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Boussinesq's equation for water waves : the soliton resolution conjecture for Sector IV

Charlier, Christophe LU and Lenells, Jonatan LU (2025) In Advanced Nonlinear Studies 25(1). p.106-151
Abstract

We consider the Boussinesq equation on the line for a broad class of Schwartz initial data relevant for water waves. In a recent work, we identified ten main sectors describing the asymptotic behavior of the solution, and for each of these sectors we gave an exact expression for the leading asymptotic term in the case when no solitons are present. In this paper, we derive an asymptotic formula in Sector IV, characterized by xt ∈ (√13, 1), in the case when solitons are present. In particular, our results provide an exact expression for the soliton-radiation interaction to leading order and a verification of the soliton resolution conjecture for the Boussinesq equation in Sector IV.

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author
and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Boussinesq equation, long-time asymptotics, Riemann-Hilbert problem
in
Advanced Nonlinear Studies
volume
25
issue
1
pages
46 pages
publisher
De Gruyter
external identifiers
  • scopus:105003027260
ISSN
1536-1365
DOI
10.1515/ans-2023-0154
language
English
LU publication?
yes
id
57eed880-e200-44f2-a646-35b0dbca66f2
date added to LUP
2025-08-26 13:13:08
date last changed
2025-08-26 13:14:19
@article{57eed880-e200-44f2-a646-35b0dbca66f2,
  abstract     = {{<p>We consider the Boussinesq equation on the line for a broad class of Schwartz initial data relevant for water waves. In a recent work, we identified ten main sectors describing the asymptotic behavior of the solution, and for each of these sectors we gave an exact expression for the leading asymptotic term in the case when no solitons are present. In this paper, we derive an asymptotic formula in Sector IV, characterized by <sup>x</sup><sub>t</sub> ∈ (<sup>√1</sup><sub>3</sub>, 1), in the case when solitons are present. In particular, our results provide an exact expression for the soliton-radiation interaction to leading order and a verification of the soliton resolution conjecture for the Boussinesq equation in Sector IV.</p>}},
  author       = {{Charlier, Christophe and Lenells, Jonatan}},
  issn         = {{1536-1365}},
  keywords     = {{Boussinesq equation; long-time asymptotics; Riemann-Hilbert problem}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{106--151}},
  publisher    = {{De Gruyter}},
  series       = {{Advanced Nonlinear Studies}},
  title        = {{Boussinesq's equation for water waves : the soliton resolution conjecture for Sector IV}},
  url          = {{http://dx.doi.org/10.1515/ans-2023-0154}},
  doi          = {{10.1515/ans-2023-0154}},
  volume       = {{25}},
  year         = {{2025}},
}