# Periodic travelling wave solutions of discrete nonlinear Klein-Gordon
lattices

Dirk Hennig  Nikos I. Karachalios111Corresponding author. E-mail<EMAIL_ADDRESS>Department of Mathematics, University of Thessaly, Lamia GR 35100, Greece

###### Abstract

We prove the existence of periodic travelling wave solutions for general
discrete nonlinear Klein-Gordon systems, considering both cases of hard and
soft on-site potentials. In the case of hard on-site potentials we implement a
fixed point theory approach, combining Schauder’s fixed point theorem and the
contraction mapping principle. This approach enables us to identify a ring in
the energy space for non-trivial solutions to exist, energy (norm) thresholds
for their existence and upper bounds on their velocity. In the case of soft
on-site potentials, the proof of existence of periodic travelling wave
solutions is facilitated by a variational approach based on the Mountain Pass
Theorem. Thresholds on the averaged kinetic energy for these solutions to
exist are also derived.

AMS Subject Classifications: 37K40; 37K60; 34C15; 34A33
Keywords: fixed point theorems; variational methods; mountain pass theorem;
nonlinear lattices; periodic travelling waves

##  I Introduction

Travelling wave solutions (TWSs) in lattice dynamical systems have attracted
considerable interest due to their fundamental importance in numerous physical
contexts such as the energy transfer in biomolecules, the mobility of
dislocations in crystalline materials and the propagation of pulses in optical
systems, to name a few Braun2004 -Satn5 . A variety of fundamental models for
the description of such phenomena has been studied, including Fermi-Pasta-
Ulam-Tsingou, discrete nonlinear Klein-Gordon (DKG) and discrete nonlinear
Schrödinger (DNLS) models Braun2004 -Kim , as well as, spatially discrete
reaction diffusion systems Zinner -Rowlands  which are relevant to pattern
formation resulting from phase transitions. Discrete systems are appropriate
when the scale of decompositions are too small to be effectively described by
continuous approximations.

The existence of solitary travelling wave solutions has been rigorously
treated with various approaches from nonlinear analysis Ambrosetti -Zeidler2 ,
based on direct and minimax variational methods FWattis –PTW1 , reduction to
finite dimensional manifolds and normal form techniques Ioos ,James  and
fixed-point methodsRosen -NJF . For a presentation of several functional-
analytic methods and corresponding background for their implementation in
nonlinear lattices we refer to the monograph Pankov1 .

On the other hand, the problem of the existence of periodic TWSs is much less
explored, in particular for second order in time lattice dynamical systems,
such as DKG lattices. In the present work, by combining a fixed-point approach
and variational methods, we study the existence of periodic TWSs for general
DKG systems with anharmonic on-site potentials in one-dimensional lattices.

In particular, we study general DKG systems described by the following set of
coupled oscillator equations

$\frac{d^{2}q_{n}}{dt^{2}}=\kappa(q_{n+1}-2q_{n}+q_{n-1})-V^{\prime}(q_{n}).$
(1.1)

The prime ′ stands for the derivative with respect to $q_{n}$, the latter
being the coordinate of the oscillator at site $n$ evolving in an anharmonic
on-site potential $V(q_{n})$. Each oscillator interacts with its neighbours to
the left and right and the strength of the interaction is regulated by the
value of the parameter $\kappa$. This system has a Hamiltonian structure
related to the energy

$H=\sum_{n}\left(\frac{1}{2}p_{n}^{2}+V(q_{n})+\frac{\kappa}{2}(q_{n+1}-q_{n})^{2}\right),$
(1.2)

and it is time-reversible with respect to the involution $p\mapsto-p$. In the
case of a finite lattice, the system (1.1) describes the dynamics of an
arbitrary number of $N+1$ oscillators, which are placed equidistantly on the
interval $\mathcal{I}=(-L,L)$ of length $2L$. The quantity $\kappa=1/2h^{2}$
serves as the discretisation parameter, with $h=2L/N$ defining the lattice
spacing. The position of the oscillators is given by the discrete spatial
coordinate

$x_{n}=-L+nh,\quad n=0,1,2,\ldots,N.$ (1.3)

For finite lattices (1.1) we impose periodic boundary conditions

$q_{n}=q_{n+N}.$ (1.4)

We consider TWSs of the form:

$q_{n}(t)=Q\left(n-c\,t\right)=Q(z),$ (1.5)

with a $2L-$periodic function $Q(z)$, $z=n-c\,t$, where
$c\in{\mathbb{R}}\setminus\\{0\\}$ is the velocity. The solutions (1.5)
satisfy the advance-delay equation

$c^{2}{Q}^{\prime\prime}(z)=\kappa(Q(z+1)-2Q(z)+Q(z-1))-V^{\prime}(Q(z)).$
(1.6)

As the system (1.1) possesses the time-reversibility symmetry it suffices to
consider $c>0$.

The presentation of the results is as follows: Section II deals with the
existence of periodic TWSs in finite lattices with hard on-site potentials.
Imposing periodic boundary conditions, the existence of periodic TWSs is
proved by a fixed-point approach, based on Schauder’s fixed point theorem and
contraction mapping principles. The Schauder’s fixed point method was made use
of in our recent works JFPT , DNBRS  in order to establish the existence of
breather solutions. To our knowledge, the extension of this approach to the
problem of the existence of TWs is novel, and as in the case of breathers, can
offer a comparatively concise method (if compared to the aforementioned ones),
for treating the advance-delay equation (1.6) in the suitable functional set-
up of Sobolev spaces of periodic functions. The combination with the
contraction mapping argument yields the identification of a ring in the energy
space where nontrivial solutions exist, together with the derivation of energy
thresholds for their existence, and an upper bound for their velocity. For the
latter, the contraction mapping argument is applied on a map defined by a
suitable auxiliary linear, non-homogeneous problem stemming from (1.6), whose
solvability is guaranteed by the Friedrichs extension theorem. We should
remark that while the current literature on the energy threshold problem
concerns mainly breather solutions for DNLS models ETH1 ,ETH2  (see also Ch1
,Ch2 ,NJF ,ETH3  and references therein), to the best of our knowledge, the
results of the present work are the first on energy thresholds for TWSs.

In section III we consider the case of soft on-site potentials. This time, for
the proof of existence of non-trivial periodic TWSs, we implement a
variational approach based on the Mountain Pass Theorem. Still in this case,
threshold values for the average kinetic energy of TWSs are derived with the
aid of the contraction mapping principle.

In both cases of potentials, the estimates on the norms of the solutions and
the various thresholds are derived under certain conditions exhibiting a
coherent dependence on the lattice parameters, the frequency of the TWs and
their velocity which is proved to be physically relevant (e.g. see the
concluding remarks of section II for the case of the hard potentials).

##  II Hard on-site potentials

In this section we consider hard on-site (uniformly convex) potentials $V(x)$
assuming that the following assumption holds on them:

A: $V:{\mathbb{R}}\rightarrow{\mathbb{R}}$ is non-negative and at least twice
continuously differentiable and is characterised by the following properties:

$\displaystyle V(0)$ $\displaystyle=$ $\displaystyle
V^{\prime}(0)=0,\,\,\,V^{\prime\prime}(0)>0,$ (2.1) $\displaystyle V(x)$
$\displaystyle=$ $\displaystyle V(-x),\;\;\mbox{for all\ $x\in\mathbb{R}$},$
(2.2)

The unique equilibrium at $x=0$ is a global minimum of $V(x)$. Further, we
assume that $V$ satisfies for some positive constants $\overline{m}$ $\alpha$,
$\beta$, and $K$, the conditions

$\displaystyle|V^{\prime\prime}(x)|$ $\displaystyle\leq$
$\displaystyle\overline{m}|x|^{\alpha},\;\;\forall x\in\mathbb{R},$ (2.3)
$\displaystyle|V^{\prime}(x_{1})-V^{\prime}(x_{2})|$ $\displaystyle\leq$
$\displaystyle K(|x_{1}|^{\beta}+|x_{2}|^{\beta})|x_{1}-x_{2}|.$ (2.4)

We consider periodic solutions $Q(z)=Q(z+2L)$ of zero-mean satisfying

$\int_{-L}^{L}Q(z)dz=\int_{-L}^{L}V^{\prime}(Q(z))dz=0,$ (2.5)

which is satisfied for even potentials (2.2) and periodic functions exhibiting
the symmetry

$Q(z-L/2)=-Q(L/2-z).$ (2.6)

We shall further discuss the conditions (2.2), (2.5) and (2.6) of the
assumption A below (see Theorem II.2). Under assumption A, we reformulate the
original problem (1.6) as a fixed point problem for a suitably defined
operator in a Banach space. Motivated by our recent works JFPT ,DNBRS
applying Schauder’s Fixed Point Theorem Zeidler  to prove the existence of
breathers in discrete nonlinear KG lattices, we show that the same Theorem is
applicable, in order to prove the existence of periodic travelling waves.

In particular, we will use the following version of Schauder’s Fixed Point
Theorem (see e.g. in Zeidler ):

######  Theorem II.1.

Let $G$ be a non-empty closed convex subset of the Banach space $X$. Suppose
that $U:\,\,G\rightarrow\,G$ is a continuous and compact map. then $U$ has a
fixed point in $G$.

###  II.1 Existence of periodic TWSs

We start by introducing appropriate function spaces on which our methods will
be employed. We consider first the space

${{\mathcal{X}}}_{0}\,=\,\left\\{\,Q\in
L^{2}_{per}(-L,L)\,|\,\,\,\int_{-L}^{L}Q(s)ds=\int_{-L}^{L}V^{\prime}(Q(s))ds=0,\,\,\,Q(s-L/2)=-Q(L/2-s)\right\\}\,,$
(2.7)

that is, the space of $L^{2}_{per}(-L,L)$, $2L-$periodic, square integrable
functions of zero mean, on the account of conditions (2.2), (2.5) and (2.6).
The space $\mathcal{X}_{0}$ is endowed with the norm,

$||Q||_{{{\mathcal{X}}}_{0}}=\left(\frac{1}{2L}\,\int_{-L}^{L}\,Q(s)^{2}ds\right)^{1/2}=\left(\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}|\hat{{Q}}_{k}|^{2}dk\right)^{1/2}.$
(2.8)

$\hat{Q}_{k}$ determines the Fourier-coefficient in the Fourier series
expansion:

$\displaystyle Q(s)$ $\displaystyle=$
$\displaystyle\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\hat{Q}_{k}\exp\left(i\Omega
ks\right),\,s\in(-L,L),\,\,\,\Omega=\frac{\pi}{L},$ (2.9)
$\displaystyle{\hat{Q}}_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{2L}\int_{-L}^{L}Q(s)\exp(-i\Omega
ks)ds,\,\,\,\hat{Q}_{k}=\overline{\hat{Q}}_{-k},$ (2.10)

of $Q(s)$ and $\overline{x}$ denotes the complex conjugate of $x$. Evidently,
$\mathcal{X}_{0}$ is a closed subspace of $L^{2}_{per}(-L,L)$. We shall also
use the spaces

