Families of stationary modes in complex potentials

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.

Observation of photonic constant-intensity waves and induced transparency in tailored non-Hermitian lattices

Light propagation is strongly affected by scattering due to imperfections in the complex medium. It has been recently theoretically predicted that a scattering-free transport through an inhomogeneous medium is achievable by non-Hermitian tailoring of the complex refractive index. Here, we implement photonic constant-intensity waves in an inhomogeneous, linear, discrete mesh lattice. By extending the existing theoretical framework, we experimentally show that a driven non-Hermitian tailoring allows us to control the propagation and diffraction of light even in highly disordered systems. In this vein, we demonstrate the transmission of shape-preserving beams and the seemingly undistorted propagation of light excitations across a strongly inhomogeneous non-Hermitian photonic lattice that can be realized by coupled optical fiber loops. Our results lead to a deeper understanding of non-Hermitian wave control and further contribute to the development of non-Hermitian photonics.

Asymmetric dissipative solitons in a waveguide lattice with non-uniform gain-loss distributions

We address the existence and stability of two types of asymmetric dissipative solitons, including fundamental and dipole solitons, supported by a waveguide lattice with non-uniform gain-loss distributions. Fundamental solitons exist only when the linear gain width is greater than or equal to the linear loss width, while dipole solitons exist only when the linear gain width is less than or equal to the linear loss width. With an increase in the relative gain depth, the effective width of the soliton gradually decreases. In addition, we find that both asymmetric fundamental and dipole solitons are stable in a considerable part of their lower edge of existence regions, and solitons beyond this range are unstable.

Equal-intensity waves in non-Hermitian media

A novel type of waves is examined in the context of non-Hermitian photonics. We can identify a class of complex guided structures that support localized paraxial solutions whose intensity distribution is exactly the same as the intensity of a corresponding solution in homogeneous media (free or bulk space). In other words, intensity-wise the two solutions are identical and their phase is different by a factor exp[iθ(x,y)]. The non-Hermitian potential is determined by the phase θ, as well as the amplitude and phase of the bulk space solution that contributes to the imaginary and real part of the potential, respectively. That way we can connect the plane waves and Gaussian beams of free space to constant-intensity waves and what we call the equal-intensity waves (EI waves) in non-Hermitian media. Such a relation allows us to study three different physical problems: Propagating EI waves inside random media, interface lattice solitons, and moving solitons in photonic waveguide structures with free-space characteristics. The relation of EI waves to unidirectional invisibility and Bohmian photonics is also examined.

Attraction centers and parity-time-symmetric delta-functional dipoles in critical and supercritical self-focusing media

We introduce a model based on the one-dimensional nonlinear Schrödinger equation with critical (quintic) or supercritical self-focusing nonlinearity. We demonstrate that a family of solitons, which are unstable in this setting against the critical or supercritical collapse, is stabilized by pinning to an attractive defect, that may also include a parity-time (PT)-symmetric gain-loss component. The model can be realized as a planar waveguide in nonlinear optics, and in a super-Tonks-Girardeau bosonic gas. For the attractive defect with the delta-functional profile, a full family of the pinned solitons is found in an exact analytical form. In the absence of the gain-loss term, the solitons' stability is investigated in an analytical form too, by means of the Vakhitov-Kolokolov criterion; in the presence of the PT-balanced gain and loss, the stability is explored by means of numerical methods. In particular, the entire family of pinned solitons is stable in the quintic (critical) medium if the gain-loss term is absent. A stability region for the pinned solitons persists in the model with an arbitrarily high power of the self-focusing nonlinearity. A weak gain-loss component gives rise to intricate alternations of stability and instability in the system's parameter plane. Those solitons which are unstable under the action of the supercritical self-attraction are destroyed by the collapse. On the other hand, if the self-attraction-driven instability is weak and the gain-loss term is present, unstable solitons spontaneously transform into localized breathers, while the collapse does not occur. The same outcome may be caused by a combination of the critical nonlinearity with the gain and loss. Instability of the solitons is also possible when the PT-symmetric gain-loss term is added to the subcritical nonlinearity. The system with self-repulsive nonlinearity is briefly considered too, producing completely stable families of pinned localized states.

