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

2024/09/03 by Zhaoyu Wang, Xuan Zhou, Wang, Zhaoyu +3
Mathematics · Physics and Astronomy · #33E17 #34M55 #35B40 #35Q15 #35Q53 #37K15 #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #FOS: Mathematics #Quantum chaos and dynamical systems #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2409.01505

openalex publication_date 2024/09/03 · openalex created_date 2024/09/29 · openalex updated_date 2026/07/28

Abstract

The Degasperis-Procesi (DP) equation amp;u_t-u_txx+3κu_x+4uu_x=3u_x u_xx+uu_xxx, 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 Painlevé \uppercase\expandafter\romannumeral2 equation in two transition zones |ξ+(3)/(8)|t^2/30 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.

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