Interaction of solitons in discrete nonlinear Schrodinger equation with localized impurities

Abstract

Soliton is a localized wave that propagates as a distinct entity while maintaining its properties, i.e., shape and velocity, across a long distance without dispersing or changing significantly over time. Currently, there has been a rising concern among researchers about investigating the impact of collisions between solitons and external potentials (impurities), particularly driven by the Nonlinear Schrödinger Equation (NLSE). In the case of discrete systems of one-dimensional NLSE, the behaviour and propagation of solitons through a series of discrete sites or points were investigated in this study. First, the scattering behaviour of the discrete soliton in the self-action mode was analyzed in the absence of external potential. Then, the scattering of discrete soliton when interacting with localized impurities, namely the Gaussian and Delta potentials, was examined. The Discrete Cubic-Quintic NLSE was the main equation used to describe the phenomenon. This equation is not integrable and does not have analytical solutions, but it can be reduced to the systems of ordinary differential equations via the variational approach where the equations of soliton parameters for the width, center-of-mass position, linear and quadratic phase-front corrections were derived to describe the soliton evolutions during the scattering process. These equations play typical roles in describing the interaction phenomena of the discrete solitons with the localized impurities. Therefore, the investigation consisted of two methods: the variational approximation (VA) method and the direct numerical method. The results from the VA method were verified through direct numerical simulations of the governing equation, with the condition that the soliton had been initially set a distance from the external potentials. Upon consideration of nonlinearities characterized by cubic and quintic types, it was figured out that soliton’s initial velocity and propagation distance were increased with increasing value of the linear phase-front correction, whereas in the presence of external potentials, the soliton was found to be reflected, transmitted or exhibited combination of both outcomes with different potential strengths. The results proposed that the VA method proves to be a reliable and valuable technique for examining the scattering phenomena of discrete NLSE with localized impurities.