2. Description of the system

The nonlinear Klein–Gordon systems on $d$-dimensional infinite lattices are given by the following set of coupled oscillator equations

(2.1)\begin{equation} \displaystyle\frac{d^{2} u_n}{{\rm d}t^{2}}=\kappa\,(\Delta_d u)_n- (U^{\prime}(u))_n,\quad n\in {\mathbb{Z}^{d}},\end{equation}

where $u_n$ is the displacement of an oscillator from its equilibrium position. The operator $(\Delta _d u)_n$ is the $d$-dimensional discrete Laplacian

\[ (\Delta_d u)_{n\in\mathbb{Z}^{d}}=\displaystyle\sum_{j} (u_{n+j}-2 u_n+u_{n-j}), \]

where $j$ are the $d$ unit vectors belonging to the $d$ axes of $\mathbb {Z}^{d}$. The function $U(x)$ is the on-site potential and prime $^{\prime }$ stands for the derivative with respect to the argument $x$. Each oscillator interacts with all of its next neighbours and the strength of the interaction is determined by the value of the parameter $\kappa$.

This system has a Hamiltonian structure related to the energy

\[ H=\displaystyle\sum_{n\in {\mathbb{Z}^{d}}}\left(\displaystyle\frac{1}{2}p_n^{2}+(U(u))_n\right)+ \frac{\kappa}{2}\sum_{n\in {\mathbb{Z}^{d}}}\sum_{j}(u_{n+j}-u_{n})^{2}, \]

and it is time-reversible with respect to the involution $p\mapsto -p$.

Discrete breathers can be characterized as follows:

(2.2)\begin{align} u_n(t+T)& =u_n(t),\quad p_n(t+T)=p_n(t),\ n\in {\mathbb{Z}^{d}}, \end{align}

(2.3)\begin{align} \lim_{|n|\rightarrow \infty}u_n& =0,\quad \lim_{|n|\rightarrow \infty}p_n=0, \end{align}

Furthermore, $u_n(t)$ has zero time average, i.e.

(2.4)\begin{equation} \displaystyle\int_0^{T} u_n(t)\,{\rm d}t=0,\quad n\in {\mathbb{Z}^{d}}.\end{equation}

The existence theorem will be proved under the following assumption:

• • A: The anharmonic on-site potential $U:{\mathbb {R}}\,\rightarrow \,{\mathbb {R}}$ possesses a minimum at $x=0$ and is at least twice continuously differentiable with

  (2.5)\begin{equation} U(0)=U^{\prime}(0)=0,\quad U^{\prime \prime}(0)=\omega_0^{2}\ge 0.\end{equation}

The solutions of the system obtained when linearizing equations (2.1) around the equilibrium $u_n=0$ are superpositions of plane wave solutions (phonons)

\[ u_n(t)= \exp(i(kn-\omega t)),\quad k\in [-\pi,\pi]^{d}, \]

with frequencies

\[ \omega^{2}(k) =\omega_0^{2}+4\kappa\displaystyle\sum_{j=1}^{d} \sin^{2} \left(\displaystyle\frac{k_j}{2}\right),\quad k_j\in[-\pi,\pi],\ j=1,\ldots,d. \]

Note that $U''(0)=\omega _0$ can be zero. An example is $U(x)=(1/\beta )x^{\beta }$, $\beta > 2$. The (extended) plane wave solutions disperse. Therefore, the frequency $\Omega$ of a localized time-periodic solution must satisfy the non-resonance condition $\Omega \neq |\omega (k)|/m$ for any integer $m\ge 1$. This requires $\Omega ^{2} >\omega _0^{2}+4\kappa d$ as a necessary condition for the existence of localized time-periodic solutions of system (2.1).

We write for the anharmonic part of the on-site potential:

(2.6)\begin{equation} V(x)=U(x)-\displaystyle\frac{\omega^{2}_0}{2}x^{2},\end{equation}

and assume

(2.7)\begin{equation} |V'(x)|\leq \overline{K}|x|^{1+\alpha},\quad |V''(x)|\leq \overline{K}_0|x|^{\alpha},\ \forall x\in\mathbb{R}, \end{equation}

and that the function $V^{\prime }(x)$ is one-to-one on $\mathbb {R}$.

3. Existence of exponentially localized breathers

a. Functional setting and proof of the main result.

In the following, we prove the existence of localized periodic solutions of system (2.1) on the infinite $d$-dimensional lattice. To this end, some appropriate function spaces are introduced, on which, the original problem is presented as a fixed-point problem for a corresponding operator. Utilizing Schauder's Fixed-Point Theorem, we establish the existence of exponentially localized solutions, extending the approach followed in [Reference Hennig and Karachalios33] to the higher-dimensional systems.

In order to obtain the required spatial localization of the solutions, we introduce suitable weighted function spaces. First, we consider the exponentially weighted Hilbert space of square-summable sequences, ${l}^{2}_w(\mathbb {Z}^{d})$, defined as

