The closeness of localised structures between the Ablowitz-Ladik lattice and Discrete Nonlinear Schr\"odinger equations II: Generalised AL and DNLS systems

TL;DR

This paper demonstrates that small amplitude localized solutions of the integrable Ablowitz-Ladik lattice persist in nonintegrable generalizations of the system, including power-law and saturable nonlinearities, over significant times.

Contribution

It establishes a closeness result showing the continuous dependence of solutions between integrable and nonintegrable discrete nonlinear Schrödinger systems, supported by numerical simulations.

Findings

Small amplitude solutions persist in nonintegrable systems over large times

Numerical simulations confirm analytical predictions

Peregrine soliton behavior is preserved in generalizations

Abstract

The Ablowitz-Ladik system, being one of the few integrable nonlinear lattices, admits a wide class of analytical solutions, ranging from exact spatially localised solitons to rational solutions in the form of the spatiotemporally localised discrete Peregrine soliton. Proving a closeness result between the solutions of the Ablowitz-Ladik and a wide class of Discrete Nonlinear Schr\"odinger systems in a sense of a continuous dependence on their initial data, we establish that such small amplitude waveforms may be supported in the nonintegrable lattices, for significant large times. The nonintegrable systems exhibiting such behavior include a generalisation of the Ablowitz-Ladik system with a power-law nonlinearity and the Discrete Nonlinear Schr\"odinger with power-law and saturable nonlinearities. The outcome of numerical simulations illustrates in an 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.