Global existence for the derivative nonlinear Schrödinger equation by the method of inverse scattering

By Jiaqi Liu, Peter Perry, Catherine Sulem. 2015

73 citations recorded by the source corpus.

ABSTRACT We develop inverse scattering for the derivative nonlinear Schrödinger equation (DNLS) on the line using its gauge equivalence with a related nonlinear dispersive equation. We prove Lipschitz continuity of the direct and inverse scattering maps from the weighted Sobolev spaces H2,2(ℝ) to itself. These results immediately imply global existence of solutions to the DNLS for initial data in a spectrally determined (open) subset of H2,2(ℝ) containing a neighborhood of 0. Our work draws ideas from the pioneering work of Lee and from more recent work of Deift and Zhou on the nonlinear Schrödinger equation.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f68a44e18a187cba238987444a7f046e4bf81691

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

1. Dmitry E. Pelinovsky
2. Yusuke Shimabukuro
3. R. Jenkins

In Statistical and Nonlinear Physics

1. Dmitry E. Pelinovsky
2. Yusuke Shimabukuro
3. Catherine Sulem
4. Peter Perry
5. Jiaqi Liu
6. R. Jenkins
7. F. Merle

Citations in the corpus

• Existence of global solutions to the derivative NLS equation with the inverse scattering transform method (2016, methodology): “The results of [23] are formulated in space H2(R) ∩ L2,2(R), which is embedded into space H2(R) ∩H1,1(R).”
• Soliton Resolution for the Derivative Nonlinear Schrödinger Equation (2017, methodology): “Our work builds on three previous papers [22, 23, 17] where we respectively considered global well-posedness in the soliton-free sector, large-time asymptotics in the soliton-free sector, and global well-posedness on the dense and open subset described in what follows.”