Duality of the Existence of Localized Modes (Low- and High-Frequency Modes) in Pure Nonlinear Lattices ---Sine Lattices, Discrete Sine-Gordon Lattices, the Ablowitz-Ladik Lattices and One-Dimensional Latticeswith Quartic Anharmonicity---

Abstract

Dual nature of the existence of localized modes in pure nonlinear lattices is shown as the intrinsic properties of the lattices that cannot be shared with their continuum counterpart. This is a coexistence, for given nonlinearity, of low-and high-frequency localized modes appearing above and below, respectively, of the frequency band of the corresponding linear lattices. As a typical example, this is illustrated for breatherlike mode solutions to sine-lattice equations and discrete sine-Gordon equations using approximate analytical solutions. The duality concept is put into firmer basis by showing the coexistence of such two types of localized modes for exact envelope soliton solutions to the Ablowitz-Ladik equations and for approximate analytical solutions with numerical testing for moving nonlinear localized modes in one-dimensional lattices with hard quartic anharmonicity.