Soliton collisions of a discrete Ablowitz–Ladik equation with variable coefficients for an electrical/optical system

Abstract

A discrete Ablowitz–Ladik equation with variable coefficients, which can describe certain phenomena in an electrical/optical system, is analytically studied in this paper. Bright one- and two-soliton solutions are derived from the bilinear forms for such an equation. Soliton propagation and collision are graphically presented and analyzed with the choice of the functions \(\upsilon _{n}(t)\), \(\gamma (t)\) and \(\Lambda (t)\), which are respectively the space-time modulated inhomogeneous frequency shift, time-modulated effective gain/loss term and coefficient of the tunnel coupling between sites, where n and t are the lattice site and scaling time, respectively. With \(\upsilon _{n}(t)\), \(\gamma (t)\) and \(\Lambda (t)\) being the constants, the one soliton is shown to maintain its original amplitude and width during the propagation, and head-on collision between the two solitons is graphically illustrated with the amplitude of each soliton unchanging during the collision. Amplitudes and travelling directions of the one and two solitons are seen to be influenced by \(\gamma (t)\) and \(\Lambda (t)\), respectively. It is shown that \(\upsilon _{n}(t)\) does not affect the propagation and collision features of the solitons.