${{\mathcal{X}}}_{1}\,=\,\left\\{\,Q\in
H^{1}_{per}(-L,L)\,|\,\,\,\int_{-L}^{L}Q(s)ds=\int_{-L}^{L}V^{\prime}(Q(s))ds=0,\,\,\,Q(s-L/2)=-Q(L/2-s)\,\right\\}\,,$
(2.11)

endowed with the scalar product and induced norm

$\displaystyle(U,Q)_{\mathcal{X}_{1}}=\int_{-L}^{L}U^{\prime}(s)Q^{\prime}(s)d{s},\;\;||Q||^{2}_{\mathcal{X}_{1}}=\int_{-L}^{L}Q^{\prime}(s)^{2}d{s}=\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}(k\Omega)^{2}|\hat{{Q}}_{k}|^{2}.$
(2.12)

It can be easily checked that the Poincaré inequality

$\displaystyle\int_{-L}^{L}Q(s)^{2}ds\leq{C}(L)\int_{-L}^{L}Q^{\prime}(s)^{2}ds,$
(2.13)

holds with the optimal constant ${C}(L)={L^{2}}/{\pi^{2}}$ when the Hilbert
space $\mathcal{X}_{1}$ is used. Note that the validity of the Poincaré
inequality (2.13) allows for the usage of the homogeneous norm (2.12). Of use
will also be the space

${{\mathcal{X}}}_{2}\,=\,\left\\{\,Q\in
H^{2}_{per}(-L,L)\,|\,\,\,\int_{-L}^{L}Q(s)ds=\int_{-L}^{L}V^{\prime}(Q(s))ds=0,\,\,\,Q(s-L/2)=-Q(L/2-s)\,\right\\}.$
(2.14)

Due to the Poincaré inequality (2.13), for ${{\mathcal{X}}}_{2}$ we shall use
the homogeneous norm

$\displaystyle||Q||_{\mathcal{X}_{2}}^{2}=\int_{-L}^{L}Q^{\prime\prime}(s)^{2}d{s}=\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}(k\Omega)^{4}|\hat{{Q}}_{k}|^{2}.$
(2.15)

It can be readily seen that $\mathcal{X}_{2}$ is a closed subspace of
$H_{per}^{2}(-L,L)$. Furthermore, $\mathcal{X}_{2}$ is compactly embedded in
$\mathcal{X}_{0}$ ($\mathcal{X}_{2}\Subset\mathcal{X}_{0}$).

In some cases, we will facilitate variational methods. In particular,
solutions of (1.6) will be considered as critical points of the action
functional $S:\,\mathcal{X}_{1}\rightarrow{\mathbb{R}}$ given by

$S(Q)=\int_{-L}^{L}\left[\frac{1}{2}\left(cQ^{\prime}(z)\right)^{2}-V(Q(z))-\frac{1}{2}\kappa\,[Q(z+1)-Q(z)]^{2}\right]dz.$

Related with the action functional is the (total) energy functional $E$ on
$\mathcal{X}_{1}$:

$E(Q)=\int_{-L}^{L}\left[\frac{1}{2}\left(cQ^{\prime}(z)\right)^{2}+V(Q(z))+\frac{\kappa}{2}\,[Q(z+1)-Q(z)]^{2}\right]dz,$
(2.16)

which will be also involved in the derivation of several bounds for the
solutions.

Let us denote by $C_{0,*}$ the constant of the embedding
$\mathcal{X}_{2}\Subset\mathcal{X}_{0}$, and by $C_{2,*}$ the constant of the
embedding $\mathcal{X}_{2}\subset L^{\infty}([-L,L])$. We now proceed to the
statement and proof of the result on the existence of periodic TWSs:

######  Theorem II.2.

Let assumption A hold. If

$\frac{4\kappa}{c^{2}}<\Omega^{2},$ (2.17)

then there exists a periodic TWS $q_{n}(t)=Q\left(n-\,ct\right)\equiv Q(z)$ of
(1.1) with $Q\in H^{2}(-L,L)$ and

$||Q||_{\mathcal{X}_{0}}\leq\left(\frac{c^{2}\Omega^{2}-4\kappa}{{K}C_{3,*}^{\beta}}\right)^{1/\beta}:=\mathscr{R}_{max},$
(2.18)

where the constant $C_{3,*}$ is given by

$C_{3,*}:=\frac{C_{2,*}}{C_{0,*}},$ (2.19)

such that

$Q(z+2L)=Q(z),\,\,\forall z\in{\mathbb{R}}.$ (2.20)

Proof: We seek $2L-$periodic functions $Q\in H^{2}(-L,L)$ satisfying
(2.5-(2.6) and solve Eq. (1.6). For the following discussions it is suitable
to re-write Eq. (1.6) as:

${Q}^{\prime\prime}(z)-\frac{\kappa}{c^{2}}(Q(z+1)-2Q(z)+Q(z-1))=-\frac{1}{c^{2}}V^{\prime}(Q(z)).$
(2.21)

Thus only the right-hand side of (2.21) features terms nonlinear in $Q$.
Ultimately, (2.21) shall be expressed as a fixed point equation in $Q$.

To proceed, we recall first some basic facts for the intersection of Banach
spaces (interB, , Lemma 2.3.1 and Theorem 2.7.1): For two Banach spaces $X$
and $Y$, the intersection $X\cap Y$ is a Banach space endowed with the norm

$\displaystyle||x||_{X\cap Y}=||x||_{X}+||x||_{Y},\;\;\mbox{for all $x\in
X\cap Y$.}$ (2.22)

Clearly, from (2.22), we have that

$\displaystyle||x||_{X}\leq||x||_{X\cap
Y},\;\;\mbox{and}\;\;||x||_{Y}\leq||x||_{X\cap Y},\;\;\mbox{for all $x\in
X\cap Y$.}$ (2.23)

Moreover, in the particular case where $Y$ is continuously embedded in $X$
with an embedding constant $c_{0}$, that is $||x||_{x}\leq c_{0}||x||_{Y}$,
for all $x\in Y$, then, due to (2.22) and (2.23),

$\displaystyle||x||_{X}\leq||x||_{X\cap Y}\leq
c_{0}^{*}||x||_{Y},\;\;\mbox{for all $x\in Y$,}\;\;c_{0}^{*}=1+c_{0}.$ (2.24)

Hence, if $x$ is in a closed ball of $Y$ of radius $r$, then due to (2.24),
$x$ is in the closed ball of $X\cap Y$ of radius $c_{0}^{*}r$, and in the
closed ball of the same radius of $X$. With these preparations we apply the
above for $X=\mathcal{X}_{0}$ and $Y=\mathcal{X}_{2}$, and we define a convex
set of $\mathcal{X}_{0}\cap\mathcal{X}_{2}$, as follows: First, we consider
arbitrary elements $P\in\mathcal{X}_{0}\cap\mathcal{X}_{2}$, such that

$||P||_{\mathcal{X}_{2}}\leq\varrho.$ (2.25)

Since $P\in\mathcal{X}_{2}\cap\mathcal{X}_{0}$ and
$\mathcal{X}_{2}\Subset\mathcal{X}_{0}$ with compact embedding, it holds by
application of (2.24), that

$\displaystyle||P||_{\mathcal{X}_{0}}\leq||P||_{\mathcal{X}_{0}\cap\mathcal{X}_{2}}\leq
C_{0,*}||P||_{\mathcal{X}_{2}}\leq C_{0,*}\varrho,$ (2.26)

Then, we set for simplicity,

$\displaystyle R=C_{0,*}\varrho,$ (2.27)

and we consider the convex set of $\mathcal{X}_{0}\cap\mathcal{X}_{2}$ as

${\mathcal{Y}}_{0}\,=\,\left\\{\,P\in\mathcal{X}_{0}\cap\mathcal{X}_{2}\,\,:\,\,||P||_{\mathcal{X}_{0}}\leq
R\,\right\\}.$ (2.28)

Evidentially, the set $\mathcal{Y}_{0}$ is well defined and non-empty due to
(2.25) and (2.26). As we will prove below, $R$ (and thus, $\varrho$, due to
(2.27)), will be explicitly determined, giving rise to the estimate (2.18) for
the TWs.

We relate the left-hand side of (2.21) to the linear mapping:
$M\,:\,\mathcal{X}_{2}\,\rightarrow\,\mathcal{X}_{0}$:

$M(P)={P}^{\prime\prime}(z)-\frac{\kappa}{c^{2}}(P(z+1)-2P(z)+P(z-1)).$ (2.29)

As a next step, we establish the invertibility of this mapping and derive an
upper bound for the norm of its inverse.

Applying the operator $M$ to the Fourier elements $\exp(i\,\Omega ks)$ in
(2.9), results in

$M\exp(i\,\Omega ks)=\nu_{k}\exp(i\,\Omega ks),$ (2.30)

where

$\nu_{k}=-\Omega^{2}\,k^{2}+4\,\frac{\kappa}{c^{2}}\sin^{2}\left(\frac{\Omega}{2}k\right).$
(2.31)

The hypothesis (2.17) guarantees that $\nu_{k}\neq 0$, for all
$k\in{\mathbb{Z}}\setminus\\{0\\}$. Therefore, the mapping $M$ possesses an
inverse obeying $M^{-1}\exp(i\,\Omega ks)=(1/\nu_{k})\exp(i\,\Omega ks)$. For
the norm of the linear operator
$M^{-1}:\,\,\mathcal{X}_{0}\rightarrow\mathcal{X}_{2}$, one gets, by using the
hypothesis (2.17), the upper bound:

$\displaystyle||M^{-1}||_{\mathcal{X}_{0},\mathcal{X}_{2}}=\sup_{0\neq
Q\in\mathcal{X}_{0}}\frac{||M^{-1}\,Q||_{\mathcal{X}_{2}}}{||Q||_{\mathcal{X}_{0}}}$
(2.32) $\displaystyle=$ $\displaystyle\sup_{0\neq
Q\in\mathcal{X}_{0}}\frac{\left(\sum_{l}^{\prime}(\Omega\,l)^{4}\left|\frac{1}{\nu_{l}}\hat{Q}_{l}\exp(i\,l\Omega
s)\right|^{2}\right)^{1/2}}{||Q||_{\mathcal{X}_{0}}}$ $\displaystyle\leq$
$\displaystyle\sup_{l\in{\mathbb{Z}}\setminus\\{0\\}}\frac{\sqrt{(\Omega\,l)^{4}}}{|\nu_{l}|}\,\cdot\sup_{0\neq
Q\in\mathcal{X}_{0}}\frac{\left(\sum_{l}^{\prime}|\hat{Q}_{l}|^{2}\right)^{1/2}}{||Q||_{\mathcal{X}_{0}}}$
$\displaystyle=$
$\displaystyle\sup_{l\in{\mathbb{Z}}\setminus\\{0\\}}\frac{{(\Omega\,l)^{2}}}{|-\Omega^{2}\,l^{2}+\frac{4\kappa}{c^{2}}\sin^{2}\left(\frac{\Omega}{2}l\right)|}$
$\displaystyle\leq$
$\displaystyle\frac{\Omega^{2}}{\,\Omega^{2}-\frac{4\kappa}{c^{2}}},$

which verifies the boundedness of $M^{-1}$; note that we have used the
notation $\sum_{l}^{\prime}=\sum_{l\in{\mathbb{Z}}\setminus\\{0\\}}$.
Obviously, the same estimate (2.32) verifies the boundedness of $M^{-1}$ as an
operator $M^{-1}:\mathcal{X}_{2}\cap\mathcal{X}_{0}\rightarrow\mathcal{X}_{0}$
On the other hand, we remark for later use, that if we consider only the
lower-order terms of the estimate (2.32), we get that

