Generalized Neighbor-Interaction Models Induced by Nonlinear Lattices

TL;DR

This paper derives and analyzes generalized nonlinear lattice models from the nonlinear Schrödinger equation with periodic potentials, revealing complex neighbor interactions and quasi-linear regimes with stable localized solutions.

Contribution

It introduces a class of nonlinear lattice models with complex neighbor interactions derived from nonlinear Schrödinger equations with periodic coefficients.

Findings

Existence of quasi-linear regimes where pulse dynamics follow linear Schrödinger behavior

Analysis of modulational stability of the derived models

Identification of stable localized solitary wave solutions

Abstract

It is shown that the tight-binding approximation of the nonlinear Schr\"odinger equation with a periodic linear potential and periodic in space nonlinearity coefficient gives rise to a number of nonlinear lattices with complex, both linear and nonlinear, neighbor interactions. The obtained lattices present non-standard possibilities, among which we mention a quasi-linear regime, where the pulse dynamics obeys essentially the linear Schr{\"o}dinger equation. We analyze the properties of such models both in connection with their modulational stability, as well as in regard to the existence and stability of their localized solitary wave solutions.