Multi-pole solutions and their asymptotic analysis of the focusing Ablowitz-Ladik equation

For the focusing Ablowitz-Ladik equation, the double- and triple-pole solutions are derived from its multi-soliton solutions via some limit technique. Also, the asymptotic analysis is performed for such two multi-pole solutions (MPSs) by considering the balance between exponential and algebraic terms. Like the continuous nonlinear Schrodinger equation, the discrete MPSs describe the elastic interactions of multiple solitons with the same amplitudes. But in contrast to the common multi-soliton solutions, most asymptotic solitons in the MPSs are localized in the curves of the nt plane, and thus they have the time-dependent velocities. In addition, the solitons' relative distances grow logarithmically with vertical bar t vertical bar, while the separation acceleration magnitudes decrease exponentially with their distance.