The derivative nonlinear Schrödinger equation: Global well-posedness and soliton resolution

Jenkins, Robert; Liu, Jiaqi; Perry, Peter; Sulem, Catherine

doi:10.1090/qam/1553

Authors: Robert Jenkins, Jiaqi Liu, Peter Perry and Catherine Sulem
Journal: Quart. Appl. Math. 78 (2020), 33-73
MSC (2010): Primary 35B30, 35Q55, 35C08
DOI: https://doi.org/10.1090/qam/1553
Published electronically: September 16, 2019
MathSciNet review: 4042219
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Abstract: We review recent results on global well-posedness and long-time behavior of smooth solutions to the derivative nonlinear Schrödinger (DNLS) equation. Using the integrable character of DNLS, we show how the inverse scattering tools and the method of Zhou [SIAM J. Math. Anal. 20 (1989), pp. 966–986] for treating spectral singularities lead to global well-posedness for general initial conditions in the weighted Sobolev space $H^{2,2}(\mathbb {R})$. For generic initial data that can support bright solitons but exclude spectral singularities, we prove the soliton resolution conjecture: the solution is asymptotic, at large times, to a sum of localized solitons and a dispersive component, Our results also show that soliton solutions of DNLS are asymptotically stable .

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