Existence of global solutions for the nonlocal derivation nonlinear Schr\"{o}dinger equation by the inverse scattering transform method

TL;DR

This paper proves the existence of global solutions for a nonlocal derivative nonlinear Schrödinger equation using the inverse scattering transform, establishing a bijection between potential and reflection coefficient in weighted Sobolev spaces.

Contribution

It introduces a method to demonstrate global solutions for the integrable nonlocal derivative nonlinear Schrödinger equation via inverse scattering and Riemann-Hilbert problem techniques.

Findings

Established bijectivity between potential and reflection coefficient.

Proved global existence of solutions in weighted Sobolev space.

Applied inverse scattering transform to a nonlocal nonlinear PDE.

Abstract

We address the existence of global solutions to the initial value problem for the integrable nonlocal derivative nonlinear Schr\"{o}dinger equation in weighted Sobolev space $H^{2}(\mathbb{R})\cap H^{1,1}(\mathbb{R})$. The key to prove this result is to establish a bijectivity between potential and reflection coefficient by using the inverse scattering transform method in the form of the Riemann-Hilbert problem.