$L^{2}$-Sobolev space bijectivity and existence of global solutions for the matrix nonlinear Schr\"{o}dinger equations

TL;DR

This paper proves global well-posedness for matrix nonlinear Schrödinger equations with complex initial data by establishing bijectivity of scattering transforms in Sobolev spaces.

Contribution

It introduces a novel approach to demonstrate the bijectivity of scattering transforms for matrix NLS equations in Sobolev spaces, ensuring global solutions.

Findings

Global well-posedness in $H^{1,1}(\

\mathbb{R})$ for matrix NLS equations.

Bijectivity of direct and inverse scattering transforms in Sobolev spaces.

Abstract

We consider the Cauchy problem to the general defocusing and focusing $p\times q$ matrix nonlinear Schr\"{o}dinger (NLS) equations with initial data allowing arbitrary-order poles and spectral singularities. By establishing the $L^{2}$-Sobolev space bijectivity of the direct and inverse scattering transforms associated with a $(p+q)\times(p+q)$ matrix spectral problem, we prove that both defocusing and focusing matrix NLS equations are globally well-posed in the weighted Sobolev space $H^{1,1}(\mathbb{R})$.