Robust [Formula: see text] symmetry of two-dimensional fundamental and vortex solitons supported by spatially modulated nonlinearity

The real spectrum of bound states produced by [Formula: see text]-symmetric Hamiltonians usually suffers breakup at a critical value of the strength of gain-loss terms, i.e., imaginary part of the complex potential. The breakup essentially impedes the use of [Formula: see text]-symmetric systems for various applications. On the other hand, it is known that the [Formula: see text] symmetry can be made unbreakable in a one-dimensional (1D) model with self-defocusing nonlinearity whose strength grows fast enough from the center to periphery. The model is nonlinearizable, i.e., it does not have a linear spectrum, while the (unbreakable) [Formula: see text] symmetry in it is defined by spectra of continuous families of nonlinear self-trapped states (solitons). Here we report results for a 2D nonlinearizable model whose [Formula: see text] symmetry remains unbroken for arbitrarily large values of the gain-loss coefficient. Further, we introduce an extended 2D model with the imaginary part of potential ~xy in the Cartesian coordinates. The latter model is not a [Formula: see text]-symmetric one, but it also supports continuous families of self-trapped states, thus suggesting an extension of the concept of the [Formula: see text] symmetry. For both models, universal analytical forms are found for nonlinearizable tails of the 2D modes, and full exact solutions are produced for particular solitons, including ones with the unbreakable [Formula: see text] symmetry, while generic soliton families are found in a numerical form. The [Formula: see text]-symmetric system gives rise to generic families of stable single- and double-peak 2D solitons (including higher-order radial states of the single-peak solitons), as well as families of stable vortex solitons with m = 1, 2, and 3. In the model with imaginary potential ~xy, families of single- and multi-peak solitons and vortices are stable if the imaginary potential is subject to spatial confinement. In an elliptically deformed version of the latter model, an exact solution is found for vortex solitons with m = 1.

Edge and bulk dissipative solitons in modulated PT-symmetric waveguide arrays

We address dissipative soliton formation in modulated parity-time (PT)-symmetric continuous waveguide arrays composed from waveguides with amplifying and absorbing sections, whose density gradually increases (due to decreasing waveguide separation) either towards the center of the array or towards its edges. In such a structure, the level of gain/loss at which PT-symmetry gets broken depends on the direction of increase of waveguide density. Breakup of PT-symmetry occurs when eigenvalues of modes localized in the region where waveguide density is largest collide and move into a complex plane. In this regime of broken symmetry, the inclusion of focusing Kerr-type nonlinearity of the material and weak two-photon absorption allows to arrest the growth of amplitude of amplified modes and may lead to the appearance of stable attractors either in the center or at the edge of the waveguide array, depending on the type of array modulation. Such solitons can be stable; they acquire specific triangular shapes and notably broaden with increase of gain/loss level. Our results illustrate how spatial array modulation that breaks PT-symmetry "locally" can be used to control the specific location of dissipative solitons forming in the array.

Impact of near-𝒫𝒯 symmetry on exciting solitons and interactions based on a complex Ginzburg-Landau model

We theoretically report the influence of a class of near-parity-time-(𝒫𝒯-) symmetric potentials on solitons in the complex Ginzburg-Landau (CGL) equation. Although the linear spectral problem with the potentials does not admit entirely-real spectra due to the existence of spectral filtering parameter α₂ or nonlinear gain-loss coefficient β₂, we do find stable exact solitons in the second quadrant of the (α₂, β₂) space including on the corresponding axes. Other fascinating properties associated with the solitons are also examined, such as the interactions and energy flux. Moreover, we study the excitations of nonlinear modes by considering adiabatic changes of parameters in a generalized CGL model. These results are useful for the related experimental designs and applications.