$\displaystyle||M^{-1}||_{\mathcal{X}_{0},\mathcal{X}_{0}}\leq\frac{c^{2}}{c^{2}\Omega^{2}-4\kappa}.$
(2.33)

Associated with the right-hand side of (2.21) we introduce the nonlinear
operator $N\,:\,\mathcal{Y}_{0}\,\rightarrow\,\mathcal{Y}_{0}$, as

$N(P)=-\frac{1}{c^{2}}V^{\prime}(P).$ (2.34)

From the definition of the operator $N$ in (2.34), the necessity for the
conditions (2.2),(2.5) and (2.6) becomes evident, as they guarantee the
closeness of the operator $N$ in $\mathcal{Y}_{0}$. To verify continuity of
the operator $N$ on $\mathcal{X}_{0}$, for any $P\in\mathcal{Y}_{0}$, (which
implies by the definition of $\mathcal{Y}_{0}$ that
$P\in\mathcal{X}_{0}\cap\mathcal{X}_{2}$), we prove that $N$ is Frechet
differentiable with a locally bounded derivative. Using (2.3) we have

$N^{\prime}(P):\,h\in\mathcal{X}_{0}\,\mapsto
N^{\prime}(P)[h]=-\frac{1}{c^{2}}V^{\prime\prime}(P)h\in\mathcal{X}_{0}.$
(2.35)

Then, since $P\in\mathcal{X}_{2}\cap\mathcal{X}_{0}$ and
$\mathcal{X}_{2}\subset L^{\infty}([-L,L])$ with continuous embedding, we have
that

$\displaystyle||N^{\prime}(P)[h]||_{\mathcal{X}_{0}}$ $\displaystyle=$
$\displaystyle||-\frac{1}{c^{2}}V^{\prime\prime}(P)\,h||_{\mathcal{X}_{0}}$
(2.36) $\displaystyle=$
$\displaystyle\frac{1}{c^{2}}\left(\frac{1}{2L}\int_{-L}^{L}|V^{\prime\prime}(P(z))\,h(z)|^{2}dz\right)^{1/2}\leq\frac{1}{c^{2}}\left(\overline{m}^{2}\max_{-L\leq
z\leq L}|P(z)|^{2\alpha}\frac{1}{2L}\int_{-L}^{L}|h(z)|^{2}dz\right)^{1/2}$
$\displaystyle\leq$ $\displaystyle\frac{\overline{m}}{c^{2}}\,\max_{-L\leq
z\leq L}|P(z)|^{\alpha}\,||h||_{\mathcal{X}_{0}}$ $\displaystyle\leq$
$\displaystyle\frac{\overline{m}}{c^{2}}C_{2,*}^{\alpha}||P||_{\mathcal{X}_{2}}^{\alpha}||h||_{\mathcal{X}_{0}}\leq
A(\overline{m},c^{2},C_{2,*},\varrho)||h||_{\mathcal{X}_{0}},$

for the constant

$A=A(\overline{m},c^{2},C_{2,*},\varrho)=\frac{\overline{m}}{c^{2}}C_{2,*}^{\alpha}\varrho^{\alpha},$

proving the local boundedness of the differential. Then, one concludes that
the Frechet derivative is locally bounded as

$||N^{\prime}(P)||_{{\cal{L}}(\mathcal{X}_{0},\mathcal{X}_{0})}\leq A.$ (2.37)

With the aid of the locally bounded derivative, we can prove the local
Lipschitz continuity of $N$ as follows:

$\displaystyle||N(P)-N(Q)||_{\mathcal{X}_{0}}$ $\displaystyle\leq$
$\displaystyle\sup_{S\in[P,Q]}||N^{\prime}(S)||_{{\cal{L}}(\mathcal{X}_{0},\mathcal{X}_{0})}||P-Q||_{\mathcal{X}_{0}}$
(2.38) $\displaystyle\leq$ $\displaystyle A||P-Q||_{\mathcal{X}_{0}}.$

As the range of $N$ is concerned, by using condition (2.4), we derive that for
any $P\in\mathcal{Y}_{0}$,

$\displaystyle||N(P)||_{\mathcal{X}_{0}}=\frac{1}{c^{2}}||\,-V^{\prime}(P)\,||_{\mathcal{X}_{0}}$
$\displaystyle=$
$\displaystyle\frac{1}{c^{2}}\left(\frac{1}{2L}\int_{-L}^{L}|-V^{\prime}(P(z))|^{2}du\right)^{1/2}\leq\frac{1}{c^{2}}\left(\frac{1}{2L}\int_{-L}^{L}K^{2}|P(z)|^{2(\beta+1)}dz\right)^{1/2}$
(2.39) $\displaystyle\leq$ $\displaystyle\frac{K}{c^{2}}\,\max_{-L\leq z\leq
L}|P(z)|^{\beta}||P||_{\mathcal{X}_{0}}$ $\displaystyle\leq$
$\displaystyle\frac{K}{c^{2}}C_{2,*}^{\beta}||P||_{\mathcal{X}_{2}}^{\beta}||P||_{\mathcal{X}_{0}}$
$\displaystyle\leq$
$\displaystyle\frac{K}{c^{2}}C_{2,*}^{\beta}C_{0,*}||P||_{\mathcal{X}_{2}}^{\beta+1}$
$\displaystyle\leq$
$\displaystyle\frac{K}{c^{2}}C_{2,*}^{\beta}C_{0,*}\varrho^{\beta+1}=\frac{K}{c^{2}}\left(\frac{C_{2,*}}{C_{0,*}}\right)^{\beta}R^{\beta+1}.$

Thus, we proved in (2.39), that for any $P\in\mathcal{Y}_{0}$,

$\displaystyle||N(P)||_{\mathcal{X}_{0}}\leq\frac{KC_{3,*}^{\beta}}{c^{2}}R^{\beta+1},$
(2.40)

with the constant $C_{3,*}=\frac{C_{2,*}}{C_{0,*}}$, as defined in (2.19).

At last, we express the problem (2.21) as a fixed point equation in terms of a
mapping $\mathcal{Y}_{0}\,\rightarrow\,\mathcal{Y}_{0}$:

$Q=M^{-1}\circ N(Q)\equiv\mathscr{L}(Q).$ (2.41)

Using (2.33) and (2.39), we have

$\displaystyle||\mathscr{L}(Q)||_{\mathcal{X}_{0}}$ $\displaystyle=$
$\displaystyle||M^{-1}(\,N(Q))||_{\mathcal{X}_{0}}\leq||M^{-1}||_{\mathcal{X}_{0},\mathcal{X}_{0}}||\,N(Q)||_{\mathcal{X}_{0}}$
(2.42) $\displaystyle\leq$
$\displaystyle\frac{{K}C_{3,*}^{\beta}}{c^{2}\Omega^{2}-4\kappa}R^{\beta+1}\leq
R,$

assuring by assumptions (2.17) and (2.18), that indeed,

$\mathscr{L}(\mathcal{Y}_{0})\subseteq\mathcal{Y}_{0}.$ (2.43)

Furthermore, since it holds that $\mathscr{L}(Q)\in\mathcal{X}_{2}$ for all
$Q\in\mathcal{Y}_{0}\subseteq\mathcal{X}_{0}$, one has
$\mathscr{L}(\mathcal{Y}_{0})\subseteq\mathcal{X}_{2}\cap\mathcal{Y}_{0}$, and
as the embedding of $\mathcal{X}_{2}$ in $\mathcal{X}_{0}$ is compact, the
operator $\mathscr{L}$ is compact. It remains to prove that $\mathscr{L}$ is
continuous on $\mathcal{Y}_{0}$: For arbitrary
$P_{1},P_{2}\in\mathcal{Y}_{0}$, we have

$\displaystyle||\mathscr{L}(P_{1})-\mathscr{L}(P_{2})||_{\mathcal{X}_{0}}$
$\displaystyle=$
$\displaystyle||M^{-1}(N(P_{1}))-M^{-1}(N(P_{2}))||_{\mathcal{X}_{2}}\leq||M^{-1}||_{\mathcal{X}_{0},\mathcal{X}_{0}}\,||N(P_{1})-N(P_{2})||_{\mathcal{X}_{0}}$
(2.44) $\displaystyle\leq$
$\displaystyle\frac{Ac^{2}}{c^{2}\Omega^{2}-4\kappa}||P_{1}-P_{2}||_{\mathcal{X}_{0}}<\epsilon,$

if

$||P_{1}-P_{2}||_{\mathcal{X}_{0}}<\delta=\frac{c^{2}\Omega^{2}-4\kappa}{Ac^{2}}\epsilon,$
(2.45)

for any given $\epsilon>0$, verifying that $\mathscr{L}(Q)$ is continuous on
$\mathcal{Y}_{0}$. Thus, all the assumptions of Schauder’s fixed point theorem
are satisfied, and hence, the fixed point equation $Q=\mathscr{L}(Q)$ has at
least one solution. $\square$

######  Remark II.1.

(Regularity of travelling waves). By the Sobolev embeddings, the obtained
$H^{2}$-travelling wave solutions $Q$ are $C^{1}$. Therefore, due to Eq. (1.6)
it holds that ${Q}^{\prime\prime}\in C^{1}$ and conclusively $Q\in C^{3}$,
that is, they are classical solutions.

###  II.2 Existence of non-trivial TWSs and an energy threshold

In this section, we consider the problem of existence and non-existence of
non-trivial TWSs with frequencies satisfying the strengthened condition

$\displaystyle\Omega^{2}>\frac{4\kappa+KC_{3,*}^{\beta}}{c^{2}},$ (2.46)

if compared to (2.17). The motivation for assuming (2.46), is explained in
Theorem II.3 concluding the section. We start by stating a result on non-
existence of non-trivial TWSs with frequencies satisfying (2.46).

######  Proposition II.1.

Suppose that conditions A and (2.46) hold and that

$||Q||_{\mathcal{X}_{0}}\leq
R<\left(\frac{c^{2}\Omega^{2}-4\kappa}{{K}C_{3,*}^{\beta}}\right)^{1/(1+\beta)}:=\mathscr{R}_{crit}.$
(2.47)

Then the equation $Q=\mathscr{L}(Q)$ has only the trivial solution, that is,
there exist no non-trivial TWSs.

