Standing localized modes in nonlinear lattices

The theory of standing localized modes in discrete nonlinear lattices is presented. We start from a rather general model describing a chain of particles subjected to an external (on-site) potential with cubic and quartic nonlinearities (the so-called Klein-Gordon model), and, using the approximation based on the discrete nonlinear Schro$iumldinger equation, derive a system of two coupled nonlinear equations for slowly varying envelopes of two counterpropagating waves of the same frequency. We show that spatially localized modes exist in the frequencywave number domain where the lattice displays modulational instability; two families of localized modes are found for this case as separatrix solutions of the effective equations for the wave envelopes. When the nonlinear plane wave in the lattice is stable to small modulations of its amplitude, nonlinear localized modes appear as dark solitons associated with the so-called extended modulational instability. These localized modes may be treated as domain walls or kinks connecting two standing plane-wave modes of the similar structure. We investigate analytically and numerically the special family of such localized solutions that, in the vicinity of the zero-dispersion point, cover exactly the case of the so-called self-induced gap solitons recently introduced by Kivshar [Phys. Rev. Lett. 70, 3055 (1993)]. Application of the theory to the case of parametrically driven damped lattices is also briefly discussed, and it is mentioned that some of the solutions considered in the present paper may be extended to include the case of localized modes in driven damped lattices, provided the mode frequency and amplitude are fixed by the parameters of the external parameters of the external parametric ac force.