Efficient determination of critical parameters of nonlinear Schr\"{o}dinger equation with point-like potential using generalized polynomial chaos methods

TL;DR

This paper introduces an efficient spectral method using generalized polynomial chaos to accurately determine the critical velocity of soliton interactions with a point-like potential in the nonlinear Schrödinger equation, reducing computational complexity.

Contribution

The paper presents a novel application of generalized polynomial chaos to find critical parameters in nonlinear Schrödinger equations with singular potentials, achieving spectral convergence.

Findings

Method accurately finds critical velocity with spectral convergence.

Computational complexity is significantly reduced.

Numerical results confirm effectiveness for various potential strengths.

Abstract

We consider the nonlinear Schr\"{o}dinger equation with a point-like source term. The soliton interaction with such a singular potential yields a critical solution behavior. That is, for the given value of the potential strength and the soliton amplitude, there exists a critical velocity of the initial soliton solution, around which the solution is either trapped by or transmitted through the potential. In this paper, we propose an efficient method for finding such a critical velocity by using the generalized polynomial chaos method. For the proposed method, we assume that the soliton velocity is a random variable and expand the solution in the random space using the orthogonal polynomials. The proposed method finds the critical velocity accurately with spectral convergence. Thus the computational complexity is much reduced. Numerical results for the smaller and higher values of the potential strength confirm the spectral convergence of the proposed method.