Nonlinear Schrodinger-Helmholtz Equation as Numerical Regularization of the Nonlinear Schrodinger Equation

TL;DR

This paper introduces a regularized nonlinear Schrödinger-Helmholtz equation as a numerical tool to better understand blow-up solutions of the classical NLS, establishing existence, uniqueness, and convergence properties.

Contribution

It develops a new regularized system for the NLS, proving local and global well-posedness, and demonstrates its convergence to the classical NLS as the regularization parameter approaches zero.

Findings

Existence and uniqueness of local solutions for certain nonlinear powers.

Existence and uniqueness of global solutions under specific conditions.

Convergence of the regularized system to the classical NLS as regularization diminishes.

Abstract

A regularized $\alpha-$system of the Nonlinear Schr\"{o}dinger Equation (NLS) with $2\sigma$ nonlinear power in dimension $N$ is studied. We prove existence and uniqueness of local solution in the case $1 \le \sigma <\frac{4}{N-2}$ and existence and uniqueness of global solution in the case $1 \le \sigma < \frac{4}{N}$. When $\alpha \to 0^+$, this regularized system will converge to the classical NLS in the appropriate range. In particular, the purpose of this numerical regularization is to shed light on the profile of the blow up solutions of the original Nonlinear Schr\"{o}dinger Equation in the range $\frac{2}{N}\le \sigma <\frac{4}{N}$, and in particular for the critical case $\sigma = \frac{2}{N}$.