The closeness of localized structures between the Ablowitz-Ladik lattice and discrete nonlinear Schrödinger equations: Generalized AL and DNLS systems

Abstract

The Ablowitz-Ladik system, being one of the few integrable nonlinear lattices, admits a wide class of analytical solutions, ranging from exact spatially localized solitons to rational solutions in the form of the spatiotemporally localized discrete Peregrine soliton. Proving a closeness result between the solutions of the Ablowitz-Ladik system and a wide class of Discrete Nonlinear Schrödinger systems in a sense of a continuous dependence on their initial data, we establish that such small amplitude waveforms may be supported in nonintegrable lattices for significantly large times. Nonintegrable systems exhibiting such behavior include a generalization of the Ablowitz-Ladik system with power-law nonlinearity and the discrete nonlinear Schrödinger equation with power-law and saturable nonlinearities. The outcome of numerical simulations illustrates, in excellent agreement with the analytical results, the persistence of small amplitude Ablowitz-Ladik analytical solutions in all the nonintegrable systems considered in this work, with the most striking example being that of the Peregine soliton. © 2022 Author(s).