(3.1)\begin{equation} l^{2}_w=\left\{u_n\in \mathbb{R}:||u||_{l_w^{2}}^{2}:=\displaystyle\sum_{n}\,w_n|u_n|^{2}\,\right\},\end{equation}

with exponential weight $w_n=\exp (\lambda |n|)$ and $\lambda \ge 0$. Then, we denote by

\[ {X}_{0} = \left\{ u\in L^{2}_{per}([0,T];{l}_w^{2})\,:\,\displaystyle\int_0^{T} u_n(t)\,{\rm d}t=0,\ n\in {\mathbb{Z}^{d}} \right\}, \]

the space of $T-$periodic square-integrable functions in time with zero time average, with values in $l^{2}_w$. Evidently, $X_0$ is a closed convex subspace of $L^{2}_{per}([0,\,T];{l}_w^{2})$. We also consider the Sobolev space

\[ X_2=\left\{u\in H^{2}_{per}([0,T];{l}_w^{2}):\displaystyle\int_0^{T} u_n(t)\,{\rm d}t=0,\quad n\in {\mathbb{Z}^{d}}\right\}, \]

containing $T-$periodic functions of time assuming values in $l_w^{2}$ which, together with their weak derivatives up to second order are in $X_0$. The above spaces are endowed with the following norms:

\begin{align*} || u ||_{X_0}^{2}& =\displaystyle\frac{1}{T}\displaystyle\int_{0}^{T}\,|| u(t)||^{2}_{{l}_w^{2}}\,{\rm d}t,\\ || u ||_{X_2}^{2}& =\displaystyle\frac{1}{T}\displaystyle\int_{0}^{T}\,(|| u(t)||^{2}_{{l}_w^{2}} +|| D{u}(t)||^{2}_{{l_w}^{2}}+|| D^{2}{u}(t)||^{2}_{{l^{2}_w}})\,{\rm d}t. \end{align*}

For an element $u\in X_2$, we consider the Fourier-series expansion of $u_n(t)$ with respect to time $t$ and space variable $n$, determined by

(3.2)\begin{align} u_n(t)& =\displaystyle\sum_{m\in {\mathbb{Z}}{\setminus} \{ 0 \}}\hat{u}_{n,m} \exp( i\Omega m t),\,t\in [0,T],\quad {\hat{u}}_{n,m}=\displaystyle\frac{1}{T} \displaystyle\int_{0}^{T} u_n(t)\exp({-}i \Omega m t)\,{\rm d}t, \end{align}

(3.3)\begin{align} \hat{u}_{n,m}& =\overline{\hat{u}}_{n,-m}, \end{align}

and

(3.4)\begin{align} u_n(t)& =\displaystyle\frac{1}{(2\pi)^{d}}\displaystyle\int_{0}^{2\pi}\cdot{\cdot} \cdot\int_{0}^{2\pi} \tilde{u}_k(t)\exp(ikn)\,{\rm d}k_1 \cdot{\cdot} \cdot {\rm d}k_d,\quad\tilde{u}_k(t)=\displaystyle\sum_{n\in {\mathbb{Z}^{d}}}u_n(t)\exp({-}i\,k\,n), \end{align}

(3.5)\begin{align} \tilde{u}_{k}& =\overline{\tilde{u}}_{{-}k}. \end{align}

Then, using (3.2)–(3.5), we have the following representation of $u$:

(3.6)\begin{equation} u_n(t)=\displaystyle\sum_{m\in {\mathbb{Z}}{\setminus} \{0\}}\displaystyle\frac{1}{(2\pi)^{d}} \displaystyle\int_{0}^{2\pi}\cdot{\cdot} \cdot\int_{0}^{2\pi} {\hat{\tilde{u}}}_{k,m}\exp(ikn)\,{\rm d}k_1 \cdot{\cdot} \cdot \,{\rm d}k_d\exp(i\Omega mt).\end{equation}

In the following, we facilitate two versions of Schauder's fixed-point Theorem [Reference Schauder58] given by

We need the following results for compact operators on infinite-dimensional Banach spaces which can be either linear or nonlinear (see [Reference Zeidler64]). The proofs generalize the corresponding results for linear compact operators (see [Reference Conway16, Reference Zeidler63]), to the nonlinear ones, as it is evident that the linearity or nonlinearity of the operator is not involved in the arguments.

We now present the statement and proof of the main result.

Theorem III.2 Let condition A hold and suppose

(3.7)\begin{equation} |\Omega|> \sqrt{\omega_0^{2}+4d \kappa}. \end{equation}

Then, there exists at least one non-zero sequence $x\equiv \{x_n\}_{n\in {\mathbb {Z}^{d}}} \in X_2$ and $||x||_{X_0}\le R$, where

(3.8)\begin{equation} R\le \left[\displaystyle\frac{\Omega^{2}-(\omega_0^{2}+4{\rm d}\kappa)}{\overline{K}}\right]^{1/\alpha}:=R_{\mathrm{max}}. \end{equation}

The sequence $x$ is an exponentially localized, time-periodic solution of system (2.1) with period $T=\frac {2\pi }{\Omega }$.

Proof. We shall provide two alternatives of the proof by applying the two versions of the Schauder's fixed- point Theorem III.1. For this purpose, it is convenient to rewrite Equation (2.1) using (2.6) as:

(3.9)\begin{equation} \ddot{u}_n+\omega_0^{2} u_n- \kappa\,(\Delta_d u)_n={-}(V^{\prime}(u))_n.\end{equation}

Thus, only the right-hand side of (3.9) features terms nonlinear in $u$. Ultimately, we shall express (3.9) as a fixed-point equation in $u$.

