Soliton resolution and asymptotic stability of $N$-loop-soliton solutions for the Ostrovsky-Vakhnenko equation
| Autor: | Ma, Ruihong, Fan, Engui |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | The Ostrovsky-Vakhnenko (OV) equation \begin{align*} &u_{txx}-3\kappa u_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. Comment: 41 pages |
| Databáze: | arXiv |
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