Continuous families of non-Hermitian surface solitons.

We show that surface solitons form continuous families in one-dimensional complex optical potentials of a certain shape. This result is illustrated by non-Hermitian gap-surface solitons at the interface between a uniform conservative medium and a complex periodic potential. Surface soliton families are parameterized by a real propagation constant. The range of possible propagation constants is constrained by the relation between the continuous spectrum of the uniform medium and the bandgap structure of the periodic potential.

Nonlinear Schrödinger equations with amplitude-dependent Wadati potentials.

Complex Wadati-type potentials of the form V(x)=-w^{2}(x)+iw_{x}(x), where w(x) is a real-valued function, are known to possess a number of intriguing features, unusual for generic non-Hermitian potentials. In the present paper, we introduce a class of nonlinear Schrödinger-type problems which generalize the Wadati potentials by assuming that the base function w(x) depends not only on the transverse spatial coordinate, but also on the amplitude of the field. Several examples of prospective physical relevance are discussed, including models with the nonlinear dispersion or with the derivative nonlinearity. The numerical study indicates that the generalized model inherits the remarkable features of standard Wadati potentials, such as the existence of continuous soliton families, the possibility of symmetry-breaking bifurcations when the model obeys the parity-time symmetry, the existence of constant-amplitude waves, and the eigenvalue quartets in the linear-instability spectra. Our results deepen the current understanding of the interplay between nonlinearity and non-Hermiticity and expand the class of systems which enjoy the exceptional combination of properties unusual for generic dissipative nonlinear models.

Universal form of arrays with spectral singularities.

An array of non-Hermitian optical waveguides can operate as a laser or as a coherent perfect absorber, which corresponds to a spectral singularity of the underlying discrete complex potential. We show that all lattice potentials with spectral singularities are characterized by the universal form of the gain-and-loss distribution. Using this result, we systematically construct potentials characterized by several spectral singularities at arbitrary wavelengths, as well as potentials with second-order spectral singularities in their spectra. Higher-order spectral singularities demonstrate a greatly enhanced response to incident beams, resulting in the excitation of high-intensity lasing modes.

Rotating patterns in polariton condensates in ring-shaped potentials under a bichromatic pump.

We consider a polariton condensate in a microcavity driven by a bichromatic resonant pump formed by two vortical laser beams carrying different topological charges. The system is additionally confined in a ring-shaped potential. We show that in this system, steadily rotating nonlinear localized modes can be excited, whose angular rotation frequency is determined by optical frequencies and topological charges of the pump beams. When pump frequencies approach eigenfrequencies of the modes of the ring potential, resonant growth of peak amplitude of the excited states occurs. Repulsive polariton-polariton interactions lead to tilting of the resonance curves and appearance of bistability of rotating patterns.

Stable Multiring and Rotating Solitons in Two-Dimensional Spin-Orbit-Coupled Bose-Einstein Condensates with a Radially Periodic Potential.

We consider two-dimensional spin-orbit-coupled atomic Bose-Einstein condensate in a radially periodic potential. The system supports different types of stable self-sustained states including radially symmetric vorticity-carrying modes with different topological charges in two spinor components that may have multiring profiles and at the same time remain remarkably stable for repulsive interactions. Solitons of the second type show persistent rotation with constant angular frequency. They can be stable for both repulsive and attractive interatomic interactions. Because of the inequivalence between clockwise and counterclockwise rotation directions introduced by spin-orbit coupling, the properties of such solitons strongly differ for positive and negative rotation frequencies. The collision of solitons located in the same or different rings is accompanied by a change of the rotation frequency that depends on the phase difference between colliding solitons.

Spectral singularities of a potential created by two coupled microring resonators.

Two microring resonators, one with gain and one with loss, coupled to each other and to a bus waveguide, create an effective non-Hermitian potential for light propagating in the waveguide. Due to geometry, the coupling for each microring resonator yields two counter-propagating modes with equal frequencies. We show that such a system enables implementation of many types of scattering peculiarities. The spectral singularities, which are either the second or fourth order, separate parameter regions where the spectrum is either purely real or composed of complex eigenvalues; hence, they represent the points of the phase transition. By modifying the gain-loss relation for the resonators, such an optical scatterer can act as a laser, as a coherent perfect absorber, be unidirectionally reflectionless or transparent, and support bound states either growing or decaying in time. These characteristics are observed for a discrete series of the incident-radiation wavelengths.

