A Riemann-Hilbert Approach to the Kundu-Eckhaus Equation on the half-Line

TL;DR

This paper applies the Fokas unified transform method to analyze the initial-boundary value problem of the Kundu-Eckhaus equation on the half-line, expressing solutions via a Riemann-Hilbert problem and exploring spectral function relations.

Contribution

It introduces a Riemann-Hilbert problem formulation for the Kundu-Eckhaus equation on the half-line using the Fokas method, including spectral function relations.

Findings

Solution expressed via a matrix Riemann-Hilbert problem

Spectral functions satisfy a global relation

Framework applicable to initial-boundary value problems

Abstract

In this paper, we consider the initial-boundary value problem of the Kundu-Eckhaus equation on the half-line by using of the Fokas unified transform method. Assuming that the solution $u(x,t)$ exists, we show that it can be expressed in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter $\lambda$. Moreover, we also get there exist spectral functions are not independent and they are satisfying the so-called global relation.