Solution of the Schrödinger equation using quadratic B-Spline collocation on non-uniform grids

Abstract

This paper applies orthogonal collocation based on quadratic B-spline basis functions to solve the Schrödinger equation on uniform and non-uniform grids. The method is well suited for solving different cases of soliton solutions for which the solution behaviour is very complicated. The solutions are found to be consistent with the exact solutions for various non-uniform cases. The overall numerical scheme is proven to be unconditionally stable and utilizes minimal memory storage due to the sparse matrix systems associated with B-spline basis functions. This could prove particularly useful in machine learning applications where repetitive calculations are needed and large amounts of data need to be processed. We found that using non-uniform grids as compared to uniform grids had a profound effect in capturing the correct solution dynamics especially in the 3-soliton case.