Anharmonic Localized Modes in Physical and Biological Systems

Abstract

Recent developments in theoretical and numerical studies of anharmonic lattices have shown the existence of immobile (stationary) localized modes and mobile ones under certain conditions irrespective of the space dimensionalilty of the lattice systems.1 In particular, the ubiquity of the immobile anharmonic localized modes has been shown by the fact that they can exist both in ordered and disordered lattices and molecular systems.2 Conceptually, the anharmonic localized modes, immobile or mobile, can be interpreted as vibrational modes caused by the intrinsic nonlinearity in lattices or molecules with frequencies appearing outside the frequency band or spectra of the corresponding harmonic systems. Namely, they may be regarded as (approximate or exact) normal modes of nonlinear systems. Moreover, it has also been shown that certain mobile anharmonic localized modes go over to mKdV-type solitons in the zero-frequency and zero-wavenumber limit.3 These results may lead to the presumption that the physical concept contained in the anharmonic localized modes may be wider than that of solitons, for which the existence in physically meaningful models has been generally limited to one-dimensional systems, and many soliton-generating models constitute spatially continuous systems, except for a few typical cases such as the Toda lattice. By their nature, the concept of anharmonic localized modes are useful to study the properties of large-amplitude collective modes or highly excited states of microscopic systems with structure characterized by the spatial discreteness. Here, the cost of the conceptual generality is paid by less mathematical rigor and beauty for their theories as compared with the conventional soliton theory.