Soliton solutions of derivative nonlinear Schrödinger equations: Conservative schemes and numerical simulation

Abstract

In this paper, we numerically study soliton solutions of derivative nonlinear Schrödinger equations based on several conservative finite difference methods. All schemes own second-order accuracy with the convergence order $O({\tau }^2+h^2)$ in the discrete $L^{\infty }$-norm, where $h$ denotes the spatial step size and $\tau$ denotes the temporal step size. We show that difference schemes preserve some discrete counterparts of continuous conservation laws, and all these schemes are solvable. Extensive numerical examples with soliton solutions are carried out to verify the theoretical results. These results manifest that our schemes have potential application to soliton propagation in optical fibers.