Long time asymptotic for the Wadati-Konno-Ichikawa equation with finite density initial data

Abstract

In this work, we investigate the Cauchy problem of the Wadati-Konno-Ichikawa (WKI) equation with finite density initial data. Employing the $\bar{\partial }$-generalization of Deift-Zhou nonlinear steepest descent method, we derive the long time asymptotic behavior of the solution $q(x,t)$ in space-time soliton region. 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 $t^{-\frac{1}{2}}$ leading order term on continuous spectrum with residual error up to $O(t^{-\frac{3}{4}})$. Our results also confirm the soliton resolution conjecture for the WKI equation with finite density initial data.

40 pages, 5 figures

References in corpus (9)

• Long-time Asymptotics for the NLS equation via dbar methods
• Global well-posedness for the derivative nonlinear Schrödinger equation
• GHW-based Necessary and Sufficient Conditions for Support Constrained Generator Matrices
• Soliton Resolution for the Short-pluse Equation
• On asymptotic approximation of the modified Camassa-Holm equation in different space-time solitonic regions
• Soliton resolution for the Hirota equation with weighted Sobolev initial data
• Soliton resolution for the complex short pulse equation with weighted Sobolev initial data
• Soliton resolution for a coupled generalized nonlinear Schrödinger equations with weighted Sobolev initial data
• Global solution of the Wadati-Konno-Ichikawa equation with small initial data