $V(q)$
is discrete, then the solution $y(t)$ converges locally
to some stationary state in the sense that
$\Vert \phi(x,t)-\phi_{(q^*)}(x)\Vert_R+\br \pi(x,t)\br_R$
vanishes and $\dot q(t)\to 0$ as $t\to\infty$.
In Theorem \re{A} we prove
the same kind of convergence
provided we substitute $\dot q(t)\to v$ and consider the fields
close to the particle.
Soliton-like asymptotics of type (\re{4}) are proved
in \ci{MNPZ} for some translation invariant 1D completely integrable
equations and in \ci{FK} for
some class of 1D first and second order nonlinear
translation invariant wave equations.
Soliton-like asymptotics are also proved for small
perturbations of soliton-like solutions
to 2D and 3D nonlinear Schr\"odinger
equations with a potential term with power decay at infinity
\ci{SW} and to 1D nonlinear
translation invariant Schr\"odinger equations \ci{BP}.
%%%%%%%%%%%%%%%%%% Section 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conservation laws and relaxation
of the acceleration}
We consider the Cauchy problem for the Hamiltonian system (\re{1}),
which we write as
\be\la{2.1}
\dot y(t)=F(y(t))~, \quad y(0)=y^0~,
\ee
where $y(t)=(\phi(t), q(t), \pi(t), p(t))$,
$y^0=(\phi^0~, q^0~, \pi^0~, p^0)
\in {\cal E}$ and for $y=(\phi(x),q,\pi(x),p)\in {\cal E}$
we denote
\be\la{2.1'}
F(y)=\Big(
\pi(x),~~p/(1+p^2)^{1/2},~~\Delta\phi(x)-\rho(x-q),~~
\int d^3x \phi(x)\nabla\rho(x-q)
\Big)~.
\ee
All derivatives in (\re{2.1}) and (\re{2.1'})
are understood in the sense of distributions.
To define what we mean by a
solution $y(t)\in C(\R, {\cal E})$,
we introduce first a suitable space
of test functions and of distributions.
\begin{definition}\la{D}
${\cal D}$ denotes the space
$D\oplus\R^3\oplus D\oplus\R^3$~, where
$D=C_0^\infty(\R^3)$ is the space of real test functions.
${\cal D}^*$ denotes the dual space
$D^*\oplus\R^3\oplus D^*\oplus\R^3$~, where
$D^*$ is the space of real distributions on $\R^3$~.
The pairing between ${\cal D}^*$ and ${\cal D}$ is written as
$<\cdot,\cdot>$~.
\end{definition}
Note that $F(y)\in{\cal D}^*$ for $y\in {\cal E}$~.
\begin{definition}\la{Eq}
$y(t)\in C(\R, {\cal E})$ is said to be a solution to
Eq. (\re{2.1}), equivalently to
the system (\re{1}), iff for all $t\in\R$ and for every $w\in{\cal D}$
\be\la{2.1''}
-=\int\limits_0^t ds .
\ee
\end{definition}
The following lemma states existence and some properties of the solution
to the Cauchy problem (\re{2.1}).
\begin{lemma}\label{existence} Let $(C)$ hold.
Then\\
(i) For every $y^0\in {\cal E}$ the Cauchy problem (\re{2.1}) has a unique
solution $y(t)\in C(\R, {\cal E})$~.\\
(ii) For every $t\in\R$ the map $w_t: y_0\mapsto y(t)$
is continuous on
${\cal E}$~.\\
(iii) The energy and total momentum are conserved, i.e. for every $t\in\R$
\be\la{MC}
h(y(t))=h(y^0)~~~and~~P(y(t))=P(y^0)~.
\ee
\end{lemma}
{\bf Proof} We refer to \ci[Lemma 2.1]{KSK}
where all items are proved except for
total
momentum conservation.
For smooth initial data $\phi^0,\pi^0$ of compact support
momentum conservation follows by partial integration.
This conservation extends to all of ${\cal E}$ by (ii) and because smooth
initial data of compact support are dense in $D^{1,2}\oplus L^2$~.
