Numerical investigation of the solutions of Schrodinger equation with exponential cubic B-spline finite element method

TL;DR

This paper presents a numerical method using exponential cubic B-splines and Crank-Nicolson discretization to solve the nonlinear Schrödinger equation, demonstrating high accuracy and efficiency through various soliton-related test problems.

Contribution

The paper introduces a novel exponential cubic B-spline collocation method combined with Crank-Nicolson for solving the nonlinear Schrödinger equation, with validation on multiple soliton scenarios.

Findings

High accuracy in soliton solution computations

Effective conservation of physical quantities

Comparable or improved results over existing methods

Abstract

In this paper, we investigate the numerical solutions of the cubic nonlinear Schrodinger equation via the exponential B-spline collocation method. Crank-Nicolson formulas are used for time discretization of the target equation. A linearization technique is also employed for the numerical purpose. Four numerical examples related to single soliton, collision of two solitons that move in opposite directions, the birht of standing and mobile solitons and bound state solution are considered as the test problems. The accuracy and the efficiency of the purposed method are measured by max error norm and conserved constants. The obtained results are compared with the possible analytical values and those in some earlier studies.