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sine-Gordon Equation: From Discrete to Continuum

  • M. Chirilus-BrucknerEmail author
  • C. Chong
  • J. Cuevas-Maraver
  • P. G. Kevrekidis
Chapter
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 10)

Abstract

In the present chapter, we consider two prototypical Klein–Gordon models: the integrable sine-Gordon equation and the non-integrable ϕ 4 model. We focus, in particular, on two of their principal solutions, namely the kink-like heteroclinic connections and the time-periodic, exponentially localized in space breather waveforms. Two limits of the discrete variants of these models are contrasted: on the one side, the analytically tractable original continuum limit, and on the opposite end, the highly discrete, so-called anti-continuum limit of vanishing coupling. Numerical computations are used to bridge these two limits, as regards the existence, stability and dynamical properties of the waves. Finally, a recent variant of this theme is presented in the form of \(\mathcal{P}\mathcal{T}\)-symmetric Klein–Gordon field theories and a number of relevant results are touched upon.

Keywords

ϕ4 model Anti-continuum limit Breathers Continuum and discrete models Kinks Klein–Gordon lattices Klein–Gordon PDEs Nanopteron PT-symmetry 

Notes

Acknowledgements

PGK gratefully acknowledges support from grants NSF-CMMI-1000337, NSF-DMS-1312856, US-AFOSR FA-9550-12-1-0332, BSF-2010239 and from the ERC through an IRSES grant. The work of MCB was partially supported by the Deutsche Forschungsgemeinschaft (DFG) under the grant CH 957/1-1. We acknowledge Aslihan Demirkaya for her technical assistance.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • M. Chirilus-Bruckner
    • 1
    Email author
  • C. Chong
    • 2
  • J. Cuevas-Maraver
    • 3
    • 4
  • P. G. Kevrekidis
    • 2
  1. 1.School of Mathematics & Statistics F07University of SydneySydneyAustralia
  2. 2.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstUSA
  3. 3.Nonlinear Physics Group, Departamento de Física Aplicada IUniversidad de Sevilla. Escuela Politécnica Superior.SevillaSpain
  4. 4.Instituto de Matemáticas de la Universidad de Sevilla (IMUS)SevillaSpain

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