The Effects of Impurities on Discrete Nonlinear Schrödinger Equation

Understanding the effect that impurities may have on the soliton propagation process, particularly during the interaction process involving the Nonlinear Schrödinger Equation (NLSE), has become a major research focus in recent years. This paper studied the phenomenon of soliton scattering when it interacts with a localized impurity of the Delta potential under the discrete case of NLSE. Using an analytical approach, i.e., the variational approximation (VA) method, the equations of soliton parameters for the width, center-of-mass position, and linear and quadratic phase-front corrections are derived in order to describe the soliton evolutions throughout the scattering process. The VA method results were validated by the direct numerical simulation of the discrete NLSE, provided that the soliton is initially set at a distance from the Delta potential. When the nonlinearity was taken to be of the cubic and quintic types, it was shown that the soliton of the Discrete Cubic-Quintic NLSE could be either reflected or transmitted by the Delta potential with different potential strengths and a constant soliton’s initial velocity. The results suggested that the VA method is an effective and useful approach to investigate the scattering process of discrete NLSE in the presence of impurities.

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