I. First version of the proof:

We relate the left-hand side of (3.9) to the linear mapping: $M\,:\,X_2\,\rightarrow \,X_{0}$:

\[ M(u_n)=\ddot{u}_n+\omega_0^{2} u_n- \kappa\,(\Delta_d u)_n. \]

Then, applying the operator $M$ to the Fourier elements $\exp (ikn)\exp (i\,\Omega m t)$ in the representation (3.6), we get that

\[ M\exp(ikn)\exp(i\,\Omega mt)=\nu_m(k) \exp(ikn)\exp(i\,\Omega mt), \]

where

\[ \nu_m(k)={-} \Omega^{2}\,m^{2}+\omega_0^{2}+4\kappa\displaystyle\sum_{j=1}^{d}\sin^{2}\left(\displaystyle\frac{k_j}{2}\right),\quad m\in \mathbb{Z} {\setminus} \{ 0 \}. \]

Since by assumption (3.7), $\Omega ^{2}>\omega _0^{2}+4{\rm d}\kappa$, it is guaranteed that $\nu _m(k) \ne 0$, for all $m\in {\mathbb {Z}} {\setminus} \{ 0 \}$ and for all $k\in [0,\,2\pi ]^{d}$, and that the mapping $M$ possesses an inverse $M^{-1}$ obeying $M^{-1}\exp (i\,(\Omega mt+kn))=(1/\nu _m(k))\exp (i\,(\Omega mt+kn))$. First, we write the norm of the linear operator $M^{-1}:\,X_{0} \rightarrow X_2$,

(3.10)\begin{align} || M^{{-}1} ||_{X_{0},X_2}^{2}& =\sup_{_{||u||_{X_0}=1}}|| M^{{-}1}\,u ||_{X_2}^{2}\nonumber\\ & =\sup_{_{||u||_{X_0}=1}}\displaystyle\frac{1}{T}\displaystyle\int_0^{T}\,[|| M^{{-}1} u(t)||^{2}_{{l}_w^{2}} +|| D{M^{{-}1}u(t)} ||^{2}_{{l}_w^{2}}+|| D^{2}{M^{{-}1}u(t)} ||^{2}_{{l}_w^{2}}]\,{\rm d}t\nonumber\\ & =\sup_{||u||_{X_0}=1}\frac{1}{T}\int_0^{T}\,\displaystyle\sum_{n\in \mathbb{Z}^{d}} w_n \left(\left| \sum_{m^{\prime}} \frac{1}{(2\pi)^{d}} \int_{0}^{2\pi}\cdot{\cdot}\right.\right.\nonumber\\ & \left.\left.\cdot\int_{0}^{2\pi}\frac{\hat{\tilde{u}}_{k,m}}{\nu_m(k)}\exp(ikn)\exp(i\Omega m t) \,{\rm d}k_1 \cdot{\cdot} \cdot \,{\rm d}k_d \right|^{2}\right.\nonumber\\ & \quad+\left|\sum_{m^{\prime}}\frac{1}{(2\pi)^{d}} \int_{0}^{2\pi}\cdot{\cdot} \cdot\int_{0}^{2\pi} \frac{i \Omega m \hat{\tilde{u}}_{k,m}}{\nu_m(k)} \exp(ikn)\exp(i\Omega m t) \,{\rm d}k_1 \cdot{\cdot} \cdot \,{\rm d}k_d \right|^{2} \nonumber\\ & \quad\left.+\left|\sum_{m^{\prime}}\frac{1}{(2\pi)^{d}} \int_{0}^{2\pi}\cdot{\cdot} \cdot\int_{0}^{2\pi} \frac{(i \Omega m)^{2} \hat{\tilde{u}}_{k,m}}{\nu_m(k)} \exp(ikn)\exp(i\Omega m t) \,{\rm d}k_1 \cdot{\cdot} \cdot \,{\rm d}k_d \right|^{2}\right)\,{\rm d}t. \end{align}

Then, by using (3.10), $|| M^{-1} ||_{X_{0},\,X_2}^{2}$ can be estimated from above, as follows:

(3.11)\begin{align} || M^{{-}1} ||_{X_{0},X_2}^{2}& \le \sup_{m\in {\mathbb{Z}}{\setminus} \{ 0 \}} \sup_{k\in [0,2\pi]^{d}} \displaystyle\frac{1 +(\Omega m)^{2}+(\Omega m)^{4}}{|\nu_m(k)|^{2}} \nonumber\\ & \quad\cdot \sup_{||u||_{X_0}=1} \frac{1}{T} \displaystyle\int_0^{T}\,\displaystyle\sum_{n\in \mathbb{Z}^{d}} w_n \left| \sum_{m^{\prime}} \frac{1}{(2\pi)^{d}} \int_{0}^{2\pi}\cdot{\cdot} \cdot\right.\nonumber\\ & \left.\quad \int_{0}^{2\pi} \hat{\tilde{u}}_{k,m} \exp(ikn)\exp(i\Omega m t) \,{\rm d}k_1 \cdot{\cdot} \cdot \,{\rm d}k_d \right|^{2} \,{\rm d}t\nonumber\\ & \le\frac{1+\Omega^{2}+\Omega^{4}}{(\Omega^{2}-(\omega_0^{2}+4{\rm d}\kappa))^{2}} \sup_{||u||_{X_0}=1} ||u||_{X_0}^{2} = \frac{1+\Omega^{2}+\Omega^{4}}{(\Omega^{2}-(\omega_0^{2} +4{\rm d}\kappa))^{2}}\nonumber\\ & \le \frac{(1+\Omega^{2})^{2}}{(\Omega^{2}-(\omega_0^{2}+4{\rm d}\kappa))^{2}}<\infty, \end{align}

verifying the boundedness of $M^{-1}$. Note that we have used the notation $\sum \nolimits _{l^{\prime }} = \sum \nolimits _{l\in {\mathbb {Z}}{\setminus} \{ 0 \}}$. For later use, we note that

(3.12)\begin{equation} || M^{{-}1} ||_{{X}_{0},{X}_0}\le \displaystyle\frac{1}{\Omega^{2}-(\omega_0^{2}+4{\rm d}\kappa)}.\end{equation}