{\hfill$\Box$}
\medskip
We restate the relaxation of the acceleration
\ci{KSK}.
\begin{pro}\la{rel}
Let all assumptions of Theorem \re{A} be fulfilled. Then
\be\la{2.5}
\lim\limits_{t\to\infty}\ddot q(t)= 0.
\ee
\end{pro}
{\bf Proof}
The system (\re{1}) is identical to the system
$(2)$ of \ci{KSK} with $V(q)\equiv 0$. The zero potential
satisfies the condition $(P_w)$ from the Remark at
the end of \ci[Section 4]{KSK}. Thus the convergence follows from
\ci[Lemma 4.1]{KSK}~~.
{\hfill$\Box$}
%%%%%%%%%%%%%%%%%%%%% Section 3 %%%%%%%%%%%%%%%%%%%%%
\section{Canonical transformation and reduced system}
Since the total momentum is conserved, it is natural to use
$P$ as a new coordinate. To maintain the symplectic structure we
have to canonically
complete the coordinate transformation.
\begin{definition}\la{TD}
Let $T:{\cal E}\to{\cal E}$ be defined by
\be\la{3.1}
T:y=(\phi,q,\pi,p)\mapsto Y=( \Phi(x),~Q,~\Pi(x),~P)
=(\phi(q+x),~q,~\pi(q+x),~P(\phi,q,\pi,p))~~,
\ee
where $P(\phi,q,\pi,p)$ is the total momentum (\re{5}).
\end{definition}
{\bf Remarks} i) $T$ is continuous on ${\cal E}$ and Fr\'echet
differentiable at points $y=(\phi,q,\pi,p)$ with sufficiently
smooth $\phi(x),\pi(x)$, but it is not everywhere differentiable.
ii)
In the $T$-coordinates the solitons (\re{7})
are stationary
except for the
coordinate $Q$,
\be\la{3.2}
Ty_v(t)=(\phi_v(x),~vt+q,~\pi_v(x),~P(v))
\ee
with $P(v)$ the total momentum of the soliton as defined in (\re{6'}).
\medskip
Let $H(Y)=h(T^{-1}Y)$ for $Y=(\Phi,Q,\Pi,P)\in{\cal E}$~. Then
\begin{eqnarray*}
&&H(\Phi,Q,\Pi,P)=H_P(\Phi,\Pi)=h(\Phi(x-Q),~Q,~\Pi(x-Q),
~P+\int~d^3x~\Pi(x)~\nabla\Phi(x))
\\
&&=\int~d^3x~\Big(\fr 12 |\Pi(x)|^2+\fr 12|\nabla\Phi(x)|^2+
\Phi(x)\rho(x)\Big)
+\Bigg(1+\Big( P+\int~d^3x~\Pi(x)~\nabla\Phi(x)\Big)^2\Bigg)^{1/2}~~.
\end{eqnarray*}
The functionals $H(Y)$ and $h(y)$ are
Fr\'echet-differentiable on the
phase space ${\cal E}$.
\begin{pro}\la{CTS}
Let $y(t)\in C(\R,{\cal E})$ be a solution to the
system (\re{1}).
Then $Y(t)=Ty(t)\in C(\R,{\cal E})$ is a solution to the
Hamiltonian system
\beqn\la{3.4}
&&\dot\Phi=\fr{\de H}{\de\Pi}~~,~~~
\dot\Pi=-\fr{\de H}{\de\Phi}~~,\nonumber\\
&&\dot Q=\fr{\pa H}{\pa P}~~,~~~
\dot P=-\fr{\pa H}{\pa Q}~~,
\eeqn
understood in the sense of distributions, compare with (\re{2.1''}).
\end{pro}
{\bf Proof} The equations for $\dot \Phi$, $\dot \Pi$
and $\dot Q$ can be checked by direct computation, while the one for $\dot P$
follows from Lemma \re{existence}.
{\hfill$\Box$}
\medskip\\
$Q$ is a cyclic
coordinate.
Hence
the system (\re{3.4})
is equivalent to a reduced Hamiltonian system for
$\Phi$ and $\Pi$ only, which can be written as
\be\la{3.5}
\dot\Phi=\fr{\de H_P}{\de\Pi}~,~~~~~~
\dot\Pi=-\fr{\de H_P}{\de\Phi}~~.