Proof: If (2.47) holds, we get
$||\mathscr{L}(Q)||_{\mathcal{X}_{0}}=||M^{-1}(N(Q))||_{\mathcal{X}_{0}}\leq||M^{-1}||_{\mathcal{X}_{0},\mathcal{X}_{0}}\cdot||N(Q)||_{\mathcal{X}_{0}}<1$.
Thus $\mathscr{L}$ is a contraction, and the Contraction Mapping Theorem
implies that there is a unique function $Q$ that solves the equation
$Q=\mathscr{L}(Q)$. Since $\mathscr{L}(0)=0$, this unique solutions is the
trivial one.

Note that due to (2.1) the potential $V(x)$ can be expressed as
$V(x)=(1/2)x^{2}+W(x)$, with $W(x)\geq 0$ for all $x\in\mathbb{R}$, satisfying
(2.3) and (2.4) (see also (DNBRS, , Eq. (2.7) and Eq. (2.8))). Furthermore,
let us observe that as the (conserved) energy functional (2.16) is coercive,
it can be used to bound the $L^{2}(-L,L)$-norm of the TWSs as follows

$\displaystyle E(Q)$ $\displaystyle=$
$\displaystyle\int_{-L}^{L}\left[\frac{1}{2}\left(Q^{\prime}(z)\right)^{2}+\frac{1}{2}Q^{2}(z)+W(Q(z))+\frac{\kappa}{2}\,[Q(z+1)-Q(z)]^{2}\right]dz$
(2.48) $\displaystyle=$
$\displaystyle\int_{-L}^{L}\left[\frac{1}{2}\left(Q^{\prime}(z)\right)^{2}+W(Q(z))+\frac{\kappa}{2}\,[Q(z+1)-Q(z)]^{2}\right]dz+\frac{1}{2}||Q||_{L^{2}(-L,L)}^{2},$

so that

$||Q||_{\mathcal{X}_{0}}\leq\sqrt{2E}.$ (2.49)

In conclusion, if the energy of the lattice system is less than
$E_{crit}(c,\kappa,K,\Omega)=\sqrt{2R_{crit}}$, no TWSs of given values for
$c,\kappa,K,\Omega$ exist. $\square$

Combining the existence Theorem II.2 and Proposition II.1 we identify a ring
in the phase space $\mathcal{X}_{0}$ with quantified radii in which the non-
trivial travelling wave solutions with frequencies satisfying the enhanced
condition (2.46), exist. This is illustrated in the cartoon of Fig. 1.

######  Theorem II.3.

Let the assumption A and the condition (2.46) hold. The system (1.1) possesses
non-trivial TWSs only if

$\mathscr{R}_{crit}=\left(\frac{c^{2}\Omega^{2}-4\kappa}{{K}C_{3,*}^{\beta}}\right)^{1/(1+\beta)}\leq||Q||_{\mathcal{X}_{0}}\leq\mathscr{R}_{max}=\left(\frac{c^{2}\Omega^{2}-4\kappa}{{K}C_{3,*}^{\beta}}\right)^{1/\beta}.$
(2.50)

Proof: On the one hand, Theorem II.2 establishes the existence of solutions
when $||Q||_{\mathcal{X}_{0}}\leq\mathscr{R}_{max}$. On the other hand,
according to Proposition II.1, if $||Q||_{\mathcal{X}_{0}}<\mathscr{R}_{crit}$
only the trivial solution exists. Thus, a non-trivial solution exists only if

$\displaystyle\mathscr{R}_{crit}<||Q||_{\mathcal{X}_{0}}\leq\mathscr{R}_{max},$

that is, when (2.50) is satisfied. For the latter to be valid, we require
$\mathscr{R}_{crit}<\mathscr{R}_{max}$, motivating the condition (2.46) on the
frequencies of the TWSs. $\square$

---
Figure 1: Illustration of the statement of Theorem II.3: Non-trivial
travelling wave solutions exist in the ring
$\mathbf{E}=\overline{\mathcal{B}(0,\mathscr{R}_{max})}\setminus\overline{\mathcal{B}(0,\mathscr{R}_{crit})}$
of the space $\mathcal{X}_{0}$ [light (green) coloured area ($\mathbf{E}$)].
The radius $\mathscr{R}_{crit}$ of the closed ball
$\overline{\mathcal{B}(0,\mathscr{R}_{crit})}$ follows from the contraction
mapping Theorem (see Proposition II.1) and defines a threshold for non-
existence of non-trivial solutions [in the darker (red) coloured area
($\mathbf{N}-\mathbf{E})$]. The radius $\mathscr{R}_{max}$ of the closed ball
$\overline{\mathcal{B}(0,\mathscr{R}_{max})}$ results from the Schauder’s
fixed point theorem (see Theorem II.1).

###  II.3 Upper bound for the velocity

By using a fixed point approach, we derive upper bounds for the velocity $c$
of TWSs. For this purpose, we rewrite equation (1.6) in the following operator
form:

$\displaystyle-Q^{\prime\prime}(z)={\frac{1}{c^{2}}\left\\{\kappa\left([Q(z)-Q(z-1)]-[Q(z+1)-Q(z)]\right)+V^{\prime}(Q(z))\right\\}}.$
(2.51)

We recall some basic auxiliary results, starting with the Friedrichs extension
Theorem.

######  Theorem II.4.

Let $\mathcal{L}_{0}:D(\mathcal{L}_{0})\subseteq{X}_{0}\rightarrow{X}_{0}$ be
a linear symmetric operator on the Hilbert space ${X}_{0}$ with its domain
$D(\mathcal{L}_{0})$ being dense in ${X}_{0}$. Assume that there exists a
constant $c>0$ such that

$\displaystyle(\mathcal{L}_{0}v,v)_{{X}_{0}}\geq
c||v||^{2}_{{X}_{0}}\;\;\mbox{for all}\;\;v\in D(\mathcal{L}_{0}).$

Then $\mathcal{L}_{0}$ has a self-adjoint extension
$\mathcal{L}:D(\mathcal{L})\subseteq{X}_{1}\subseteq{X}_{0}\rightarrow{X}_{0}$where
${X}_{1}$ denotes the energetic Hilbert space endowed with the energetic
scalar product $(v,w)_{{X}_{1}}=(\mathcal{L}v,w)_{{X}_{0}}$ for all
$u,v\in{X}_{1}$ and the energetic norm
$||v||_{{X}_{1}}^{2}=(\mathcal{L}v,v)_{{X}_{0}}$. Furthermore, the operator
equation

$\displaystyle\mathcal{L}v=f,f\in{X}_{0},$

has a unique solution $v\in D(\mathcal{L})$. In addition, if
$\hat{\mathcal{L}}:{X}_{1}\rightarrow{X}_{1}^{*}$ denotes the energetic
extension of $\mathcal{L}$, then $\hat{\mathcal{L}}$ is the canonical
isomorphism from ${X}_{1}$ to its dual ${X}_{1}^{*}$ and the operator equation

$\displaystyle\hat{\mathcal{L}}v=f,f\in{X}_{1}^{*},$

has also a unique solution $v\in{X}_{1}$.

With Theorem II.4 in hand, we discuss the left-hand side of Eq. (2.51). It is
well known that Theorem II.4 is applicable to the operator
$\mathcal{L}_{0}:D(\mathcal{L}_{0})\subseteq L^{2}(-L,L)\rightarrow
L^{2}(-L,L)$, $\mathcal{L}_{0}Q=-Q^{\prime\prime}(z)$, with domain of
definition, $D(\mathcal{L}_{0})$, the space of $C^{\infty}$-functions on
$(-L,L)$. Since $D(\mathcal{L}_{0})$ is dense in $\mathcal{X}_{0}$, and
inequality (2.13) holds, the Friedrichs extension of $\mathcal{L}_{0}$ is the
operator $\mathcal{L}:D(\mathcal{L})\rightarrow\mathcal{X}_{0}$ where

$\displaystyle D(\mathcal{L})=\left\\{v\in\mathcal{X}_{1}\;:\;\mathcal{L}v\in
L^{2}(-L,L)\right\\}.$

Consequently, the equation

$\displaystyle-Q^{\prime\prime}(z)=f,\;\;\mbox{for every}\;\;f\in
L^{2}(-L,L),$ (2.52)

has a unique solution in $D(\mathcal{L})$. Thus, we shall consider the right-
hand side of Eq. (2.51) as a suitable mapping on $L^{2}(-L,L)$. For its linear
part, we have the following lemma proved in (Smets, , Proposition 1, pg. 268).

######  Lemma II.1.

The linear operators

$A_{1}[Q(z)]=Q(z+1)-Q(z)=\int_{z}^{z+1}Q^{\prime}(s)ds,\;\;A_{2}[Q(z)]=Q(z)-Q(z-1)=\int_{z-1}^{z}Q^{\prime}(s)ds,$

are continuous from $\mathcal{X}_{1}$ to $L^{2}(-L,L)\cap L^{\infty}(-L,L)$
and $||A_{i}Q||_{L^{\infty}}\leq||Q||_{{\mathcal{X}_{1}}}$,
$||A_{i}Q||_{{\mathcal{X}_{0}}}\leq||Q||_{{\mathcal{X}_{1}}}$, $i=1,2$.

Using Theorem II.4, we treat the following auxiliary linear, non-homogeneous
problem

$\displaystyle-Q^{\prime\prime}(z)=\frac{1}{c^{2}}\left\\{\kappa\left([\Psi(z)-\Psi(z-1)]-[\Psi(z+1)-\Psi(z)]\right)+V^{\prime}(\Psi(z))\right\\}.$
(2.53)

for some arbitrary fixed $\Psi\in\mathcal{X}_{1}$, as an equation of the form
(2.52). In particular, we have the following result:

######  Proposition II.2.

For any $\Psi\in\mathcal{X}_{1}$, the equation (2.53) has a unique solution
$Q\in D(\mathcal{L})\subset\mathcal{X}_{1}$.

