Eur. Phys. J. Plus (2020) 135: 607
https://doi.org/10.1140/epjp/s13360-020-00596-1

Regular Article

Kuznetsov–Ma breather-like solutions in the Salerno model

¹ Department of Mathematics and Statistics, University of Massachusetts Amherst, 01003-4515, Amherst, MA, USA
² Mathematics Department, California Polytechnic State University, 93407-0403, San Luis Obispo, CA, USA
³ Grupo de Física No Lineal, Departamento de Física Aplicada I, Universidad de Sevilla. Escuela Politécnica Superior, C/ Virgen de África, 7, 41011, Sevilla, Spain
⁴ Instituto de Matemáticas de la Universidad de Sevilla (IMUS), Edificio Celestino Mutis. Avda. Reina Mercedes s/n, 41012, Sevilla, Spain
⁵ Mathematical Institute, University of Oxford, Oxford, UK
⁶ Department of Mathematics, University of the Aegean, 83200, Karlovassi, Samos, Greece

^b echarala@calpoly.edu

Received: 31 May 2020
Accepted: 7 July 2020
Published online: 28 July 2020

Abstract

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.