\ee
For every $P\in\R^3$ the functional
$H_P$ is Fr\'echet differentiable on the Hilbert space
${\cal F}=D^{1,2}\oplus L^2$~.
\begin{pro}\la{SCP}
For every $v\in V$ the functional $H_{P(v)}$ has the lower bound
\be\la{3.6}
H_{P(v)}(\Phi,\Pi)-H_{P(v)}(\phi_v,\pi_v)\geq
\fr{1-|v|}2\Big(\Vert\Phi-\phi_v\Vert^2+\br\Pi-\pi_v\br^2\Big)
\ee
on the space ${\cal F}$. Besides $(\phi_v,\pi_v)$, $H_{P(v)}$
has no other critical point in ${\cal F}$.
\end{pro}
{\bf Proof }
Denoting
$\Phi-\phi_v=\phi$ and $\Pi-\pi_v=\pi$ we have
\beqn\la{3.9}
H_{P(v)}(\phi_v+\phi,\pi_v+\pi)-H_{P(v)}(\phi_v,\pi_v)=
\int d^3x(\pi_v(x)\pi(x)+\nabla\phi_v(x)\cdot\nabla\phi(x)+\rho(x)\phi(x))
\nonumber\\
+\fr 12 \int~d^3x~(|\pi(x)|^2+|\nabla\phi(x)|^2 )
+
(1+(p_v+m)^2)^{1/2}-(1+p_v^2)^{1/2}~~,~~~~~~~~~~~~~
\eeqn
where $p_v=P(v)+\int~d^3x~\pi_v(x)~\nabla\phi_v(x)$ and
$$
m =\int~d^3x~(\pi(x)~\nabla\phi_v(x)+\pi_v(x)~\nabla\phi(x)+
\pi(x)~\nabla\phi(x))~.
$$
Soliton-like solutions (\re{3}) satisfy
\be\la{3.9'}
\pi_v(x)=-v \cdot\nabla\phi_v(x)~,~~~~~~~
-\Delta\pi_v(x)+\rho(x)=v \cdot\nabla\phi_v(x).
\ee
Setting $v=(1+ p_v^2)^{-1/2} p_v$ and inserting in the first
integral of (\re{3.9}) we obtain
\begin{eqnarray*}
&&H_{P(v)}(\phi_v+\phi,\pi_v+\pi)-H_{P(v)}(\phi_v,\pi_v)\\
&&=
\fr 12 \int~d^3x~(|\pi(x)|^2+|\nabla\phi(x)|^2 )
+(1+ p_v^2)^{-1/2} \int~d^3x~\pi(x)~p_v\cdot\nabla\phi(x)\\
&&-(1+ p_v^2)^{-1/2} p_v\cdot m+(1+(p_v+m)^2)^{1/2}-(1+p_v^2)^{1/2}~.
\end{eqnarray*}
Since the expression in the third line
is nonnegative, the
lower bound (\re{3.6}) follows by using
$|(1+ p_v^2)^{-1/2} p_v|=|v|$~.
If $(\Phi,\Pi)\in{\cal F}$ is a critical point for $H_{P(v)}$,
then
it satisfies
$$
0=\Pi(x)+(1+\ti p^2)^{-1/2}\ti p \cdot\nabla\Phi(x)~,
~~~0=-\Delta\Phi(x)+\rho(x)-(1+\ti p^2)^{-1/2}\ti p\cdot\nabla\Pi(x)~,
$$
where $\ti p=P(v)+\int~d^3x~\Pi(x)~\nabla\Phi(x)$~. This system coincides
with the system (\re{3.9'}) for soliton-like solutions
provided we set the velocity
$\ti v=(1+\ti p^2)^{-1/2}\ti p$~. Hence
$\Phi=\phi_{\ti v}$~, $\Pi=\pi_{\ti v}$ and
$P(\ti v)=P(v)$~.
Since $P(v)=\kappa(|v|) v$ with $\kappa(|v|)\geq 0$
and $|P(v)|= \kappa(|v|)|v|$
is a monotone increasing function of $|v|\in[0,1[$,
as proved in Appendix 1, we conclude that $v=\ti v$.
