Solvable model for solitons pinned to a parity-time-symmetric dipole

Formation, stability, and adiabatic excitation of peakons and double-hump solitons in parity-time-symmetric Dirac-δ(x)-Scarf-II optical potentials

We introduce a class of physically intriguing PT-symmetric Dirac-δ-Scarf-II optical potentials. We find the parameter region making the corresponding non-Hermitian Hamiltonian admit the fully real spectra, and present the stable parameter domains for these obtained peakons, smooth solitons, and double-hump solitons in the self-focusing nonlinear Kerr media with PT-symmetric δ-Scarf-II potentials. In particular, the stable wave propagations are exhibited for the peakon solutions and double-hump solitons from some given parameters even if the corresponding parameters belong to the linear PT-phase broken region. Moreover, we also find the stable wave propagations of exact and numerical peakons and double-hump solitons in the interplay between the power-law nonlinearity and PT-symmetric potentials. Finally, we examine the interactions of the nonlinear modes with exotic waves, and the stable adiabatic excitations of peakons and double-hump solitons in the PT-symmetric Kerr nonlinear media. These results provide the theoretical basis for the design of related physical experiments and applications in PT-symmetric nonlinear optics, Bose-Einstein condensates, and other relevant physical fields.

Stable flat-top solitons and peakons in the PT-symmetric δ-signum potentials and nonlinear media

We discover that the physically interesting PT-symmetric Dirac delta-function potentials can not only make sure that the non-Hermitian Hamiltonians admit fully-real linear spectra but also support stable peakons (nonlinear modes) in the Kerr nonlinear Schrödinger equation. For a specific form of the delta-function PT-symmetric potentials, the nonlinear model investigated in this paper is exactly solvable. However, for a class of PT-symmetric signum-function double-well potentials, a novel type of exact flat-top bright solitons can exist stably within a broad range of potential parameters. Intriguingly, the flat-top solitons can be characterized by the finite-order differentiable waveforms and admit the novel features differing from the usual solitons. The excitation features and the direction of transverse power flow of flat-top bright solitons are also explored in detail. These results are useful for the related experimental designs and applications in nonlinear optics and other related fields.

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.

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.

Interactions of bright and dark solitons with localized PT-symmetric potentials

We study collisions of moving nonlinear-Schrödinger solitons with a PT-symmetric dipole embedded into the one-dimensional self-focusing or defocusing medium. Accurate analytical results are produced for bright solitons, and, in a more qualitative form, for dark ones. In the former case, an essential aspect of the approximation is that it must take into regard the intrinsic chirp of the soliton, thus going beyond the framework of the simplest quasi-particle description of the soliton's dynamics. Critical velocities separating reflection and transmission of the incident bright solitons are found by means of numerical simulations, and in the approximate semi-analytical form. An exact solution for the dark soliton pinned by the complex PT-symmetric dipole is produced too.

Discrete solitons and vortices on two-dimensional lattices of PT-symmetric couplers

We introduce a 2D network built of PT-symmetric dimers with on-site cubic nonlinearity, the gain and loss elements of the dimers being linked by parallel square-shaped lattices. The system may be realized as a set of PT-symmetric dual-core waveguides embedded into a photonic crystal. The system supports PT-symmetric and antisymmetric fundamental solitons (FSs) and on-site-centered solitary vortices (OnVs). Stability of these discrete solitons is the central topic of the consideration. Their stability regions in the underlying parameter space are identified through the computation of stability eigenvalues, and verified by direct simulations. Symmetric FSs represent the system's ground state, being stable at lowest values of the power, while anti-symmetric FSs and OnVs are stable at higher powers. Symmetric OnVs, which are also stable at lower powers, are remarkably robust modes: on the contrary to other PT-symmetric states, unstable OnVs do not blow up, but spontaneously rebuild themselves into stable FSs.

Pinned modes in two-dimensional lossy lattices with local gain and nonlinearity

We introduce a system with one or two amplified nonlinear sites ('hot spots', HSs) embedded into a two-dimensional linear lossy lattice. The system describes an array of evanescently coupled optical or plasmonic waveguides, with gain applied to selected HS cores. The subject of the analysis is discrete solitons pinned to the HSs. The shape of the localized modes is found in quasi-analytical and numerical forms, using a truncated lattice for the analytical consideration. Stability eigenvalues are computed numerically, and the results are supplemented by direct numerical simulations. In the case of self-focusing nonlinearity, the modes pinned to a single HS are stable and unstable when the nonlinearity includes the cubic loss and gain, respectively. If the nonlinearity is self-defocusing, the unsaturated cubic gain acting at the HS supports stable modes in a small parametric area, whereas weak cubic loss gives rise to a bistability of the discrete solitons. Symmetric and antisymmetric modes pinned to a symmetric set of two HSs are also considered.

Families of stationary modes in complex potentials

We show that a general class of complex asymmetric potentials of the form w2(x)-iw(x)(x), where w(x) is a real function, allows for the existence of one-parametric continuous families of the stationary nonlinear modes bifurcating from the linear spectrum and propagating in Kerr media. As an example, we introduce an asymmetric double-hump complex potential and show that it supports continuous families of stable nonlinear modes.

Discrete solitons and scattering of lattice waves in guiding arrays with a nonlinear PT-symmetric defect

Discrete fundamental and dipole solitons are constructed, in an exact analytical form, in an array of linear waveguides with an embedded PT-symmetric dimer, which is composed of two nonlinear waveguides carrying equal gain and loss. Fundamental solitons in tightly knit lattices, as well as all dipole modes, exist above a finite threshold value of the total power. However, the threshold vanishes for fundamental solitons in loosely knit lattices. The stability of the discrete solitons is investigated analytically by means of the Vakhitov-Kolokolov (VK) criterion, and, in the full form, via the computation of eigenvalues for perturbation modes. Fundamental and dipole solitons tend to be stable at smaller and larger values of the total power (norm), respectively. The increase of the strength of the coupling between the two defect-forming sites leads to strong expansion of the stability areas. The scattering problem for linear lattice waves impinging upon the defect is considered too.

Quasi-energies, parametric resonances, and stability limits in ac-driven PT-symmetric systems

We introduce a simple model for implementing the concepts of quasi-energy and parametric resonances (PRs) in systems with the PT symmetry, i.e., a pair of coupled and mutually balanced gain and loss elements. The parametric (ac) forcing is applied through periodic modulation of the coefficient accounting for the coupling of the two degrees of freedom. The system may be realized in optics as a dual-core waveguide with the gain and loss applied to different cores, and the thickness of the gap between them subject to a periodic modulation. The onset and development of the parametric instability for a small forcing amplitude (V1) is studied in an analytical form. The full dynamical chart of the system is generated by systematic simulations. At sufficiently large values of the forcing frequency, ω, tongues of the parametric instability originate, with the increase of V1, as predicted by the analysis. However, the tongues following further increase of V1 feature a pattern drastically different from that in usual (non-PT) parametrically driven systems: instead of bending down to larger values of the dc coupling constant, V0, they maintain a direction parallel to the V1 axis. The system of the parallel tongues gets dense with the decrease of ω, merging into a complex small-scale structure of alternating regions of stability and instability. The cases of ω-->0 and ω-->∞ are studied analytically by means of the adiabatic and averaging approximation, respectively. The cubic nonlinearity, if added to the system, alters the picture, destabilizing many originally robust dynamical regimes, and stabilizing some which were unstable.