Solvability of the Zakharov-Shabat systems with meromorphic potentials by quadrature

TL;DR

This paper proves that two-dimensional Zakharov-Shabat systems with meromorphic, reflectionless potentials are always solvable by quadrature, facilitating analytical solutions for certain integrable PDEs via inverse scattering transform.

Contribution

It establishes a necessary and sufficient condition for the quadrature solvability of ZS systems with meromorphic potentials, extending previous results to a broader class of potentials.

Findings

ZS systems are integrable by quadrature if potentials are reflectionless.

Reflectionless, meromorphic potentials are characterized as the key to solvability.

Results generalize previous work on Schrödinger equations to ZS systems.

Abstract

We study the solvability of the general two-dimensional Zakharov-Shabat (ZS) systems with meromorphic potentials by quadrature. These systems appear in application of the inverse scattering transform (IST) to an important class of nonlinear partial differential equations (PDEs) called integrable systems. Their solvability by quadrature is a key to obtain analytical expressions for solutions to the initial value problems of the integrable PDEs by using the IST. We prove that the ZS systems are always integrable in the sense of differential Galois theory, i.e., solvable by quadrature, if and only if the meromporphic potentials are reflectionless, under the condition that the potentials are absolutely integrable on $\mathbb{R}\setminus(-R_0,R_0)$ for some $R_0>0$. Similar statements were previously proved to be true by the author for a limited class of potentials and the linear Schr\"odinger equations.