The Cauchy problem for the Degasperis-Procesi Equation: Painlev\'e Asymptotics in Transition Zones

The Degasperis-Procesi (DP) equation {align} &u_t-uₜₓₓ+3κ u_x+4uu_x=3u_x uₓₓ+uuₓₓₓ, {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 3×3 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): ξ>3 \ ∪ \(x,t): ξ<-3/8 \ and the solitonless region \(x,t): -3/8<ξ< 0 \ ∪ \(x,t): 0≤ ξ <3 \ where ξ:=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} {2} equation in two transition zones |ξ+3/8|t2/3<C and |ξ -3|t2/3<C with $C>0$ lying between the solitonic region and solitonless region. Our results are established by performing the ∂-generalization of the Deift-Zhou nonlinear steepest descent method and applying a double scaling limit technique to an associated vector Riemann-Hilbert problem.