{\hfill$\Box$}
\medskip\\
{\bf Remark} Proposition \re{CTS}
is not really needed for the proof of Theorem \re{A}.
However it shows directly that $(\phi_v,\pi_v)$ is
a critical point,
using (\re{3.5}) and (\re{3.2}),
and suggests
an investigation of the stability through a lower bound as in (\re{3.6}).
In Appendix 2
we sketch the derivation of Proposition \re{CTS}
for sufficiently smooth solutions
based only on the invariance of the symplectic structure.
We expect a similar proposition to
hold for other translation
invariant systems similar to (\re{1}).
%%%%%%%%%%%%%%%%% Section 4 %%%%%%%%%%%%%%%%
\section{Orbital stability of solitons}
We follow \ci{BG} and deduce
orbital stability from the conservation of
$H_P$ together with its lower bound (\re{3.6}).
\begin{pro}\la{OSS}
Let us fix some $v\in V$ and $ q\in\R^3$. Let
$y(t)=(\phi(t),q(t),\pi(t),p(t))\in C(\R,{\cal E})$
be a solution to the system (\re{1}) with initial state
$y(0)=y^0=(\phi^0,q^0,\pi^0,p^0)\in{\cal E}$ and denote
\be\la{4.2}
\de=\Vert\phi^0(x)-\phi_v(x-q)\Vert+
\br\pi^0(x)-\pi_v(x-q)\br+|p^0-p_v|~.
\ee
Then for every
$\ve>0$ there exists a $\de(\ve)>0$ such that
\be\la{4.3}
\Vert\phi(q(t)+x,t)-\phi_v(x)\Vert+
\br\pi(q(t)+x,t)-\pi_v(x)\br+|p(t)-p_v|\leq\ve ~~for~~all~~t\in\R
\ee
provided $\de\leq\de(\ve)$.
\end{pro}
{\bf Proof} We denote by $P^0$ the total momentum
of the
considered solution $y(t)$.
There exists
a soliton-like solution (\re{7}) corresponding to some $\ti v\in V$
and having the same total momentum
$P(\ti v)=P^0$~. Then (\re{4.2}) implies
$|P^0-P(v)|=|P(\ti v)-P(v)|
={\cal O}(\de)$~, hence also
$|\ti v-v|={\cal O}(\de)$ and
$$
\Vert\phi^0(x)-\phi_{\ti v}(x-q)\Vert+
\br\pi^0(x)-\pi_{\ti v}(x-q)\br+|p^0-p_{\ti v}|={\cal O}(\de)~.
$$
Therefore denoting $(\Phi^0,Q^0,\Pi^0,P^0)=Ty^0$ we have
\be\la{4.5}
H_{P(\ti v)}(\Phi^0~,~\Pi^0)-H_{P(\ti v)}(\phi_{\ti v}~,~p_{\ti v})=
{\cal O}(\de)~.
\ee
Total momentum and energy conservation (\re{MC}) imply for
$(\Phi(t),Q(t),\Pi(t),P^0)=Ty(t)$
$$
H_{P(\ti v)}(\Phi(t),~\Pi(t))=H(Ty(t))=
H_{P(\ti v)}(\Phi^0~,~\Pi^0){\rm ~~~for~~}t\in\R~.
$$
Hence (\re{4.5}) and (\re{3.6}) with $\ti v$ instead of $v$
imply
\be\la{4.6}
\Vert \Phi(t)-\phi_{\ti v}\Vert+\br \Pi(t)-\pi_{\ti v}\br={\cal O}(\de)
\ee
uniformly in $t\in\R$~. On the other hand, total momentum conservation
implies
$$
p(t)=P(\ti v)+<\Pi(t),\nabla\Phi(t)>{\rm ~~~for~~}t\in\R~.
$$
Therefore (\re{4.6}) leads to
\be\la{4.7}
|p(t)-p_{\ti v}|={\cal O}(\de)
\ee
uniformly in $t\in\R$~.
Finally (\re{4.6}), (\re{4.7}) together imply (\re{4.3}) because
$|\ti v-v|={\cal O}(\de)$~.
