Envelope solitary waves on two-dimensional lattices with in-plane displacements

Abstract

We investigate envelope solitary waves on square lattices with two degrees of freedom and nonlinear nearest and next-nearest neighbor interactions. We consider solitary waves which are localized in the direction of their motion and periodically modulated along the perpendicular direction. In the quasimonochromatic approximation and low-amplitude limit a system of two coupled nonlinear Schrödinger equations (CNLS) is obtained for the envelopes of the longitudinal and transversal displacements. For the case of bright envelope solitary waves the solvability condition is discussed, also with respect to the modulation. The stability of two special solution classes (type-I and type-II) of the CNLS equations is tested by molecular dynamics simulations. The shape of type-I solitary waves does not change during propagation, whereas the width of type-II excitations oscillates in time.

Combined breathing–kink modes in the FPU lattice

“…Moving breathing-kink modes have been observed in several numerical simulations; see for instance, the work of Bickham et al [1], Gaididei et al [2], Huang et al [3] and Wang [4]. In this paper we provide an explanation for the phenomenon in terms of an asymptotic reduction of the governing lattice equations to the mKdV equation, which is an integrable system known to support both travelling kink and breather solutions.…”

“…They found that both components of the displacement vector have the form of moving breather-kink modes which appear to travel through the lattice without suffering any appreciable loss of energy; they also find modes in which one displacement has the form of a moving breather-kink mode and the other is a classical breather; see Figs. 2, 3 and 8 of [2]. Kourakis and Shukla have noted the existence of breather-kink modes in models of dusty plasma crystals, though they named them 'asymmetric bright envelope solutions'; see Figs.…”

“…Gaididei et al [2] considered a two-dimensional lattice with inplane displacements, and reduced the system to two coupled nonlinear Schrödinger equations. They found that both components of the displacement vector have the form of moving breather-kink modes which appear to travel through the lattice without suffering any appreciable loss of energy; they also find modes in which one displacement has the form of a moving breather-kink mode and the other is a classical breather; see Figs.…”

Oscillating multidromion excitations in higher-dimensional nonlinear lattice with intersite and external on-site potentials using symbolic computation

“…[36,37] Furthermore, several authors have shown the existence of soliton-like excitations in 2D square, triangular and hexagonal lattices. [38][39][40][41][42][43][44][45][46][47][48][49][50][51] It has been reported in Ref. [52] that the nonlinear modulation of N-dimensional lattice system with cubic and quartic interatomic potentials are governed by the generalized (N + 1)-dimensional Davey-Stewartson equations using the weakly nonlinear theory.…”