Proof: Equation (2.53) can be rewritten in the form

$\displaystyle-Q^{\prime\prime}(z)=\frac{1}{c^{2}}\left\\{\kappa{\left([A_{2}[\Psi(z)]-A_{1}[\Psi(z)]\right)}+V^{\prime}(\Psi(z))\right\\}:=\mathcal{F}[\Psi(z)].$
(2.54)

Due to the continuous embedding $\mathcal{X}_{1}\subset L^{r}(-L,L)$ for any
$1\leq r\leq\infty$ and condition (2.4), we have that for some constant
$C=C(K,\beta)>0$,

$\displaystyle||V^{\prime}(\Psi)||^{2}_{\mathcal{X}_{0}}=\int_{-L}^{L}(V^{\prime}(\Psi(z)))^{2}dz\leq
K^{2}\int_{-L}^{L}|\Psi(z)|^{2(\beta+1)}dz\leq
C||\Psi||_{\mathcal{X}_{1}}^{2(\beta+1)}.$

Then, for the right-hand side of Eq. (2.54) we get

$\displaystyle||\mathcal{F}[\Psi]||_{\mathcal{X}_{0}}$ $\displaystyle\leq$
$\displaystyle\frac{1}{c^{2}}\left\\{\kappa||A_{2}[\Psi]-A_{1}[\Psi]||_{\mathcal{X}_{0}}+||V^{\prime}(\Psi(z))||_{{\mathcal{X}_{0}}}\right\\}$
(2.55) $\displaystyle\leq$
$\displaystyle\frac{1}{c^{2}}\left\\{2\kappa||\Psi||_{\mathcal{X}_{1}}+C||\Psi||_{\mathcal{X}_{1}}^{2(\beta+1)}\right\\}.$

Thus, ${\mathcal{F}[\Psi]\in L^{2}(-L,L)}$, and by virtue of Theorem II.4 Eq.
(2.54) has a unique solution $Q\in D(\mathcal{L})$. $\square$

We now proceed with the implementation of the fixed point argument. To this
aim we consider for some $R>0$ the closed ball of $\mathcal{X}_{1}$,
$\mathcal{B}_{R}:=\left\\{\psi\in\mathcal{X}_{1}\,:\,||\psi||_{{\mathcal{X}_{1}}}\leq
R\right\\}$. Proposition II.2 shows that the map
$\mathcal{T}:\mathcal{X}_{1}\rightarrow\mathcal{X}_{1}$ defined as

$\displaystyle\mathcal{T}[\Psi]=Q,$

where $Q$ is the unique solution of the auxiliary problem (2.54), is well
defined. Hence we introduce $\Psi_{1},\Psi_{2}\in\mathcal{B}_{R}$ such that
$Q=\mathcal{T}[\Psi_{1}]\;\;\mbox{and}\;\;P=\mathcal{T}[\Psi_{2}].$ Then the
difference $Y=Q-P$ satisfies the equation

$\displaystyle-Y^{\prime\prime}(z)$ $\displaystyle=$
$\displaystyle\mathcal{F}[\Psi_{1}(z)]-\mathcal{F}[\Psi_{2}(z)]$ (2.56)
$\displaystyle=$
$\displaystyle\frac{1}{c^{2}}\left\\{\kappa\left(A_{1}[\Psi_{2}(z)]-A_{1}[\Psi_{1}(z)]+A_{2}[\Psi_{1}(z)]-A_{2}[\Psi_{2}(z)]\right)\right\\}$
$\displaystyle+$
$\displaystyle\left.V^{\prime}(\Psi_{1}(z))-V^{\prime}(\Psi_{2}(z))\right\\}.$

From Lemma II.1 the linear operators $A_{i}:\mathcal{X}_{1}\rightarrow
L^{2}(-L,L)\cap L^{\infty}(-L,L)$ are globally Lipschitz,

$\displaystyle||A_{i}\psi_{1}-A_{i}\psi_{2}||_{\mathcal{X}_{0}}\leq||\psi_{1}-\psi_{2}||_{\mathcal{X}_{1}},$
(2.57)
$\displaystyle||A_{i}\psi_{1}-A_{i}\psi_{2}||_{L^{\infty}}\leq||\psi_{1}-\psi_{2}||_{\mathcal{X}_{1}},\,\,\,{{{i=1,2.}}}$

To estimate the difference of the remaining terms in Eq. (2.56), we use (2.4)
and the embedding $\mathcal{X}_{1}\subset L^{\infty}(-L,L)$ with its embedding
constant $C_{*}$,

$\displaystyle\int_{-L}^{L}|V^{\prime}(\Psi_{1}(z))-V^{\prime}(\Psi_{2}(z))|^{2}dz$
$\displaystyle\leq$ $\displaystyle
K^{2}\int_{-L}^{L}(|\Psi_{1}(z)|^{\beta}+|\Psi_{2}(z)|^{\beta})^{2}|\Psi_{1}(z)-\Psi_{2}(z)|^{2}dz$
(2.58) $\displaystyle\leq$ $\displaystyle K^{2}\max_{-L\leq z\leq
L}(|\Psi_{1}(z)|^{\beta}+|\Psi_{2}(z)|^{\beta})\int_{-L}^{L}|\Psi_{1}(z)-\Psi_{2}(z)|^{2}dz$
$\displaystyle=$ $\displaystyle
2K^{2}C_{*}^{2\beta}R^{2\beta}||\Psi_{1}(z)-\Psi_{2}(z)||_{\mathcal{X}_{0}}^{2}.$

Hence, for the right-hand side of (2.56) we get

$\displaystyle||\mathcal{F}[\Psi_{1}(z)]-\mathcal{F}[\Psi_{2}(z)]||_{\mathcal{X}_{0}}\leq
M\||\Psi_{1}-\Psi_{2}||_{\mathcal{X}_{1}},$ (2.59)

where the constant $M$ is given by

$\displaystyle M=\frac{2}{c^{2}}\left(\kappa+KC_{*}R^{\beta}\right).$ (2.60)

Next, by multiplying (2.56) in the $L^{2}(-L,L)$-scalar product and using the
Cauchy-Schwarz inequality and Young’s inequality, we get the estimate

$\displaystyle||Y||_{\mathcal{X}_{1}}^{2}$ $\displaystyle\leq$
$\displaystyle||\mathcal{F}[\Psi_{1}(z)]-\mathcal{F}[\Psi_{2}(z)]||_{\mathcal{X}_{0}}\,||Y||_{\mathcal{X}_{0}}$
(2.61) $\displaystyle\leq$ $\displaystyle
M\sqrt{C(L)}||\Psi_{1}-\Psi_{2}||_{\mathcal{X}_{1}}\,||Y||_{\mathcal{X}_{1}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}||Y||_{\mathcal{X}_{1}}^{2}+\frac{M^{2}C(L)}{2}||\Psi_{1}-\Psi_{2}||_{\mathcal{X}_{1}}^{2}.$

Note that the Poincaré inequality (2.13) has been used. From (2.61) we derive

$\displaystyle||Y||_{\mathcal{X}_{1}}^{2}=||\mathcal{T}[\Psi_{1}]-\mathcal{T}[\Psi_{2}]||_{\mathcal{X}_{1}}^{2}\leq
M^{2}C(L)||\Psi_{1}-\Psi_{2}||_{\mathcal{X}_{1}}^{2}.$ (2.62)

We conclude that if the Lipschitz constant satisfies

$\displaystyle M\sqrt{C(L)}<1,$ (2.63)

then the map $\mathcal{T}:\mathcal{B}_{R}\rightarrow\mathcal{B}_{R}$ is a
contraction. Hence the map $\mathcal{T}$ satisfies the assumptions of the
Banach Fixed Point Theorem and has a unique fixed point. By the assumptions we
have that $\mathcal{F}(0)=0$. Therefore we deduce that if (2.63) holds, then
the unique fixed point is the trivial one. _Thus, nontrivial solutions exist
only if ( 2.63) is violated, that is, when_

$\displaystyle M\sqrt{C(L)}>1.$ (2.64)

Regarding the upper bound for the velocity we summarise in

######  Theorem II.5.

An upper bound for the velocity $c$ of nontrivial periodic TWSs
$q_{n}(t)=Q(n-ct)=Q(z)$ of prescribed norm $||Q||_{\mathcal{X}_{1}}\leq R$ to
the system (1.1), on the periodic lattice $-L\leq n\leq L$, is given by

$\displaystyle c^{2}<2(\kappa+KC_{*}R^{\beta})C(L).$ (2.65)

#### Remarks on the physical significance of the estimates for the TWSs and
their velocity.

The estimates on the TWSs proved in Theorems II.2, II.3 and II.5 implicate a
coherent dependence on the lattice parameters, the frequency $\Omega$ and $R$
and the velocity $c$. Motivated by the discussion of (DNBRS, , Section A, pg.
9), we aim to discuss the potential physical relevance of these estimates:

  1. 1.

For fixed $\overline{m}$ and $\kappa$, we observe that

$\displaystyle\lim_{\Omega\rightarrow\infty}\mathscr{R}_{max}$
$\displaystyle=$
$\displaystyle\lim_{\Omega\rightarrow\infty}\mathscr{R}_{crit}=\infty,\;\;\mbox{for
fixed $c$},$ $\displaystyle\lim_{c\rightarrow\infty}\mathscr{R}_{max}$
$\displaystyle=$
$\displaystyle\lim_{c\rightarrow\infty}\mathscr{R}_{crit}=\infty,\;\;\mbox{for
fixed $\Omega$}.$

Both limits are physically relevant in the sense that in the limit of
arbitrary large frequency or velocity, a type of “energy” of the solution,
measured herein in the norm of $\mathcal{X}_{0}$, should become also
arbitrarily large. This behavior can be relevant to energy localization
phenomena His  or the notion of quasi-collapse Kim . Note that in the second
limit as $c\rightarrow\infty$, the growth of the norm in $\mathcal{X}_{0}$,
implies due to the Poincaré inequality (2.13) (or due to Sobolev embeddings),
the growth of the kinetic energy of the TWSs, which is consistent with the
growth of $c$.

  2. 2.

The above coherent dependence on the lattice parameters is also evident in the
derived upper bound for the velocity (2.65). Theorem II.5 justifies that TWs
of given “energy” measured by the norm of $\mathcal{X}_{1}$ can evolve with
velocity satisfying the upper-bound (2.65).

##  III Soft on-site potentials

In this section we study the KG lattice with soft on-site potentials of the
form

$V(x)=-\frac{\omega_{0}^{2}}{2}x^{2}+\frac{a}{p+1}x^{p+1},\,\,\,a>0,\,\,p>1,$
(3.1)

where we assume that $p+1=2r>0$, with integer $r$. That is, $p$ is an odd
integer so that the on-site potentials $V(x)$ possess the reflection symmetry
$V(x)=V(-x)$. The standard quartic double-well potential is obtained for
$p=3$. We prove the existence of periodic TWSs on finite lattices with imposed
periodic boundary conditions utilising the Mountain Pass Theorem.

###  III.1 Periodic TWSs: Existence by the Mountain Pass Theorem

We consider periodic TWSs satisfying (2.5), that is, solutions $Q(z)$
performing oscillations about $Q=0$ so that the associated energy $E>0$.
Hence, the Poincaré inequality applies. These periodic TWSs, as solutions of
(1.6), are critical points of the action functional
$S:\,\mathcal{X}_{1}\rightarrow{\mathbb{R}}$ given by

$S(Q)=\int_{-L}^{L}\left[\frac{c^{2}}{2}\left(Q^{\prime}(z)\right)^{2}+\frac{\omega_{0}^{2}}{2}Q^{2}(z)-\frac{a}{p+1}Q^{p+1}(z)-\frac{\kappa}{2}[Q(z+1)-Q(z)]^{2}\right]dz.$

We get

$\displaystyle<S^{\prime}(Q),P>$ $\displaystyle=$
$\displaystyle\int_{-L}^{L}\left[c^{2}Q^{\prime}(z)P^{\prime}(z)+\omega_{0}^{2}Q(z)P(z)-aQ^{p}(z)P(z)\right.$
(3.2) $\displaystyle+$
$\displaystyle\left.\kappa[Q(z+1)-2Q(z)+Q(z-1)]P(z)\right]dz,\,\,\,P,Q\in\mathcal{X}_{1},$

where $<\cdot,\cdot>$ is the standard duality bracket between
$\mathcal{X}_{1}$ and its dual $\mathcal{X}_{1}^{*}$; by the definition of the
derivative $S^{\prime}$, for any $Q\in\mathcal{X}_{1}$, the functional
$S^{\prime}(Q):\mathcal{X}_{1}\rightarrow\mathbb{R}$ is a linear functional
acting on any $P\in\mathcal{X}_{1}$ as $S^{\prime}(Q)[P]=<S^{\prime}(Q),P>$
Chow .

