Lyapunov-Sylvester Operators for Numerical Solutions of NLS Equation

TL;DR

This paper introduces a numerical method utilizing Lyapunov-Sylvester operators to solve the 2D nonlinear Schrödinger equation with singular potential, ensuring stability and convergence.

Contribution

The paper develops a novel numerical scheme based on Lyapunov-Sylvester algebraic operators for the 2D NLS equation with singular potential, proving its invertibility and stability.

Findings

Method is consistent, convergent, and stable.

Lyapunov-Sylvester operators are invertible using new topological methods.

Numerical solutions effectively handle singular potentials.

Abstract

In the present paper a numerical method is developed to approximate the solution of two-dimensional NLS equation in the presence of a singular potential. The method leads to Lyapunov-Syslvester algebraic operators that are shown to be invertible using original topological and differential calculus issued methods. The numerical scheme is proved to be consistent, convergent and stable using the based on Lyapunov criterion, lax equivalence theorem and the properties of the Lyapunov-Syslvester operators.