Spontaneous symmetry breaking in Schrödinger lattices with two nonlinear sites

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 Bose-Einstein condensates (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, 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.

One- and two-dimensional solitons supported by singular modulation of quadratic nonlinearity

“…Similar to other models with two separated symmetric peaks of the local χ (2) [22] and χ (3) [11,33,34] nonlinearity strength , the asymmetric solitons are generated from the symmetric ones by the spontaneous-symmetrybreaking bifurcation [27], which occurs with the increase of separation 2∆ between the peaks, if other parameters are fixed. The bifurcation is illustrated by Fig.…”

“…In that case, the strongly concentrated nonlinearity may be described by the delta-function [9,33,35]. Another variety of the strongly localized self-focusing χ (3) nonlinearity is a discrete linear lattice with one [10] or two nonlinear [11] sites embedded into it, or externally coupled to the lattice [12].…”

Transfer and scattering of wave packets by a nonlinear trap

“…In either case, full analytical solutions have been found for symmetric, asymmetric, and antisymmetric localized modes trapped by the set of two δ functions. In addition to that, full symmetric and asymmetric solutions can be found [23] for a similar discrete model, with two cubic-nonlinear sites embedded in a linear host chain, which was introduced (without consideration of spontaneous symmetry breaking) in Refs. [24].…”

Switch between the Types of the Symmetry Breaking Bifurcation in Optically Induced Photorefractive Rotational Double-Well Potential

“…Among many kinds of nonlinear systems, double-well potential (DWP) or dual-core system is the most essential model employed to study the phase transition and SSB of the nonlinear states [2][3][4][5][6][7][8][9]. In DWP system, the important process relating to the phase transition and SSB is symmetric breaking bifurcation (SBB), which determines the process of symmetric states transiting to the asymmetric ones [10][11][12][13][14][15][16][17]. There are two kinds of SBB, subcritical and super critical, which are tantamount to the phase transition of the first and second kind respectively.…”