Initial Boundary-Value Problems for Derivative Nonlinear Schroedinger Equation. Justification of Two-Step Algorithm

Abstract

We investigate two different initial boundary-value problems for derivative nonlinear Schrodinger equation. The boundary conditions are Dirichlet ¨ or generalized periodic ones. We propose a two-step algorithm for numerical solving of this problem. The method consists of Backlund type transformations ¨ and difference scheme. We prove the convergence and stability in C and H1 norms of Crank–Nicolson finite difference scheme for the transformed problem. There are no restrictions between space and time grid steps. For the derivative nonlinear Schrodinger equation, the proposed numerical algorithm converges ¨and is stable in C1 norm.

Keywords

• derivative nonlinear Schrodinger equation
• initial boundary-value ¨ problem
• Backlund transformations
• Crank–Nicolson finite difference scheme
• convergence and stability of difference schemes