To prove the existence of TWSs we facilitate the Mountain Pass Theorem (MPT).
We recall [Chow , Definition 4.1, p. 130] (Palais-Smale (PS) condition) and
[Chow , Theorem 6.1, p. 140] (Mountain Pass Theorem (MPT) of Ambrosetti-
Rabinowitz Ambrosetti ).

######  Definition III.1.

Let $X$ be a Banach space and ${\bf E}:X\rightarrow\mathbb{R}$ be $C^{1}$. We
say that ${\bf E}$ satisfies condition (PS) if, for any sequence
$\\{u_{n}\\}\in X$ such that $|{\bf E}(u_{n})|$ is bounded and ${\bf
E}^{\prime}(u_{n})\rightarrow 0$ as $n\rightarrow\infty$ in $X^{*}$, there
exists a convergent subsequence.

######  Theorem III.1.

Let ${\bf E}:X\rightarrow\mathbb{R}$ be $C^{1}$ and satisfy (A) ${\bf
E}(0)=0$, (B) $\exists\rho>0$, $\alpha>0$: $||u||_{X}=\rho$ implies ${\bf
E}(u)\geq\alpha$, (C) $\exists u_{1}\in X$: $||u_{1}||_{X}\geq\rho$ and ${\bf
E}(u_{1})<\alpha$. Define

$\Gamma=\left\\{\gamma\in
C^{0}([0,1],X)\,:\,\gamma(0)=0,\,\gamma(1)=u_{1}\right\\}.$ (3.3)

Let $F_{\gamma}=\\{\gamma(t)\in X\,:\,0\leq t\leq 1\\}$ and
$\mathcal{L}=\\{F_{\gamma}\,:\,\gamma\in\Gamma\\}.$ If ${\bf E}$ satisfies
(PS), then

$\beta:=\inf_{F_{\gamma}\in\mathcal{L}}\,\sup\\{{\bf E}(v)\,:\,v\in
F_{\gamma}\\}\geq\alpha,$ (3.4)

is a critical value of the functional ${\bf E}$.

We proceed by proving the validity of the assumptions of the MPT:

######  Lemma III.1.

Assume either
(i) $\omega_{0}^{2}\geq 4\kappa$,
or
(ii) $\omega_{0}^{2}<4\kappa$ and $c^{2}>C(L)(4\kappa-\omega_{0}^{2})$.

Then the functional $S$ satisfies the PS-condition.

Proof: Assume $S(Q_{m})$ is bounded, i.e. $|S(Q_{m})|\leq M$ for all
$m\in{\mathbb{N}}$, and

$S^{\prime}(Q_{m})\rightarrow 0,\;\;\mbox{as $m\rightarrow\infty$ in
$\mathcal{X}_{1}^{*}$}.$ (3.5)

Recall also that

$|<S^{\prime}(Q_{m}),Q_{m}>|\leq||S^{\prime}(Q_{m})||_{\mathcal{X}_{1}^{*}}||Q_{m}||_{X_{1}},$
(3.6)

by the standard inequality for the duality bracket (Zeidler2, , Sec. 21.5, pg.
251). Then, using (3.5),we deduce that for any $\epsilon>0$, there exists
$N(\epsilon)\in\mathbb{N}$ such that

$||S^{\prime}(Q_{m})||_{\mathcal{X}_{1}^{*}}\leq\epsilon,\;\;\mbox{for
$m>N(\epsilon)$}.$ (3.7)

Applying (3.7) for $\epsilon\leq 1$ and using the inequality (3.6), we deduce
that

$|<S^{\prime}(Q_{m}),Q_{m}>|\leq||Q_{m}||_{\mathcal{X}_{1}},\;\;\mbox{for
$m>N(\epsilon)$}.$ (3.8)

Hence, for chosen $b\in(1/(p+1),1/2)$, we deduce from (3.8), that

$b\,|<S^{\prime}(Q_{m}),Q_{m}>|\leq||Q_{m}||_{\mathcal{X}_{1}}<||Q_{m}||_{\mathcal{X}_{1}}+1,\;\;\mbox{for
$m>N(\epsilon)$}.$ (3.9)

We will use (3.9) to prove that $Q_{m}$ is bounded in $\mathcal{X}_{1}$, as
follows: First, we derive the inequality

$\displaystyle 1+M+||Q_{m}||_{\mathcal{X}_{1}}$ $\displaystyle\geq$
$\displaystyle M-b<S^{\prime}(Q_{m}),Q_{m}\geq
S(Q_{m})-b<S^{\prime}(Q_{m}),Q_{m}>$ (3.10) $\displaystyle=$
$\displaystyle\left(\frac{1}{2}-b\right)\int_{-L}^{L}\left[c^{2}\left(Q^{\prime}_{m}(z)\right)^{2}+\omega_{0}^{2}Q_{m}^{2}(z)-\kappa[Q_{m}(z+1)-Q_{m}(z)]^{2}\right]dz$
$\displaystyle-$
$\displaystyle\frac{a}{p+1}(1-(p+1)b)\int_{-L}^{L}Q_{m}^{p}(z)dz$
$\displaystyle\geq$
$\displaystyle\left(\frac{1}{2}-b\right)\left[c^{2}||Q^{\prime}_{m}||_{L^{2}}^{2}+(\omega_{0}^{2}-4\kappa)||{Q}_{m}||_{L^{2}}^{2}\right].$

If assumption (i) holds, then

$1+M+||Q_{m}||_{\mathcal{X}_{1}}\geq\left(\frac{1}{2}-b\right)c^{2}||{Q}_{m}||_{\mathcal{X}_{1}}^{2},$
(3.11)

and if assumption (ii) holds, then

$1+M+||Q_{m}||_{\mathcal{X}_{1}}\geq\left(\frac{1}{2}-b\right)\left(c^{2}-C(L)(4\kappa-\omega_{0}^{2})\right]||{Q}_{m}||_{\mathcal{X}_{1}}^{2},$
(3.12)

implying that $Q_{m}$ is bounded in $\mathcal{X}_{1}$. Hence, there is a
subsequence of $Q_{m}$ (not relabeled) and a $Q\in\mathcal{X}_{1}$ such that
$Q_{m}\rightarrow Q$ weakly in $\mathcal{X}_{1}$, so that by Sobolev compact
embedding, one has the strong convergence $Q_{m}\rightarrow Q$ in
$L^{2}(-L,L)$ (and in $C([-L,L])$). Using Hölder’s inequality and the
embedding $\mathcal{X}_{1}\subset L^{\infty}(-L,L)$, we obtain

$\displaystyle||Q_{m}-Q||_{\mathcal{X}_{1}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{c^{2}}\left(<S^{\prime}(Q_{m})-S^{\prime}(Q),Q_{m}-Q>\right.$
(3.13) $\displaystyle-$
$\displaystyle\frac{1}{c^{2}}\int_{-L}^{L}\left[\left(\omega_{0}^{2}(Q_{m}(z)-Q(z))^{2}-\kappa[Q_{m}(z+1)-Q(z+1)-(Q_{m}(z)-Q(z))]^{2}\right.\right.$
$\displaystyle-$
$\displaystyle\left.\left.a(Q_{m}(z)-Q(z))^{p+1}\right]dz\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{c^{2}}|<S^{\prime}(Q_{m})-S^{\prime}(Q),Q_{m}-Q>|$
$\displaystyle+$
$\displaystyle\frac{1}{c^{2}}\left((\omega_{0}^{2}+4\kappa)||Q_{m}-Q||_{L^{2}}+a||Q_{m}^{p}-Q^{p}||_{L^{2}(-L,L)}\right)||Q_{m}-Q||_{L^{2}(-L,L)}.$

The first term on the right-hand side of (3.13) converges to zero because by
assumption $<S^{\prime}(Q_{m})-S^{\prime}(Q),Q_{m}-Q>\rightarrow 0$ as
$m\rightarrow\infty$. The last converges to zero by strong convergence. Thus,
$||Q_{m}-Q||_{\mathcal{X}_{1}}=0$ so that $(Q_{m})_{m\in{\mathbb{Z}}}$ has a
strongly convergent subsequence and the proof is finished. $\square$

######  Lemma III.2.

The functional $S$ is $C^{1}$ on $\mathcal{X}_{1}$.

Proof: The functional $S$ can be expressed as

$S(q)=\frac{c^{2}}{2}(Q,Q)+\Gamma(Q),$ (3.14)

where

$\Gamma(Q)=\int_{-L}^{L}\left[\frac{\omega_{0}^{2}}{2}Q^{2}(z)-\frac{a}{p+1}Q^{p+1}(z)-\frac{\kappa}{2}(Q(z+1)-Q(z))^{2}\right]dz.$
(3.15)

Continuity of the quadratic term $(Q,Q)$ is obvious. By using The embedding
$\mathcal{X}_{1}\subset L^{\infty}(-L,L)$ which implies that
$Q^{p}\in\mathcal{X}_{0}$ and the Poincaré inequality (2.13) we get the
estimate,

$||Q^{p}||^{2}_{\mathcal{X}_{0}}=\int_{-L}^{L}|Q(z)|^{2p}dz\leq
C(L)||Q||_{\mathcal{X}_{1}}^{2p}.$ (3.16)

Then, we have that

$\left|\Gamma(Q)\right|\leq\frac{\omega_{0}^{2}}{2}||Q||_{L^{2}}^{2}+\frac{a}{p+1}C(L)||Q||_{\mathcal{X}_{1}}^{p+1}+2\kappa||Q||_{L^{2}}^{2}\leq
C(L)\left(\left(\frac{\omega_{0}^{2}}{2}+2\kappa\right)||Q||_{\mathcal{X}_{1}}^{2}+\frac{a}{p+1}||Q||_{\mathcal{X}_{1}}^{p+1}\right)<\infty.$
(3.17)

The Gateaux derivative of $\Gamma$ exists and is given by

$\displaystyle<\Gamma^{\prime}(Q),h>$ $\displaystyle=$
$\displaystyle\int_{-L}^{L}\left[\omega_{0}^{2}Q(z)-aQ^{p}(z)\right.$ (3.18)
$\displaystyle+$
$\displaystyle\left.\kappa(A_{1}[Q(z)]-A_{2}[Q(z)])\right]h(z)dz.$

To prove that $\Gamma^{\prime}$ is continuous, we let
$||h||_{\mathcal{X}_{1}}\leq 1$ and $Q_{m}\rightarrow Q$ in $\mathcal{X}_{1}$.
Then

