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

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

2008-11-23

arxiv.org/abs/0811.3729v2

Physics

Mathematical Physics

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

Scientific paper

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

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