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\begin{document}
\begin{titlepage}
\begin{center}
{\large \bf Traveling Waves in Lattice \\[1mm]
Dynamical Systems \\[10mm] }
A.\ A.\ Pankov and K.\ Pf\/l\"uger
\end{center}
\vspace{1cm}
A. Pankov: Mathematisches Institut, Universit{\" a}t Giessen, Arndtstrasse 2,
35392 Giessen. Germany\\On leave from:
Department of Mathematics, Vinnista State Pedagogical University, Ukraine\\
email: pankov@emath.ams.org \\[2mm]
K. Pf\/l\"uger: Institut f\"ur Mathematik I, Freie Universit\"at Berlin,
Arnimalle 3,
14195 Berlin, Germany \\ email: pflueger@math.fuberlin.de
{\bf Keywords:} Lattice systems, traveling waves
{\bf AMS Subject Classification:}
58E05, 34C25
\begin{abstract}
For a class of 1dimensional lattice dynamical systems we prove the existence
of periodic traveling waves with prescribed speed and arbitrary period. Then
we study asymptotic behaviour of such waves for big values of period and
show that they converge, in an appropriate topology, to a solitary
traveling wave.
\end{abstract}
\end{titlepage}
\section{Introduction}
This paper concerns traveling waves for lattices of particles
with nearest neighbour interaction. Corresponding equations of motion are
of the form
\begin{equation}\label{eq11}
\ddot{q}_n=V'(q_{n+1}q_n)V'(q_nq_{n1}),\quad n\in\Z,
\end{equation}
where $q_n$ denotes the coordinate of the $n$\/th particle and $V(r)$ is the
potential of interaction. Equation (\ref{eq11}) can be written as an infinite
dimensional Hamiltonian system with Hamiltonian
$$
H=\sum_{n\in\Z}(\frac{1}{2}p_n^2+V(q_{n+1}q_n)),
$$
where $p_n=\dot{q}_n$ is the momentum.
After the famous work by E.~Fermi, J.~Pasta and S.~Ulam \cite{FPU}, such
systems are of common interest due to their unexpected properties. First
of all, the wellknown Toda lattice defined by
$$
V(r)=ab^{1}(\exp (br)+br1)
$$
is a completely integrable system and its dynamics is understood very well
\cite{To89}. There is also a number of other potentials, like the power
potential, the LennardJones potential, etc., which are of interest for
many applications. However, until the beginning of the 90's dynamical properties
of
equation (\ref{eq11}) with potentials of such kind were studied mainly by
means of numerical simulations (see, e.g., \cite{DW,Ei}).
In the 90's, due to the implementation of variational methods, the situation was
changed and a number of rigorous results was obtained in the case of the
general equation (\ref{eq11}). In \cite{AC,AG1,AG2,AGT,AS,RS} periodic motions
for lattice systems were studied (even for time dependent and spatially
inhomogeneous cases).
In the case of a general superquadratic potential, the first rigorous study of
traveling waves, i.e. solutions of the form $q_n(t)=u(nct)$, was carried
out in \cite{FW94}. Making use of a constrained minimization approach, those
authors obtained an existence result for solitary traveling waves with
prescribed averaged potential energy. Then the wave speed $c$ is
determined in terms of the corresponding Lagrange multiplier.
A more direct approach to the existence problem for solitary waves has been
suggested in a recent paper \cite{SW97}. Here the Mountain Pass Theorem
\cite{W} was applied to prove the existence of solitary waves with prescribed
wave speed $c$ lying in a natural range.
We have also to point out a recent paper \cite{FP} containing, among other
things, a number of qualitative properties of lattice solitary waves, mainly
in the case of near sonic speed in the terminology of that paper
(in our notations $cc_0>0$ is small enough, $c_0^2$ is the coefficient
in the linear term of the righthand side of (\ref{eq11}) ).
In the present paper we study both solitary and time periodic traveling
waves. Our starting point is the following observation. For the Toda
lattice, the explicit expression for $k$periodic waves contains, as the
limit case $k=\infty$, the Toda soliton, i.e. periodic waves give rise to
solitary ones in the large wave length limit \cite{To89}. For general
potentials we first prove the existence of $k$periodic traveling waves, i.e.
waves whose profile function $u(t)$ has $k$periodic derivative.
Of course, for the corresponding
lattice solution $\dot{q}_n(t)=cu'(nct)$ is $k/c$periodic in time,
but only almost
periodic with respect to the spatial variable $n$ in general. Such a solution
is periodic in $n$ only in the case of rational $k$. Then we examine the
behavior of such waves in the large wave length limit, $k\to\infty$, and
prove that (after translations and passage to a subsequence) they converge
to a solitary traveling wave in an appropriate local topology. Remark that
in the case of LennardJones type potentials an existence result for periodic
traveling waves was obtained in \cite{Val}. Under some additional, not so
restrictive, assumptions we study socalled ground waves, i.e. waves with
minimal possible averaged action. Here we use subsequently a constrained
minimization approach known as the Nehari variational principle, or the
Nehari manifold approach. We prove the existence of periodic ground waves
and improve for such waves the above mentioned result on large wave length
limit. Namely, in this case the limit is a solitary ground wave and
convergence takes place on the entire period intervals, depending on $k$, i.e.
in a sense globally. Roughly speaking, this means that a single bump of
a periodic wave has almost the same shape as a solitary one whenever the
wave length is large enough. Let us remark also that here we exploit the same
ideas as in our previous papers and, therefore, the present paper may be
considered as a natural continuation of \cite{PP1,PP2}.
