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 schr\u00f6dinger equation of high order $$i\u00a0\frac{\u2202u}{\u2202t} + (\u20131)^m \frac{\u2202^m}{\u2202x^m} \Bigg( a(x) \frac{\u2202^mu}{\u2202x^m} \Bigg) +\u00a0\u03b2\f(x)q(|u|^2)u + f (x; t)u = g(x; t)$$ 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.\u00a0