Smooth parametric dependence of asymptotics of the semiclassical focusing NLS

TL;DR

This paper proves that the asymptotic solutions of the semiclassical focusing NLS depend smoothly on a parameter, supported by numerical evidence, enhancing understanding of parameter influence in nonlinear wave behavior.

Contribution

It establishes the smooth dependence of semiclassical focusing NLS asymptotics on a parameter, a novel result in the analysis of nonlinear wave equations.

Findings

Asymptotic solutions depend smoothly on the parameter

Numerical results support theoretical estimates

Enhanced understanding of parameter effects in NLS

Abstract

We consider the one dimensional focusing (cubic) Nonlinear Schr\"odinger equation (NLS) in the semiclassical limit with exponentially decaying complex-valued initial data, whose phase is multiplied by a real parameter. We prove smooth dependence of the asymptotic solution on the parameter. Numerical results supporting our estimates of important quantities are presented.