Boussinesq’s equation for water waves: the soliton resolution conjecture for Sector IV
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 asymptot...
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De Gruyter
2025-01-01
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| Series: | Advanced Nonlinear Studies |
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| Online Access: | https://doi.org/10.1515/ans-2023-0154 |
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| author | Charlier Christophe Lenells Jonatan |
| author_facet | Charlier Christophe Lenells Jonatan |
| author_sort | Charlier Christophe |
| collection | DOAJ |
| description | 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)
$\frac{x}{t}\in \left(\frac{1}{\sqrt{3}},1\right)$
, 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. |
| format | Article |
| id | doaj-art-e05584c5b2cb423cbdcdb355083a4e2a |
| institution | OA Journals |
| issn | 2169-0375 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advanced Nonlinear Studies |
| spelling | doaj-art-e05584c5b2cb423cbdcdb355083a4e2a2025-08-20T02:17:46ZengDe GruyterAdvanced Nonlinear Studies2169-03752025-01-0125110615110.1515/ans-2023-0154Boussinesq’s equation for water waves: the soliton resolution conjecture for Sector IVCharlier Christophe0Lenells Jonatan1Centre for Mathematical Sciences, Lund University, 22100Lund, SwedenDepartment of Mathematics, KTH Royal Institute of Technology, 10044Stockholm, SwedenWe 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) $\frac{x}{t}\in \left(\frac{1}{\sqrt{3}},1\right)$ , 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.https://doi.org/10.1515/ans-2023-0154boussinesq equationlong-time asymptoticsriemann–hilbert problem35c0835g2535q1537k4076b15 |
| spellingShingle | Charlier Christophe Lenells Jonatan Boussinesq’s equation for water waves: the soliton resolution conjecture for Sector IV Advanced Nonlinear Studies boussinesq equation long-time asymptotics riemann–hilbert problem 35c08 35g25 35q15 37k40 76b15 |
| title | Boussinesq’s equation for water waves: the soliton resolution conjecture for Sector IV |
| title_full | Boussinesq’s equation for water waves: the soliton resolution conjecture for Sector IV |
| title_fullStr | Boussinesq’s equation for water waves: the soliton resolution conjecture for Sector IV |
| title_full_unstemmed | Boussinesq’s equation for water waves: the soliton resolution conjecture for Sector IV |
| title_short | Boussinesq’s equation for water waves: the soliton resolution conjecture for Sector IV |
| title_sort | boussinesq s equation for water waves the soliton resolution conjecture for sector iv |
| topic | boussinesq equation long-time asymptotics riemann–hilbert problem 35c08 35g25 35q15 37k40 76b15 |
| url | https://doi.org/10.1515/ans-2023-0154 |
| work_keys_str_mv | AT charlierchristophe boussinesqsequationforwaterwavesthesolitonresolutionconjectureforsectoriv AT lenellsjonatan boussinesqsequationforwaterwavesthesolitonresolutionconjectureforsectoriv |