The Cauchy problem for the Degasperis-Procesi Equation: Painlevé Asymptotics in Transition Zones

Abstract

The Degasperis-Procesi (DP) equation \begin{align} &u_t-u_{txx}+3κu_x+4uu_x=3u_x u_{xx}+uu_{xxx}, \nonumber \end{align} serving as an asymptotic approximation for the unidirectional propagation of shallow water waves, is an integrable model of the Camassa-Holm type and admits a $3\times 3$ matrix Lax pair. In our previous work, we obtained the long-time asymptotics of the solution $u(x,t)$ to the Cauchy problem for the DP equation in the solitonic region $\{(x,t):\hat{I}¾>3\}\cup \{(x,t):\hat{I}¾<-\frac{3}{8}\}$ and the solitonless region $\{(x,t):-\frac{3}{8}<\hat{I}¾<0\}\cup \{(x,t):0\le \hat{I}¾<3\}$ where $\hat{I}¾:=\frac{x}{t}$. In this paper, we derive the leading order approximation to the solution $u(x,t)$ in terms of the solution for the Painlevé \uppercase\expandafter{\romannumeral2} equation in two transition zones $\left|\hat{I}¾+\frac{3}{8}\right|t^{2/3}<C$ and $\left|\hat{I}¾-3\right|t^{2/3}<C$ with $C>0$ lying between the solitonic region and solitonless region. Our results are established by performing the $\bar{\partial }$-generalization of the Deift-Zhou nonlinear steepest descent method and applying a double scaling limit technique to an associated vector Riemann-Hilbert problem.

48 pages