Long-time asymptotics of the nonlinear SchrÖdinger equation shock problem

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.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/56049/1/20179_ftp.pd

Unified Approach to KdV Modulations

We develop a unified approach to integrating the Whitham modulation equations. Our approach is based on the formulation of the initial value problem for the zero dispersion KdV as the steepest descent for the scalar Riemann-Hilbert problem, developed by Deift, Venakides, and Zhou, 1997, and on the method of generating differentials for the KdV-Whitham hierarchy proposed by El, 1996. By assuming the hyperbolicity of the zero-dispersion limit for the KdV with general initial data, we bypass the inverse scattering transform and produce the symmetric system of algebraic equations describing motion of the modulation parameters plus the system of inequalities determining the number the oscillating phases at any fixed point on the $x, t$ - plane. The resulting system effectively solves the zero dispersion KdV with an arbitrary initial data.Comment: 27 pages, Latex, 5 Postscript figures, to be submitted to Comm. Pure. Appl. Mat