{\hfill$\Box$}
%%%%%%%%%%%%%%%%%%% Section 5 %%%%%%%%%%%%%%%%%%%
\section{Soliton-like asymptotics}
We combine orbital stability and relaxation of the acceleration
to prove Theorem \re{A}.
\begin{pro}\la{5.1}
Let the assumptions of Theorem \re{A} be fulfilled.
Then for every $\de>0$ there exist a $t_*=t_*(\de)$ and a solution
$y_*(t)=(\phi_*(x,t),q_*(t),\pi_*(x,t),p_*(t))
\in C([t_*,\infty),{\cal E})$ to the system (\re{1}) such that
i) $y_*(t)$ coincides with $y(t)$ in some future cone,
\beqn
q_*(t)=&q(t)~~&for~~~t\geq t_*~,\la{5.2'}\\
\phi_*(x,t)=&\phi(x,t)~~&for~~~|x-q(t_*)|0$ there exists a $\de>0$
such that
$$
\Vert\phi_*(q_*(t)+x,t)-\phi_v(x)\Vert+
\br\pi_*(q_*(t)+x,t)-\pi_v(x)\br+|\dot q_*(t)-v|
\leq\ve{\rm ~~for~~}t>t_*~.
$$
Therefore, using (\re{5.2'}) and
(\re{5.2}), for every $R>0$
\begin{eqnarray*}
&\Vert\phi (q(t)+x,~t)-\phi_v(x)\Vert_R~+~
\br\pi (q(t)+x,~t)-\pi_v(x)\br_R~+~|\dot q(t)-v|~&\\
=&\Vert\phi_*(q_*(t)+x,t)-\phi_v(x)\Vert_R+~
\br\pi_*(q_*(t)+x,t)-\pi_v(x)\br_R+~|\dot q_*(t)-v|&\leq\ve,
~t>t_*+R.
\end{eqnarray*}
Since $\ve >0$
is arbitrary, we conclude (\re{10}).
{\hfill$\Box$}
\medskip\\
{\bf Proof of Proposition \re{5.1}}
The Kirchhoff formula asserts that
\be\la{5.5}
\phi(x,t)=\phi_r(x,t)+\phi_0(x,t)
{\rm ~~for~~}|x-q^0|0$ there exists
$t_\ve$
such that
\be\la{5.7}
|\ddot q(t)|\leq\ve{\rm ~~~for~~}t\geq t_\ve
{\rm ~~~and~~}t_\ve\to\infty{\rm ~~as~~}\ve\to 0~.
\ee
Let us define
$$
t_{0,\ve}=t_\ve+R_\rho~~,~~~~
t_{1,\ve}=t_{0,\ve}+R_\rho~~,~~~~
t_{2,\ve}=t_{1,\ve}+2R_\rho/(1-q_1)~.
$$
Then we modify $q(t)$ by
\beqn\la{5.8}
q_\ve(t)=
\left\{
\ba{lll}
q(t)&{\rm ~~for~~}&t\geq t_{0,\ve}~,\\
q_\ve+v_\ve(t-t_{0,\ve})&
{\rm ~~for~~}&t\leq t_{0,\ve}~,
\ea
\right.
\eeqn
where $q_\ve=q(t_{0,\ve})$ and $v_\ve=\dot q(t_{0,\ve})$~.
Then $q_\ve(t)\in C^1(\R)$ and
(\re{5.7}) implies that
\be\la{5.9}
|\ddot q_\ve(t)|\leq\ve{\rm ~~~for~~all~~}t\in\R.
\ee
Let us modify the initial values $\phi^0(x)\in D^{1,2}~,~\pi^0(x)\in L^2$
by
cutting off a large ball with the center at the point $q_\ve$~.
\begin{lemma}\la{mid}
For every $\ve>0$ there exist
$\phi^0_\ve(x)\in D^{1,2}~,~\pi^0_\ve(x)\in L^2$
such that
\beqn\la{5.10}
\phi^0_\ve(x)=
\left\{
\ba{l}
\phi^0(x),\\
0~,
\ea
\right.
