Domain walls in two-component dynamical lattices.

Abstract

We introduce domain-wall (DW) states in the bimodal discrete nonlinear Schrödinger equation, in which the modes are coupled by cross-phase modulation (XPM). The results apply to an array of nonlinear optical waveguides carrying two different polarizations of light, or two different wavelengths, with anomalous intrinsic diffraction controlled by direction of the light beam, and to a string of drops of a binary Bose-Einstein condensate, trapped in an optical lattice. By means of continuation from various initial patterns taken in the anticontinuum (AC) limit, we find a number of different solutions of the DW type, for which different stability scenarios are identified. In the case of strong-XPM coupling, DW configurations contain a single mode at each end of the chain. The most fundamental solution of this type is found to be always stable. Another solution, which is generated by a different AC pattern, demonstrates behavior which is unusual for nonlinear dynamical lattices: it is unstable for small values of the coupling constant C (which measures the ratio of the nonlinearity and coupling lengths), and becomes stable at larger C. Stable bound states of DWs are also found. DW configurations generated by more sophisticated AC patterns are identified as well, but they are either completely unstable, or are stable only at small values of C. In the case of weak XPM, a natural DW solution is the one which contains a combination of both polarizations, with the phase difference between them 0 and pi at the opposite ends of the lattice. This solution is unstable at all values of C, but the instability is very weak for large C, indicating stabilization as the continuum limit is approached. The stability of DWs is also verified by direct simulations, and the evolution of unstable DWs is simulated too; in particular, it is found that, in the weak-XPM system, the instability may give rise to a moving DW. The DW states can be observed experimentally in the same parameter range where discrete solitons have been found in the lattice setting.