Phase transition through the splitting of self-dual spectral singularity in optical potentials: erratum.

Some errors and particular conclusions drawn in our Letter [Opt. Lett.42, 5206 (2017)OPLEDP0146-959210.1364/OL.42.005206] are corrected.

Coherent perfect absorber and laser for nonlinear waves in optical waveguide arrays.

A localized non-Hermitian potential can operate as a coherent perfect absorber or as a laser for nonlinear waves. The effect is illustrated for an array of optical waveguides, with the central waveguide being either active or absorbing. The arrays situated to the left and to the right of the center can have different characteristics. The result is generalized to setups with the central waveguide carrying additional nonlinear dissipation or gain and to the two-dimensional arrays with embedded one-dimensional absorbing or lasing subarrays.

Transverse instability of dark solitons in spin-orbit coupled polariton condensates.

We consider dark solitons and their stability in spin-orbit coupled polariton condensates. The system supports spinor solitons of two types: conventional (symmetric) dark solitons and asymmetric half-dark solitons. They demonstrate essentially different behavior upon variation of the strength of spin-orbit coupling. One-dimensional spin-orbit coupled dark solitons are usually unstable, while half-dark solitons can be stable. Two-dimensional dark solitons at early stages of the development of transverse instabilities turn into asymmetric snaking patterns and later into sets of vortex-antivortex solitons with notably different shapes. Depending on the sign of spin-orbit coupling, two distinct instability scenarios are possible for such solitons, in which vortices in one component correspond to vortices or antivortices in other component. The decay of two-dimensional half-dark solitons results in the formation of half-vortex chains.

Coherent perfect absorption of nonlinear matter waves.

Coherent perfect absorption is the complete extinction of incoming radiation by a complex potential in a physical system supporting wave propagation. The concept was proven for linear waves in a variety of systems including light interacting with absorbing scatterers, plasmonic metasurfaces, and graphene films, as well as sound waves. We extend the paradigm to coherent perfect absorption of nonlinear waves and experimentally demonstrate it for matter waves in an atomic Bose-Einstein condensate. Coherent absorption of nonlinear matter waves is achieved easier than its linear analogs because the strength of two-body interactions offers additional freedom for control. Implementation of the coherent perfect absorber of Bose-Einstein condensates paves the way toward broad exploitation of the phenomenon in nonlinear optics, exciton-polariton condensates, acoustics, and other areas of nonlinear physics. It also opens perspectives for designing atom lasers.

Odd-Time Reversal PT Symmetry Induced by an Anti-PT-Symmetric Medium.

We introduce an optical system (a coupler) obeying parity-time (PT) symmetry with odd-time reversal, T^{2}=-1. It is implemented with two birefringent waveguides embedded in an anti-PT-symmetric medium. The system possesses properties that are untypical for most physical systems with the conventional even-time reversal. Having a symmetry-protected degeneracy of the linear modes, the coupler allows for the realization of a coherent switch operating with a superposition of binary states that are distinguished by their polarizations. When a Kerr nonlinearity is taken into account, each linear state, being doubly degenerated, bifurcates into several distinct nonlinear modes, some of which are dynamically stable. The nonlinear modes are characterized by amplitude and by polarization and come in PT-conjugate pairs.

Phase transition through the splitting of self-dual spectral singularity in optical potentials.

We consider optical media, which feature antilinear symmetries. We show that (i) spectral singularities of such media (if any) are always self-dual, i.e., correspond to coherent perfect absorber lasers; (ii) under the change of a system parameter, the self-dual spectral singularity may split into a pair of isolated complex conjugate eigenvalues, which corresponds to an unconventional and overlooked, in the most of the previous studies, scenario of the phase transition (known as parity-time [PT]-symmetry breaking in systems obeying PT symmetry); (iii) if the antilinear symmetry is local, i.e., does not involve any spatial reflection, then no spectral singularity is possible. Our findings are illustrated with several examples including a PT-symmetric bilayer and other complex potentials discussed in recent literature.

CPT-symmetric coupler with intermodal dispersion.

