Robert Jenkins

Regularization of a sharp shock by the defocusing nonlinear Schrödinger equation

The defocusing nonlinear Schrodinger (NLS) equation is studied for a family of step-like initial data with piecewise constant amplitude and phase velocity with a single jump discontinuity at the origin. Riemann-Hilbert and steepest descent techniques are used to study the long time/zero-dispersion limit of the solution to NLS associated to this family of initial data. We show that the initial discontinuity is regularized in the long time/zero-dispersion limit by the emergence of five distinct regions in the

Soliton Resolution for the Derivative Nonlinear Schrödinger Equation

(x, t)

Semiclassical Soliton Ensembles for the Three-Wave Resonant Interaction Equations

half-plane. These are left, right, and central plane waves separated by a rarefaction wave on the left and a slowly modulated elliptic wave on the right. Rigorous derivations of the leading order asymptotic behavior and error bounds are presented

Long time asymptotic behavior of the focusing nonlinear Schrödinger equation

We study the derivative nonlinear Schrödinger equation for generic initial data in a weighted Sobolev space that can support bright solitons (but exclude spectral singularities). Drawing on previous well-posedness results, we give a full description of the long-time behavior of the solutions in the form of a finite sum of localized solitons and a dispersive component. At leading order and in space-time cones, the solution has the form of a multi-soliton whose parameters are slightly modified from their initial values by soliton–soliton and soliton–radiation interactions. Our analysis provides an explicit expression for the correction dispersive term. We use the nonlinear steepest descent method of Deift and Zhou (Commun Pure Appl Math 56:1029–1077, 2003) revisited by the

On asymptotic stability of N-solitons of the defocusing nonlinear Schrodinger equation

{overline{partial}}

Global Well-posedness and soliton resolution for the Derivative Nonlinear Schr\"{o}dinger equation

∂¯-analysis of McLaughlin and Miller (IMRP Int Math Res Pap 48673:1–77, 2006) and Dieng and McLaughlin (Long-time asymptotics for the NLS equation via dbar methods. Preprint, arXiv:0805.2807, 2008), and complemented by the recent work of Borghese etxa0al. (Ann Inst Henri Poincaré Anal Non Linéaire, https://doi.org/10.1016/j.anihpc.2017.08.006, 2017) on soliton resolution for the focusing nonlinear Schrödinger equation. Our results imply that N-soliton solutions of the derivative nonlinear Schrödinger equation are asymptotically stable.

On the Asymptotic Stability of {N}-Soliton Solutions of the Defocusing Nonlinear Schrödinger Equation

The three-wave resonant interaction equations are a non-dispersive system of partial differential equations with quadratic coupling describing the time evolution of the complex amplitudes of three resonant wave modes. Collisions of wave packets induce energy transfer between different modes via pumping and decay. We analyze the collision of two or three packets in the semiclassical limit by applying the inverse-scattering transform. Using WKB analysis, we construct an associated semiclassical soliton ensemble, a family of reflectionless solutions defined through their scattering data, intended to accurately approximate the initial data in the semiclassical limit. The map from the initial packets to the soliton ensemble is explicit and amenable to asymptotic and numerical analysis. Plots of the soliton ensembles indicate the space–time plane is partitioned into regions containing either quiescent, slowly varying, or rapidly oscillatory waves. This behavior resembles the well-known generation of dispersive shock waves in equations such as the Korteweg–de Vries and nonlinear Schrödinger equations, although the physical mechanism must be different in the absence of dispersion.