Stability Satisfied Numerical Approximates to the Non-analytical Solutions of the Cubic Schr\"odinger Equation

TL;DR

This paper introduces a sinc-based differential quadrature method to numerically solve the cubic Schrödinger equation, effectively approximating solutions and analyzing stability and conservation properties.

Contribution

The paper develops a novel sinc-based differential quadrature algorithm for solving the cubic Schrödinger equation, including stability analysis and conservation verification.

Findings

The method accurately approximates analytical solutions where available.

Numerical solutions preserve conserved quantities effectively.

Stability analysis guides appropriate time step selection.

Abstract

The time dependent complex Schr\"odinger equation with cubic nonlinearity is solved by constructing differential quadrature algorithm based on sinc functions. Reduction to a coupled system of real equations enables to approach the space derivative terms by the proposed method. The resulted ordinary differential equation system is integrated with respect to the time variable by using a bunch explicit methods of lower and higher orders. Some initial boundary value problems containing some analytical and non-analytical initial data are solved for experimental illustrations. The computational errors between the analytical and numerical solutions are measured by the discrete maximum error norm in case the analytical solution exists. The two conserved quantities are calculated by using the numerical results in all cases. The matrix stability analysis is implemented to control the time step size.