Solitons in a one-dimensional discrete system with external potential

Researchers are currently investigating the impact of collisions between solitons and external potentials, particularly the Nonlinear Schrödinger Equation (NLSE), which governs the soliton interaction with the potential in the presence of nonlinearity. In the case of discrete systems of one-dimensional NLSE, the behaviour and propagation of discrete solitons through a series of discrete sites or points are described. In this paper, we investigated the interaction of a discrete soliton with a localized impurity, namely the Gaussian external potential, using two methods: the variational approximation (VA) method and the direct numerical method. Discrete Cubic-Quintic NLSE is the main equation used to describe the phenomenon. This equation is non-integrable and does not have analytical solutions, but it can be reduced to ordinary differential equations using the variational approach. The VA method is applied by taking into account the Gaussian profile as the ansatz, which then produces evolution equations for soliton solutions of width, linear and quadratic phase-front correction and center-of-mass position. These equations play typical roles in describing the scattering process of the discrete soliton with the localized impurity. The direct numerical method of discrete NLSE is then performed to compare and validate the accuracy of the approximation results from the VA method. Considering different values of potential strengths and a fixed soliton initial velocity, it is found that the soliton can be transmitted, trapped or reflected after the interaction with external Gaussian potential. The findings imply that the VA method is a successful and practical way for examining the scattering of soliton in discrete NLSE on external potential.