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Inverse problem for the divisor of the good Boussinesq equation

Andrey Badanin, Evgeny Korotyaev

TL;DR

This work addresses the inverse spectral problem for a non-self-adjoint third-order operator with 3-point Dirichlet conditions on a circle, linked to the good Boussinesq equation. The authors build spectral data consisting of eigenvalues $\mu_n$ and norming constants $h_{s,n}$ via a McKean-type transformation to a Schrödinger-type problem and establish a local analytic bijection from the coefficient pair ${\mathfrak u}=(p,q)$ to these data near the zero-coefficient state. They develop a comprehensive analytic framework: fundamental matrices, monodromy, eigenfunctions, and gradients of the characteristic function $\Delta$, together with detailed asymptotics for eigenvalues and norming constants and their Frechét derivatives. The results yield precise gradient formulas, asymptotic expansions, and a rigorous pathway for solving the inverse problem in the small-data regime, with implications for the finite- and infinite-gap theory of the Boussinesq flow. The approach integrates monodromy techniques, theta-function perspectives, and PT87-type inverse spectral theory to establish the analytic structure of the mapping from coefficients to spectral data. The findings provide a foundational step toward extending finite-gap methods to infinite-gap settings for the good Boussinesq equation.

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.

Inverse problem for the divisor of the good Boussinesq equation

TL;DR

This work addresses the inverse spectral problem for a non-self-adjoint third-order operator with 3-point Dirichlet conditions on a circle, linked to the good Boussinesq equation. The authors build spectral data consisting of eigenvalues and norming constants via a McKean-type transformation to a Schrödinger-type problem and establish a local analytic bijection from the coefficient pair to these data near the zero-coefficient state. They develop a comprehensive analytic framework: fundamental matrices, monodromy, eigenfunctions, and gradients of the characteristic function , together with detailed asymptotics for eigenvalues and norming constants and their Frechét derivatives. The results yield precise gradient formulas, asymptotic expansions, and a rigorous pathway for solving the inverse problem in the small-data regime, with implications for the finite- and infinite-gap theory of the Boussinesq flow. The approach integrates monodromy techniques, theta-function perspectives, and PT87-type inverse spectral theory to establish the analytic structure of the mapping from coefficients to spectral data. The findings provide a foundational step toward extending finite-gap methods to infinite-gap settings for the good Boussinesq equation.

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.
Paper Structure (37 sections, 374 equations)

This paper contains 37 sections, 374 equations.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Eigenvalues
  4. Norming constant
  5. Inverse results
  6. Monodromy matrices and their properties
  7. Fundamental matrix
  8. Derivatives of the fundamental matrix
  9. Asymptotics of the fundamental matrix
  10. Traces of the monodromy matrix
  11. The fundamental matrix at the three-point eigenvalues
  12. The unperturbed case
  13. Asymptotics of the fundamental matrix at the eigenvalues
  14. Eigenfunctions
  15. Derivatives of the fundamental solutions
  16. ...and 22 more sections