# Universality in the profile of the semiclassical limit solutions to the focusing Nonlinear Schroedinger equation at the first breaking curve - Nonlinear Sciences > Exactly Solvable and Integrable Systems

Universality in the profile of the semiclassical limit solutions to the focusing Nonlinear Schroedinger equation at the first breaking curve - Nonlinear Sciences > Exactly Solvable and Integrable Systems - Download this document for free, or read online. Document in PDF available to download.

Abstract: We consider the semiclassical zero-dispersion limit of the one-dimensionalfocusing Nonlinear Schroedinger equation NLS with decaying potentials. If apotential is a simple rapidly oscillating wave the period has the order of thesemiclassical parameter epsilon with modulated amplitude and phase, thespace-time plane subdivides into regions of qualitatively different behavior,with the boundary between them consisting typically of collection of piecewisesmooth arcs breaking curves. In the first region the evolution of thepotential is ruled by modulation equations Whitham equations, but for everyvalue of the space variable x there is a moment of transition breaking, wherethe solution develops fast, quasi-periodic behavior, i.e., the amplitudebecomes also fastly oscillating at scales of order epsilon. The very firstpoint of such transition is called the point of gradient catastrophe. We studythe detailed asymptotic behavior of the left and right edges of the interfacebetween these two regions at any time after the gradient catastrophe. The mainfinding is that the first oscillations in the amplitude are of nonzeroasymptotic size even as epsilon tends to zero, and they display two separatenatural scales; of order epsilon in the parallel direction to the breakingcurve in the x,t-plane, and of order epsilon lnepsilon in a transversaldirection. The study is based upon the inverse-scattering method and thenonlinear steepest descent method.

Author: ** M. Bertola, A. Tovbis**

Source: https://arxiv.org/