Throughout this paper we consider only increasing waves,
i.e. waves with monotone profile function $u(t)$. However,
similar results hold for decreasing waves as well.
\section{Variational framework}
Solitary waves of equation (\ref{eq11}), i.\ e.\ solutions of the form
$q_n(t) = u(n  ct)$ are solutions of the equation
\beq\label{eq12}
c^2 u''(t) = V'(u(t+1)  u(t))  V'(u(t)  u(t1)) \, .
\eeq
In addition to this infinite lattice, we also consider the corresponding
problem on a periodic lattice consisting of $2k$ particles, denoted by
$k, (k1), \ldots, k$, where $k$ and $k$ are treated as the same point.
This leads to the same equation as (\ref{eq11}) (resp. (\ref{eq12})),
but with an additional periodicity condition.
Equation (\ref{eq12}) is equivalent to the following variational problem:
Let $X$ be the Hilbert space
$$
X = \{ u \in H^1_{loc} \mid u' \in L^2(\R), \; u(0) = 0 \}
$$
with norm $\, \ u \^2 = \int_{\R}  u'(t) ^2 dt $.
On $X$ we define the operator $A$ by
$$
A(u)(t) = u(t+1)  u(t) = \int_{t}^{t+1} u'(s) \, ds \, .
$$
Then a critical point of the functional
$$
\Phi(u) = \int_{\R} [\frac{c^2}{2} u'(t)^2  V(Au(t))] \, dt
$$
on $X$ is a solution of (\ref{eq11}).
The corresponding periodic problem is given by
$$
\Phi_k(u) = \int_{k}^{k} [\frac{c^2}{2} u'(t)^2  V(Au(t))] \, dt,
$$
on the Hilbert space of periodic functions
$$
X_k = \{ u \in H^1_{loc}(\R) \mid u'(t+2k) = u'(t), \; u(0) = 0 \}
$$
with norm $\, \ u \^2_k = \int_{k}^{k}  u'(t) ^2 dt $.
We denote by $\\cdot\_*$ and $\\cdot\_{*,k}$ the norms in dual spaces
$X^*$ and $X^*_k$ respectively.
Note that $\Phi$ and $\Phi_k$ are Hamiltonian actions for $u$.
In the following it is important that increasing waves are fixed
points of the operator $P : X_k \to X_k $, defined
by $P (u)(t) = \int_0^t  u' (s)  ds $.
First we recall the following estimates for $A$ which are proved in \cite{SW97},
Proposition 1, in the case $k=+\infty$ (the case $k<+\infty$ is similar) :
\begin{lemma}\label{t11}
$A\,$ is a bounded operator from $X\,$ to $\,L^2(\R) \cap L^{\infty}(\R) \,$
(resp. from
$\, X_k \,$ to $\, L^2(k, k) \cap L^{\infty}(k, k) $) satisfying
$\, \ Au \_{\infty} \leq \ u \$, $ \ Au \_{L^2} \leq \ u \ \,$ for all
$\,u \in X\,$
(resp. $\ Au \_{\infty} \leq \ u \_k$, $\ Au \_{L^2} \leq \ u \_k \,$
for all
$\,u \in X_k$).
\qed
\end{lemma}
In the sequel, we assume that the potential $V\,$ satisfies the following:
\begin{description}
\item[(i) ] $ V(r) = \frac{c_0^2}{2} r^2 + W(r)$,
where $c > c_0 \geq 0 $, $W \in C^1$, $W(0) = 0$, $W'(r) = o(r)$ as $r \to 0$.
\item[(ii)] $W(r_0) > 0 $ for some $r_0 > 0$, and there exists some $\mu > 2$
such that
$0 \leq \mu W(r) \leq W'(r)r $ for all $r \geq 0$.
\end{description}
Since we only consider monotone waves (i.\ e.\ $Au(t) \geq 0$), we may also
assume that $W(r) = 0 $ if $r \leq 0$, or we may modify $W$ to be symmetric
around 0. We note that Assumption $(ii)$ can be written as a differential
inequality
$$
r^{\mu + 1} \, \frac{d}{dr} (r^{\mu} W(r)) \geq 0, \quad r > 0 \, .
$$
Integration of this inequality shows $W(r) \geq a_0 r^{\mu}$ for $r > r_0$ and
$a_0 = r_0^{\mu}W(r_0)$. Together with Assumption $(i)$ this implies
\beq\label{eq13}
W(r) \geq a_1(r^{\mu}  r^2) \quad \mbox{ for } \; r \geq 0
\eeq
with some constant $a_1 > 0$.
