Publications [#139789] of Stephanos Venakides

Papers Published

1. R. Buckingham, S. Venakides, Long-time asymptotics of the nonlinear {S}chr\"odinger equation shock problem, Communications on Pure and Applied Mathematics, vol. 60 no. 9 (2007), pp. 1349--1414, ISSN 0010-3640
   (last updated on 2007/12/17)

   Abstract:
   The long-time asymptotics of two colliding plane waves governed by the focusing nonlinear Schrödinger equation are analyzed via the inverse scattering method. We find three asymptotic regions in space-time: a region with the original wave modified by a phase perturbation, a residual region with a one-phase wave, and an intermediate transition region with a modulated two-phase wave. The leading-order terms for the three regions arecomputedwith error estimates using the steepest-descent method for Riemann-Hilbert problems. The nondecaying initial data requires a new adaptation of this method. A new breaking mechanisminvolvingacomplexconjugatepairofbranch points emerging from the real axis is observed between the residual and transition regions. Also, the effect of the collision is felt in the plane-wave state well beyond the shock fr ont at large times.