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A proof of the soliton resolution conjecture for the Benjamin--Ono equation

Louise Gassot, Patrick Gérard, Peter D. Miller

TL;DR

The paper proves a soliton resolution conjecture for the Benjamin--Ono equation for a broad class of initial data by linking long-time dynamics to the spectral data of the Lax operator. It combines an explicit solution representation with a detailed analysis of the distorted Fourier transform to decompose solutions into finitely many solitons and a radiative term that evolves by the linearized flow. The soliton parameters are directly tied to negative eigenvalues of the Lax operator, while radiation is captured by distorted Fourier modes of the initial data. The result extends known multi-soliton cases by including a nontrivial radiation component and provides a rigorous framework connecting integrable structure, spectral data, and nonlinear long-time asymptotics for BO. This approach may inform future work on soliton resolution in other integrable or near-integrable PDEs and offers sharp tools for analyzing radiation in dispersive systems.

Abstract

We give a proof of the soliton resolution conjecture for the Benjamin--Ono equation, namely every solution with sufficiently regular and decaying initial data can be written as a finite sum of soliton solutions with different velocities up to a radiative remainder term in the long--time asymptotics. We provide a detailed correspondence between the spectral theory of the Lax operator associated to the initial data and the different terms of the soliton resolution expansion. The proof is based on a new use of a representation formula of the solution due to the second author, and on a detailed analysis of the distorted Fourier transform associated to the Lax operator.

A proof of the soliton resolution conjecture for the Benjamin--Ono equation

TL;DR

The paper proves a soliton resolution conjecture for the Benjamin--Ono equation for a broad class of initial data by linking long-time dynamics to the spectral data of the Lax operator. It combines an explicit solution representation with a detailed analysis of the distorted Fourier transform to decompose solutions into finitely many solitons and a radiative term that evolves by the linearized flow. The soliton parameters are directly tied to negative eigenvalues of the Lax operator, while radiation is captured by distorted Fourier modes of the initial data. The result extends known multi-soliton cases by including a nontrivial radiation component and provides a rigorous framework connecting integrable structure, spectral data, and nonlinear long-time asymptotics for BO. This approach may inform future work on soliton resolution in other integrable or near-integrable PDEs and offers sharp tools for analyzing radiation in dispersive systems.

Abstract

We give a proof of the soliton resolution conjecture for the Benjamin--Ono equation, namely every solution with sufficiently regular and decaying initial data can be written as a finite sum of soliton solutions with different velocities up to a radiative remainder term in the long--time asymptotics. We provide a detailed correspondence between the spectral theory of the Lax operator associated to the initial data and the different terms of the soliton resolution expansion. The proof is based on a new use of a representation formula of the solution due to the second author, and on a detailed analysis of the distorted Fourier transform associated to the Lax operator.
Paper Structure (31 sections, 21 theorems, 206 equations)

This paper contains 31 sections, 21 theorems, 206 equations.

Table of Contents

  1. Introduction and main result
  2. A more precise formulation of the main result
  3. The Hardy space and the Lax operator
  4. The main result in more detail
  5. Strategy of proof
  6. Reduction to the case $t\to +\infty$
  7. Sufficient conditions based on weak convergence
  8. Soliton limit
  9. Radiation limit
  10. The soliton limit
  11. Setting the radiation limit
  12. Choosing the unknown function
  13. Making the distorted Fourier transform appear
  14. A priori estimates
  15. $L^2$ bounds
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $u_0\in H^1(\mathbb{R})$ be a real--valued function such that $xu_0\in H^1(\mathbb{R})$. Assume moreover that there exist $c_0\in \mathbb{R}$ and $v_0\in L^2(\mathbb{R})$ such that Then there exist a nonnegative integer $N$, elements $p_1,\dots ,p_N$ of $\mathbb{C}_+$, with $\mathrm{Im}(p_1)<\dots <\mathrm{Im}p_N$, and real--valued functions $u_\infty ^\pm$ in $H^1(\mathbb{R})$ such that in

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 2.1: Soliton resolution and spectral data
  • Corollary 2.2
  • Remark 2.3
  • Theorem 3.1: Explicit formulaGerard22
  • Lemma 3.2: Sufficient conditions for Theorem \ref{['thm:main']}
  • proof
  • Remark 3.3
  • Lemma 4.1
  • proof
  • ...and 29 more