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Discrete Breathers in \(\phi ^4\) and Related Models

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A Dynamical Perspective on the ɸ4 Model

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 26))

Abstract

In this chapter, we touch upon the wide topic of discrete breathers with a special emphasis on the prototypical system of interest, namely the \(\phi ^4\) model. We start by introducing the model and discussing some of the application areas/motivational aspects of exploring time periodic, spatially localized structures, such as the discrete breathers. Our main emphasis is on the existence, and especially on the stability features of such solutions. We explore their spectral stability numerically, as well as in special limits (such as the vicinity of the so-called anti-continuum limit of vanishing coupling) analytically. We also provide and explore a simple, yet powerful stability criterion involving the sign of the derivative of the energy versus frequency dependence of such solutions. We then turn our attention to nonlinear stability, bringing forth the importance of a topological notion, namely the Krein signature. Furthermore, we briefly touch upon linearly and nonlinearly unstable dynamics of such states. Some special aspects/extensions of such structures are only touched upon, including moving breathers and dissipative variations of the model and some possibilities for future work are highlighted. While this chapter by no means aspires to be comprehensive, we hope that it provides some recent developments (a large fraction of which is not included in time-honored discrete breather reviews) and associated future possibilities.

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Notes

  1. 1.

    These lattices have been traditionally denoted as Fermi–Pasta–Ulam, forgetting the outstanding role of Mary Tsingou who was the person responsible for all the numerical simulations in these first computations of nonlinear lattice dynamics [11].

  2. 2.

    Notice that, strictly speaking, we are dealing with a Neimark-Sacker bifurcation (Hopf bifurcation of periodic orbits). However, through a slight abuse of the relevant terminology, we denote it simply as a Hopf bifurcation.

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Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1809074 (P.G.K.). J.C.-M. thanks financial support from MAT2016-79866-R project (AEI/FEDER, UE). P.G.K. also gratefully acknowledges support from the US-AFOSR under Grant No. FA9550-17-1-0114. We are grateful to J.F.R. Archilla for his useful comments.

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Authors and Affiliations

  1. Grupo de Física No Lineal, Universidad de Sevilla, Departamento de Física Aplicada I, Escuela Politécnica Superior, 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. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, 01003-4515, USA

    Panayotis G. Kevrekidis

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  1. Jesús Cuevas–Maraver
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Correspondence to Jesús Cuevas–Maraver .

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

    Panayotis G. Kevrekidis

  2. Grupo de Física No Lineal, Universidad de Sevilla, Departamento de Física Aplicada I, Escuela Politécnica Superior, Sevilla, Spain

    Jesús Cuevas-Maraver

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Cuevas–Maraver, J., Kevrekidis, P.G. (2019). Discrete Breathers in \(\phi ^4\) and Related Models. In: Kevrekidis, P., Cuevas-Maraver, J. (eds) A Dynamical Perspective on the ɸ4 Model. Nonlinear Systems and Complexity, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-030-11839-6_7

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