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 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 $\{ (x,t): \xi>3 \} \cup \{ (x,t): \xi<-\frac{ 3}{ 8} \}$ and the solitonless region $\{ (x,t): -\frac{ 3}{ 8}<\xi< 0 \} \cup \{ (x,t): 0\leq \xi <3 \}$ where $\xi:=\frac{ x}{ t}$. In this paper, we derive the leading order approximation to the solution $u(x,t)$ in terms of the solution for the Painlev\'{ e} \uppercase\expandafter{ \romannumeral2} equation in two transition zones $\left|\xi+\frac{ 3}{ 8}\right|t^{ 2/3}<C$ and $\left|\xi -3\right|t^{ 2/3}<C$ 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.
Categories: math.AP