Two-dimensional flat-band solitons in superhoneycomb lattices.

Flat-band periodic materials are characterized by a linear spectrum containing at least one band where the propagation constant remains nearly constant irrespective of the Bloch momentum across the Brillouin zone. These materials provide a unique platform for investigating phenomena related to light localization. Meantime, the interaction between flat-band physics and nonlinearity in continuous systems remains largely unexplored, particularly in continuous systems where the band flatness deviates slightly from zero, in contrast to simplified discrete systems with exactly flat bands.

Vortex solitons in topological disclination lattices.

The existence of thresholdless vortex solitons trapped at the core of disclination lattices that realize higher-order topological insulators is reported. The study demonstrates the interplay between nonlinearity and higher-order topology in these systems, as the vortex state in the disclination lattice bifurcates from its linear topological counterpart, while the position of its propagation constant within the bandgap and localization can be controlled by its power. It is shown that vortex solitons are characterized by strong field confinement at the disclination core due to their topological nature, leading to enhanced stability.

Observation of Thouless pumping of light in quasiperiodic photonic crystals.

Topological transport is determined by global properties of physical media where it occurs and is characterized by quantized amounts of adiabatically transported quantities. Discovered for periodic potential, it was also explored in disordered and discrete quasiperiodic systems. Here, we report on experimental observation of pumping of a light beam in a genuinely continuous incommensurate photorefractive quasicrystal emulated by its periodic approximants.

Observation of nonlinear fractal higher order topological insulator.

Higher-order topological insulators (HOTIs) are unique materials hosting topologically protected states, whose dimensionality is at least by 2 lower than that of the bulk. Topological states in such insulators may be strongly confined in their corners which leads to considerable enhancement of nonlinear processes involving such states. However, all nonlinear HOTIs demonstrated so far were built on periodic bulk lattice materials.

Antiferromagnetic Ising model in a triangular vortex lattice of quantum fluids of light.

Vortices are topologically distinctive objects appearing as phase twists in coherent fields of optical beams and Bose-Einstein condensates. Structured networks and artificial lattices of coupled vortices could offer a powerful platform to study and simulate interaction mechanisms between constituents of condensed matter systems, such as antiferromagnetic interactions, by replacement of spin angular momentum with orbital angular momentum. Here, we realize such a platform using a macroscopic quantum fluid of light based on exciton-polariton condensates.

Topological solitons in coupled Su-Schrieffer-Heeger waveguide arrays.

We investigate the formation of multipole topological solitons at the edges of two and three coupled parallel Su-Schrieffer-Heeger (SSH) waveguide arrays. We show that independent variations of waveguide spacing in the unit cells (dimers) in coupled waveguide arrays result in the emergence at their edges of several topological edge states with different internal symmetries. The number of emerging edge states is determined by how many arrays are in topologically nontrivial phase.

Stable Vortex Solitons Sustained by Localized Gain in a Cubic Medium.

We propose a simple dissipative system with purely cubic defocusing nonlinearity and nonuniform linear gain that can support stable localized dissipative vortex solitons with high topological charges without the utilization of competing nonlinearities and nonlinear gain or losses. Localization of such solitons is achieved due to an intriguing mechanism when defocusing nonlinearity stimulates energy flow from the ringlike region with linear gain to the periphery of the medium where energy is absorbed due to linear background losses. Vortex solitons bifurcate from linear gain-guided vortical modes with eigenvalues depending on topological charges that become purely real only at specific gain amplitudes.

Robust Three-Dimensional High-Order Solitons and Breathers in Driven Dissipative Systems: A Kerr Cavity Realization.

We present a general approach to excite robust dissipative three-dimensional and high-order solitons and breathers in passively driven nonlinear cavities. Our findings are illustrated in the paradigmatic example provided by an optical Kerr cavity with diffraction and anomalous dispersion, with the addition of an attractive three-dimensional parabolic potential. The potential breaks the translational symmetry along all directions, and impacts the system in a qualitatively unexpected manner: three-dimensional solitons, or light bullets, are the only existing and stable states for a given set of parameters.

Theory of nonlinear corner states in photonic fractal lattices.

