The Salerno model is a discrete variant of the celebrated nonlinear Schrödinger (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 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.

中文翻译：

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萨勒诺模型中的库兹涅佐夫–马式呼吸解决方案

Salerno模型是著名的非线性Schrödinger（NLS）方程的离散变体，它通过适当地调整相关的同伦参数而插值在离散NLS（DNLS）方程和完全可积分的Ablowitz–Ladik（AL）模型之间。尽管AL模型具有称为Kuznetsov-Ma（KM）呼吸器的显式时间周期解，但是到目前为止，还没有研究超出可积极限的时间周期解的存在。因此，本工作的目的是阐明萨勒诺模型的时间周期解的存在性和稳定性。特别是，我们通过采用伪弧长连续算法来更改模型的同伦参数，在伪算法中，定点迭代可确定时间周期解。我们表明，解决方案转变为具有小但非衰减的远场振荡的时间周期模式。值得注意的是，我们的数值结果支持了以前未知的时间周期解的存在即使在使用Floquet理论探索稳定性的可积情况下。还讨论了这些模式向DNLS极限的延续。