We consider a one-dimensional discrete nonlinear Schr & ouml;dinger (DNLS) model with Kerr-type on-site nonlinearity, where the nearest-neighbor coupling constants take two different values ordered in a three-periodic sequence. The existence of localized edge states in the linear limit (Su-Schrieffer-Heeger trimer) is known to depend on the precise location of the edge. Here, we show that for a termination that does not support linear edge states, an arbitrarily weak on-site nonlinearity will induce an edge mode with asymptotic exponential localization, bifurcating from a linear band edge. Close to the gap edge, the shape of the mode can be analytically described in a continuum approximation as one half of a standard gap soliton. The linear stability properties of nonlinear edge modes are also discussed.

We report new methods to control the soliton generation in ring microresonators via the soliton blockade and symmetry breaking in the bidirectionally pumped and coupled ring systems.

We consider a two-dimensional octagonal-diamond network with a fine-tuned diagonal coupling inside the diamond-shaped unit cell. Its linear spectrum exhibits coexistence of two dispersive bands (DBs) and two flat bands (FBs), touching one of the DBs embedded between them. Analogous to the kagome lattice, one of the FBs will constitute the ground state of the system for a proper sign choice of the Hamiltonian. The system is characterized by two different flat-band fundamental octagonal compactons, originating from the destructive interference of fully geometric nature. In the presence of a nonlinear amplitude (on-site) perturbation, the singleoctagon linear modes continue into one-parameter families of nonlinear compact modes with the same amplitude and phase structure. However, numerical stability analysis indicates that all strictly compact nonlinear modes are unstable, either purely exponentially or with oscillatory instabilities, for weak and intermediate nonlinearities and sufficiently large system sizes. Stabilization may appear in certain ranges for finite systems and, for the compacton originating from the band at the spectral edge, also in a regime of very large focusing nonlinearities. In contrast to the kagome lattice, the latter compacton family will become unstable already for arbitrarily weak defocusing nonlinearity for large enough systems. We show analytically the existence of a critical system size consisting of 12 octagon rings, such that the ground state for weak defocusing nonlinearity is a stable single compacton for smaller systems, and a continuation of a nontrivial, noncompact linear combination of single compacton modes for larger systems. Investigating generally the different nonlinear localized (noncompact) mode families in the semi-infinite gap bounded by this FB, we find that, for increasing (defocusing) nonlinearity the stable ground state will continuously develop into an exponentially localized mode with two main peaks in antiphase. At a critical nonlinearity strength a symmetry-breaking pitchfork bifurcation appears, so that the stable ground state is single peaked for larger defocusing nonlinearities. We also investigate numerically the mobility of localized modes in this regime and find that the considered modes are generally immobile both with respect to axial and diagonal phase-gradient perturbations.

We consider a system of generalized coupled Discrete Nonlinear Schrödinger (DNLS) equations, derived as a tight-binding model from the Gross-Pitaevskii-type equations describing a zigzag chain of weakly coupled condensates of exciton-polaritons with spin-orbit (TE-TM) coupling. We focus on the simplest case when the angles for the links in the zigzag chain are ±π/4 with respect to the chain axis, and the basis (Wannier) functions are cylindrically symmetric (zero orbital angular momenta). We analyze the properties of the fundamental nonlinear localized solutions, with particular interest in the discrete gap solitons appearing due to the simultaneous presence of spin–orbit coupling and zigzag geometry, opening a gap in the linear dispersion relation. In particular, their linear stability is analyzed. We also find that the linear dispersion relation becomes exactly flat at particular parameter values, and obtain corresponding compact solutions localized on two neighboring sites, with spin-up and spin-down parts π/2 out of phase at each site. The continuation of these compact modes into exponentially decaying gap modes for generic parameter values is studied numerically, and regions of stability are found to exist in the lower or upper half of the gap, depending on the type of gap modes.

We examine the existence and properties of certain discrete breathers for a discrete nonlinear Schrodinger model where all but one site are placed in a ring and coupled to the additional central site. The discrete breathers we focus on are stationary solutions mainly localized on one or a few of the ring sites and possibly also the central site. By numerical methods, we trace out and study the continuous families the discrete breathers belong to. Our main result is the discovery of a split bifurcation at a critical value of the coupling between neighboring ring sites. Below this critical value, families form closed loops in a certain parameter space, implying that discrete breathers with and without central-site occupation belong to the same family. Above the split bifurcation the families split up into several separate ones, which bifurcate with solutions with constant ring amplitudes. For symmetry reasons, the families have different properties below the split bifurcation for even and odd numbers of sites. It is also determined under which conditions the discrete breathers are linearly stable. The dynamics of some simpler initial conditions that approximate the discrete breathers are also studied and the parameter regimes where the dynamics remain localized close to the initially excited ring site are related to the linear stability of the exact discrete breathers.