For the treatment of the nonlinear terms, we assign the nonlinear operator $N\,:\,X_{0}\,\rightarrow \,X_{0}$ given by

\[ (N(u))_n={-}(V^{\prime}(u))_n, \]

to the right-hand side of (3.9). We demonstrate that the operator $N$ is continuous on $X_{0}$. To this end, we prove that $N$ is Frechet differentiable at any $u$, with bounded derivative. We have that

\[ N^{\prime}(u):\,h\in X_{0}\,\mapsto N^{\prime}(u)[h] ={-}V^{\prime\prime}(u)h \in X_{0}, \]

and by using condition (2.7), we may derive the estimate

\begin{align*} || N^{\prime}(u)[h]||_{X_{0}}^{2}& =\displaystyle\frac{1}{T} \displaystyle\int_0^{T}\displaystyle\sum_{n \in \mathbb{Z}^{d}}{w_n} \left|(V^{\prime\prime}(u))_n(t))\,h_n(t)\right|^{2}\,{\rm d}t\\ & \le\frac{1}{T} \int_0^{T}\sum_{n \in \mathbb{Z}^{d}} w_n \overline{K}_0^{2}|u_{n}(t)|^{2\alpha}\,|h_n(t)|^{2}\,{\rm d}t\\ & \le\overline{K}_0^{2} \sup_{n\in {\mathbb{Z}^{d}}} \max_{t\in [0,T]} |u_n(t)|^{2\alpha}\,|| h||^{2}_{X_0}=A^{2} || h||^{2}_{X_0}, \end{align*}

where $A^{2}=\overline {K}_0^{2} \sup _{n\in {\mathbb {Z}^{d}}} \max _{t\in [0,T]} |u_n(t)|^{2\alpha }$. Hence,

(3.13)\begin{equation} || N^{\prime}(u)||_{{\mathcal{L}}(X_{0},X_{0})}\le A,\end{equation}

implying the (uniform) boundedness of the differential. Let us now use as the closed convex subset $Y_0$ of $X_0$, its closed ball centred at $0$ of radius $R$,

\[ {Y}_{0} = \left\{ u\in X_0:|| u||_{X_0}\le R\right\}. \]

Using again assumption (2.7), for the range of $N$ on $Y_0$, we get the bound

(3.14)\begin{align} || N(u) ||_{X_{0}}^{2}& =\displaystyle\frac{1}{T}\displaystyle\int_0^{T} \displaystyle\sum_{n \in \mathbb{Z}^{d}}{w_n}| (V^{\prime}(u))_n(t)) |^{2}\,{\rm d}t\nonumber\\ & \le\overline{K}^{2}\displaystyle\frac{1}{T}\displaystyle\int_0^{T}\displaystyle\sum_{n \in \mathbb{Z}^{d}}w_n|u_n(t)|^{2(\alpha+1)}\,{\rm d}t\nonumber\\ & \le\overline{K}^{2}||u||_{{X}_0}^{2(\alpha+1)}\le\overline{K}^{2} R^{2(\alpha+1)},\quad \forall u\in Y_0. \end{align}

For the application of Theorem III.1, our final step is to express problem (3.9) as a fixed-point equation in terms of a mapping $S:Y_0\,\rightarrow \, Y_{0}$:

(3.15)\begin{equation} x=M^{{-}1}\circ N(x)\equiv S(x).\end{equation}

Clearly, $S$ is continuous on $X_{0}$, as relations (3.11) and (3.13) establish that its constituents $M^{-1}$ and $N$ are continuous. Next, we show that $S({Y}_0)\subseteq {Y}_0$. Using (3.11) and (3.14), we have

(3.16)\begin{align} || S(x)||_{{X}_0}& =|| M^{{-}1}(\,N(x))||_{{X^{0}}}\le || M^{{-}1} ||_{{X}_0,X_2}\cdot|| \,N(x)||_{{X}_0}\nonumber\\ & \le\displaystyle\frac{\overline{K}}{\Omega^{2}-(\omega_0^{2}+4{\rm d}\kappa)} ||x||_{x_0}^{\alpha+1},\quad \forall x\in X_0, \end{align}

which implies that $S$ is bounded on $X_0$. Then, for all $x\in Y_0$, estimate (3.16) implies that

(3.17)\begin{equation} || S(x)||_{{X}_0}\le\displaystyle\frac{\overline{K}}{\Omega^{2}-(\omega_0^{2}+4{\rm d}\kappa)} R^{\alpha+1}\le R,\quad \forall x\in Y_0,\end{equation}

assuring by assumption (3.8), that indeed

\[ S({Y}_0)\subseteq {Y}_0. \]

As $M^{-1}$ maps $X_0$ to $X_2 \Subset X_{0}$ (compactly embedded), is compact, while $N:X_0\rightarrow X_0$ is bounded and continuous. We also have that $S(Y_0) \subseteq Y_0 \cap X_2$. Therefore, Lemma III.2 implies that the map $S=M^{-1}\circ N$, viewed as a map $S: Y_0\subseteq X_0\mapsto Y_0 \subseteq X_0$, is compact. The first version of Schauder's fixed point theorem implies then, that the fixed-point equation $x = S(x)$ has at least one solution.

It remains to show the existence of at least one non-trivial fixed-point solution.

Consider the operator $S$. Suppose that the kernel of the operator $S-I$ is trivial. Then, for every $x \in Y_0{\setminus} \left \{ 0\right \} \subseteq X_0 {\setminus} \left \{ 0\right \}$, there is $y\neq 0$, $y\in X_0$ solving the inhomogeneous system

(3.18)\begin{equation} S(x)-x=y\neq 0. \end{equation}

This is equivalent to $S(x)=x+y$ for all $x \in Y_0{\setminus} \left \{ 0\right \} \subseteq X_0 {\setminus} \left \{ 0\right \}$. Since $||S(x)||_{X_0}=||x+y||_{X_0}$ and $S: Y_0\subseteq X_0\mapsto Y_0 \subseteq X_0$, we have that $x+y$ in $Y_0$.