$\displaystyle\left|<\Gamma^{\prime}(Q_{m})-\Gamma^{\prime}(Q),h>\right|$
$\displaystyle=$
$\displaystyle\left|\int_{-L}^{L}\left[\omega_{0}^{2}(Q_{m}(z)-Q(z))-a(Q_{m}^{p}(z)-Q^{p}(z))\right.\right.$
$\displaystyle+$
$\displaystyle\left.\left.\kappa(A_{1}[Q_{m}(z)]-A_{2}[Q_{m}(z)]-(A_{1}[Q(z)]-A_{2}[Q(z)]))\right]h(z)dz\right|$
$\displaystyle\leq$
$\displaystyle\omega_{0}^{2}||Q_{m}-Q||_{L^{2}}^{2}+aC\left(||Q_{m}||_{L^{\infty}},||Q||_{L^{\infty}}\right)||Q_{m}-Q||_{L^{2}(-L,L)}^{2}+2\kappa||Q_{m}-Q||_{L^{2}(-L,L)}^{2}$
$\displaystyle=$
$\displaystyle\left(\omega_{0}^{2}+aC\left(||Q_{m}||_{L^{\infty}},||Q||_{L^{\infty}}\right)+2\kappa\right)||Q_{m}-Q||_{L^{2}(-L,L)}^{2}\leq
C\epsilon,$

for $m$ sufficiently large. Hence

$\left|<\Gamma^{\prime}(Q_{m})-\Gamma^{\prime}(Q),h>\right|\rightarrow
0\,\,\,{\rm as}\,\,\,m\rightarrow\infty,$

and the proof of the lemma is completed. $\square$

Obviously, $S(0)=0$, therefore condition (A) of the MPT is satisfied. For the
remaining conditions (B) and (C), the proofs are given as follows:

Proof of (B): We distinguish the cases (i) $\omega_{0}^{2}\geq 4\kappa$, and
(ii) $\omega_{0}^{2}<4\kappa$, $c^{2}>C(L)(4\kappa-\omega_{0}^{2})$.

(i) With the aid of the estimate

$\int_{-L}^{L}\left[\frac{a}{p+1}Q^{p+1}(z)+\frac{\kappa}{2}[Q(z+1)-Q(z)]^{2}\right]dz\leq\frac{a}{p+1}C(L)||Q||_{\mathcal{X}_{1}}^{p+1}+2\kappa||Q||_{L^{2}(-L,L)}^{2},$

we get the sufficient condition

$\displaystyle\frac{c^{2}}{2}||Q||_{\mathcal{X}_{1}}^{2}>\frac{a}{p+1}C(L)||Q||_{\mathcal{X}_{1}}^{p+1},$

for $S(Q)$ being positive. Conclusively, for $||Q||_{\mathcal{X}_{1}}$ small
enough, say $||Q||_{\mathcal{X}_{1}}=\rho$, and

$0<\rho<\left(\frac{(p+1)c^{2}}{2aC(L)}\right)^{1/{(p-1)}}$ (3.19)

there is a $\alpha>0$ with $S(Q)\geq\alpha$ for all
$||Q||_{\mathcal{X}_{1}}=\rho$.

(ii) In this case we use the estimate

$\int_{-L}^{L}\left[\frac{a}{p+1}Q^{p+1}(z)+\frac{\kappa}{2}[Q(z+1)-Q(z)]^{2}\right]dz\leq\frac{a}{p+1}C(L)||Q||_{\mathcal{X}_{1}}^{p+1}+2\kappa
C(L)||Q||_{\mathcal{X}_{1}}^{2}.$

Using the Poincaré inequality we derive the following condition for $S>0$:

$\displaystyle\frac{1}{2}\left(c^{2}-(4\kappa-\omega_{0}^{2})C(L)\right)||Q||_{\mathcal{X}_{1}}^{2}>\frac{a}{p+1}C(L)||Q||_{\mathcal{X}_{1}}^{p+1}.$

Hence, for $||Q||_{\mathcal{X}_{1}}$ small enough, say
$||Q||_{\mathcal{X}_{1}}=\rho$, and

$0<\rho<\left(\frac{(p+1)}{2aC(L)}(c^{2}-(4\kappa-\omega_{0}^{2}))\right)^{1/{(p-1)}},$
(3.20)

there is a $\alpha>0$ with $S(Q)\geq\alpha$ for all
$||Q||_{\mathcal{X}_{1}}=\rho$.
Proof of (C): For $||Q||_{\mathcal{X}_{0}}\neq 0$ we observe that

$S(tQ)=\int_{-L}^{L}\left[\frac{c^{2}}{2}t^{2}\left(Q^{\prime}(z)\right)^{2}+\frac{\omega_{0}^{2}}{2}t^{2}Q^{2}(z)-\frac{a}{p+1}t^{p+1}Q^{p+1}(z)-\frac{\kappa}{2}t^{2}[Q(z+1)-Q(z)]^{2}\right]dz\rightarrow-\infty,$
(3.21)

as $t\rightarrow\infty$. To summarise, by virtue of the MPT we state

######  Theorem III.2.

Let either
(i) $\omega_{0}^{2}\geq 4\kappa$,
or
(ii) $\omega_{0}^{2}<4\kappa$ and $c^{2}>C(L)(4\kappa-\omega_{0}^{2})$.

Then the system (1.1) on the periodic lattice $-L\leq n\leq L$ with an on-site
potential (3.1) has at least one nontrivial periodic TWS.

###  III.2 Thresholds for the average kinetic energy of TWSs of prescribed
speed

For the derivation of lower bounds of the average kinetic energy of TWSs we
utilise the fixed point method outlined in Section II. In the present case,
using Theorem II.4, we treat the following auxiliary linear, non-homogeneous
problem

$\displaystyle-Q^{\prime\prime}(z)=\frac{\kappa}{c^{2}}\left\\{\left(\Psi(z)-\Psi(z-1)\right)-\left(\Psi(z+1)-\Psi(z)\right)-\omega_{0}^{2}\Psi(z)+a\left(\Psi(z)\right)^{p}\right\\},$
(3.22)

for some arbitrary fixed $\Psi\in\mathcal{X}_{1}$, as an equation of the form
(2.52). We have the following result.

######  Proposition III.1.

For any $\Psi\in\mathcal{X}_{1}$, the equation (3.22) has a unique solution
$Q\in D(\mathcal{L})\subset\mathcal{X}_{1}$.

Proof: We reformulate Eq. (3.22) as

$\displaystyle-Q^{\prime\prime}(z)=\frac{1}{c^{2}}\left\\{\kappa{\left(A_{2}\left[\Psi(z)\right]-A_{1}\left[\Psi(z)\right]\right)}-\omega_{0}^{2}\Psi(z)+a\left(\Psi(z)\right)^{p}\right\\}:=\mathcal{F}\left[\Psi(z)\right].$
(3.23)

We use again the estimate (3.17) for $\Psi$,

$\displaystyle||\Psi^{p}||^{2}_{\mathcal{X}_{0}}=\int_{-L}^{L}|\Psi(z)|^{2p}dz\leq
C(L)||\Psi||_{\mathcal{X}_{1}}^{2p}.$

Then, for the right-hand side of Eq. (3.23) we get

$\displaystyle||\mathcal{F}[\Psi]||_{\mathcal{X}_{0}}$ $\displaystyle\leq$
$\displaystyle\frac{1}{c^{2}}\left\\{\kappa||A_{2}[\Psi]-A_{1}[\Psi]||_{\mathcal{X}_{0}}+C(L)\omega_{0}^{2}||\Psi||_{\mathcal{X}_{1}}+C_{{{0}}}a||\Psi||^{2p}_{\mathcal{X}_{1}}\right\\}$
(3.24) $\displaystyle\leq$
$\displaystyle\frac{1}{c^{2}}\left\\{2\kappa||\Psi||_{\mathcal{X}_{1}}+C_{1}\omega_{0}^{2}||\Psi||_{\mathcal{X}_{1}}+C_{{{0}}}a||\Psi||^{2p}_{\mathcal{X}_{1}}\right\\}.$

Thus, $\mathcal{F}[\Psi]\in L^{2}(-L,L)$, and due to Theorem II.4, Eq. (3.23)
has a unique solution $Q\in D(\mathcal{L})$. $\square$

We proceed along the lines in Section II.3 by adapting the steps for the use
of the fixed point argument to the present case, that is replacing the hard
potential $V(Q)$ in the expressions (2.56) by the soft potential (3.1).

In particular, we need the following estimate of the difference of the power
nonlinearity terms

$\displaystyle\int_{-L}^{L}|s_{1}^{p}-s_{2}^{p}|^{2}$ $\displaystyle\leq$
$\displaystyle
p^{2}\int_{-L}^{L}\left\\{\int_{0}^{1}|\xi|^{p-1}|s_{1}-s_{2}|d\theta\right\\}^{2},$
(3.25)

where $s_{1},\;s_{2}\in\mathbb{R}$, and $\xi=\theta s_{1}+(1-\theta)s_{2}$,
$\theta\in(0,1)$. Then, we have that

$\displaystyle\int_{-L}^{L}\left|[\Psi_{1}(z)]^{p}-[\Psi_{2}(z)]^{p}\right|^{2}dz$
$\displaystyle\leq$ $\displaystyle
p^{2}||\xi||_{L^{\infty}}^{2(p-1)}\int_{-L}^{L}|\Psi_{1}(z)-\Psi_{2}(z)|^{2}dz$
(3.26) $\displaystyle\leq$ $\displaystyle
p^{2}||\xi||_{L^{\infty}}^{2(p-1)}C\int_{-L}^{L}|\Psi_{1}^{\prime}(z)-\Psi_{2}^{\prime}(z)|^{2}dz,$

for $\xi(z)=\theta\Psi_{1}(z)+(1-\theta)\Psi_{2}(z)$. For the norm
$||\xi||_{L^{\infty}}$ we have the estimate

$\displaystyle||\xi||_{L^{\infty}}\leq\theta||\Psi_{1}||_{L^{\infty}}+(1-\theta)||\Psi_{2}||_{L^{\infty}}\leq\theta
C_{*}||\Psi_{1}||_{\mathcal{X}_{1}}+(1-\theta)C_{*}||\Psi_{2}||_{\mathcal{X}_{1}},$
(3.27)

where the constant $C_{*}>0$ denotes the optimal constant of the embedding
$\mathcal{X}_{1}\subset L^{\infty}(-L,L)$. Therefore, since
$\Psi_{1},\Psi_{2}\in\mathcal{B}_{R}$, we have that $||\xi||_{L^{\infty}}\leq
C_{*}R$. Thus by using (3.26) and (3.27) we deduce the inequality

$\displaystyle||[\Psi_{1}(z)]^{p}-[\Psi_{2}(z)]^{p}||_{\mathcal{X}_{0}}\leq
p\sqrt{C}(C_{*}R)^{p-1}||\Psi_{1}-\Psi_{2}||_{\mathcal{X}_{1}}.$