The question whether a nonlinear localized mode (discrete soliton/breather) can be mobile in a lattice has a standard interpretation in terms of the Peierls-Nabarro (PN) potential barrier. For the most commonly studied cases, the PN barrier for strongly localized solutions becomes large, rendering these essentially immobile. Several ways to improve the mobility by reducing the PN-barrier have been proposed during the last decade, and the first part gives a brief review of such scenarios in 1D and 2D. We then proceed to discuss two recently discovered novel mobility scenarios. The first example is the 2D Kagome lattice, where the existence of a highly degenerate, flat linear band allows for a very small PN-barrier and mobility of highly localized modes in a small-power regime. The second example is a 1D waveguide array in an active medium with intrinsic (saturable) gain and damping, where exponentially localized, travelling discrete dissipative solitons may exist as stable attractors. Finally, using the framework of an extended Bose-Hubbard model, we show that while quantum fluctuations destroy the mobility of slowly moving, strongly localized classical modes, coherent mobility of rapidly moving states survives even in a strongly quantum regime

We examine the existence and properties of Charge Flipping Vortices (CFVs), rtices which periodically flip the topological charge, in three-site (trimer) and six-site (hexamer) discrete nonlinear Schrödinger (DNLS) lattices. We demonstrate numerically that CFVs exist as exact quasiperiodic solutions in continuous families which connect two different stationary solutions without topological charge, and that it is possible to interpret the dynamics of certain CFVs as the result of perturbations of these stationary solutions. The CFVs are calculated with high numerical accuracy and we may therefore accurately determine many of their properties, such as their energy and linear stability, and the CFVs are found to be stable over large parameter regimes.

We also show that, like in earlier studies for lattices with a multiple of four sites, trimer and hexamer CFVs can be obtained by perturbing stationary constant amplitude vortices with certain linear eigenmodes. However, in contrast to the former case where the perturbation could be infinitesimal, the magnitude of the perturbations for trimers and hexamers must overcome a quite large threshold value. These CFVs may be interpreted as exact quasiperiodic CFVs, with a small perturbation applied to. The concept of a charge flipping energy barrier is introduced and discussed.

REFERENCES

[1]P. Jason, M. Johansson, Phys. Rev. E. 91, 022910 (2015).

We examine the existence and properties of charge flipping vortices (CFVs), vortices which periodically flip the topological charge, in three-site (trimer) and six-site (hexamer) discrete nonlinear Schrodinger lattices. We demonstrate numerically that CFVs exist as exact quasiperiodic solutions in continuous families which connect two different stationary solutions without topological charge, and that it is possible to interpret the dynamics of certain CFVs as the result of perturbations of these stationary solutions. The CFVs are calculated with high numerical accuracy and we may therefore accurately determine many of their properties, such as their energy and linear stability, and the CFVs are found to be stable over large parameter regimes. We also show that, like in earlier studies for lattices with a multiple of four sites, trimer and hexamer CFVs can be obtained by perturbing stationary constant amplitude vortices with certain linear eigenmodes. However, in contrast to the former case where the perturbation could be infinitesimal, the magnitude of the perturbations for trimers and hexamers must overcome a quite large threshold value. These CFVs may be interpreted as exact quasiperiodic CFVs, with a small perturbation applied. The concept of a charge flipping energy barrier is introduced and discussed.

We discuss the properties of nonlinear localized modes in sawtooth lattices, in the framework of a discrete nonlinear Schrodinger modelwith general on-site nonlinearity. Analytic conditions for existence of exact compact three-site solutions are obtained, and explicitly illustrated for the cases of power-law (cubic) and saturable nonlinearities. These nonlinear compact modes appear as continuations of linear compact modes belonging to a flat dispersion band. While for the linear system a compact mode exists only for one specific ratio of the two different coupling constants, nonlinearity may lead to compactification of otherwise noncompact localized modes for a range of coupling ratios, at some specific power. For saturable lattices, the compactification power can be tuned by also varying the nonlinear parameter. Introducing different on-site energies and anisotropic couplings yields further possibilities for compactness tuning. The properties of strongly localized modes are investigated numerically for cubic and saturable nonlinearities, and in particular their stability over large parameter regimes is shown. Since the linear flat band is isolated, its compact modes may be continued into compact nonlinear modes both for focusing and defocusing nonlinearities. Results are discussed in relation to recent realizations of sawtooth photonic lattices.

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