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Solitary Waves in the Nonlinear Dirac Equation

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Nonlinear Systems, Vol. 1

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

In the present work, we consider the existence, stability, and dynamics of solitary waves in the nonlinear Dirac equation. We start by introducing the Soler model of self-interacting spinors, and discuss its localized waveforms in one, two, and three spatial dimensions and the equations they satisfy. We present the associated explicit solutions in one dimension and numerically obtain their analogues in higher dimensions. The stability is subsequently discussed from a theoretical perspective and then complemented with numerical computations. Finally, the dynamics of the solutions is explored and compared to its non-relativistic analogue, which is the nonlinear Schrödinger equation.

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Notes

  1. 1.

    With the notation \((N+1)D\) we want to denote that the system possesses \(N+1\) dimensions, with N spatial ones plus time.

  2. 2.

    Notice that this value of N is not related to the dimension of the NLD, although the same symbol is used in both cases.

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Acknowledgements

The research of Andrew Comech was carried out at the Institute for Information Transmission Problems of the Russian Academy of Sciences at the expense of the Russian Foundation for Sciences (project 14-50-00150). J.C.-M. thanks financial support from MAT2016-79866-R project (AEI/FEDER, UE). P.G.K. gratefully acknowledges the support of NSF-PHY-1602994, the Alexander von Humboldt Foundation, the Stavros Niarchos Foundation via the Greek Diaspora Fellowship Program, and the ERC under FP7, Marie Curie Actions, People, International Research Staff Exchange Scheme (IRSES-605096). This work was supported in part by the U.S. Department of Energy.

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  1. Grupo de Física No Lineal, Departamento de Física Aplicada I, Escuela Politécnica Superior, Universidad de Sevilla, C/ Virgen de África, 7, 41011, Sevilla, Spain

    Jesús Cuevas-Maraver

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

    Jesús Cuevas-Maraver

  3. Université Bourgogne Franche-Comté, 25030, Besançon CEDEX, France

    Nabile Boussaïd

  4. Department of Mathematics, Texas A&M University, College Station, TX, 77843-3368, USA

    Andrew Comech & Ruomeng Lan

  5. IITP, Moscow, 127994, Russia

    Andrew Comech

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

    Panayotis G. Kevrekidis

  7. Los Alamos National Laboratory, Center for Nonlinear Studies and Theoretical Division, Los Alamos, NM, 87545, USA

    Avadh Saxena

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  1. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    Victoriano Carmona

  2. Departamento de Física Aplicada I, Universidad de Sevilla, Sevilla, Spain

    Jesús Cuevas-Maraver

  3. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    Fernando Fernández-Sánchez

  4. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    Elisabeth García- Medina

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Cuevas-Maraver, J., Boussaïd, N., Comech, A., Lan, R., Kevrekidis, P.G., Saxena, A. (2018). Solitary Waves in the Nonlinear Dirac Equation. In: Carmona, V., Cuevas-Maraver, J., Fernández-Sánchez, F., García- Medina, E. (eds) Nonlinear Systems, Vol. 1. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-66766-9_4

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