It follows from these assumptions and from Lemma \ref{t11}, that $\Phi$ and
$\Phi_k$ are differentiable; the derivative of $\Phi_k$ is given by
$$
\langle \Phi_k'(u), v \rangle = \int_{k}^{k}
[c^2 u'(t) v'(t)  V'(Au(t))Av(t)] \,dt.
$$
\begin{remark}
Assuming that $W(r_0)>0$ for some $r_0<0$ and $0\leq\mu W(r)\leq W'(r)r$ for
$r\leq 0$, one can treat the case of decreasing waves along the same lines as
it is done here for increasing waves.
\end{remark}
\section{Periodic lattices}
In this section we prove the existence of a sequence of periodic traveling
waves
by showing that $\Phi_k$ satisfies the assumptions of the Mountain Pass
Theorem.
Moreover, we derive several a priori estimates which enable us to pass to the
limit $k \to \infty$ in the next section.
\begin{lemma}\label{t12}
There exist $\delta, \rho > 0$ independent of $k$ such that $ \Phi_k(u)  \geq
\delta $
if $\ u \_k = \rho$. Furthermore, there are functions $e_k \in P(X_k)$
satisfying
$\Phi_k(e_k) = \Phi_1(e_1) \leq 0 \,$ and $\ e_k \_k > \rho $.
\end{lemma}
{\bf Proof:}
Assumption $(i)$ implies that, given $\eps > 0$, there exists $\rho >0$ such
that
$ W(r)  \leq \eps r^2$ if $r \leq \rho$. If $\Au\_{\infty}\leq\rho$,
this and Lemma~\ref{t11} imply
\beqas
\Phi_k(u) & \geq & \int_{k}^k [\frac{c^2}{2} u'^2  \frac{c_0^2}{2}  Au ^2

\eps  Au ^2] dt \\
& \geq & \frac{c^2c_0^22\eps}{2} \, \ u \_k^2 ,
\eeqas
and $\eps, \, \rho$ can be choosen such that the right hand side is strictly
positive
for $\ u \_k = \rho $.
To construct the functions $e_k$, we first choose a function $v \in P(X_1)$
satisfying
$$
v(t) = 0 \; \mbox{ for } \; 0 \leq t \leq 1, \quad v'(1) = 0
$$
($v'$ is nonzero only on intervals $[2l1, 2l], \, l \in \Z$) and $V(Av(t_0)) >
0$
for some $t_0$. Since $Av \geq 0$, we see from (\ref{eq13}) that
\beqas
\Phi_1(\tau v) & \leq & \tau^2 \int_{1}^1 [\frac{c^2}{2}v'^2 +
(a_1\frac{c_0^2}{2})
 Av ^2] dt  \tau^{\mu} a_1 \int_{1}^1  Av ^{\mu} dt \\
& \leq &
\tau^2 \left( \frac{c^2}{2} \v\_1^2 + a_1 \ A v \^2_{L^2} \right)
 \tau^{\mu} a_1 \ Av \_{L^{\mu}}^{\mu} .
\eeqas
This shows $\Phi_1(\tau v) \to \infty $ as $ \tau \to \infty$, so we can
fix $e_1 = \tau_0 v$ satisfying $\Phi_1(e_1) \leq 0 $ and $\e_1 \_k > \rho$.
Then we define $e_k \in X_k$ by
$$
e_k(t) = e_1(t) \quad \mbox{for } \; t \leq 1, \quad e_k' (t) = 0 \quad
\mbox{for } \; 1 \leq t \leq k
$$
and extend $e_k'$ periodically to $\R$ (then $e_k'$ is nonzero only on
intervals $[2kl1, 2kl], l \in \Z$).
It follows that
$\ e_k \_k = \ e_1 \_1 $, and we have
$$
e_k(t+1)  e_k(t) = \left\{ \ba{cl} e_1(t+1)  e_1(t) & , t \in [2, 0] \\
0 & , t \in [k , k] \setminus [2, 0] \ea \right. \, .
$$
Consequently
\beqas
\int_{k}^{k} V (Ae_k(t)) dt
& = &
\int_{2}^{1} V (e_1(t+1)  e_1(t)) dt + \int_{1}^{0} V (e_1(t+1)  e_1(t)) dt
\\
& = &
\int_{0}^{1} V (e_1(t1)  e_1(t2)) dt + \int_{1}^{0} V (e_1(t+1)  e_1(t)) dt
\\
& = &
\int_{1}^{1} V (e_1(t+1)  e_1(t)) dt,
\eeqas
since the difference $e_1(t+1)  e_1(t) $ is 2periodic.
In particular, we have
\beq\label{eq14}
\Phi_k(e_k) = \Phi_1(e_1) \leq 0 \quad \mbox{ and } \quad
\Phi_k(s e_k) = \Phi_1(s e_1) \quad \mbox{for } \; s \in \R.
\eeq
\qed
Now we can prove the following theorem:
\begin{theo}\label{t13}
There exists a sequence $u_k \in P(X_k)$ of nontrivial critical points of
$\Phi_k$ such that the critical values $\alpha_k = \Phi_k(u_k)$ satisfy
\beq\label{eq15}
0 < \delta \leq \alpha_k \leq M
\eeq
with constants $\delta$ and $M\,$ independent of $k$.
