Discrete Breathers in Nonlinear Lattices: A Review and Recent Results

Abstract

Localization phenomena in systems of many (often infinite) degrees of freedom have attracted attention in solid state physics, nonlinear optics, superconductivity and quantum mechanics. The type of localization we are concerned with here is dynamic and refers to oscillations occurring not because of the presence of some defect, but due to the interaction between nonlinearity and resonances. In particular, we shall describe an entity called discrete breathers, which represent localized periodic oscillations in nonlinear lattices. As suggested by other authors in this volume, this type of behavior may be observed in density fluctuations of stars rotating in a galaxy in the discrete or continuum approximation. Since the reader may not be too familiar with these concepts, we have chosen first to review the history of discrete breathers in the second half of last century and then present an account of our recent results on the efficient computation of breathers in multi-dimensional lattices using homoclinic orbits. This allows us to make a much more detailed study and classification of discrete breathers than had previously been possible, as well as accurately follow their existence and stability properties as certain physical parameters of the problem are varied.