Wave transmission in nonlinear lattices

The interplay of nonlinearity with lattice discreteness leads to phenomena and propagation properties quite distinct from those appearing in continuou s nonlinear systems. For a large variety of condensed matter and optics app lications the continuous wave approximation is not appropriate. In the pres ent review we discuss wave transmission properties in one dimensional nonli near lattices. Our paradigmatic equations are discrete nonlinear Schrodinge r equations and their study is done through a dynamical systems approach. W e focus on stationary wave properties and utilize well known results from t he theory of dynamical systems to investigate various aspects of wave trans mission and wave localization. We analyze in detail the more general dynami cal system corresponding to the equation that interpolates between the non- integrable discrete nonlinear Schrodinger equation and the integrable Albow itz-Ladik equation. We utilize this analysis in a nonlinear Kronig-Penney m odel and investigate transmission and band modification properties. We disc uss the modifications that are effected through an electric held and the no nlinear Wannier-Stark localization effects that are induced. Several applic ations are described, such as polarons in one dimensional lattices, semicon ductor superlattices and one dimensional nonlinear photonic band gap system s. (C) 1999 Elsevier Science B.V. All rights reserved.