\end{theo}
{\bf Proof:}
Since the embedding $X_k \to L^p(k, k)$ is compact, Lemma \ref{t11}
and standard arguments (see Proposition~B.35 and the proof of Theorem~2.15)
in \cite{RA} show
that $\Phi_k$ satisfies the PalaisSmale condition. Lemma \ref{t12} shows that
$\Phi_k$ has the mountain pass geometry and $\Phi_k(P_k(u)) \leq \Phi_k(u)$,
$P_k(0) = 0, \, P_k(e_k) = e_k$. Consequently, the BrezisNirenberg version of
the Mountain Pass Lemma (cf. \cite{BN}, Theorem 10) shows that
\beqas
\alpha_k & = &
\inf_{\gamma \in \Gamma_k} \sup_{s \in [0,1]} \Phi_k(\gamma(s)),
\quad \mbox{with} \\
\Gamma_k & = &
\{ \gamma \in C([0,1], X_k) \mid \gamma(0) = 0, \gamma(1) \leq 0, \,
\ \gamma(1) \_k > \rho \}
\eeqas
are critical values of $\Phi_k$ with corresponding critical points
$u_k \in \overline{P(X_k)}$. Since $P(X_k)$ is closed, $u_k \in P(X_k)$.
Moreover, from (\ref{eq14}) we get
$$
\alpha_k \leq \max_{s \in [0,1]} \Phi_k(s e_k) = \max_{s \in [0,1]} \Phi_1(s
e_1) = M
$$
and $M\,$ is independent of $k$.
\qed
\section{Passage to the limit}
To pass to the limit, we need a cutoff operator $T_k : X_k \to X$.
which is defined as follows:
$$
T_k u (t) = \left\{ \ba{cl} u(t) &,  t  \leq k+1 \\
u(\pm (k+1)) &, t \geq k+1 \ea \right. \; ,
\qquad u \in X_k.
$$
Obviously, we have $(T_k u)' \in L^2(\R) $ and
$$
\ T_k u \^2_X = \int_{\infty}^{\infty}  (T_k u)' ^2 dt =
\int_{k1}^{k+1}  u' ^2 dt \leq 2 \ u \^2_k \, ,
$$
so $T_k $ is a bounded linear operator with norm independent of $k$.
Remark that
\beq
(T_k A u)(t) = (AT_ku)(t) = Au(t) \quad \mbox{ if} \quad t \leq k,
\label{eq16}
\eeq
and for increasing waves we have
\beq
0 \leq (A T_k(u))(t) \leq A(u(t)) \quad \mbox{ for all} \quad t.
\label{eq17}
\eeq
\begin{lemma}\label{t21}
Any nontrivial critical point $u$ of $\, \Phi_k$ satisfies
$0 < \eps_1 \leq \ u \_k \leq C_k$, and $0< \eps_2 \leq \Phi_k(u)$
with $\eps_1, \, \eps_2$ independent of $k$
and $C_k$ only depending on the critical value. \\
The same assertions hold for critical points $u \in X$ of $\, \Phi$.
\end{lemma}
{\bf Proof.}
To show the upper bounds for $\ u \_k$, note that
\beqas
c_k & = & \Phi_k(u)  \frac{1}{2} \langle \Phi_k'(u), u \rangle \\
& = &
\int_{k}^k [\frac{1}{2} W'(Au) Au  W(Au)] dt \\
& \geq &
\frac{\mu  2}{2} \int_{k}^k W(Au) dt ,
\eeqas
by Assumption $(ii)$. Therefore
\beqas
\frac{c^2  c_0^2}{2} \, \ u \_k^2 & \leq & \Phi_k(u) + \int_{k}^k W(Au) dt
\\
& \leq & \frac{\mu}{\mu  2} \, c_k .
\eeqas
For the lower bounds, we assume on the contrary that there exists a
sequence of nontrivial critical points $u_{k_n} \in X_{k_n}$ such that
$\u_{k_n}\_{k_n} \leq 1/n$. Since $\ A u_{k_n} \_{\infty} \leq 1/n$
(by Lemma \ref{t11}), Assumption $(i)$ shows
$W'(A u_{k_n}) A u_{k_n} \leq \eps_n A u_{k_n}^2, \, \eps_n \to 0$.
Hence
\beqas
c^2 \ u_{k_n} \^2_{k_n}
& = &
\int_{k_n}^{k_n} [c_0^2  A u_{k_n}^2 + W'(Au_{k_n})Au_{k_n}] dt \\
& \leq &
(c_0^2 + \eps_n) \, \ u_{k_n} \_{k_n}^2 \, .
\eeqas
Since $c^2 > c_0^2$, this is a contradiction.
Finally we obtain the lower bounds for $\Phi_k(u)$ as follows.
