Soliton resolution for the Ostrovsky–Vakhnenko equation

Abstract

We consider the Cauchy problem of the Ostrovsky–Vakhnenko (OV) equation expressed in the new variables $(y,\tau )$ $q^3-q{(logq)}_{y\tau }-1=0$ with Schwartz initial data $q_0\left(y\right)>0$ which supports smooth and single-valued solitons. It is shown that the solution to the Cauchy problem for the OV equation can be characterized by a 3 × 3 matrix Riemann–Hilbert (RH) problem. Furthermore, by employing the $\bar{\partial }$-steepest descent method to deform the RH problem into solvable models, we derive the soliton resolution for the OV equation across two space–time regions: $y/\tau >0$ and $y/\tau <0$. This result also implies that the $N$-soliton solutions of the OV equation in variables $(y,\tau )$ are asymptotically stable.