Inverse problem for the divisor of the good Boussinesq equation

TL;DR

This paper addresses the inverse spectral problem for a third-order operator related to the Boussinesq equation, establishing a bijective correspondence between coefficients and spectral data near zero.

Contribution

It provides an explicit solution to the inverse problem for a third-order operator with three-point Dirichlet conditions, including the construction of an analytic bijection.

Findings

The spectral data uniquely determine the operator coefficients near zero.

The mapping from coefficients to spectral data is analytic and bijective.

The results connect the divisor of Floquet solution poles with the spectrum of the Dirichlet problem.

Abstract

A third-order operator with periodic coefficients is an L-operator in the Lax pair for the Boussinesq equation on a circle. The projection of the divisor of the Floquet solution poles for this operator coincides with the spectrum of the three-point Dirichlet problem. The sign of the norming constant of the three-point problem determines the sheet of the Riemann surface on which the pole lies. We solve the inverse problem for a third-order operator with three-point Dirichlet conditions when the spectrum and norming constant are known. We construct a mapping from the set of coefficients to the set of spectral data and prove that this mapping is an analytic bijection in the neighborhood of zero.