First, from Assumption $(ii)$ we have
$$
\Phi_k(u) \geq \frac{c^2}{2} \ u \_k^2  \frac{c_0^2}{2} \ Au \_{L^2}^2
 \frac{1}{\mu} \int_{k}^k W'(Au)Au \, dt \, .
$$
Then, by Lemma \ref{t11}, we get
\beqas
\Phi_k(u)
& = &
\Phi_k(u)  \frac{1}{\mu} \langle \Phi_k'(u), u \rangle \\
& \geq &
(c^2  c_0^2)(\frac{1}{2}  \frac{1}{\mu}) \, \ u \_k^2 \\
& \geq &
(c^2  c_0^2)(\frac{1}{2}  \frac{1}{\mu}) \, \eps_1 \, .
\eeqas
Obviously, these lower bounds are independent of $k$, since they only
depend on the geometry of the functional. Moreover, all the calculations
above can be repeated for critical points $u \in X$ of $\Phi$.
\qed
{\bf Remark.}
Actually in the proof we have only used the fact that $\langle \Phi_k'(u), u
\rangle = 0$,
so the statements of this Lemma remain true under this weaker assumption.
For $\zeta \in \R, \, r > 0$ we denote by $K_r(\zeta)$ the interval with
center $\zeta$ and length $r$.
\begin{lemma}\label{t22}
Let $u_k \in X_k$ be a sequence such that $(\Phi_k'(u_k),u_k) \to 0$
and
$\ u_k \_k \leq C$. Then either $\ u_k \_k \to 0$, or there exist
$r, \eta > 0$ and a sequence $\zeta_k \in \R$ such that, along a subsequence,
$$
\int_{K_r(\zeta_k)}  A (T_k u_k) ^2 dt \geq \eta.
$$
\end{lemma}
{\bf Proof.}
First note that Lemma \ref{t11} implies that $A (T_k u_k)$ is a bounded
sequence in $H^1(\R)$. Now assume that
$$
\lim_{k\to \infty} \sup_{\zeta \in \R} \int_{K_r(\zeta)} A (T_k u_k)^2 dt = 0.
$$
Then the concentration lemma of Lions (\cite{Li84}, Lemma 1.1) shows that
$\ A (T_k u_k) \_{L^s(\R)} \to 0 $ for all $s > 2$.
Since $\\Phi_k'(u_k)\_k \to 0$, there is a sequence $\eps_k \to 0$ such
that
$$
\int_{k}^k [c^2 (u_k')^2  c_0^2Au_k^2  W'(Au_k)Au_k] \, dt = \eps_k \ u_k
\_k \, .
$$
Again Lemma \ref{t11} shows $\Au_k\_{\infty} \leq C$ and from Assumption
$(i)$ we deduce that for $r \leq C$, $\eps > 0$ there exist $C_{\eps} > 0$,
$\alpha > 2$ such that $W'(r)r \leq \eps r^2 + C_{\eps}r^{\alpha}$.
We get
\beqas
c^2 \ u_k \_k^2
& \leq &
\int_{k}^k [(c_0^2 + \eps) Au_k^2 + Au_k^{\alpha}] dt + \eps_k \ u_k \_k
\\
& = &
(c_0^2+\eps) \Au_k\_{L^2}^2 + \int_{k}^k A (T_k u_k)^{\alpha} dt + \eps_k
\ u_k \_k \\
& \leq &
(c_0^2+\eps) \u_k\_k^2 + \A (T_k u_k) \_{L^{\alpha}(\R)}^{\alpha} +
\eps_k \ u_k \_k \, .
\eeqas
Choosing $\eps$ small enough, this shows $\u_k\_k \to 0 $ as desired.
\qed
In what follows, we frequently consider convergence of some subsequences
of a sequence without explicit change of indices.
\begin{theo}\label{t23}
Let $u_k \in P(X_k)$ be a sequence such that the values
$\Phi_k(u_k)$ is uniformly bounded and $\\Phi'_k(u_k)\_k \to 0$.
Then there exists a sequence $\zeta_k \in \R$ such that a subsequence of
$T_k(u_k(\cdot + \zeta_k)  u_k(\zeta_k))$ converges weakly and locally in $X$
to a nontrivial solution $u \in P(X)$ of (\ref{eq12}). In particular, it is
so if $u_k \in P(X_k)$ is a sequence of periodic traveling waves such
that $\alpha_k=\Phi_k(u_k)$ satisfy the estimate (\ref{eq15}).
\end{theo}
{\bf Proof.}
From (\ref{eq15}) and Lemma \ref{t21} we conclude that $\ u_k
\_k$ is bounded
both from above and below. Hence Lemma \ref{t22} and (\ref{eq17}) imply
that there exist $r$, $\eta$ and $\zeta_k$ such that
$$
\int_{K_r(\zeta_k)}  A u_k ^2 dt \geq \eta
$$
for a subsequence. Now set $\tilde{u}_k(t) = u_k(t + \zeta_k)  u_k(\zeta_k)$.
Then $\tilde{u}_k \in P(X_k)$, $\ \tilde{u}_k \_k = \ u_k \_k$, and since
$\Phi_k$
is invariant under shifts and addition of constants, $\Phi_k(\tilde{u}_k) =
\Phi_k(u_k)$,
and $\Phi_k'( \tilde{u}_k) = 0$. Moreover, $T_k \tilde{u}_k$ converges weakly in
$X$
to some function $u$, and since $AT_k\tilde{u}_k$ is bounded in $H^1(\R)$, we
see
that $V'(AT_k\tilde{u}_k) \to V'(Au) $ strongly in $L^{\infty}_{loc}(\R)$.