A dual-core waveguide with balanced gain and loss in different arms and with intermodal coupling is considered. The system is not invariant under the conventional PT symmetry, but obeys CPT symmetry where an additional spatial inversion C corresponds to swapping the coupler arms. We show that second-order dispersion of coupling allows for unbroken CPT symmetry and supports propagation of stable vector solitons along the coupler. Small-amplitude solitons are found in explicit form. The combined effect of gain-and-loss and dispersive coupling results in several interesting features which include a separation between the components in different arms, nontrivial dependence of stability of a soliton on its velocity, and the existence of more complex stationary two-hump solutions. Unusual decay dynamics of unstable solitons are also discussed.

Bloch Oscillations in Optical and Zeeman Lattices in the Presence of Spin-Orbit Coupling.

We address Bloch oscillations of a spin-orbit coupled atom in periodic potentials of two types: optical and Zeeman lattices. We show that in optical lattices the spin-orbit coupling allows controlling the direction of atomic motion and may lead to complete suppression of the oscillations at specific values of the coupling strength. In Zeeman lattices the energy bands are found to cross each other at the boundaries of the Brillouin zone, resulting in period doubling of the oscillations. In all cases, the oscillations are accompanied by rotation of the pseudospin, with a dynamics that is determined by the strength of the spin-orbit coupling. The predicted effects are discussed also in terms of a Wannier-Stark ladder, which in optical lattices consist of two mutually shifted equidistant subladders.

Two-dimensional solitons in conservative and parity-time-symmetric triple-core waveguides with cubic-quintic nonlinearity.

We analyze a system of three two-dimensional nonlinear Schrödinger equations coupled by linear terms and with the cubic-quintic (focusing-defocusing) nonlinearity. We consider two versions of the model: conservative and parity-time (PT) symmetric. These models describe triple-core nonlinear optical waveguides, with balanced gain and losses in the PT-symmetric case. We obtain families of soliton solutions and discuss their stability. The latter study is performed using a linear stability analysis and checked with direct numerical simulations of the evolutional system of equations. Stable solitons are found in the conservative and PT-symmetric cases. Interactions and collisions between the conservative and PT-symmetric solitons are briefly investigated, as well.

Tunable nonlinear double-core PT-symmetric waveguides.

We propose a scheme for creating a tunable, highly nonlinear defect in a one-dimensional photonic crystal with an embedded atomic cell filled by a mixture of two isotopes of three-level atoms. The defect creates an effective double-core waveguide. The induced refractive index for a probe field can be made parity-time (PT) symmetric by means of a proper combination of a control field and a Stark field; hence the complex defect potential obtained contains double-well real and linear imaginary parts. We also show that it is possible to form various stable nonlinear defect modes supported by the focusing nonlinearity of the system.

Tunable nonlinear parity-time-symmetric defect modes with an atomic cell.

We propose a scheme of creating a tunable highly nonlinear defect in a one-dimensional photonic crystal. The defect consists of an atomic cell filled in with two isotopes of three-level atoms. The probe-field refractive index of the defect can be made parity-time (PT) symmetric, which is achieved by proper combination of a control field and of Stark shifts induced by a far-off-resonance field. In the PT-symmetric system, families of stable nonlinear defect modes can be formed by the probe field.

Solitons in a medium with linear dissipation and localized gain.

We present a variety of dissipative solitons and breathing modes in a medium with localized gain and homogeneous linear dissipation. The system possesses a number of unusual properties, like exponentially localized modes in both focusing and defocusing media, existence of modes in focusing media at negative propagation constant values, simultaneous existence of stable symmetric and antisymmetric localized modes when the gain landscape possesses two local maxima, as well as the existence of stable breathing solutions.

Dissipative double-well potential: nonlinear stationary and pulsating modes.

The analysis of nonlinear modes in a complex absorbing double-well potential supported by linear gain is presented. Families of the nonlinear modes and their bifurcations are found numerically by means of the properly modified "shooting" method. Linear stability and dynamics of the modes are studied. It is shown that no stable modes exist in the case of attractive nonlinearity, while stable modes, including nonsymmetric ones, are found when the nonlinearity is repulsive. Varying a control parameter (e.g., the height of barrier between the wells) results in switching from one mode to another. Apart from stationary modes we have found pulsating solutions emergent from unstable modes.