Modulation theory for self-focusing in the nonlinear Schr\"{o}dinger-Helmholtz equation

TL;DR

This paper applies modulation theory to the Schr"{o}dinger-Helmholtz equation, demonstrating that its regularization prevents singularity formation in the critical nonlinear Schr"{o}dinger equation, supported by numerical simulations.

Contribution

It introduces a modulation approach to analyze how the Schr"{o}dinger-Helmholtz regularization inhibits singularities in the critical NLS.

Findings

Regularization extends global existence of solutions

Prevents singularity formation in critical NLS

Numerical simulations confirm theoretical predictions

Abstract

The nonlinear Schr\"{o}dinger-Helmholtz (SH) equation in $N$ space dimensions with $2\sigma$ nonlinear power was proposed as a regularization of the classical nonlinear Schr\"{o}dinger (NLS) equation. It was shown that the SH equation has a larger regime ($1\le\sigma<\frac{4}{N}$) of global existence and uniqueness of solutions compared to that of the classical NLS ($0<\sigma<\frac{2}{N}$). In the limiting case where the Schr\"{o}dinger-Helmholtz equation is viewed as a perturbed system of the classical NLS equation, we apply modulation theory to the classical critical case ($\sigma=1,\:N=2$) and show that the regularization prevents the formation of singularities of the NLS equation. Our theoretical results are supported by numerical simulations