Now for any function $\vi \in C^{\infty}(\R)$ with
$\vi(0)= 0, \, \vi' \in C_0^{\infty}(\R)$ (these functions are dense in $X$), we
have
\beqas
\langle \Phi'(u), \vi \rangle
& = &
\lim_{k \to \infty} \int_{\R} [c^2 (T_k \tilde{u}_k)' \vi' 
V'(AT_k \tilde{u}_k) A\vi] \, dt \\
& = &
\lim_{k \to \infty} \int_{k}^k [c^2 \tilde{u}_k' \vi' 
V'(A \tilde{u}_k) A\vi] \, dt \\
& = & 0 .
\eeqas
Hence $u$ is a solution of (\ref{eq12}) and clearly belongs to $P(X)$.
Since $u_k$ is a solution of (\ref{eq12}) with right hand side converging in
$L^{\infty}_{loc}$, we also have $u_k'' \to u''$ in $L^{\infty}_{loc}$,
and $u_k \to u$ locally in $X$.
\qed
\begin{theo}\label{t24}
Let $u_k \in X_k$ be a uniformly bounded sequence satisfying
$\\Phi_k'(u_k)\_k \to 0$ and
$\alpha_k = \Phi_k(u_k) \to \alpha > 0$. Assume that $W$ is in $C^2$.
Then there exist critical points $u^i \in X$ of $\Phi$,
$i = 1, \ldots, \ell$, and $\zeta_k^i \in \R$ such that
$$
\ u_k  \sum_{i=1}^{\ell} u^i(\cdot + \zeta_k^i) \_k \to 0, \quad
\sum_{i=1}^{\ell} \Phi(u^i) = \alpha \, .
$$
\end{theo}
{\bf Proof.}
By Theorem \ref{t23},
there exist $\zeta_k \in \R$ such
that $\tilde{u}_k(t) = u_k(t + \zeta_k)  u_k(\zeta_k)$ satisfies
$T_k \tilde{u}_k \to u$ locally and weakly in $X$, where a solution of
(\ref{eq12}).
Now let $\vi_k \in C^{\infty}(\R)$ be a sequence with the following properties:
$\vi_k(0) = 0, \, \vi_k' \in C_0^{\infty}(\R)$, supp$(\vi'_k) \subset [k+1,
k1]$,
and $\vi_k \to u$ strongly in $X$. Let $\bar{\vi}_k\in X_k$ be the extention
of ${\vi_k}_{[k,k]}$.
Since $ \Phi$ is $C^1$, we get
for an arbitrary $\eta \in X_k$:
\beqas
\langle \Phi_k' (\bar\vi_k), \eta \rangle
& = &
\langle \Phi' (\vi_k), T_k\eta \rangle \\
& \leq &
\eps_k \ T_k \eta \ \; \leq \; 2 \eps_k \ \eta \_k \, ,
\eeqas
with $\eps_k \to 0$, i.\ e.\ $\\Phi'_k (\bar\vi_k)\_{*,k} \to 0$.
Set $v_k = \tilde{u}_k \bar{\vi}_k$. We want to show that
$\\Phi_k' (v_k)\_{*,k} \to 0$. First note that
\beqa\label{aa}
\langle \Phi_k' (v_k), \eta \rangle
& = &
\langle \Phi_k' (\tilde{u}_k)  \Phi_k' (\bar{\vi}_k), \eta \rangle \\
& &
+ \int_{k}^{k} \left[ W'(A\tilde{u}_k)  W'(A\vi_k) 
W'(A\tilde{u}_kA\vi_k) \right] A\eta dt \, .\nonumber
\eeqa
Let $B=[k_0,k_0]$ and $B_k=[k,k]\setminus B$. Since $\vi_k\to u$
in $X$, for any $\eps > 0$ we can fix $k_0>0$ such that
$(A\vi_k)(t)\leq\eps$ whenever $k\geq k_0$ and $t\not\in B$. Hence,
$$
\Big \int_{B_k} W'(A\vi_k) A\eta dt \Big \leq \eps \ \eta \_k \, .
$$
Since $W \in C^2$, we have
$$
\left \int_{B_k} [ W'(A\tilde{u}_kA\vi_k)  W'(A\tilde{u}_k) ] A \eta dt
\right \leq
\int_{B_k} A\vi_k \, 1 + W''(A\tilde{u}_k) \,  A \eta dt \qquad \mbox{ }
$$
\beqas
\mbox{ } \qquad \mbox{ }
& \leq &
\ A\eta \_{L^2} \left( \ A \vi_k \_{L^2(B_k)} +
\ W''(A\tilde{u}_k) \_{L^{\infty}(B_k)} \ A \vi_k \_{L^2(B_k)} \right) \\
& \leq & C \cdot\eps \ \eta \_k \, .