Recall that $S(x)=M^{-1}(N(x))$ where $M^{-1}$ is linear and $N$ is nonlinear. Since the function $-V^{\prime }(x)$ is one-to-one on $\mathbb {R}$, the operator $N$ is one-to-one also, implying, in conjunction with $M^{-1}(x)\neq 0$ for $x\neq 0$, that the compact operator $S=M^{-1}N$ is one-to-one on $Y_0$. Let $Z_0=S(Y_0)\subset Y_0$. Then $T:Y_0 \mapsto Z_0$, $T(x)=S(x)$ is bijective, that is, invertible. But as $Y_0$ is infinite dimensional and $T:\,Y_0\subseteq X_0\mapsto Z_0\subseteq Y_0$ is compact, from Lemma III.3 follows that $T$ is not invertible. Conclusively, the kernel of $S-I$ is not trivial, so that there is $x\neq 0$ solving $S(x)-x=0$. Hence, the fixed-point equation (3.15) possesses at least one non-trivial solution.

II. Second version of the proof:

For the application of the second version of Schauder's fixed-point theorem, we verify that the range of $S$ is contained in a compact subset of $Y_0$. We consider the space

\[ {X}_{0,0} = \left\{ u\in L^{2}_{per}([0,T];{l}^{2})\,\vert\,\displaystyle\int_0^{T} u_n(t)\,{\rm d}t=0,\quad n\in {\mathbb{Z}^{d}} \right\}. \]

Representing $N(u) \in Y_0$ in terms of its spatial and temporal Fourier-transforms as

(3.19)\begin{equation} (N(u))_n(t)=\displaystyle\sum_{m\in {\mathbb{Z}}{\setminus} \{ 0 \}} \displaystyle\frac{1}{(2\pi)^{d}} \displaystyle\int_{0}^{2\pi}\cdot{\cdot} \cdot\int_{0}^{2\pi} \hat{\tilde{N}}_{k,m}\exp(ikn)\,{\rm d}k_1 \cdot{\cdot} \cdot \,{\rm d}k_d \exp(i\Omega m t),\end{equation}

the Fourier coefficients of $M^{-1}(N(u))$ fulfil for all $u\in Y_0$, for all $k\in [0,\,2\pi ]^{d}$, and for all $m\in {\mathbb {Z}}{\setminus} \{ 0 \}$, the estimate

(3.20)\begin{align} & \left|\displaystyle\frac{\hat{{\tilde{N}}}_{k,m}}{- \Omega^{2}\,m^{2}+\omega_0^{2}+4\kappa \displaystyle\sum_{j=1}^{d}\sin^{2}\left(\frac{k_j}{2}\right)}\right|^{2}\le \frac{\sum_{m\in {\mathbb{Z}}{\setminus} \{ 0 \}} \frac{1}{(2\pi)^{d}} \displaystyle\int_{0}^{2\pi}\cdot{\cdot} \cdot\int_{0}^{2\pi}\left|\hat{{\tilde{N}}}_{k,m}\right|^{2} \,{\rm d}k_1 \cdot{\cdot} \cdot \,{\rm d}k_d}{m^{4}(\Omega^{2}-(\omega_0^{2}+4{\rm d}\kappa))^{2}}\nonumber\\ & \quad=\displaystyle\frac{\parallel N(u) \parallel_{{X}_{0,0}}^{2}}{m^{4}(\Omega^{2}-(\omega_0^{2}+4{\rm d}\kappa))^{2}}. \end{align}

Since $l_w^{2}\subset l^{2}$ and the inequality $||u||_{l^{2}} \le ||u ||_{l_w^{2}}$ holds, we get

(3.21)\begin{equation} \left|\displaystyle\frac{\hat{{\tilde{N}}}_{k,m}}{- \Omega^{2}\,m^{2}+\omega_0^{2}+4\kappa \displaystyle\sum_{j=1}^{d} \sin^{2}\left(\frac{k_j}{2}\right)}\right|^{2}\le \frac{{\overline{K}}^{2}R^{2(\alpha+1)}}{m^{4}(\Omega^{2}-(\omega_0^{2}+4{\rm d}\kappa))^{2}}\le \left(\frac{R}{m^{2}(1+\Omega^{2})^{2}}\right)^{2}.\end{equation}

Hence, we conclude that $S$ maps $Y_0$ into the subset

\begin{align*} Y_{0,c}& =\left\{u=\{u\}_{n\in {\mathbb{Z}^{d}}} \in Y_0,\,u_n(t)=\displaystyle\sum_{m\in {\mathbb{Z}}{\setminus} \{ 0 \}} \displaystyle\frac{1}{(2\pi)^{d}} \displaystyle\int_{0}^{2\pi}\cdot{\cdot} \cdot\right.\nonumber\\ & \left.\quad \int_{0}^{2\pi}\hat{{\tilde{u}}}_{k,m}\,\exp(ikn)\,{\rm d}k_1 \cdot{\cdot} \cdot {\rm d}k_d\exp( i \Omega m t)\ \mbox{:}\right.\\ & \quad\left.\left|\hat{{\tilde{u}}}_{k,m}\right|\le \frac{R}{m^{2}(1+\Omega^{2})}\right\}, \end{align*}

which is compact in $Y_0$. That is, the operator $S$ maps closed convex subsets $Y_0\subset X_0 \subset L_{per}^{2}((0,\,T);l^{2})$ into compact subsets $Y_{0,c}$ of $Y_0$. The second version of Schauder's fixed-point theorem implies then, that the fixed-point equation $u = S(u)$ has at least one non-trivial solution, and the proof is finished.$\Box$

b. A localization ring in $X_0$: upper and lower bounds for the norms of non-trivial solutions.

