Band-Theoretical Interpretation of Stationary Localized Modes in Pure Nonlinear Lattices-Ubiquity of Existence and Similarity to Force-Constant-Defect Modes-

Abstract

A lattice Green's function theory is formulated to give a band-theoretical interpretation of stationary nonlinear localized modes (SNLM) in pure nonlinear lattices. In one-dimensional lattices and in its simplest form in higher-dimensional cases, the theory yields the eigenfrequency ω of the SNLM in terms of the frequency ω 0 ( q ) ( q is the wave vector) of the corresponding linear lattice in the form ω=ω 0 ( q 0 + i K ), where q 0 is an edge point of the Brillouin zone of ω 0 ( q ), and K ≡ K ( A ) is a parameter depending on the localized mode amplitude A. Such a result and more general ones involving lattice Green's functions lead to the ubiquity of the existence of the SNLM irrespective of the lattice dimensionality as approximate nonlinear modes in nonintegable lattices and its similarity to force-constant defect modes. Stationary solitons in the Ablowitz-Ladik lattice can be considered as an example of the SNLM in integrable lattices.