Vector solitons in nonparity-time-symmetric complex potentials

The existence and stability of vector solitons in non-parity-time (PT)-symmetric complex potentials are investigated. We study the vector soliton family, in which the propagation constants of the two components are different. It is found that vector solitons can be stable below and above the phase transition of the non-PT-symmetric complex potentials. Below the phase transition, vector solitons are stable in the low power region. Above the phase transition, there are two continuous stable intervals in the existence region. The profiles of two components of these vector solitons show the asymmetry and we also study the transverse power flow in the two components of these vector solitons in the non-PT-symmetric complex potentials.

Wave propagation through disordered media without backscattering and intensity variations

A fundamental manifestation of wave scattering in a disordered medium is the highly complex intensity pattern the waves acquire due to multi-path interference. Here we show that these intensity variations can be entirely suppressed by adding disorder-specific gain and loss components to the medium. The resulting constant-intensity waves in such non-Hermitian scattering landscapes are free of any backscattering and feature perfect transmission through the disorder. An experimental demonstration of these unique wave states is envisioned based on spatially modulated pump beams that can flexibly control the gain and loss components in an active medium.

Waveguides with Absorbing Boundaries: Nonlinearity Controlled by an Exceptional Point and Solitons

We reveal the existence of continuous families of guided single-mode solitons in planar waveguides with weakly nonlinear active core and absorbing boundaries. Stable propagation of TE and TM-polarized solitons is accompanied by attenuation of all other modes, i.e., the waveguide features properties of conservative and dissipative systems. If the linear spectrum of the waveguide possesses exceptional points, which occurs in the case of TM polarization, an originally focusing (defocusing) material nonlinearity may become effectively defocusing (focusing). This occurs due to the geometric phase of the carried eigenmode when the surface impedance encircles the exceptional point. In its turn, the change of the effective nonlinearity ensures the existence of dark (bright) solitons in spite of focusing (defocusing) Kerr nonlinearity of the core. The existence of an exceptional point can also result in anomalous enhancement of the effective nonlinearity. In terms of practical applications, the nonlinearity of the reported waveguide can be manipulated by controlling the properties of the absorbing cladding.

The nonlinear Schrödinger equation with generalized nonlinearities and PT-symmetric potentials: Stable solitons, interactions, and excitations

We investigate the nonlinear Schrödinger (NLS) equation with generalized nonlinearities and complex non-Hermitian potentials and present the novel parity-time-( PT-) symmetric potentials for the NLS equation with power-law nonlinearities supporting some bright solitons. For distinct types of PT-symmetric potentials including Scarf-II, Hermite-Gaussian, and asymptotically periodic potentials, we, respectively, explore the phase transitions for the linear Hamiltonian operators. Moreover, we analytically find stable bright solitons in the generalized NLS equations with several types of PT-symmetric potentials, and their stability is corroborated by the linear stability spectrum and direct wave-propagation simulations. Interactions of two solitons are also explored. More interestingly, we find that the nonlinearity can excite the unstable linear modes (i.e., possessing broken linear PT-symmetric phase) to stable nonlinear modes. The results may excite potential applications in nonlinear optics, Bose-Einstein condensates, and relevant fields.

