Global Existence for the Derivative Nonlinear Schrodinger Equation by the Method of Inverse Scattering

Abstract

We develop inverse scattering for the derivative nonlinear Schrodinger 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 $H^{2,2}(\mathbb{R})$ to itself. These results immediately imply global existence of solutions to the DNLS for initial data in a spectrally determined (open) subset of $H^{2,2}(\mathbb{R})$ containing a neighborhood of 0.

62 pages, 2 figures. Revised according to referee comments with section added on Beals-Coifman solutions. To appear in Communications in Partial Differential Equations

References in corpus (2)

• Long-time Asymptotics for the NLS equation via dbar methods
• Existence of global solutions to the derivative NLS equation with the inverse scattering transform method

Cited by in corpus (7)

• Stability of multi-solitons for the derivative nonlinear Schr{ö}dinger equation
• Stability of the traveling waves for the derivative Schrödinger equation in the energy space
• Local structure of singular profiles for a Derivative Nonlinear Schrödinger Equation
• Existence of global solutions to the derivative NLS equation with the inverse scattering transform method
• Long-Time Behavior of Solutions to the Derivative Nonlinear Schrödinger Equation for Soliton-Free Initial Data
• Stability of the sum of two solitary waves for (gDNLS) in the energy space
• Global well-posedness of the derivative nonlinear Schrödinger equation with periodic boundary condition in $H^{\frac{1}{2}}$