Dynamics of Partial Differential Equations

Volume 14 (2017)

Number 3

The derivative NLS equation: global existence with solitons

Pages: 271 – 294

DOI: https://dx.doi.org/10.4310/DPDE.2017.v14.n3.a3

Authors

Dmitry E. Pelinovsky (Department of Mathematics, McMaster University, Hamilton, Ontario, Canada)

Aaron Saalmann (Mathematisches Institut, Universität zu Köln, Germany)

Yusuke Shimabukuro (Institute of Mathematics, Academia Sinica, Taipei, Taiwan)

Abstract

We prove the global existence result for the derivative NLS equation in the case when the initial datum includes a finite number of solitons. This is achieved by an application of the Bäcklund transformation that removes a finite number of zeros of the scattering coefficient. By means of this transformation, the Riemann–Hilbert problem for meromorphic functions can be formulated as the one for analytic functions, the solvability of which was obtained recently. A major difficulty in the proof is to show invertibility of the Bäcklund transformation acting on weighted Sobolev spaces.

Keywords

derivative nonlinear Schrödinger equation, global existence, Bäcklund transformation, inverse scattering transform, solitons

2010 Mathematics Subject Classification

35P25, 35Q55, 37K40

Full Text (PDF format)

A.S. gratefully acknowledges financial support from the projects “Quantum Matter and Materials” and SFB-TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics” (Cologne University, Germany).

Received 31 May 2017

Published 8 September 2017