CRANK-NICOLSON DIFFERENCE SCHEME FOR THE DERIVATIVE NONLINEAR SCHRÖDINGER EQUATION WITH THE RIESZ SPACE FRACTIONAL DERIVATIVE

Abstract

<abstract> This paper studied the Crank-Nicolson(CN) difference scheme for the derivative nonlinear Schrödinger equation with the Riesz space fractional derivative, which generalized the classical Schrödinger equation that was used as a model in quantum mechanics. The existence of this difference solution is proved by the Brouwer fixed point theorem. Since the difference solution of the equation satisfies the mass conservation law, the corresponding convergence is also investigated in the $ L_2 $ norm, which turns out to be the second order accuracy in both temporal and space directions. Especially when the fractional order equals to two, all those results are in accordance with the conclusions for the difference solution developed for the non-fractional derivative Schrödinger equation. Finally, some numerical examples are carried out and further verified the theoretical results. </abstract>