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Nonlinear supratransmission in fractional wave systems

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Abstract

In this work, we review various nonlinear systems with fractional derivatives of the Riesz type in space. Concretely, we consider partial differential equations with fractional Laplacians, which consider potentials of the Fermi–Pasta–Ulam, sine-Gordon, Klein–Gordon and double sine-Gordon. These regimes have the important feature that some energy functional is available in the fractional scenario. For each of these cases, we propose numerical techniques to approximate their solutions, and some discrete functionals are provided in order to approximate the energy dynamics of the systems. The methodologies are energy-preserving (conservative) techniques which are employed then to investigate the presence of nonlinear supratransmission in those systems. It is well known that such process is present in continuous Fermi–Pasta–Ulam, sine-Gordon, Klein–Gordon and double sine-Gordon regimes when the derivatives are of integer order. As one of the most important outcomes of this survey, the computer simulations confirm that this process is also present when the derivatives are of fractional order.

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Acknowledgements

The authors want to thank the anonymous reviewers and the associate editor in charge of handling this manuscript, for all the invaluable suggestions and comments.

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Authors and Affiliations

  1. Departamento de Ciencias Exactas y Tecnología, Universidad de Guadalajara, Enrique Díaz de León 1144, Colonia Paseos de la Montaña, Lagos de Moreno, Jalisco, Mexico

    Luis E. Piña-Villalpando & Evguenii Kurmyshev

  2. Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, 20131, Aguascalientes, AGS, Mexico

    Luis E. Piña-Villalpando & Jorge E. Macías-Díaz

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  1. Luis E. Piña-Villalpando
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  2. Jorge E. Macías-Díaz
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Correspondence to Jorge E. Macías-Díaz.

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Piña-Villalpando, L.E., Macías-Díaz, J.E. & Kurmyshev, E. Nonlinear supratransmission in fractional wave systems. J Math Chem 57, 790–811 (2019). https://doi.org/10.1007/s10910-018-0983-5

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  • DOI: https://doi.org/10.1007/s10910-018-0983-5

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