Soliton resolution for the Wadati-Konno-Ichikawa equation with weighted Sobolev initial data

TL;DR

This paper uses the $ar{ ext{}}$-steepest descent method to analyze the long-term behavior of solutions to the Wadati-Konno-Ichikawa equation with weighted Sobolev initial data, confirming the soliton resolution conjecture.

Contribution

It proves the soliton resolution conjecture for the WKI equation, detailing the asymptotic decomposition into soliton and radiation components with precise error estimates.

Findings

Confirmed soliton resolution with discrete and continuous spectrum contributions.

Derived $t^{-1/2}$ decay rate for the radiation term.

Established residual error bounds up to $O(t^{-3/4})$.

Abstract

In this work, we employ the $\bar{\partial}$-steepest descent method to investigate the Cauchy problem of the Wadati-Konno-Ichikawa (WKI) equation with initial conditions in weighted Sobolev space $\mathcal{H}(\mathbb{R})$. The long time asymptotic behavior of the solution $q(x,t)$ is derived in a fixed space-time cone $S(y_{1},y_{2},v_{1},v_{2})=\{(y,t)\in\mathbb{R}^{2}: y=y_{0}+vt, ~y_{0}\in[y_{1},y_{2}], ~v\in[v_{1},v_{2}]\}$. Based on the resulting asymptotic behavior, we prove the soliton resolution conjecture of the WKI equation which includes the soliton term confirmed by $N(\mathcal{I})$-soliton on discrete spectrum and the $t^{-\frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-\frac{3}{4}})$.