A Conservative Finite Difference Scheme for Simulation of Soliton in Inhomogeneous Medium with Nonlocal Nonlinearity

Abstract

We construct a finite difference scheme to solve nonlocal nonlinear Schrödinger (NNLS) equation which models the soliton propagation in nonlocal Kerr medium with transversal linear refractive index variation. This scheme is constructed based on Crank-Nicolson method with a special treatment for the nonlinear terms. This method leads to a system of nonlinear equation which can be solved by iteration method. It is shown that this numerical scheme preserves the energy conservation law. This energy conservation law shows that the constructed numerical scheme is unconditionally stable. Then we implement our scheme to simulate the propagation of soliton in a triangular waveguide with nonlocal nonlinearity. The simulations results show that in a uniform medium with nonlocal nonlinearity, the soliton experiences self-bending. Effect of such self-bending is more pronounced if the nonlocality is increased. If a triangular linear refractive index profile is introduced in a local medium then the soliton may experience an oscillation behavior. Such oscillation is disturbed when the medium also exhibits a nonlocal nonlinear response. For a relatively high nonlocality, the soliton will not oscillate in the waveguide and it will be forced to exit from the waveguide.