\pi^0_\ve(x)=
\left\{
\ba{lll}
\pi^0(x)&{\rm ~~for~~}&|x-q_\ve|>t_\ve~,\\
0&{\rm ~~for~~}&|x-q_\ve|t_\ve-1}~d^3x~
(|\nabla\phi^0(x)|^2+|\pi^0(x)|^2)\to 0
{\rm ~~~as~~}\ve\to 0,
$$
because of $t_\ve\to\infty$ and $|q_\ve|={\cal O}(q_1\cdot t_\ve)$
where $00,
\ee
where
\beqn
\phi_{r,\ve}(x,t)&=&
-\int~\fr{d^3y}{4\pi|x-y|}\rho(y-q_\ve(t-|x-y|))~,\la{5.12}\\
\phi_{0,\ve}(x,t)&=&
\fr 1{4\pi t}\int_{S_t(x)}~d^2y~\pi^0_\ve(y)+
\fr\pa{\pa t}
\left(
\fr 1{4\pi t}\int_{S_t(x)}~d^2y~\phi^0_\ve(y)
\right).\la{5.12'}
\eeqn
Then $\phi_\ve(x,t)$ is a solution to the wave equation
\be\la{5.13}
\ddot \phi_\ve(x,t)=\Delta \phi_\ve(x,t)-
\rho(x-q_\ve(t)){\rm ~~for~~}t>0~.
\ee
By (\re{5.12}), (\re{5.6}), and (\re{5.8}) we have
\beqn
&\phi_{r,\ve}(x,t)=
\phi_r(x,t)~~~~~~~~~~~~&{\rm for~~}|x-q_\ve|t-t_\ve~.~~~~
\la{5.14'}
\eeqn
Now, let us fix $T>0$~. Then $|\dot q_\ve(t)-v_\ve|={\cal O}(\ve)$
uniformly in $t_\ve\leq t\leq t_\ve+T$ due to (\re{5.9}).
Therefore (\re{5.12}) and (\re{5.14'}) imply
\be\la{5.15}
\sup\limits_{x\in\R^3,~00}\Big(
\Vert \phi_{0,\ve}(\cdot,t)\Vert+
\br\dot\phi_{0,\ve}(\cdot,t)\br\Big)\to 0
{\rm ~~~as~~}\ve\to 0~.
\ee
Finally, we define
\be\la{5.17'}
y_*(t)=(\phi_\ve(\cdot,t),q(t),
\dot \phi_\ve(\cdot,t), p(t))
{\rm ~~~for~~}t>t_*=t_{2,\ve}~.
\ee
It is easy to check that the time $t_*$ and the function
$y_*(t)$ for $t\geq t_*$
satisfy all requirements of Proposition \re{5.1}
with $v(\de)=v_\ve$ and $q(\de)=q_\ve$
provided
one chooses $\ve>0$ sufficiently small.
Firstly, $y_*(t)\in C([t_*,\infty),{\cal E})$ is a solution
to the system (\re{1})
for $t>t_*$~. Indeed, (\re{5.14}), (\re{5.16})
and (\re{5.5}), (\re{5.11'})
imply for large enough $t_{1,\ve}$
\be\la{5.18}
\phi_\ve(x,t)=\phi(x,t)
{\rm ~~~for~~}|x-q_\ve|t-t_{1,\ve}$ and $t>t_{2,\ve}$ we have $\rho(x-q(t))=0$
and
$q_\ve(t)=q(t)$, hence $y_*(t)$ is a solution to the
system
(\re{1}) also in this region due to (\re{5.13}).
Secondly, (\re{5.2}) and (\re{5.2'}) follow from (\re{5.18}) and (\re{5.8}),
and (\re{5.3}) follows from
(\re{5.17'}) and (\re{5.11'}) due to
(\re{5.15}) and (\re{5.17}).
{\hfill$\Box$}
%%%%%%%%%%%%%%%%%%%% Appendix 1 %%%%%%%%%%%%%%%%%%%%
\section{Appendix 1. Soliton-like solutions}
{\bf 1.} {\it For every $v\in V$ the function $\phi_v$
in the soliton-like solution (\re{3}), (\re{7})
is given by Eq. (\re{3'})}.\\
{\bf Proof} The system (\re{3.9'}) for a soliton-like solution
reads $(v\cdot\nabla)^2\phi_v(x)=\Delta\phi_v(x)+\rho(x)$~, which
through
Fourier transform becomes
\be\la{A1}
\hat\phi_v(k)=\hat\rho(k)/(k^2-(v\cdot k)^2)~.
