Semiclassical Asymptotics of the Focusing Nonlinear Schrodinger Equation for Square Barrier Initial Data

Abstract

The small dispersion limit of the focusing nonlinear Schroödinger equation (fNLS) exhibits a rich structure with rapid oscillations at microscopic scales. Due to the non self-adjoint scattering problem associated to fNLS, very few rigorous results exist in the semiclassical limit. The asymptotics for reectionless WKB-like initial data was worked out in [KMM03] and for the family q(x, 0) = sech^(1+(i/∈)μ in [TVZ04]. In both studies the authors observed sharp breaking curves in the space-time separating regions with disparate asymptotic behaviors. In this paper we consider another exactly solvable family of initial data, specifically the family of centered square pulses, q(x; 0) = qx[-L,L] for real amplitudes q. Using Riemann- Hilbert techniques we obtain rigorous pointwise asymptotics for the semiclassical limit of fNLS globally in space and up to an O(1) maximal time. In particular, we find breaking curves emerging in accord with the previous studies. Finally, we show that the discontinuities in our initial data regularize by the immediate generation of genus one oscillations emitted into the support of the initial data. This is the first case in which the genus structure of the semiclassical asymptotics for fNLS have been calculated for non-analytic initial data.