Soliton resolution and asymptotic stability of $N$-loop-soliton solutions for the Ostrovsky-Vakhnenko equation
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913345140948992 |
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| author | Ma, Ruihong Fan, Engui |
| author_facet | Ma, Ruihong Fan, Engui |
| contents | The Ostrovsky-Vakhnenko (OV) equation \begin{align*} &u_{txx}-3κu_x+3u_xu_{xx}+uu_{xxx}=0 \end{align*} is a short wave model of the well-known Degasperis-Procesi equation and admits a $3\times 3$ matrix Lax pair. In this paper, we study the soliton resolution and asymptotic stability of $N$-loop soliton solutions for the OV equation with Schwartz initial data that supports soliton solutions. It is shown that the solution of the Cauchy problem can be characterized via a $3\times 3$ matrix Riemann-Hilbert (RH) problem in a new scale. Further by deforming the RH problem into solvable models with $\bar\partial$-steepest descent method, we obtain the soliton resolution to the OV equation in two space-time regions $x/t>0$ and $x/t<0$. This result also implies that $N$-loop soliton solutions of the OV equation are asymptotically stable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_13657 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Soliton resolution and asymptotic stability of $N$-loop-soliton solutions for the Ostrovsky-Vakhnenko equation Ma, Ruihong Fan, Engui Mathematical Physics The Ostrovsky-Vakhnenko (OV) equation \begin{align*} &u_{txx}-3κu_x+3u_xu_{xx}+uu_{xxx}=0 \end{align*} is a short wave model of the well-known Degasperis-Procesi equation and admits a $3\times 3$ matrix Lax pair. In this paper, we study the soliton resolution and asymptotic stability of $N$-loop soliton solutions for the OV equation with Schwartz initial data that supports soliton solutions. It is shown that the solution of the Cauchy problem can be characterized via a $3\times 3$ matrix Riemann-Hilbert (RH) problem in a new scale. Further by deforming the RH problem into solvable models with $\bar\partial$-steepest descent method, we obtain the soliton resolution to the OV equation in two space-time regions $x/t>0$ and $x/t<0$. This result also implies that $N$-loop soliton solutions of the OV equation are asymptotically stable. |
| title | Soliton resolution and asymptotic stability of $N$-loop-soliton solutions for the Ostrovsky-Vakhnenko equation |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2310.13657 |