Bifurcation of internal solitary wave modes from the essential spectrum

Abstract

The bifurcation of internal modes from the phonon band in models supporting solitary wave solutions is currently one of the exciting phenomena in the field. We will present a number of analytical and semianalytical 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's function method. It is shown that these methods give equivalent results. Moreover, they are corroborated by mixed analytical-numerical computations based on the recently developed discrete Evans's function method. The extent to which these predictions survive to strong discretizations is discussed. The results will be presented in the context of both the sine-Gordon and the ${\ensuremath{\varphi}}^{4}$ models.