\eeqas
Since $A\tilde{u}_k \to Au$ in $L^{\infty}_{loc}$ we see that
for $B$ fixed above
$$
\left \int_{B} \left[ W'(A\tilde{u}_k)  W'(A\vi_k)  W'(A\tilde{u}_kA\vi_k)
\right] A\eta dt \right \leq C \cdot\eps \ \eta \_k,
$$
provided $k$ is large enough.
Therefore, due to (\ref{aa}), $\\Phi_k' (v_k)\_k \to 0$.
Now consider
\beqa
\Phi_k(\tilde{u}_k)
& = &
\Phi_k(v_k) + \Phi_k(\bar{\vi}_k) + (v_k, \bar{\vi}_k)_k \nonumber \\
& &
\int_{k}^k [V(Av_k  A\vi_k)  V(Av_k)  V(A\vi_k)] \, dt . \label{eq24}
\eeqa
Since $v_k = \tilde{u}_k  \bar{\vi}_k$ tends locally to zero
and $\vi_k\to u$ in $X$,
we see that $(v_k, \bar{\vi}_k)_k \to 0 $ as $k \to \infty$.
The integral in (\ref{eq24}) can be estimated by similar arguments as above
(using $L^{\infty}_{loc}$ estimates), so that we finally get
\beq
\label{eq25}
\alpha = \lim_{k \to \infty} \Phi_k(\tilde{u}_k) = \Phi(u) +
\lim_{k \to \infty} \Phi_k(v_k), \quad\mbox{i. e.} \; \; \Phi_k(v_k)
\to \alpha  \Phi(u) \, .
\eeq
Since $\\Phi_k'(v_k)\_k \to 0$, either $\Phi_k(v_k) \to 0$ and $\ v_k \_k
\to 0$,
or $\Phi_k(v_k) \geq \eps_2 > 0$ and $\ v_k \_k \geq \eps_1 > 0$ (cf. Remark
following Lemma \ref{t21}). In the first case we are done, with $l=1$ and
$u^1=u$. In the second case
we have $0 < \eps_2 \leq \Phi(u) < \alpha$. Now we can repeat the
arguments above, with
$u_k$ replaced by $v_k$ and $\alpha$ replaced by $\alpha  \Phi(u)$,
to get $u^2$. After a finite number of such steps (not greater then
$\alpha/\eps_2$) we conclude.
\qed
\section{Ground Waves}
In this section we present another approach to construct travelling
wave solutions of (\ref{eq11}), which in addition have minimal energy
among all nontrivial solutions. Here we use Nehari's variational principle
and instead of Assumption (ii) we assume
the following:
\begin{description}
\item[(N) ] $W\in C^2$, $W(r_0) > 0$ for some $r_0 > 0$, and there
exists $\theta \in (0,1)$
such that $0 \leq r^{1}W'(r) \leq \theta W''(r)$ holds for all $r > 0$.
\end{description}
\begin{remark}
In the case of decreasing waves one has to assume instead of (N) that
$W(r_0)>0$ for some $r_0<0$ and $0\leq r^{1}W'(r)\leq\theta W"(r)$ for
$r\leq 0$, with some $\theta\in (0,1)$.
\end{remark}
As before, since we only consider increasing waves, we may assume that
$W(r) \equiv 0$ for $r \leq 0$, or that $W$ is symmetric around 0.
We note that Assumption (N) implies (ii) with $ \mu = (1+\theta)/\theta$.
Now let
$$
I_k(u) = \langle \Phi_k' (u), u \rangle = c^2 \, \ u \_k^2  \int_{k}^k
V'(Au)Au \, dt
$$
and define the Nehari manifold by
$$
S_k = \{ u \in P(X_k) \setminus \{ 0 \} \mid I_k(u) = 0 \} \, .
$$
Note that Assumption (N) implies that $W'(r)/r$ increases monotonically to
$+\infty$ on $(0,+\infty)$.
Hence, for any $u \in P(X_k) \setminus \{ 0 \}$ there exists a unique
$\tau > 0$ such that
$\tau u\in S_k$. Indeed, we have obviously
$$
I_k(t u)= t^2(c^2\u\^2_k c^2_0\Au\^2_{L^2}
t^{1}\int_{k}^k W'(tAu)Au \, dt\, .
$$
Due to Lemma~\ref{t11}, $c^2\u\^2_k c^2_0\Au\^2_{L^2} >0$ and from
monotonicity
of $W'(r)/r$ we get the required immediately.
Now we want to solve the periodic minimization problem
\beq\label{eq33}
m_k = \inf \{ \Phi_k(u) \mid u \in S_k \}
\eeq
and the corresponding problem for the infinite lattice
\beq\label{eq34}
m = \inf \{ \Phi(u) \mid u \in S \}
\eeq
(where $S \subset P(X)$ is defined in the same way as $S_k$ above).
Note that on $S_k$ the functional has the form
\beq\label{eq32}
\Phi_k(u) = \int_{k}^k [\frac{1}{2} \, W'(Au) Au  W(Au)] \, dt \, .
\eeq
First we make the following useful observation:
\begin{lemma}\label{t31}
Under Assumptions (i) and (N), $\alpha_k = m_k$.
