Solvability of the Korteweg-de Vries equation under meromorphic initial conditions by quadrature

TL;DR

This paper investigates the conditions under which the Schrödinger equation associated with the Korteweg-de Vries (KdV) equation is solvable by quadrature, revealing that reflectionless meromorphic potentials are exactly those for which integrability holds.

Contribution

It establishes a precise criterion linking reflectionless meromorphic potentials to differential Galois integrability of the Schrödinger equation in the context of the KdV inverse scattering transform.

Findings

Schrödinger equation is integrable by quadrature if and only if the potential is reflectionless.

The integrability criterion applies to meromorphic potentials that are absolutely integrable outside a finite interval.

Rational potentials decaying at infinity are not integrable unless they meet the specified conditions.

Abstract

We study the solvability of the Korteweg-de Vries equation under meromorphic initial conditions by quadrature when the inverse scattering transform (IST) is applied. It is a key to solve the Schr\"odinger equation appearing in the Lax pair in application of the IST. We show that the Schr\"odinger equation is always integrable in the sense of differential Galois theory, i.e., solvable by quadrature, if and only if the meromporphic potential is reflectionless, under the condition that the potential is absolutely integrable on $\mathbb{R}\setminus(-R_0,R_0)$ for some $R_0>0$.This statement was previously proved to be true by the author for a limited class of potentials. We also show that the Schr\"odinger equation is not integrable in this sense for rational potentials that decay at infinity but do not satisfy the weak condition.