Long‐time asymptotics of the nonlinear Schrödinger equation shock problem

Abstract

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 are computed with 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 mechanism involving a complex conjugate pair of branch 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 front at large times. © 2007 Wiley Periodicals, Inc.