Gap Solitons in Nonlinear Polyatomic Chains

Abstract

Dynamics of gap solitons in various kinds of polyatomic chains are analyzed comprehensively. First, we consider a case that the periodic modulation of the anharmonic lattice parameters is small (shallow grating) and obtain coupled mode models between the forward and backward propagating waves at the Bragg wavenumber. Depending on the period of the chain N (>1) and on the nonlinearity, we derive four types of coupled mode equations. Moving localized solutions for gap solitons are obtained analytically. It is found that owing to the quadratic nonlinearity, static dc waves should be taken into account, which leads to the concept of “ dynamical rectification ”. The theoretical results for gap solitons are checked by numerical simulations. Secondly, we consider the case of large modulation of the lattice parameters (deep grating). We develop a theory of the nonlinearity-induced carrier-wave modulations of the lattice mode dynamics (Bloch wave) and derive the effective nonlinear Schrödinger equation. Numerical simulations of the standing and moving solitons and their collision revealed almost elastic interactions of the gap soliton.