Asymptotics of the divisor for the good Boussinesq equation

TL;DR

This paper analyzes the asymptotic behavior of the divisor associated with the good Boussinesq equation, focusing on spectral estimates and their implications for inverse spectral problems.

Contribution

It provides the first asymptotic estimates of the divisor for the good Boussinesq equation, advancing the understanding of its inverse spectral problem.

Findings

Spectral estimates in terms of small operator coefficients

Norming constants asymptotics

Foundation for inverse problem solutions

Abstract

We consider a third order operator under the three-point Dirichlet condition. Its spectrum is the so-called auxiliary spectrum for the good Boussinesq equation, as well as the Dirichlet spectrum for the Schr\"odinger operator on the unit interval is the auxiliary spectrum for the periodic KdV equation. The auxiliary spectrum is formed by projections of the points of the divisor onto the spectral plane. We estimate the spectrum and the corresponding norming constants in terms of small operator coefficients. This work is the first in a series of papers devoted to solving the inverse problem for the Boussinesq equation.