The proof of Theorem III.4 provides an upper bound for the norm of the breather solution. Strengthening the condition (3.7) to

(3.22)\begin{equation} \Omega^{2}>\omega_0^{2}+4{\rm d}\kappa +\overline{K}, \end{equation}

enables for the identification of a ring in $X_0$ containing non-trivial solutions with frequencies satisfying (3.22). As in [Reference Hennig and Karachalios33], we re-examine the definition of the map $S$ in (3.15) and estimate (3.17) in order to derive a contraction regime for the map $S$. It follows that if we require

\[ \frac{\overline{K}}{\Omega^{2}-(\omega_0^{2}+4{\rm d}\kappa)} R^{\alpha+1}<1, \]

then, we get that when

(3.23)\begin{equation} R\le \left[\displaystyle\frac{\Omega^{2}-(\omega_0^{2}+4{\rm d}\kappa)}{\overline{K}}\right]^{\frac{1}{1+\alpha}}:=R_{\mathrm{crit}}, \end{equation}

there exists as the unique solution, only the trivial one, $u=0$. To be consistent with the bound $R_{\mathrm {max}}$ defined in (3.8), and in order to exclude the trivial solution, we need to assume the enhanced condition on the frequency (3.22). Then, under the extra condition (3.22), it holds that $R_{\mathrm {crit}}< R_{\mathrm {max}}$ and non-trivial solutions with frequencies satisfying (3.22) are located in the ring $\mathbf {R}_E$ of $X_0$, determined by

(3.24)\begin{equation} \mathbf{R}_{X_0}=\left\{u\in X_0:R_{\mathrm{crit}}\leq ||u||_{X_0}\leq R_{\mathrm{max}}\right\}. \end{equation}

4. Breathers in systems of coupled KG lattices

The fixed-point method can also be extended to prove the existence of (exponentially localized) breather solutions in systems of $N$ diffusively coupled nonlinear KG lattices, of the form

(4.1)\begin{equation} \displaystyle\frac{d^{2} u_n^{l}}{{\rm d}t^{2}}= \kappa\,(\Delta_d u^{l})_n- U^{\prime}(u_n^{l})+\eta (u_n^{l+1}+u_n^{l-1}-2u_n^{l}),\quad n\in {\mathbb{Z}^{d}},\ 1\le l\le N,\end{equation}

where $\eta$ is the strength of the linear (diffusive) lattice-lattice interaction. For $\eta =0$ system (4.1) decomposes into $N$, $d-$dimensional KG lattices each of them determined by (2.1).

We have the following statement:

Theorem IV.1 Let condition A hold and suppose

(4.2)\begin{equation} |\Omega|> \sqrt{\omega_0^{2}+4d \kappa+4\eta}. \end{equation}

Then, there exist $N$ non-zero sequences $x^{l}\equiv \{x_n^{l}\}_{n\in {\mathbb {Z}^{d}}} \in X_2,$ $1 \le l \le N$, and $||x^{l}||_{X_0}\le R,$ where

\[ R\le \left(\displaystyle\frac{\Omega^{2}-4{\rm d}\kappa-4\eta}{\overline{K}}\right)^{1/\alpha}. \]

The sequences $x^{l}$ are solutions of system (4.1). They are exponentially localized along the $N$ KG lattices, and time-periodic with period $T=\frac {2\pi }{\Omega }$.

We use the notation from § 3 and denote by $X_0^{N}$ and $X_2^{N}$ the extended function spaces $X_0^{N}=X_0 \times \ldots \times X_0$, $X_2^{N}=X_2 \times \ldots \times X_2$ on which we express system (4.1) as an operator equation $M_cu=N(u)$. The linear and nonlinear operator is determined by

(4.3)\begin{equation} M_c u_n^{l}=\ddot{u}_n^{l}+\omega_0^{2}-\kappa\,(\Delta_d u^{l})_n-\eta (u_n^{l+1}+u_n^{l-1}-2u_n^{l}),\quad n\in \mathbb{Z}^{d},\ 1\le l\le N,\end{equation}

and

(4.4)\begin{equation} (N(u))_n^{l}={-}U^{\prime}(u_n^{l}),\quad n\in \mathbb{Z}^{d},\ 1\le l\le N,\end{equation}

respectively. We use the Fourier representation

(4.5)\begin{equation} u_n^{l}(t)=\displaystyle\sum_{m \neq 0} \displaystyle\frac{1}{(2\pi)^{1+d}}\displaystyle\int_0^{2\pi} \int_{0}^{2\pi}\cdot{\cdot} \cdot\int_{0}^{2\pi} \check{u}_{k,m}^{j} \exp(i\,kn)\exp(i\,jl)\,{\rm d}k_1 \cdot{\cdot} \cdot {\rm d}k_ddj\,\exp(i\,m\Omega t). \end{equation}

Applying the linear operator $M_c$ to the Fourier elements $\exp (i\,kn)\exp (i\,jl)\exp (i\,m\Omega t)$, gives

(4.6)\begin{align} M_c\exp(i\,kn)\exp(i\,jl)\exp(i\,m\Omega t)& =\left(-(m\Omega)^{2}-4\kappa \displaystyle\sum_{p=1}^{d} \sin^{2}\left(\displaystyle\frac{k_p}{2}\right)-4\eta \sin^{2}\left(\frac{j}{2}\right)\right)\nonumber\\& \quad \exp(i\,kn)\exp(i\,jl)\exp(i\,m\Omega t).\end{align}

