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

2026/01/31 by Louise Gassot, Patrick Gérard, Peter D. Miller · 11 citations
Mathematics · #math.AP #msc:37K10 #msc:35B40 #msc:35Q51

paper · pdf

29 pages, no figure

arxiv created 2026/08/06 · arxiv updated 2026/08/07

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.

Citations

• Scattering of the defocusing Calogero--Moser derivative nonlinear Schrödinger equation
• Orbital stability of Benjamin--Ono multisolitons
• The Benjamin-Ono Initial-Value Problem for Rational Data with Application to Long-Time Asymptotics and Scattering
• Soliton resolution for Calogero--Moser derivative nonlinear Schrödinger equation
• Justification of the Benjamin-Ono equation as an internal water waves model
• Sharp well-posedness for the Benjamin--Ono equation
• Soliton resolution for the energy-critical nonlinear wave equation in the radial case
• Soliton resolution for energy-critical wave maps in the equivariant case
• Soliton resolution for critical co-rotational wave maps and radial cubic wave equation
• Soliton resolution for the focusing modified KdV equation
• Complete integrability of the Benjamin--Ono equation on the multi-soliton manifolds
• Soliton resolution for the radial critical wave equation in all odd space dimensions
• Soliton Resolution for the Derivative Nonlinear Schrödinger Equation
• Long Time Asymptotic Behavior of the Focusing Nonlinear Schrodinger Equation
• Long time asymptotic behavior of the focusing nonlinear Schrödinger equation
• The Cauchy problem for the Benjamin-Ono equation in L² revisited
• Internal waves of permanent form in fluids of great depth

Cited by

• Splitting methods for Intermediate Long Wave and perturbed Benjamin--Ono models