Inverse scattering transform for an integrable inhomogeneous time shifted nonlocal lattice

Abstract

In this paper, we develop the inverse scattering transform for an inhomogeneous time shifted nonlocal lattice equation, also known as the inhomogeneous discrete time shifted nonlocal nonlinear Schrödinger equation, and obtain the multi-type soliton solutions under zero boundary condition. According to the analyticity, asymptotic behavior and symmetries of eigenfunctions and scattering coefficients as well as the distribution of discrete eigenvalues, the Riemann-Hilbert problem is constructed and solved and the reconstruction formula of potential used to construct soliton solutions under the reflectionless condition is derived. Specially, according to the time shifted nonlocal condition, we deduce the specific constraint on the real function $\gamma \left(t\right)$ in the equation, which is a constraint on the time evolution of the scattering coefficients and eigenfunctions.