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 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.

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