Then the Lipschitz constant $M$ in the equation equivalent to (2.60) in II.3
is determined by

$\displaystyle
M=\frac{1}{c^{2}}\left({{{2}}}\kappa+{{{C(L)}}}\omega_{0}^{2}+p\sqrt{C(L)}aC_{*}^{p-1}R^{p-1}\right).$
(3.28)

Using again the fixed point argument we end up with the following statement:

If

$M\sqrt{C(L)}<1$ (3.29)

holds, the unique fixed point is the trivial one. _Nontrivial solutions exist
only if_

$\displaystyle M\sqrt{C(L)}>1.$ (3.30)

From Eq. (3.28) we derive the following condition for the existence of
nontrivial solutions:

$\displaystyle
R^{2}>\left[\frac{c^{2}-\sqrt{C(L)}({{{2}}}\kappa+{{{C(L)}}}\omega_{0}^{2})}{C(L)}\right]^{\frac{2}{p-1}}\left(\frac{1}{paC_{*}^{p-1}}\right)^{\frac{2}{p-1}}:=T_{\mathrm{thresh}}.$
(3.31)

We conclude:

######  Theorem III.5.

Consider the system (1.1) on the periodic lattice of $2L$ particles, $-L\leq
n\leq L$ with a soft on-site potential (3.1) with reflection symmetry
$V(x)=V(-x)$. Every nontrivial periodic travelling wave solution
$q_{n}(t)=Q(n-ct)=Q(z)$ with speed $c$ satisfying

$\displaystyle
c^{2}>\sqrt{C(L)}({{{2}}}\kappa+{{{C(L)}}}\omega_{0}^{2})=c_{\mathrm{crit}}^{2},$
(3.32)

must have average kinetic energy
$T(Q)=\frac{1}{2}\int_{-L}^{L}U^{\prime}(z)^{2}dz$ satisfying the lower bound

$\displaystyle T_{\mathrm{thresh}}<2T(Q).$ (3.33)

The relation (3.33) can be regarded as a threshold value criterion for the
average kinetic energy in order that travelling waves of speed $c>c^{*}$
exist.

We remark that further useful quantifications of the norm and energy
thresholds can be provided as explicit values of the optimal constants $C_{*}$
and $C_{1,*}$ of the Sobolev embeddings used in our proofs (see in bsc ).

Acknowledgment The authors are grateful to the referees for their valuable
comments and suggestions which improved considerably the presentation of the
manuscript.

Data Availability Statement
The article has no associated data.

Authors Declarations
The authors have no conflicts to disclose.

Authors Contributions Statement
All authors contributed equally to the study conception, design and writing of
the manuscript. Material preparation, data collection and analysis were
performed equally by all authors. All authors read and approved the final
manuscript.

## References

  * (1) O.M. Braun and Y.S. Kivshar, The Frenkel-Kontorova Model: Concepts, Methods and Applications, Texts and monographs in Physics (Springer-Verlag, Berlin Heidelberg, 2004).
  * (2) P.G. Kevredikis, The Nonlinear Discrete Schrödinger Equation: Mathematical Analysis, Numerial Computations, and Physical Perspectives (Springer-Verlag, Berlin, Heidelberg, 2009).
  * (3) D.B. Duncan, J.C. Eilbeck, H. Feddersen and J.A.D. Wattis. Solitons on lattices, Phys. D 68 (1993), 1-11.
  * (4) J.C. Eilbeck and M. Johansson. The Discrete Nonlinear Schrödinger Equation-20 Years on. “Localization and Energy transfer in Nonlinear Systems, eds L. Vázquez, R.S. MacKay, M.P. Zorzano. World Scientific, Singapore, (2003), 44-67.
  * (5) T.R.O. Melvin, A.R. Champneys, P.G. Kevrekidis and J. Cuevas, _Radiationless travelling waves in saturable nonlinear Schrödinger lattices_ , Phys. Rev. Lett. 97 (2006), 124101\.
  * (6) G. Friesecke and J.A. D. Wattis, Existence theorem for travelling waves on lattices, Comm. Math. Phys. 161 (1994), 391–418.
  * (7) M. Willem and D. Smets, Solitary waves with prescribed speed on lattices, J. Funct. Anal. 149 (1997), 266–275.
  * (8) A. Pankov and K. Pflüger. Travelling waves in Lattice Dynamical systems, Math. Meth. Appl. Sc. 23 (2000), 1223-1235.
  * (9) A. Pankov and N. Zakharchenko. On some discrete variational problems, Special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday. Acta Appl. Math. 65 (2001), 295–303.
  * (10) A. Pankov, Traveling waves in Fermi-Pasta-Ulam chains with nonlocal interaction, Discrete Contin. Dyn. Syst. Ser. S 12 (2019), 2097–2113.
  * (11) A. Pankov, Solitary waves on nonlocal Fermi-Pasta-Ulam lattices: exponential localization, Nonlinear Anal. Real World Appl. 50 (2019), 603-612.
  * (12) A. Pankov and V. Rothos, Traveling waves in Fermi-Pasta-Ulam lattices with saturable nonlinearities, Discrete Contin. Dyn. Syst. 30 (2011), 835–849.
  * (13) M. Herrmann, Periodic travelling waves in convex Klein-Gordon chains, Nonlinear Anal. 71 (2009), 5501-5508.
  * (14) M. Fe${\check{c}}$kan and V.M. Rothos, Traveling waves of discrete nonlinear Schrödinger equations with nonlocal interactions, Appl. Anal. 89 (2010), 1387-1411.
  * (15) C. F. Kreiner and J. Zimmer, Travelling wave solutions for the discrete sine-Gordon equation with nonlinear pair interaction, Nonlinear Analysis 70 (2009), 3146-3158.
  * (16) G. Iooss and K. Kirchgässner, Travelling waves in a chain of coupled nonlinear oscillators, Comm. Math. Phys. 211 (2000), 439–464.
  * (17) G. James, Nonlinear waves in Newton’s cradle and the discrete p-Schrödinger equation, Math. Models Methods Appl. Sci. 21 (2011), 2335-2377.
  * (18) R. Mirollo and N. Rosen, Existence, uniqueness, and nonuniqueness of single-wave-form solutions to Josephson junction systems, SIAM J. Appl. Math. 60 (2000), 1471–1501.
  * (19) G. Katriel, Existence of travelling waves in discrete sine-Gordon rings, SIAM J. Math. Anal. 36 (2005), 1434–1443.
  * (20) J. Cuevas, J. C. Eilbeck and N. I. Karachalios, A lower bound for the power of periodic solutions of the defocusing discrete nonlinear Schrödinger equation, Dyn. Partial Differ. Equ. 5 (2008), no. 1, 69–85.
  * (21) J. Cuevas, J. C. Eilbeck and N. I. Karachalios, Thresholds for breather solutions of the discrete nonlinear Schrödinger equation with saturable and power nonlinearity, Discrete Contin. Dyn. Syst. 21 (2008), no. 2, 445–475.
  * (22) J. Cuevas, N.I. Karachalios and F. Palmero, Energy thresholds for the existence of breather solutions and travelling waves on lattices, Appl. Anal. 89 (2010), 1351–1385.
  * (23) A.A. Pankov, Travelling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices, Imperial College Press, London, (2005).
  * (24) G. James and D. Pelinovsky, Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potential, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2014), no. 2165, 20130462.
  * (25) D. Hennig, Localised time-periodic solutions of discrete nonlinear Klein-Gordon systems with convex on-site potentials, J. Fixed Point Theory Appl. 23 (2021), 31-38.
  * (26) D. Hennig and N. I. Karachalios, Existence of exponentially spatially localized breather solutions for lattices of nonlinearly coupled particles: Schauder’s fixed point theorem approach, J. Math. Phys. 62 123506 (2021).
  * (27) S. Flach, K. Kladko and R. S. MacKay, Energy thresholds for discrete breathers in one-, two-, and three-dimensional lattices Phys. Rev. Lett. 78 (1997), 1207–1210.
  * (28) M. Weinstein, Excitation thresholds for nonlinear localized modes on lattices, Nonlinearity 12 (1997), 673–-691.
  * (29) N. I. Karachalios, B. Sánchez-Rey, P. G. Kevrekidis and J. Cuevas, Breathers for the discrete nonlinear Schrödinger equation with nonlinear hopping, J. Nonlinear Sci. 23 (2013), 205–239.
  * (30) T. Cretegny, T. Dauxois, S. Ruffo and A. Torcini, Localization and equipartition of energy in $\beta$-FPU chain: Chaotic breathers, Phys. D 121 (1998), 109–126.
  * (31) P. L. Christiansen, Yu. B. Gaididei, V. K. Mezentsev, S. L. Musher, K. Ø. Rasmussen, J. Juul Rasmussen, I. V . Ryzhenkova and S. K. Turitsyn, Discrete Localized States and Localization Dynamics in Discrete Nonlinear Schrödinger Equations, Phys. Scr. Vol. T67 (1996), 160–166.
  * (32) B. Zinner, G. Harris, and W. Hudson, Traveling wavefronts for the discrete Fisher’s equation, J. Differential Equations 105 (1993), 46–92.
  * (33) S.-N. Chow, J. Mallet-Paret, and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations 149 (1998), 248–291.
  * (34) J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, J. Dynam. Differential Equations 11 (1999), 49–127
  * (35) A. Carpio, S.J. Chapman, S. Hastings, and J.B. McLeod, Wave solutions for a discrete reaction-diffusion equation, European J. Appl. Math. 11 (2000), 399–412.
  * (36) P.W. Bates, X. Chen, and A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal. 35, 520 (2003).
  * (37) X. Chen, J.-S. Guo and C.-C. Wu, Traveling waves in discrete periodic media for bistable dynamics, Arch. Rational Mech. Anal. 189 (2008), 189–236.
  * (38) J.S. Guo and Y.C. Lin, Traveling wave solution for a lattice dynamical system with convolution type nonlinearity, Discrete Contin. Dyn. Syst. 32 (2012), 101–124.
  * (39) D. Bettinson and G. Rowlands, Transverse stability of the one-dimensional kink solution of the discrete Cahn-Hilliard equation, Phys. Rev E 57 (1998), 169-178.
  * (40) A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.
  * (41) S.N. Chow and J.K. Hale, Methods of Bifurcation Theory, Grundlehren der mathematischen Wissenschaften–A Series of Comprehensive Studies in Mathematics, vol. 251 (Springer-Verlag, New York, 1982).
  * (42) E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed Point Theorems (Springer-Verlag New York, 1986).
  * (43) E. Zeidler, Nonlinear Functional Analysis and Its Applications II/A: Linear Monotone Operators (Springer-Verlag New York, 1990).
  * (44) J. Bergh and J. Löfström, Interpolation spaces (Springer, New-York, 1976).
  * (45) S. Saitoh, K. Watanabe, T. Yamada and W. Takahashi Reproducing Kernels of $H^{m}(a,b)$, $(m=1,2,3)$ and Least Constants in Sobolev’s Inequalities. Appl. Anal. 82 (2003), 809-820.