Initial-boundary value problem and long-time asymptotics for the Kundu--Eckhaus equation on the half-line

TL;DR

This paper analyzes the initial-boundary value problem for the Kundu--Eckhaus equation on the half-line, deriving long-time asymptotics using the Fokas method and Riemann--Hilbert problem techniques.

Contribution

It introduces a novel application of the Fokas method combined with nonlinear steepest descent to obtain asymptotic formulas for the Kundu--Eckhaus equation on the half-line.

Findings

Solution expressed via a matrix Riemann--Hilbert problem

Derived precise long-time asymptotic formulas

Established the effectiveness of the Fokas method for boundary value problems

Abstract

The initial-boundary value problem for the Kundu--Eckhaus equation on the half-line is considered in this paper by using the Fokas method. We will show that the solution $u(x,t)$ can be expressed in terms of the solution of a matrix Riemann--Hilbert problem formulated in the complex $k$-plane. Furthermore, based on a nonlinear steepest descent analysis of the associated Riemann--Hilbert problem, we can give the precise asymptotic formulas for the solution of the Kundu--Eckhaus equation on the half-line.