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

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 $3\times3$ 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 $\{(x,t): ξ>3 \} \cup \{(x,t): ξ<-\frac{3}{8} \}$ and the solitonless region $\{(x,t): -\frac{3}{8}<ξ< 0 \} \cup \{(x,t): 0\leq ξ<3 \}$ where $ξ:=\frac{x}{t}$. 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 $\left|ξ+\frac{3}{8}\right|t^{2/3}<C$ and $\left|ξ-3\right|t^{2/3}<C$ 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.