A unified approach to split absorbing boundary conditions for nonlinear Schr\"{o}dinger equations

TL;DR

This paper introduces a unified, nonlinear absorbing boundary condition method for efficiently solving nonlinear Schrödinger equations in unbounded domains, ensuring stability and accurate wave absorption.

Contribution

It develops a novel boundary condition approach by approximating kinetic energy with a one-way equation and uniting it with potential energy, enhancing numerical stability and absorption.

Findings

The method effectively absorbs outgoing waves in numerical simulations.

Numerical stability is confirmed through normal mode analysis.

The approach demonstrates advantages in stability and tractability in examples.

Abstract

An efficient method is proposed for numerical solutions of nonlinear Schr\"{o}dinger equations in an unbounded domain. Through approximating the kinetic energy term by a one-way equation and uniting it with the potential energy equation, absorbing boundary conditions are designed to truncate the unbounded domain, which are in nonlinear form and can perfectly absorb the waves outgoing from the truncated domain. We examine the stability of the induced initial boundary value problems defined on the computational domain with the boundary conditions by a normal mode analysis. Numerical examples are given to illustrate the stable and tractable advantages of the method.