\ee
{\hfill$\Box$}
\medskip\\
{\bf 2.} {\it For the total momentum $P(v)$
of the soliton-like solution (\re{7}) we have
$P(v)=\kappa(|v|) v$ with $\kappa(|v|)\geq 0$
and $|P(v)|=\kappa(|v|)|v|$
is a monotone increasing function of $|v|\in[0,1[$~.}\\
{\bf Proof} Parseval identity and (\re{A1}) imply
\beqn
P(v)&=&p_v+\int~d^3x~v\cdot\nabla\phi_v(x)~\nabla\phi_v(x)\nonumber\\
&=&\fr v{\sqrt{1-v^2}}+(2\pi)^{-3}\int~d^3k
\fr{(v\cdot k)\hat\rho(k) \ov{k\hat\rho(k)}}
{(k^2-(v\cdot k)^2)^2}~~.\nonumber
\eeqn
Hence $P(v)=\kappa(|v|) v$ with $\kappa(|v|)\geq 0$ and
for $v\not= 0$
$$
|P(v)|
=\fr {|v|}{\sqrt{1-v^2}}+\fr 1{(2\pi)^3|v|}
\int~d^3k
\fr{|(v\cdot k) \hat\rho(k)|^2}
{(k^2-(v\cdot k)^2)^2}~~.
$$
{\hfill$\Box$}
%%%%%%%%%%%%%%%%% Appendix 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Appendix 2. Invariance of symplectic structure}
The canonical equivalence of the Hamiltonian systems
(\re{1}) and (\re{3.4}) can be seen from the Lagrangian viewpoint.
We remain at the formal level. For a complete mathematical
justification we would have to develop some theory
of infinite dimensional Hamiltonian systems which is
beyond the scope of this paper.
By definition we have $H(\Phi,Q,\Pi,P)=h(\phi,q,\pi,p)$
with the arguments related through the canonical transformation $T$.
To each Hamiltonian we associate a Lagrangian through the
Legendre transformation
\begin{eqnarray*}
l(\phi,q,\dot\phi,\dot q)=&
<\pi,\dot\phi>+p\cdot\dot q-h(\phi,q,\pi,p)~~,~~~~~~
&~~~\dot\phi =\fr {\de h}{\de \pi}~~,~~~~\dot q=\fr {\pa h}{\pa p}~~,\\
L(\Phi,Q,\dot\Phi,\dot Q)=&
<\Pi,\dot\Phi>+P\cdot\dot Q-H(\Phi,Q,\Pi,P)~~,
&~~~\dot\Phi =\fr {\de H}{\de \Pi}~~,~~~\dot Q=\fr {\pa H}{\pa P}~~.
\end{eqnarray*}
These Legendre transforms are well defined because the Hamiltonian
functionals are convex in the momenta.
We claim the identity
$L(\Phi,Q,\dot\Phi,\dot Q)=l(\phi,q,\dot\phi,\dot q)$.
Clearly we have to check the invariance of the canonical 1-form,
\be\la{AA2}
<\Pi,\dot\Phi>+P\cdot\dot Q=<\pi,\dot\phi>+p\cdot\dot q~.
\ee
For this purpose we substitute
$$
\ba{clcl}
\Pi(x)&=\pi(q+x)~,&\dot\Phi(x)&=\dot\phi(q+x)+\dot q\cdot\nabla\phi(q+x)~,\\
P&=p-\int~d^3x~\dot\phi\cdot\nabla\phi~,&\dot Q&=\dot q~.
\ea
$$
The left hand side of (\re{AA2})
becomes then
$$
<\pi(q+x),\dot\phi(q+x)+\dot q\cdot\nabla\phi(q+x)>+
(p-<\pi(x),\nabla\phi(x)>)\cdot \dot q=<\pi,\dot\phi>+p\cdot \dot q~.
$$
Since $l=L$, the corresponding action functionals are identical
when transformed by $T$.
The dynamical trajectories are stationary points of the
respective action functionals. Therefore the two
Hamiltonian systems are equivalent.
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\end{document}