Christopher Chong

Standing generalized modulating pulse solutions for a nonlinear wave equation in periodic media

Standing modulating pulse solutions consist of a standing pulse-like envelope modulating an underlying spatially and temporarily oscillating carrier wave. Using spatial dynamics, invariant manifold theory and normal form theory for periodic systems we construct such solutions on large domains in time and space for a nonlinear wave equation with spatially periodic coefficients. Such solutions play an important role in theoretical scenarios where photonic crystals are used as optical storage.

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.