Nonlocal Kundu–Eckhaus equation: integrability, Riemann–Hilbert approach and Cauchy problem with step-like initial data

Abstract

The main purpose of this paper is to discuss the Cauchy problem of integrable nonlocal (reverse-space-time) Kundu–Eckhaus (KE) equation through the Riemann–Hilbert (RH) method. Firstly, based on the zero-curvature equation, we present an integrable nonlocal KE equation and its Lax pair. Then, we discuss the properties of eigenfunctions and scattering matrix, such as analyticity, asymptotic behavior, and symmetry. Finally, for the prescribed step-like initial value: \(u(z,t)=o(1)\), \(z\rightarrow -\infty \) and \(u(z,t)=R+o(1)\), \(z\rightarrow +\infty \), where \(R>0\) is an arbitrary constant, we consider the initial value problem of the nonlocal KE equation. The paramount techniques is the asymptotic analysis of the associated RH problem.