A Leap Frog Finite Difference Scheme for a Class of Nonlinear Schrödinger Equations of High Order

Abstract

In this paper, the periodic initial value problem for the following class of nonlinear schrödinger equation of high order $$i \frac{∂u}{∂t} + (-1)^m \frac{∂^m}{∂x^m} \Bigg( a(x) \frac{∂^mu}{∂x^m} \Bigg) + β\f(x)q(|u|^2)u + f (x; t)u = g(x; t)$$ is considered. A leap-frog finite difference scheme is given, and convergence and stability is proved. Finally, it is shown by a numerical example that numerical result is coincident with theoretical result.