On the long-time asymptotic behavior of the Camassa-Holm equation in space-time solitonic regions

Abstract

In this work, we are devoted to study the Cauchy problem of the Camassa-Holm (CH) equation with weighted Sobolev initial data in space-time solitonic regionswhere κ is a positive constant. Based on the Lax spectrum problem, a Riemann-Hilbert problem corresponding to the original problem is constructed to give the solution of the CH equation with the initial boundary value condition. Furthermore, by developing the ∂-generalization of Deift-Zhou nonlinear steepest descent method, different long-time asymptotic expansions of the solution q(x, t) are derived. Four asymptotic regions are divided in this work: For ξ ∈ −∞, − 1 4 ∪ (2, ∞), the phase function θ(z) has no stationary point on the jump contour, and the asymptotic approximations can be characterized with the soliton term confirmed by N (j0)-soliton on discrete spectrum with residual error up to O(t −1+2τ ); For ξ ∈ − 1 4 , 0 and ξ ∈ (0, 2), the phase function θ(z) has four and two stationary points on the jump contour, and the asymptotic approximations can be characterized with the soliton term confirmed by N (j0)-soliton on discrete spectrum and the t − 1 2 order term on continuous spectrum with residual error up to O(t −1 ). Our results also confirm the soliton resolution conjecture for the CH equation with weighted Sobolev initial data in space-time solitonic regions.