Inverse problem for the L-operator in the Lax Pair of the Boussinesq equation on the circle

TL;DR

This paper studies the inverse spectral problem for a third-order operator in the Lax pair of the Boussinesq equation on a circle, establishing an analytic and bijective spectral data mapping near zero.

Contribution

It constructs and analyzes a spectral data mapping for a third-order non-self-adjoint operator related to the Boussinesq equation, extending inverse spectral theory techniques.

Findings

Mapping from operator coefficients to spectral data is analytic near zero.

The spectral data mapping is one-to-one in a neighborhood of zero.

The work generalizes inverse spectral methods to a third-order operator on the circle.

Abstract

We consider a third-order non-self-adjoint operator, which is an $L$-operator in the Lax pair for the Boussinesq equation on the circle. We construct a mapping from the set of operator coefficients to the set of spectral data, similar to the corresponding mapping for the Hill operator constructed by E. Korotyaev. We prove that in a neighborhood of zero our mapping is analytic and one-to-one.