Bifurcation of Internal Solitary Wave Modes from the Essential Spectrum

The bifurcation of internal modes from the phonon band in models supporting solitary wave solutions has been one of the exciting new phenomena in the field. We will present a number of analytical and semi-analytical techniques for the detection, study and understanding of these modes. We will see how they appear, without threshold, due to the discretization of the continuum equations. This perturbation is viewed in terms of a singular continuum approximation and analyzed by both perturbation theory and the Evans’ function method. It is shown that these methods give equivalent results. Moreover, they are corroborated by mixed analytical-numerical computations based on the discrete Evans’ function recently developed by the first author. The extent to which these predictions survive to strong discretizations are discussed. The results will be presented in the context of both the sine-Gordon and the φ models. PACS: 45.10.Hj,63.20.Pw,45.10.-b