Long-time asymptotics for the Wadati-Konno-Ichikawa equation with the Schwartz initial data

TL;DR

This paper analyzes the long-time behavior of solutions to the Wadati-Konno-Ichikawa equation with Schwartz initial data using the nonlinear steepest descent method on Riemann-Hilbert problems, deriving explicit asymptotics.

Contribution

It applies the nonlinear steepest descent method to derive explicit long-time asymptotics for the Wadati-Konno-Ichikawa equation with Schwartz initial data.

Findings

Explicit long-time asymptotic formulas obtained

Solution behavior characterized for large times

Method demonstrates effectiveness for integrable equations

Abstract

In this work, we investigate the long-time asymptotic behavior of the Wadati-Konno-Ichikawa equation with initial data belonging to Schwartz space at infinity by using the nonlinear steepest descent method of Deift and Zhou for the oscillatory Riemann-Hilbert problem. Based on the initial value condition, the original Riemann-Hilbert problem is constructed to express the solution of the Wadati-Konno-Ichikawa equation. Through a series of deformations, the original RH problem is transformed into a model RH problem, from which the long-time asymptotic solution of the equation is obtained explicitly.