Split-step cubic B-spline collocation methods for nonlinear Schrödinger equations in one, two, and three dimensions with Neumann boundary conditions

Abstract

In this paper, split-step cubic B-spline collocation (SS3BC) schemes are constructed by combining the split-step approach with the cubic B-spline collocation (3BC) method for the nonlinear Schrödinger (NLS) equation in one, two, and three dimensions with Neumann boundary conditions. Unfortunately, neither of the advantages of the two methods can be maintained for the multi-dimensional problems, if one combines them in the usual manner. For overcoming the difficulty, new medium quantities are introduced in this paper to successfully reduce the multi-dimensional problems into one-dimensional ones, which are essential for the SS3BC methods. Numerical tests are carried out, and the schemes are verified to be convergent with second-order both in time and space. The proposed method is also compared with the split-step finite difference (SSFD) scheme. Finally, the present method is applied to two problems of the Bose-Einstein condensate. The proposed SS3BC method is numerically verified to be effective and feasible.