We study linear and nonlinear higher-order topological insulators (HOTIs) based on waveguide arrays arranged into Sierpiński gasket and Sierpiński carpet structures, both of which have non-integer effective Hausdorff dimensionality. Such fractal structures possess different discrete rotational symmetries, but both lack transverse periodicity. Their characteristic feature is the existence of multiple internal edges and corners in their optical potential landscape, and the formal absence of an insulating bulk.

Observation of π solitons in oscillating waveguide arrays.

Floquet systems with periodically varying in time parameters enable realization of unconventional topological phases that do not exist in static systems with constant parameters and that are frequently accompanied by appearance of novel types of the topological states. Among such Floquet systems are the Su-Schrieffer-Heeger lattices with periodically-modulated couplings that can support at their edges anomalous π modes of topological origin despite the fact that the lattice spends only half of the evolution period in topologically nontrivial phase, while during other half-period it is topologically trivial. Here, using Su-Schrieffer-Heeger arrays composed from periodically oscillating waveguides inscribed in transparent nonlinear optical medium, we report experimental observation of photonic anomalous π modes residing at the edge or in the corner of the one- or two-dimensional arrays, respectively, and demonstrate a new class of topological π solitons bifurcating from such modes in the topological gap of the Floquet spectrum at high powers.

Observation of nonlinear disclination states.

Introduction of controllable deformations into periodic materials that lead to disclinations in their structure opens novel routes for construction of higher-order topological insulators hosting topological states at disclinations. Appearance of these topological states is consistent with the bulk-disclination correspondence principle, and is due to the filling anomaly that results in fractional charges to the boundary unit cells. So far, topological disclination states were observed only in the linear regime, while the interplay between nonlinearity and topology in the systems with disclinations has been never studied experimentally.

Vortex solitons in moiré optical lattices.

We show that optical moiré lattices enable the existence of vortex solitons of different types in self-focusing Kerr media. We address the properties of such states both in lattices having commensurate and incommensurate geometries (i.e.

Topological edge solitons in χ waveguide arrays.

We address the formation of χ topological edge solitons emerging in a topologically nontrivial phase in Su-Schrieffer-Heeger (SSH) waveguide arrays. We consider edge solitons, whose fundamental frequency (FF) component belongs to the topological gap, while the phase mismatch determines whether the second harmonic (SH) component falls into topological or trivial forbidden gaps of the spectrum for the SH wave. Two representative types of edge solitons are found, one of which is thresholdless and bifurcates from the topological edge state in the FF component, while the other exists above a power threshold and emanates from the topological edge state in the SH wave.

Observation of Linear and Nonlinear Light Localization at the Edges of Moiré Arrays.

We observe linear and nonlinear light localization at the edges and in the corners of truncated moiré arrays created by the superposition of periodic mutually twisted at Pythagorean angles square sublattices. Experimentally exciting corner linear modes in the femtosecond-laser written moiré arrays we find drastic differences in their localization properties in comparison with the bulk excitations. We also address the impact of nonlinearity on the corner and bulk modes and experimentally observe the crossover from linear quasilocalized states to the surface solitons emerging at the higher input powers.

Rotating topological edge solitons.

We address the formation of topological edge solitons in rotating Su-Schrieffer-Heeger waveguide arrays. The linear spectrum of the non-rotating topological array is characterized by the presence of a topological gap with two edge states residing in it. Rotation of the array significantly modifies the spectrum and may move these edge states out of the topological gap.

Two-Dimensional Nonlinear Thouless Pumping of Matter Waves.

We consider theoretically the nonlinear quantized Thouless pumping of a Bose-Einstein condensate loaded in two-dimensional dynamical optical lattices. We encountered three different scenarios of the pumping: a quasilinear one occurring for gradually dispersing wave packets, transport carried by a single two-dimensional soliton, and a multisoliton regime when the initial wave packet splits into several solitons. The scenario to be realized depends on the number of atoms in the initial wave packet and on the strength of the two-body interactions.

Two-dimensional Thouless pumping of light in photonic moiré lattices.

Continuous and quantized transports are profoundly different. The latter is determined by the global rather than local properties of a system, it exhibits unique topological features, and its ubiquitous nature causes its occurrence in many areas of science. Here we report the first observation of fully-two-dimensional Thouless pumping of light by bulk modes in a purpose-designed tilted moiré lattices imprinted in a photorefractive crystal.