Families of stable solitons and excitations in the PT-symmetric nonlinear Schrödinger equations with position-dependent effective masses

Since the parity-time-([Formula: see text]-) symmetric quantum mechanics was put forward, fundamental properties of some linear and nonlinear models with [Formula: see text]-symmetric potentials have been investigated. However, previous studies of [Formula: see text]-symmetric waves were limited to constant diffraction coefficients in the ambient medium. Here we address effects of variable diffraction coefficient on the beam dynamics in nonlinear media with generalized [Formula: see text]-symmetric Scarf-II potentials. The broken linear [Formula: see text] symmetry phase may enjoy a restoration with the growing diffraction parameter. Continuous families of one- and two-dimensional solitons are found to be stable. Particularly, some stable solitons are analytically found. The existence range and propagation dynamics of the solitons are identified. Transformation of the solitons by means of adiabatically varying parameters, and collisions between solitons are studied too. We also explore the evolution of constant-intensity waves in a model combining the variable diffraction coefficient and complex potentials with globally balanced gain and loss, which are more general than [Formula: see text]-symmetric ones, but feature similar properties. Our results may suggest new experiments for [Formula: see text]-symmetric nonlinear waves in nonlinear nonuniform optical media.

2D in-band solitons in PT-symmetric waveguide arrays

We address two types of two-dimensional (2D) localized solitons in Kerr media with an imprinted quasi-one-dimensional lattice featuring a parity-time (PT) symmetry. Solitary waves originating from the edges of the Bloch bands are stable in their entire existence domains. Purely nonlinear multipeaked states propagate stably in wide parameter windows. Both types of nonlinear waves exist in the finite bandgaps of the corresponding linear system and, cross-continuously, the Bloch band (continuous spectrum) sandwiched between (or neighboring) them. To the best of our knowledge, our findings thus provide the first example of "embedded solitons" in 2D PT periodic systems.

Nonlinear light behaviors near phase transition in non-parity-time-symmetric complex waveguides

Many classes of non-parity-time (PT)-symmetric waveguides with arbitrary gain and loss distributions still possess all-real linear spectrum or exhibit phase transition. In this Letter, nonlinear light behaviors in these complex waveguides are probed analytically near a phase transition. Using multi-scale perturbation methods, a nonlinear ordinary differential equation (ODE) is derived for the light's amplitude evolution. This ODE predicts that a single class of these non-PT-symmetric waveguides supports soliton families and amplitude-oscillating solutions both above and below linear phase transition, in close analogy with PT-symmetric systems. For the other classes of waveguides, the light's intensity always amplifies under the effect of nonlinearity, even if the waveguide is below the linear phase transition. These analytical predictions are confirmed by direct computations of the full system.

Topological States in Partially-PT-Symmetric Azimuthal Potentials

We introduce partially-parity-time (pPT)-symmetric azimuthal potentials composed from individual PT-symmetric cells located on a ring, where two azimuthal directions are nonequivalent in a sense that in such potential excitations carrying topological dislocations exhibit different dynamics for different directions of energy circulation in the initial field distribution. Such nonconservative ratchetlike structures support rich families of stable vortex solitons in cubic nonlinear media, whose properties depend on the sign of the topological charge due to the nonequivalence of azimuthal directions. In contrast, oppositely charged vortex solitons remain equivalent in similar fully-PT-symmetric potentials. The vortex solitons in the pPT- and PT-symmetric potentials are shown to feature qualitatively different internal current distributions, which are described by different discrete rotation symmetries of the intensity profiles.

Symmetry breaking of solitons in two-dimensional complex potentials

Symmetry breaking is reported for continuous families of solitons in the nonlinear Schrödinger equation with a two-dimensional complex potential. This symmetry breaking is forbidden in generic complex potentials. However, for a special class of partially parity-time-symmetric potentials, it is allowed. At the bifurcation point, two branches of asymmetric solitons bifurcate out from the base branch of symmetry-unbroken solitons. Stability of these solitons near the bifurcation point are also studied, and two novel properties for the bifurcated asymmetric solitons are revealed. One is that at the bifurcation point, zero and simple imaginary linear-stability eigenvalues of asymmetric solitons can move directly into the complex plane and create oscillatory instability. The other is that the two bifurcated asymmetric solitons, even though having identical powers and being related to each other by spatial mirror reflection, can possess different types of unstable eigenvalues and thus exhibit nonreciprocal nonlinear evolutions under random-noise perturbations.