$L^2$-Sobolev space bijectivity and existence of global solutions for the matrix nonlinear Schrödinger equations

Abstract

We consider the Cauchy problem to the general defocusing and focusing $p\times q$ matrix nonlinear Schrödinger (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 $H^{1,1}(\mathbb{R})$.

48 pages