On the soliton resolution and the asymptotic stability of N-soliton solution for the Wadati-Konno-Ichikawa equation with finite density initial data in space-time solitonic regions

Abstract

In this work, we investigate the Cauchy problem of the Wadati-Konno-Ichikawa (WKI) equation with finite density initial data in space-time solitonic regions,iqt+(q1+|q|2)xx=0,q(x,0)=q0(x),limx→±∞⁡q0(x)=q±,q0(x)−q±∈H4,4(R). By analyzing the Lax spectrum problem, the solution of the original problem is convert to the solution of the corresponding Riemann-Hilbert problem for the first time with the initial boundary value condition. By developing the ∂¯-generalization of Deift-Zhou nonlinear steepest descent method, we derive the leading order approximation to the solution q(x,t) in soliton region of space-time, −(yΦ02t)=z0 for any z0∈R, and give bounds for the error decaying as |t|→∞. Based on the resulting asymptotic behavior, the asymptotic approximation of the WKI equation is characterized with the soliton term confirmed by N(I)-soliton on discrete spectrum and the t−12 leading order term on continuous spectrum with residual error up to O(t−34). Our results also confirm the soliton resolution conjecture and the asymptotic stability of N-soliton solution for the WKI equation with finite density initial data.