The Riemann-Hilbert approach to focusing Kundu-Eckhaus equation with nonzero boundary conditions

TL;DR

This paper applies the Riemann-Hilbert method to analyze the focusing Kundu-Eckhaus equation with nonzero boundary conditions, deriving soliton solutions through spectral analysis and Riemann-Hilbert problem formulation.

Contribution

It introduces a Riemann-Hilbert approach for the focusing Kundu-Eckhaus equation with nonzero boundary conditions, including explicit soliton solutions.

Findings

Derived N-soliton solutions using Riemann-Hilbert problem

Obtained explicit first-order soliton solution

Analyzed spectral properties and asymptotics

Abstract

In this article, we focus on investigating the focusing Kundu-Eckhaus equation with nonzero boundary condition. A appropriate two-sheeted Riemann surface is introduced to map the spectral parameter $k$ into a single-valued parameter $z$. Starting from the Lax pair of Kundu-Eckhaus equation,two kind of Jost solutions are construed. Further their asymptotic, analyticity, symmetries as well as spectral matrix are detailed analyzed. It is shown that the solution of Kundu-Eckhaus equation with nonzero boundary condition can characterized with a matrix Riemann-Hilbert problem. Then a formula of $N$-soliton solutions is derived by solving Riemann-Hilbert problem. As applications, the first-order explicit soliton solution is obtained.