Sine-Gordon Equation: From Discrete to Continuum
In the present Chapter, we consider two prototypical Klein-Gordon models: the integrable sine- Gordon equation and the non-integrable φ4 model. We focus, in particular, on two of their prototypical solutions, namely the kink-like heteroclinic connections and the time-periodic, exponentially localize...
| Authors: | , , , |
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| Format: | book part |
| Status: | Versión enviada para evaluación y publicación |
| Publication Date: | 2014 |
| Country: | España |
| Institution: | Universidad de Sevilla (US) |
| Repository: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/83341 |
| Online Access: | https://hdl.handle.net/11441/83341 https://doi.org/10.1007/978-3-319-06722-3__2 |
| Access Level: | Open access |
| Keyword: | ϕ4 model Anti-continuum limit Breathers Continuum and discrete models Kinks Klein–Gordon lattices Klein–Gordon PDEs Nanopteron PT-symmetry |
| Summary: | In the present Chapter, we consider two prototypical Klein-Gordon models: the integrable sine- Gordon equation and the non-integrable φ4 model. We focus, in particular, on two of their prototypical 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 PT -symmetric Klein-Gordon field theories and a number of relevant results are touched upon. |
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