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We prove the existence of exponentially and superexponentially localized breather solutions for discrete nonlinear Klein–Gordon systems. Our approach considers $d$-dimensional infinite lattice models with general on-site potentials and interaction potentials being bounded by an arbitrary power law, as well as, systems with purely anharmonic forces, cases which are much less studied particularly in a higher-dimensional set-up. The existence problem is formulated in terms of a fixed-point equation considered in weighted sequence spaces, which is solved by means of Schauder's Fixed-Point Theorem. The proofs provide energy bounds for the solutions depending on the lattice parameters and its dimension under physically relevant non-resonance conditions.

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. The proof of the existence of travelling wave solutions satisfying Dirichlet boundary conditions establishes rigorously the presence of compactons in discrete nonlinear Klein-Gordon chains. Thresholds on the averaged kinetic energy for these solutions to exist are also derived.

While the Ablowitz-Ladik lattice is integrable, the Discrete Nonlinear Schrödinger equation, which is more significant for physical applications, is not. We prove closeness of the solutions of both systems in the sense of a “continuous dependence” on their initial data in the and metrics. The most striking relevance of the analytical results is that small amplitude solutions of the Ablowitz-Ladik system persist in the Discrete Nonlinear Schrödinger one. It is shown that the closeness results are also valid in higher dimensional lattices, as well as, for generalised nonlinearities. For illustration of the applicability of the approach, a brief numerical study is included, showing that when the 1-soliton solution of the Ablowitz-Ladik system is initiated in the Discrete Nonlinear Schrödinger system with cubic or saturable nonlinearity, it persists for long-times. Thereby, excellent agreement of the numerical findings with the theoretical predictions is obtained. Regional Government of Andalusia and EU (FEDER program) project P18-RT-3480 Regional Government of Andalusia and EU (FEDER program) project US-1380977 MICINN, AEI and EU (FEDER program) project PID2019-110430GB-C21 MICINN, AEI and EU (FEDER program) project PID2020-112620GB-I00

Discrete Ginzburg-Landau (DGL) equations with non-local nonlinearities have been established as significant inherently discrete models in numerous physical contexts, similar to their counterparts with local nonlinear terms. We study two prototypical examples of non-local and local DGLs on the one-dimensional infinite lattice. For the non-local DGL, we identify distinct scenarios for the asymptotic behavior of the globally existing in time solutions depending on certain parametric regimes. One of these scenarios is associated with a restricted compact attractor according to J. K. Hale's definition. We also prove the closeness of the solutions of the two models in the sense of a "continuous dependence on their initial data" in the $l^2$ metric under general conditions on the intrinsic linear gain or loss incorporated in the model. As a consequence of the closeness results, in the dissipative regime we establish the congruence of the attractors possessed by the semiflows of the non-local and of the local model respectively, for initial conditions in a suitable domain of attraction defined by the non-local system., Comment: 18 pages, 1 figure. To appear in Nonlinear Analysis (2022)