By hypothesis (4.2), it is assured that $M_c$ possesses a left inverse $M_c^{-1}$, determined by

(4.7)\begin{equation} M_c^{{-}1}\exp(i\,kn)\exp(i\,jl)\exp(i\,m\Omega t)=\displaystyle\frac{1}{\nu_{m}^{\,l}(k)}\exp(i\,kn)\exp(i\,jl)\exp(i\,m\Omega t),\end{equation}

with

(4.8)\begin{equation} \nu_{m}^{\,l}(k)=(m\Omega)^{2}-4\kappa \displaystyle\sum_{p=1}^{d} \sin^{2}\left(\displaystyle\frac{k_p}{2}\right)-4\eta \sin^{2}\left(\frac{j}{2}\right).\end{equation}

Then, the remainder of the proof of Theorem IV.1 follows the lines in the proof of Theorem III.4.

5. Nonlinear interactions and superexponential localization

In this section, we treat higher-dimensional KG lattices with purely nonlinear interaction terms, of the form

(5.1)\begin{equation} \displaystyle\frac{d^{2} u_n}{{\rm d}t^{2}}= \displaystyle\sum_{j} W^{\prime}(u_{n+j}-u_n)-\sum_jW^{\prime}(u_n-u_{n-j}),\quad n\in {\mathbb{Z}}^{d},\end{equation}

The function $W(x)$ describes the anharmonic interaction potential between nearest neighbours. In the case $d=1$, examples of discrete KG systems for which superexponentially localized breathers (i.e. solutions decaying faster than any exponential) exist, involve anharmonic interaction forces of the form $(x_{n+1}-x_n)^{3}-(x_n-x_{n-1})^{3}$ [Reference Gorbach and Flach30, Reference Kivshar41]. Superexponentially localized travelling (solitary) waves for DGK systems with anharmonic interaction forces $|x_{n+1}-x_n|^{m} \operatorname {sign} (x_{n+1}-x_n)-|x_{n}-x_{n-1}|^{m} \operatorname {sign} (x_{n}-x_{n-1})$ with $m\ge 1$, were found in [Reference Ahnert and Pikovsky3].

Concerning the interaction potential, we make the following assumption:

• • B: The anharmonic interaction potential $W(x)$ has a minimum at $x=0$ and is at least twice continuously differentiable on $\mathbb {R}$, with $W(0)=W^{\prime }(0)=W^{\prime \prime }(0)=0$. We further assume that $W$ satisfies for some constant $\overline {K}_s>0$ and $\gamma > 0$, the relation

  (5.2)\begin{equation} |W'(x)|\leq \overline{K}_s|x|^{1+\gamma},\quad\forall x\in\mathbb{R}.\end{equation}

As a consequence of the absence of linear interaction terms, plane wave (phonon) solutions to the linearized system do not exist. That is, the linearized system has no continuous spectrum. This fact excludes resonances with the internal breather frequency, allowing for the occurrence of superexponential localization. Therefore, what used to be the non-resonance condition in the presence of a continuous spectrum in the previous sections, namely (3.7) and (4.2) for the existence of breather solutions with frequency $\Omega$, changes here to $|\Omega |\neq 0$.

We use the same functional analysis set-up as in § 3, except for

(5.3)\begin{equation} l^{2}_s=\left\{u_n\in \mathbb{R}:||u||_{l_s^{2}}^{2}:=\displaystyle\sum_{n}\,s_n|u_n-u_{n-1}|^{2}\right\},\end{equation}

with the superexponential weight

(5.4)\begin{equation} s_n=\exp(\lambda |n|\ln(1+\mu |n|)),\quad \lambda \ge 0,\,\mu\ge 0. \end{equation}

We use the spaces

\[ {X}_{0}^{s} = \left\{ u\in L^{2}_{per}([0,T];{l}_s^{2})\,:\,\displaystyle\int_0^{T} u_n(t)\,{\rm d}t=0,\quad n\in {\mathbb{Z}} \right\}, \]

and

\[ X_2^{s}=\left\{u\in H^{2}_{per}([0,T];{l}_s^{2}):\displaystyle\int_0^{T} u_n(t)\,{\rm d}t=0,\quad n\in {\mathbb{Z}}\right\}. \]

Regarding the existence of superexponentially localized breather solutions in system (5.1), we have the following theorem:

Theorem V.1 Let conditions A and B hold, and suppose

(5.5)\begin{equation} |\Omega| \neq 0.\end{equation}

System (5.1) possesses at least one superexponentially localized time-periodic solution, represented by a non-zero sequence $x\equiv \{x_n\}_{n\in {\mathbb {Z}^{d}}} \in X_2^{s}$ and $||x||_{X_0^{s}}\le R$, where

(5.6)\begin{equation} R\le \left(\displaystyle\frac{\Omega^{2}}{2^{2+\gamma}\, d\,\overline{K}_s}\right)^{1/\gamma}\end{equation}

and period $T=\frac {2\pi }{\Omega }$.

