Kuznetsov-Ma breather-like solutions in the Salerno model

TL;DR

This paper investigates the existence and stability of time-periodic solutions in the Salerno model, revealing new solutions even at the integrable limit and analyzing their behavior across different parameter regimes.

Contribution

It introduces a numerical method to find and analyze time-periodic solutions in the Salerno model, including previously unknown solutions at the integrable limit.

Findings

Existence of new time-periodic solutions at the integrable limit.

Solutions exhibit small, non-decaying far-field oscillations.

Stability of solutions analyzed using Floquet theory.

Abstract

The Salerno model is a discrete variant of the celebrated nonlinear Schr\"odinger (NLS) equation interpolating between the discrete NLS (DNLS) equation and completely integrable Ablowitz-Ladik (AL) model by appropriately tuning the relevant homotopy parameter. Although the AL model possesses an explicit time-periodic solution known as the Kuznetsov-Ma (KM) breather, the existence of time-periodic solutions away from the integrable limit has not been studied as of yet. It is thus the purpose of this work to shed light on the existence and stability of time-periodic solutions of the Salerno model. In particular, we vary the homotopy parameter of the model by employing a pseudo-arclength continuation algorithm where time-periodic solutions are identified via fixed-point iterations. We show that the solutions transform into time-periodic patterns featuring small, yet non-decaying far-field oscillations. Remarkably, our numerical results support the existence of previously unknown time-periodic solutions {\it even} at the integrable case whose stability is explored by using Floquet theory. A continuation of these patterns towards the DNLS limit is also discussed.