Integrability of the Zakharov-Shabat systems by quadrature

TL;DR

This paper investigates the integrability of two-dimensional Zakharov-Shabat systems using differential Galois theory, establishing that such systems are integrable by quadrature if and only if the potentials are reflectionless, facilitating analytical solutions.

Contribution

It provides a characterization of integrability for Zakharov-Shabat systems via differential Galois theory, linking reflectionless potentials to solvability by quadrature.

Findings

Reflectionless potentials are exactly those integrable by quadrature.

Integrability by quadrature enables analytical solutions through inverse scattering transform.

The results apply to a broad class of potentials in the Zakharov-Shabat systems.

Abstract

We study the integrability of the general two-dimensional Zakharov-Shabat systems, which appear in application of the inverse scattering transform (IST) to an important class of nonlinear partial differential equations (PDEs) called integrable systems, in the meaning of differential Galois theory, i.e., their solvability by quadrature. It becomes a key for obtaining analytical solutions to the PDEs by using the IST. For a wide class of potentials, we prove that they are integrable in that meaning if and only if the potentials are reflectionless. It is well known that for such potentials particular solutions called n-solitons in the original PDEs are yielded by the IST.