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

Abstract

Based on the ∂‾-generalization of the Deift-Zhou steepest descent method, we extend the long-time and Painlevé asymptotics for the Camassa-Holm (CH) equation to the solutions with initial data in a weighted Sobolev space H4,2(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):−∞<y<∞,t>0} is divided into four asymptotic regions: 1. Fast decay region, y/t∈(−∞,−1/4) with an error O(t−1/2); 2. Modulation-soliton region, y/t∈(2,+∞), the result can be characterized with an modulation-solitons with residual error O(t−1/2); 3. Zakhrov-Manakov region, y/t∈(0,2) and y/t∈(−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|≈2 and |y/t|≈−1/4, the asymptotic results are described by the solution of Painlevé II equation with error order O(t−1/2).