Spontaneous symmetry breaking in Schr\"{o}dinger lattices with two nonlinear sites

TL;DR

This paper studies discrete Schrödinger lattices with two nonlinear sites, revealing symmetry-breaking bifurcations, stability properties, and dynamic behaviors of localized modes in optical and BEC systems.

Contribution

It provides exact analytical solutions and stability analysis for symmetric, asymmetric, and antisymmetric modes, highlighting the nature of bifurcations in these nonlinear lattices.

Findings

Symmetry-breaking bifurcation is subcritical, becoming supercritical in broad or small ring configurations.

Antisymmetric modes do not undergo bifurcations and have stable and unstable segments.

Unstable asymmetric modes evolve into breathers, restoring symmetry dynamically.

Abstract

We introduce discrete systems in the form of straight (infinite) and ring-shaped chains, with two symmetrically placed nonlinear sites. The systems can be implemented in nonlinear optics (as waveguiding arrays) and BEC (by means of an optical lattice). A full set of exact analytical solutions for symmetric, asymmetric, and antisymmetric localized modes is found, and their stability is investigated in a numerical form. The symmetry-breaking bifurcation (SBB), through which the asymmetric modes emerge from the symmetric ones, is found to be of the subcritical type. It is transformed into a supercritical bifurcation if the nonlinearity is localized in relatively broad domains around two central sites, and also in the ring of a small size, i.e., in effectively nonlocal settings. The family of antisymmetric modes does not undergo bifurcations, and features both stable and unstable portions. The evolution of unstable localized modes is investigated by means of direct simulations. In particular, unstable asymmetric states, which exist in the case of the subcritical bifurcation, give rise to breathers oscillating between the nonlinear sites, thus restoring an effective dynamical symmetry between them.