Modulation theory for self-focusing in the nonlinear Schrödinger-Helmholtz equation

Abstract

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

new article, 20 pages, 10 figures, to appear in Numerical Functional Analysis and Optimization