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Kuznetsov–Ma breather-like solutions in the Salerno model

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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.

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Acknowledgements

J.C.-M. was supported by MAT2016-79866-R project (AEI/FEDER, UE) and by project P18-RT-3480 (Regional Government of Andalusia). PGK acknowledges from the U.S. National Science Foundation under Grants Nos. PHY-1602994 and DMS-1809074. He also acknowledges support from the Leverhulme Trust via a Visiting Fellowship and the Mathematical Institute of the University of Oxford for its hospitality during part of this work.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA, 01003-4515, USA

    J. Sullivan & P. G. Kevrekidis

  2. Mathematics Department, California Polytechnic State University, San Luis Obispo, CA, 93407-0403, USA

    E. G. Charalampidis

  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

    J. Cuevas-Maraver

  4. Instituto de Matemáticas de la Universidad de Sevilla (IMUS), Edificio Celestino Mutis. Avda. Reina Mercedes s/n, 41012, Sevilla, Spain

    J. Cuevas-Maraver

  5. Mathematical Institute, University of Oxford, Oxford, UK

    P. G. Kevrekidis

  6. Department of Mathematics, University of the Aegean, 83200, Karlovassi, Samos, Greece

    N. I. Karachalios

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  1. J. Sullivan
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Correspondence to E. G. Charalampidis.

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Sullivan, J., Charalampidis, E.G., Cuevas-Maraver, J. et al. Kuznetsov–Ma breather-like solutions in the Salerno model. Eur. Phys. J. Plus 135, 607 (2020). https://doi.org/10.1140/epjp/s13360-020-00596-1

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