Topological Floquet bound states in the continuum.

A honeycomb array of helical waveguides with zigzag-zigzag edges and a refractive index gradient orthogonal to the edges may support Floquet bound states in the continuum (BICs). The gradient of the refractive index leads to strong asymmetry of the Floquet-Bloch spectrum. The mechanism of creation of such Floquet BICs is understood as emergence of crossings and avoided crossings of the branches supported by spatially limited stripe array.

Vortex Solitons in Twisted Circular Waveguide Arrays.

We address the formation of topological states in twisted circular waveguide arrays and find that twisting leads to important differences of the fundamental properties of new vortex solitons with opposite topological charges that arise in the nonlinear regime. We find that such system features the rare property that clockwise and counterclockwise vortex states are nonequivalent. Focusing on arrays with C_{6v} discrete rotation symmetry, we find that a longitudinal twist stabilizes the vortex solitons with the lowest topological charges m=±1, which are always unstable in untwisted arrays with the same symmetry.

Light bullets in moiré lattices.

We predict that photonic moiré lattices produced by two mutually twisted periodic sublattices in a medium with Kerr nonlinearity can support stable three-dimensional (3D) light bullets localized in both space and time. The stability of light bullets and their properties are closely connected with the properties of linear spatial eigenmodes of moiré lattices that undergo localization-delocalization transition (LDT) upon the increase of the depth of one of the sublattices forming the moiré lattice, but only for twist angles corresponding to incommensurate, aperiodic moiré structures. Above the LDT threshold, such incommensurate moiré lattices support stable light bullets without an energy threshold.

Observation of nonlinearity-controlled switching of topological edge states.

We report the experimental observation of the periodic switching of topological edge states between two dimerized fs-laser written waveguide arrays. Switching occurs due to the overlap of the modal fields of the edge states from topological forbidden gap, when they are simultaneously present in two arrays brought into close proximity. We found that the phenomenon occurs for both strongly and weakly localized edge states and that switching rate increases with decreasing spacing between the topological arrays.

Quartic Kerr cavity combs: bright and dark solitons.

We theoretically investigate the dynamics, bifurcation structure, and stability of localized states in Kerr cavities driven at the pure fourth-order dispersion point. Both the normal and anomalous group velocity dispersion regimes are analyzed, highlighting the main differences from the standard second-order dispersion case. In the anomalous regime, single and multi-peak localized states exist and are stable over a much wider region of the parameter space.

Nonlinear Thouless Pumping: Solitons and Transport Breakdown.

One-dimensional topological pumping of matter waves in two overlaid optical lattices moving with respect to each other is considered in the presence of attractive nonlinearity. It is shown that there exists a threshold nonlinearity level above which the matter transfer is completely arrested. Below this threshold, the transfer of both dispersive wave packets and solitons occurs in accordance with the predictions of the linear theory; i.

Observation of Edge Solitons in Topological Trimer Arrays.

We report the experimental observation of nonlinear light localization and edge soliton formation at the edges of fs-laser written trimer waveguide arrays, where transition from nontopological to topological phases is controlled by the spacing between neighboring trimers. We found that, in the former regime, edge solitons occur only above a considerable power threshold, whereas in the latter one they bifurcate from linear states. Edge solitons are observed in a broad power range where their propagation constant falls into one of the topological gaps of the system, while partial delocalization is observed when considerable nonlinearity drives the propagation constant into an allowed band, causing coupling with bulk modes.

Valley Hall edge solitons in a photonic graphene.

We predict the existence and study properties of the valley Hall edge solitons in a composite photonic graphene with a domain wall between two honeycomb lattices with broken inversion symmetry. Inversion symmetry in our system is broken due to detuning introduced into constituent sublattices of the honeycomb structure. We show that nonlinear valley Hall edge states with sufficiently high amplitude bifurcating from the linear valley Hall edge state supported by the domain wall, can split into sets of bright spots due to development of the modulational instability, and that such an instability is a precursor for the formation of topological bright valley Hall edge solitons localized due to nonlinear self-action and travelling along the domain wall over large distances.