Soliton resolution and asymptotic stability of $N$-solitons to the Degasperis-Procesi equation on the line

Abstract

The Degasperis-Procesi (DP) equation \begin{align} &u_t-u_{txx}+3\kappa u_x+4uu_x=3u_x u_{xx}+uu_{xxx}, \nonumber \end{align} serving as a model delineating the propagation of shallow water waves, stands as a completely integrable system and admits a $3\times3$ matrix Lax pair. In this manuscript, we study the soliton resolution and large time behavior of solutions to the Cauchy problem of the DP equation with generic initial data in Schwarz space. Employing the $\bar \partial$-generalization of the Deift-Zhou nonlinear steepest descent method, we deduce different long time asymptotic expansions of the solution $u(x,t)$ in two distinct types of space-time regions. This result verifies the soliton resolution conjecture and asymptotic stability of $N$-soliton solutions for the DP equation.