Symmetry breaking, coupling management, and localized modes in dual-core discrete nonlinear-Schr\"{o}dinger lattices

TL;DR

This paper studies coupled discrete nonlinear Schrödinger equations with time-modulated coupling, revealing stability regions, symmetry-breaking bifurcations, and the effectiveness of averaging methods in modeling localized modes in bimodal BECs.

Contribution

It introduces a novel averaged model for coupled DNLSEs with modulated coupling, analyzing stability and bifurcations of localized modes in bimodal BEC systems.

Findings

Identified stability regions for symmetric and in-phase modes.

Discovered symmetry-breaking bifurcation leading to asymmetric modes.

Validated averaging method as an accurate approximation for the original system.

Abstract

We introduce a system of two linearly coupled discrete nonlinear Schr\"{o}dinger equations (DNLSEs), with the coupling constant subject to a rapid temporal modulation. The model can be realized in bimodal Bose-Einstein condensates (BEC). Using an averaging procedure based on the multiscale method, we derive a system of averaged (autonomous) equations, which take the form of coupled DNLSEs with additional nonlinear coupling terms of the four-wave-mixing type. We identify stability regions for fundamental onsite discrete symmetric solitons (single-site modes with equal norms in both components), as well as for two-site in-phase and twisted modes, the in-phase ones being completely unstable. The symmetry-breaking bifurcation, which destabilizes the fundamental symmetric solitons and gives rise to their asymmetric counterparts, is investigated too. It is demonstrated that the averaged equations provide a good approximation in all the cases. In particular, the symmetry-breaking bifurcation, which is of the pitchfork type in the framework of the averaged equations, corresponds to a Hopf bifurcation in terms of the original system.