\end{lemma}
{\bf Proof.}
Since $\alpha_k$ is a critical value with critical point $u_k \in P(X_k)$, $u_k$
belongs
to $S_k$, hence $m_k \leq \alpha_k$.
If $v \in P(X_k), \, v \not\equiv 0$, then as in the proof of Lemma \ref{t12} we
see that
$\Phi_k(\tau v) \leq 0$ for $\tau$ large enough. This shows
$\alpha_k \leq \alpha_k'$,
where $\alpha_k'$ is defined as
$$
\alpha_k' = \inf_{v \in P(X_k) \setminus \{ 0 \}} \sup_{\tau > 0} \, \Phi_k(\tau
v) \, .
$$
Now observe that for $v \in P(X_k), \, v \not\equiv 0$, there exists $\tau > 0$
such that $\Phi_k(\tau v) \geq \delta > 0$ (cf. Lemma \ref{t21}). Since
$\Phi_k(\tau v) \to \infty$ as $\tau \to \infty$, the maximum of $\Phi_k(\tau
v) $
is obtained for some $\tau_0 > 0$ and we have
\beqas
0 \; = \; \frac{d}{d\tau} \, \phi_k(\tau v)_{  \tau_0}
& = &
\int_{k}^k [c^2 \tau_0 v'^2  V'(\tau_0 Av)Av] \, dt \\
& = &
\frac{1}{\tau_0} \, I_k(\tau_0 v) \, ,
\eeqas
hence $\tau_0 v \in S_k$. This shows
$\alpha_k' = \inf_{v \in S_k } \, \Phi_k(v) = m_k$.
\qed
Since Assumption (N) implies (ii), from this Lemma and Theorem \ref{t13}
we conclude the following:
\begin{theo}\label{t32}
Under Assumptions (i) and (N), for every $k$ there exists a nontrivial
minimizer $u_k$ of (\ref{eq33}) which is a solution of (\ref{eq12}).
Moreover, the sequence $m_k$ is uniformly bounded from above and below.
\qed
\end{theo}
Remark that any minimizer of (\ref{eq33}) is a critical point of $\Phi_k$
(this follows from the Lagrange multiplier theorem and Assumption (N)).
Moreover, the existence of a minimizer
can be proved directly by standard variational methods, since $\Phi_k$ has
the necessary compactness properties.
For these ground waves we obtain the following stronger convergence result:
\begin{theo}\label{t33}
Under Assumptions (i) and (N), there exists a sequence $\zeta_k \in \R$
and a ground wave $u \in X$ such that, along a subsequence, the functions
$\tilde{u}_k = u_k( \cdot + \zeta_k)  u_k(\zeta_k)$ satisfy
$$
\lim_{k \to \infty} \ \tilde{u}_k  u \_k = 0 \, .
$$
\end{theo}
{\bf Proof.}
The existence of a nontrivial limit which is a solution of (\ref{eq12}) follows
by the same arguments as in the proof of Theorem \ref{t23}. Note that
Lemmas \ref{t31} and \ref{t21} imply the uniform boundedness of $u_k$.
To prove that the limit is actually a ground state, first note that for any
$v \in S$, $\eps > 0$, there exist $v_k \in S_k$ such that
$\Phi_k(v_k) \leq \Phi(v) + \eps$ if $k$ is sufficiently large. To show this,
take a sequence $\vi_k \in X_{k}, \; \vi_k' \in C_0^{\infty}(k+1, k1)$,
$\vi_k \to v$ in $X$, and let $\tau_k > 0$ be the unique minimal number
such that $\tau_k \vi_k \in S_k$.
>From $\Phi(\vi_k) \to \Phi(v), \, I(\vi_k) \to I(v)$,
we see that $\tau_k \to 1$ and $v_k = \tau_k \vi_k$ satisfies
$\Phi_k(v_k) = \Phi(v_k) \to \Phi(v)$.
Since $m_k \leq \Phi_k(v_k)$, this implies
$\limsup_{k \to \infty} m_k \leq \Phi(v) + \eps $ for all
$v \in S, \, \eps > 0$, hence
$$
\limsup_{k \to \infty} m_k \leq m .
$$
On the other hand consider the function $g(r) = \frac{1}{2} W'(r) r  W(r)$.
Due to Assumption (N), $g(Au) \geq 0$. Since $g(A\tilde{u}_k) \to g(Au)$
locally in $L^{\infty}$, and
$$
m_k = \int_{k}^k g(A\tilde{u}_k) dt \geq \int_{B} g(A\tilde{u}_k) dt
$$
for any bounded interval $B$ and $k$ large enough, we see
$$
\liminf_{k \to \infty} m_k \geq m.
$$
Thus, $m_k \to m = \Phi(u)$ as desired.
The last statement of Theorem \ref{t33} now follows directly from Theorem
\ref{t24}
(set $\alpha=m, \, \alpha_k = m_k$, and $\ell =1$).
\qed
{\bf Acknowledgement}. This work was carried out during the
visits of the first author to the Humboldt University, Berlin,
(September  December 1998) and the Giessen University (November 1999 
July 2000) as a
guest professor under support from the Deutsche Forschungsgemeinschaft
(DFG).
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