It is well known that typical PT -symmetric systems suffer symmetry breaking when the strength of the gain-loss terms, i.e., the coefficient in front of the non-Hermitian part of the underlying Hamiltonian, exceeds a certain critical value. In this article, we present a summary of recently published and newly produced results which demonstrate various possibilities of extending the PT symmetry to arbitrarily large values of the gain-loss coefficient. First, we recapitulate the analysis which demonstrates a possibility of the restoration of the PT symmetry and, moreover, complete avoidance of the breaking in a photonic waveguiding channel of a subwavelength width. The analysis is necessarily based on the system of Maxwell's equations, instead of the usual paraxial approximation. Full elimination of the PT -symmetry-breaking transition is found in a deeply subwavelength region. Next, we review a recently proposed possibility to construct stable one-dimensional (1D) PT -symmetric solitons in a paraxial model with arbitrarily large values of the gain-loss coefficient, provided that the self-trapping of the solitons is induced by self-defocusing cubic nonlinearity, whose local strength grows sufficiently fast from the center to periphery. The model admits a particular analytical solution for the fundamental soliton, and provides full stability for families of fundamental and dipole solitons. It is relevant to stress that this model is nonlinearizable, hence the concept of the PT symmetry in it is also an essentially nonlinear one. Finally, we report new results for unbreakable PT -symmetric solitons in 2D extensions of the 1D model: one with a quasi-1D modulation profile of the local gain-loss coefficient, and another with the fully-2D modulation. These settings admit particular analytical solutions for 2D solitons, while generic soliton families are found in a numerical form. The quasi-1D modulation profile gives rise to a stable family of single-peak 2D solitons, while their dual-peak counterparts tend to be unstable. The soliton stability in the full 2D model is possible if the local gain-loss term is subject to spatial confinement.2 exciton-polariton condensates [36]- [38], and in other physically relevant contexts. In particular, the transitions from unbroken to broken PT symmetry was observed in many experiments. One of prominent experimentally demonstrated applications of the PT symmetry in optics is unidirectional transmission of light [39].Other classical waveguiding settings also admit emulation of the PT symmetry, as demonstrated in acoustics [40] and predicted in optomechanical systems [41]. Also predicted were realizations of this symmetry in atomic Bose-Einstein condensates [42], magnetism [43], mechanical chains of coupled pendula [44], and electronic circuits [45] (in the latter case, the prediction was also demonstrated experimentally). In terms of the theoretical analysis, PT -symmetric extensions were also elaborated for 47], Burgers [48], and sine-Gordon [49] equations, as well as in a system combi...

We introduce a model of one-and two-dimensional (1D and 2D) optical media with the χ (2) nonlinearity whose local strength is subject to cusp-shaped spatial modulation, χ (2) ∼ r −α , with α > 0, which can be induced by spatially nonuniform poling. Using analytical and numerical methods, we demonstrate that this setting supports 1D and 2D fundamental solitons, at α < 1 and α < 2, respectively. The 1D solitons have a small instability region, while the 2D solitons have a stability region at α < 0.5 and are unstable at α > 0.5. 2D solitary vortices are found too. They are unstable, splitting into a set of fragments, which eventually merge into a single fundamental soliton pinned to the cusp. Spontaneous symmetry breaking of solitons is studied in the 1D system with a symmetric pair of the cusp-modulation peaks.

The real spectrum of bound states produced by -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 -symmetric systems for various applications. On the other hand, it is known that the 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) 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 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 -symmetric one, but it also supports continuous families of self-trapped states, thus suggesting an extension of the concept of the 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 symmetry, while generic soliton families are found in a numerical form. The -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.

