The Cauchy problem of the Camassa-Holm equation in a weighted Sobolev space: Long-time and Painlevé asymptotics

Based on the (partial derivative) -generalization of the Deift-Zhou steepest descent method, we extend the long-time and Painlev & eacute; asymptotics for the Camassa-Holm (CH) equation to the solutions with initial data in a weighted Sobolev space H-4,H-2(R). With a new scale (y, t) and a Riemann-Hilbert problem associated with the initial value problem, we derive different long time asymptotic expansions for the solutions of the CH equation in different space-time solitonic regions. The half-plane {(y, t) : -infinity < y < infinity, t > 0} is divided into four asymptotic regions: 1. Fast decay region, y/t is an element of (-infinity, -1/4) with an error O(t(-1/2)); 2. Modulation-soliton region, y/t is an element of (2, +infinity), the result can be characterized with an modulation-solitons with residual error O(t(-1/2)); 3. Zakhrov-Manakov region, y/t is an element of (0, 2) and y/t is an element of (-1/4, 0). The asymptotic approximation is characterized by the dispersion term with residual error O(t(-3/4)); 4. Two transition regions, |y/t| approximate to 2 and |y/t| approximate to-1/4, the asymptotic results are described by the solution of Painlev & eacute; II equation with error order O(t(-1/2)).(c) 2023 Elsevier Inc. All rights reserved.