Abstract

We consider the semiclassical (zero-dispersion) limit of the one-dimensional focusing Nonlinear Schrödinger equation (NLS) with decaying potentials. If a potential is a simple rapidly oscillating wave (the period has the order of the semiclassical parameter ε) with modulated amplitude and phase, the space-time plane subdivides into regions of qualitatively different behaviors, with the boundary between them consisting typically of collection of piecewise smooth arcs (breaking curve(s)). In the first region, the evolution of the potential is ruled by modulation equations (Whitham equations), but for every value of the space variable x there is a moment of transition (breaking), where the solution develops fast, quasi-periodic behavior, that is, the amplitude becomes also fastly oscillating at scales of order ε. The very first point of such transition is called the point of gradient catastrophe. We study the detailed asymptotic behavior of the left and right edges of the interface between these two regions at any time after the gradient catastrophe. The main finding is that the first oscillations in the amplitude are of nonzero asymptotic size even as ε tends to zero, and they display two separate natural scales: of order in the parallel direction to the breaking curve in the (x,t) plane, and of order in a transversal direction. The study is based upon the inverse-scattering method and the nonlinear steepest descent method.