Soliton resolution and asymptotic stability of N-loop-soliton solutions for the Ostrovsky-Vakhnenko equation

2023/10/19 by Ruihong Ma, Ma, Ruihong, Engui Fan +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2310.13657

openalex publication_date 2023/10/19 · openalex created_date 2023/10/24 · openalex updated_date 2026/07/28

Abstract

The Ostrovsky-Vakhnenko (OV) equation amp;u_txx-3κu_x+3u_xu_xx+uu_xxx=0 is a short wave model of the well-known Degasperis-Procesi equation and admits a 3× 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× 3 matrix Riemann-Hilbert (RH) problem in a new scale. Further by deforming the RH problem into solvable models with ∂-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.

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