Inverse scattering transform and the soliton solution of the discrete Ablowitz–Ladik equation

Abstract

This paper studies the discrete Ablowitz–Ladik equation via the Riemann-Hilbert (RH) approach. By its matrix spectral problem and Lax pair, the Jost solution and the reflection coefficients are constructed. Based on the zero curvature formulation, we assume that there are higher-order zeros for the scattering coefficient $a\left(\lambda \right)$, and construct the corresponding RH problem. The inverse scattering transform of the discrete Ablowitz–Ladik equation is presented by the matrix spectral problem, the reconstruction formula and the RH problem, which enables us to obtain the multiple-pole solutions. And then the dynamics of one-and two-soliton solutions are discussed and presented graphically. Compared with simple-pole solutions, multiple-pole solutions possess more complicated profiles.