We consider the semiclassical (zero-dispersion) limit of the one-dimensional focusing Nonlinear Schroedinger equation (NLS) with decaying potentials. If a potential is a simple rapidly oscillating wave (the period has the order of the semiclassical p
arameter epsilon) with modulated amplitude and phase, the space-time plane subdivides into regions of qualitatively different behavior, 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, i.e., the amplitude becomes also fastly oscillating at scales of order epsilon. 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 epsilon tends to zero, and they display two separate natural scales; of order epsilon in the parallel direction to the breaking curve in the (x,t)-plane, and of order epsilon ln(epsilon) in a transversal direction. The study is based upon the inverse-scattering method and the nonlinear steepest descent method.
The contents of the paper is now covered in two separate papers arXiv:0904.2188 and arXiv:0904.2602. Please refer to those. Note that you can still access the original version arXiv:0711.4082v1.
We introduce a new class of two(multi)-matrix models of positive Hermitean matrices coupled in a chain; the coupling is related to the Cauchy kernel and differs from the exponential coupling more commonly used in similar models. The correlation funct
ions are expressed entirely in terms of certain biorthogonal polynomials and solutions of appropriate Riemann-Hilbert problems, thus paving the way to a steepest descent analysis and universality results. The interpretation of the formal expansion of the partition function in terms of multicolored ribbon-graphs is provided and a connection to the O(1) model. A steepest descent analysis of the partition function reveals that the model is related to a trigonal curve (three-sheeted covering of the plane) much in the same way as the Hermitean matrix model is related to a hyperelliptic curve.
