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arXiv · 1710.03819

Soliton Resolution for the Derivative Nonlinear Schrödinger Equation

Robert Jenkins·Jiaqi Liu·Peter Perry·Catherine Sulem
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Abstract

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, revisited by the $\overline{\partial}$-analysis of {McLaughlin-Miller and Dieng-McLaughlin, and complemented by the recent work of Borghese-Jenkins-McLaughlin 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.

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Robert Jenkins, Jiaqi Liu, Peter Perry, Catherine Sulem. 2018-02-17. Soliton Resolution for the Derivative Nonlinear Schrödinger Equation. https://doi.org/10.1007/s00220-018-3138-4

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