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

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