2022/08/15 by Zhiqiang Li, Li, Zhi-Qiang, Shou‐Fu Tian +3
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fractional Differential Equations Solutions #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2208.07015
openalex publication_date 2022/08/15 · openalex created_date 2022/08/17 · openalex updated_date 2026/07/28
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 regions mt+2κqx+3qqx=2qxqxx+qqxx,~~m=q-qxx+κ,
q(x,0)=q0(x)∈ H4,2(\mathbb R),~~x∈\mathbb R, ~~tgt;0, where 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.