Symmetry breaking, coupling management, and localized modes in dual-core discrete nonlinear-Schrödinger lattices

• Published in: Journal of computational and applied mathematics Vol. 235; no. 13; pp. 3883 - 3888
• Main Authors: Susanto, H., Kevrekidis, P.G., Abdullaev, F.Kh, Malomed, Boris A.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Kidlington Elsevier B.V 01.05.2011; Elsevier
• Subjects: Combinatorics. Ordered structures; Discrete nonlinear Schrödinger equation; Discrete solitons; Exact sciences and technology; Global analysis, analysis on manifolds; Linear coupling; Mathematical analysis; Mathematics; Nonlinear algebraic and transcendental equations; Numerical analysis; Numerical analysis. Scientific computation; Order, lattices, ordered algebraic structures; Partial differential equations; Sciences and techniques of general use; Symmetry breaking; Temporal management; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Discrete nonlinear Schrödinger equation; Temporal management; Discrete solitons; Linear coupling; Symmetry breaking; Transcendental equation; Approximation; Schrödinger equation; Equation system; Non linear equation; Numerical analysis; Applied mathematics; Multiscale method; Algebraic equation; Hopf bifurcation; Numerical stability
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2011.01.034

We introduce a system of two linearly coupled discrete nonlinear Schrödinger 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.