We introduce a model of a lossy second-harmonic-generating (χ (2) ) cavity externally pumped at the third harmonic, which gives rise to driving terms of a new type, corresponding to a cross-parametric gain. The equation for the fundamental-frequency (FF) wave may also contain a quadratic self-driving term, which is generated by the cubic nonlinearity of the medium. Unlike previously studied phase-matched models of χ (2) cavities driven at the second harmonic (SH) or at FF, the present one admits an exact analytical solution for the soliton, at a special value of the gain parameter. Two families of solitons are found in a numerical form, and their stability area is identified through numerical computation of the perturbation eigenvalues (stability of the zero solution, which is a necessary condition for the soliton's stability, is investigated in an analytical form). One family is a continuation of the special analytical solution.At given values of parameters, one soliton is stable and the other one is not; they swap their stability at a critical value of the mismatch parameter. The stability of the solitons is also verified in direct simulations, which demonstrate that the unstable pulse rearranges itself into the stable one, or into a delocalized state, or decays to zero. A soliton which was given an initial boost C starts to move but quickly comes to a halt, if the boost is smaller than a critical value C cr . If C > C cr , the boost destroys the soliton (sometimes, through splitting into two secondary pulses). Interactions between initially separated solitons are investigated too. It is concluded that stable solitons always merge into a single one. In the system with weak loss, it appears in a vibrating form, slowly relaxing to the static shape. With stronger loss, the final soliton emerges in the stationary form.

We introduce multi-soliton sets in the two-dimensional medium with the χ (2) nonlinearity subject to spatial modulation in the form of a triangle of singular peaks. Various families of symmetric and asymmetric sets are constructed, and their stability is investigated. Stable symmetric patterns may be built of 1, 4, or 7 individual solitons, while stable asymmetric ones contain 1, 2, or 3 solitons. Symmetric and asymmetric patterns may demonstrate mutual bistability. The shift of the asymmetric single-soliton state from the central position is accurately predicted analytically. Vortex rings composed of three solitons are produced too. possibility to generalize the study of the onset of collapse in nonlinear media [12,13], as well as to emulate the nonlinear dynamics in a sub-1D space, with the effective dimension D = 2 (1 − α) / (2 − α) < 1 [11]. The singular modulation can be emulated by means of above-mentioned techniques, tuning them to the exact resonance in a narrow layer.Another possibility is to consider spatial modulation of the local interaction strength in media with the quadratic (χ (2) ) nonlinearity, which has well-known realizations in optics [14]- [17]. Experimental realizations of such settings are possible, in particular, using the well-elaborated technique of the quasi-phasematching [18]- [20], which can be implemented in a spatially nonuniform form, in 1D and 2D geometries alike, thus helping one to create a required profile of the χ (2) coefficient [21]-[23]. In the theoretical analysis, singular modulation of the quadratic nonlinearity in the 1D system, accounted for by a delta-function,, and localized modes (solitons) pinned to it, were introduced in Ref. [24], and a pair of modulating delta-functions was considered in Ref. [25]. A discrete version of the localized quadratic nonlinearity was elaborated in the form of a linear lattice with one or two χ (2) -nonlinear sites embedded in it [26].While the delta-like modulation of the local χ (2) coefficient may not be easily realized in the experiment, a more realistic case of the 1D singular modulation, with χ (2) ∼ |x| −α and positive α, was introduced in Ref. [27]. It was found that this modulation format supports quadratic solitons, pinned to the singular peak, for α < 1 (the pinned modes vanish at α = 1), and they are chiefly stable. A natural extension of that setting is a symmetric pair of two peaks (similar to the above-mentioned symmetric set of two delta-functions multiplying the nonlinear terms [25]). The consideration of the twin peaks has made it possible to predict effects of the spontaneously symmetry breaking [28] and formation of asymmetric two-soliton states pinned to the two peaks [27]. These results may be used for the design of steering optical beams in the form of spatial solitons.An essential advantage of using the quadratic nonlinearity with this type of the local modulation is that it may be extended to the 2D geometry, by choosing χ (2) (r) ∼ r −α (r is the radial coordinate), while any 2D singular modulation of the self...