Proof. The proof utilizes the first version of Schauder's fixed-point theorem and proceeds analogously to the proof of Theorem III.4, so that we only present the essential steps. We cast system (5.1) in the from:

(5.7)\begin{equation} \ddot{u}_n=\displaystyle\sum_{j}W^{\prime}(u_{n+j}-u_n)-\sum_{j}W^{\prime}(u_{n}-u_{n-j}),\end{equation}

so that the left-hand side of (5.7) is related to the linear mapping: $M_s\,:\,X_2^{s}\,\rightarrow \,X_{0}^{s}$:

\[ M_s(u_n)=\ddot{u}_n. \]

Then, applying the operator $M_s$ to the Fourier elements $\exp (i\,\Omega m t)$ of the representation (3.2), (3.3), we get that

\[ M_s\exp(i\,\Omega mt)=\nu_m \exp(i\,\Omega mt),\quad \nu_m={-} \Omega^{2}\,m^{2},\,\, m\in \mathbb{Z} {\setminus} \{ 0 \}. \]

The operator $M_s$ possesses an inverse $M_s^{-1}$, obeying $M_s^{-1}\exp (i\,\Omega mt)=(1/\nu _m)\exp (i\,\Omega mt)$. For the norm of the linear operator $M_s^{-1}:\,X_{0}^{s} \rightarrow X_2^{s}$ one derives the upper bound:

(5.8)\begin{equation} || M_s^{{-}1} ||_{X_{0}^{s},X_2^{s}}^{2}\le = \displaystyle\frac{1+\Omega^{2}+\Omega^{4}}{\Omega^{4}}\le \frac{(1+\Omega^{2})^{2}}{\Omega^{4}}<\infty, \end{equation}

verifying the boundedness of $M_s^{-1}$. When we consider $M_s^{-1}$ as a map $M_s^{-1}:\,X_{0}^{s} \rightarrow X_0^{s}$, we have the estimate

(5.9)\begin{equation} || M_s^{{-}1} ||_{{X}_{0}^{s},{X}_0^{s}}\le \displaystyle\frac{1}{\Omega^{2}}. \end{equation}

We associate with the right-hand side of (5.7) the nonlinear operator $N\,:\,X_{0}^{s}\,\rightarrow \,X_{0}^{s}$, as

\[ (N(u))_n= \displaystyle\sum_{j}W^{\prime}(u_{n+j}-u_n)-\sum_{j}W^{\prime}(u_{n}-u_{n-j}). \]

Continuity of the operator $N$ on $X_{0}$ is proven in an analogous vein as in the proof of Theorem III.4.

We proceed by considering the closed convex subset $Y_0^{s}$ of $X_0^{s}$ determined by its closed ball centred at $0$ of radius $R$,

\[ {Y}_{0}^{s} = \left\{ u\in X_0^{s}:|| u||_{X_0}\le R \right\}. \]

With the aid of the continuous embeddings

\[ l_s^{p}\subset l_s^{q},\,\,\,|| x||_{l_s^{q}}\le || x||_{l_s^{p}},\,\,\,1 \le p\le q \le \infty, \]

the Banach algebra property of $l_s^{2}$, i.e. $||xy||_{l_s^{2}}\le ||y||_{l_s^{2}}\,||y||_{l_s^{2}}$ for all $x,\,y \in l_s^{2}$, and the monotonicity $\exp (\lambda |n|\ln (1+\mu |n|))<\exp (\lambda |n+1|\ln (1+\mu |n+1|))$, we obtain for the range of $N$ on $Y_0^{s}$, the estimate

(5.10)\begin{align} || N(u) ||_{X_{0}^{s}}\!& =\!\left(\displaystyle\frac{1}{T}\displaystyle\int_0^{T}\displaystyle\sum_{n \in \mathbb{Z}}{w_n}\sum_{j}|W^{\prime}(u_{n+j}(t)\!-\!u_n(t)\right)\!-\!\sum_{j}W^{\prime}(u_{n}(t)-u_{n-j}(t))|^{2}\,{\rm d}t)^{1/2}\nonumber\\ & \le 2d\overline{K}_s (2R)^{\gamma} ||u||_{X_0^{s}}\le 2d\overline{K}_s (2R)^{1+\gamma},\quad \forall u\in Y_0^{s}. \end{align}

The final step of the proof is to express problem (5.7) as a fixed-point equation, in terms of a mapping $Y_0^{s}\,\rightarrow \, Y_{0}^{s}$:

\[ x=M_s^{{-}1}\circ N(x)\equiv S(x). \]

$S$ is continuous on $X_{0}^{s}$ as its ingredients $M_s^{-1}$ and $N$ are continuous. Now we confirm that $S({Y}_0^{s})\subseteq {Y}_0^{s}$. With the help of (5.9) and (5.10), one gets

(5.11)\begin{align} || S(x)||_{{X}_0^{s}}& =|| M_s^{{-}1}(\,N(x))||_{{X_0^{s}}}\le || M_s^{{-}1} ||_{{X}_0^{s},X_0^{s}}\cdot|| \,N(x)||_{{X}_0^{s}}\nonumber\\ & \le\displaystyle\frac{ 2d\overline{K}_s (2R)^{1+\gamma} }{\Omega^{2}} \le R,\quad \forall x\in Y_0^{s}, \end{align}

affirming by assumption (5.6), that

\[ S({Y}_0^{s})\subseteq {Y}_0^{s}. \]

Since $M_s^{-1}$ maps $X_0^{s}$ to $X_2^{s} \Subset X_{0}^{s}$, it follows that $S(Y_0^{s}) \subseteq Y_0^{s} \cap X_2^{s}$. Thus, when considered as a map $S: Y_0^{s}\subseteq X_0^{s}\mapsto Y_0^{s} \subseteq X_0^{s}$, $S=M_s^{-1}\circ N$ is compact. By the first version of Schauder's fixed-point theorem and the argumentation regarding the existence of non-trivial solutions in the proof of Theorem III.4, the fixed-point equation $x = S(x)$ has at least one non-zero solution.