Stability of solitons in parity-time (PT)-symmetric periodic potentials (optical lattices) is analyzed in both one-and two-dimensional systems. First we show analytically that when the strength of the gain-loss component in the PT lattice rises above a certain threshold (phase-transition point), an infinite number of linear Bloch bands turn complex simultaneously. Second, we show that while stable families of solitons can exist in PT lattices, increasing the gain-loss component has an overall destabilizing effect on soliton propagation. Specifically, when the gain-loss component increases, the parameter range of stable solitons shrinks as new regions of instability appear. Thirdly, we investigate the nonlinear evolution of unstable PT solitons under perturbations, and show that the energy of perturbed solitons can grow unbounded even though the PT lattice is below the phase transition point.

Conical diffraction in honeycomb lattices is analyzed. This phenomenon arises in nonlinear Schrödinger equations with honeycomb lattice potentials. In the tight-binding approximation the wave envelope is governed by a nonlinear classical Dirac equation. Numerical simulations show that the Dirac equation and the lattice equation have the same conical diffraction properties. Similar conical diffraction occurs in both the linear and nonlinear regimes. The Dirac system reveals the underlying mechanism for the existence of conical diffraction in honeycomb lattices.

A method for constructing optical potentials with an arbitrary distribution of gain and loss and completely real spectrum is presented. For each arbitrary distribution of gain and loss, several classes of refractive-index profiles with freely tunable parameters are obtained such that the resulting complex potentials, although being non-parity-time-symmetric in general, still feature all-real spectra for a wide range of tuning parameters. When these refractive indices are tuned below certain thresholds, phase transition can occur, where complex-conjugate pairs of eigenvalues appear in the spectrum. These non-parity-time-symmetric complex potentials generalize the concept of parity-time-symmetric potentials to allow for more flexible gain and loss distributions while still maintaining all-real spectra and the phenomenon of phase transition.Parity-time (PT ) symmetric optics is a frontier of current research. PT symmetry, first introduced by Bender and Boettcher [1] as a non-Hermitian generalization of quantum mechanics, involves the study of potentials which, though complex, still possess an all-real spectrum. This concept later spread to optics, where an even refractive index profile together with an odd gainloss landscape constitutes a PT -symmetric system and could feature all-real spectrum [2]. In this optical context, PT symmetry was demonstrated experimentally for the first time [3,4]. A distinctive phenomenon in PTsymmetric systems is phase transition, where the spectrum changes from all-real to partially complex when the gain-loss component (relative to the index of refraction) exceeds a certain threshold [1][2][3][4][5]. This phase transition has been utilized for many emerging applications of PT optics, such as single-mode PT lasers [6,7] and unidirectional reflectionless optical devices [8]. When nonlinearity is introduced into PT -symmetric systems, the interplay between PT symmetry and nonlinearity yields additional interesting properties which are being actively explored [9][10][11][12][13]. PT -symmetric systems break the boundaries between traditional conservative and dissipative systems and open up new exciting research territories. In addition, PT symmetry makes loss useful, which is quite enlightening.Optical PT symmetry requires the refractive index to be even and gain-loss profile to be odd in space, which is quite restrictive. This leads us to the natural question, how can we relax the PT -symmetry requirement of the optical potential, while still preserving the all-real spectrum? Recently, several examples of non-PT -symmetric complex potentials with real spectra have been reported [14][15][16]. In [14,15], families of non-PT potentials with real spectra were constructed by supersymmetry. In [16], a class of such non-PT potentials were constructed by relating the Schrödinger eigenvalue problem to the Zakharov-Shabat eigenvalue problem. In the latter case, the resulting non-PT potentials not only admit all-real spectra, but also support continuous families of soliton modes [16,17] and allow f...

A direct perturbation method for approximating dark soliton solutions of the nonlinear Schrödinger (NLS) equation under the influence of perturbations is presented. The problem is broken into an inner region, where the core of the soliton resides, and an outer region, which evolves independently of the soliton. It is shown that a shelf develops around the soliton that propagates with speed determined by the background intensity. Integral relations obtained from the conservation laws of the NLS equation are used to determine the properties of the shelf. The analysis is developed for both constant and slowly evolving backgrounds. A number of problems are investigated, including linear and nonlinear damping type perturbations. Keywords: perturbation theory; solitons; opticsPerturbation theory as applied to solitons that decay at infinity, i.e. so-called bright solitons, has been developed over many years (cf. Karpman & Maslov 1977;Kodama & Ablowitz 1981;Herman 1990). The analytical work employs a diverse set of methods including perturbations of the inverse scattering transform (IST), multi-scale perturbation analysis, perturbations of conserved quantities, etc.; the analysis applies to a wide range of physical problems. In optics, a central equation that describes the envelope of a quasi-monochromatic wave train is the nonlinear Schrödinger (NLS) equation, which in normalized form readswhere D, n are constants. In this paper, we consider the NLS equation in a typical nonlinear optics context, where D corresponds to the group-velocity dispersion (GVD), n > 0 is related to the nonlinear index of refraction, z is the

Continuous families of solitons in the nonlinear Schrödinger equation with non‐scriptPT‐symmetric complex potentials and general forms of nonlinearity are studied analytically. Under a weak assumption, it is shown that stationary equations for solitons admit a constant of motion if and only if the complex potential is of a special form g2(x)+ig′(x), where g(x) is an arbitrary real function. Using this constant of motion, the second‐order complex soliton equation is reduced to a new second‐order real equation for the amplitude of the soliton. From this real soliton equation, a novel perturbation technique is employed to show that continuous families of solitons bifurcate out from linear discrete modes in these non‐scriptPT‐symmetric complex potentials. All analytical results are corroborated by numerical examples.

Nonlinear dynamics of wave packets in parity-time-symmetric optical lattices near the phase-transition point is analytically studied. A nonlinear Klein-Gordon equation is derived for the envelope of these wave packets. A variety of phenomena known to exist in this envelope equation are shown to also exist in the full equation, including wave blowup, periodic bound states, and solitary wave solutions.

Nonlinear wave propagation in parity-time (PT ) symmetric localized potentials is investigated analytically near a phase-transition point where a pair of real eigenvalues of the potential coalesce and bifurcate into the complex plane. Necessary conditions for phase transition to occur are derived based on a generalization of the Krein signature. Using multi-scale perturbation analysis, a reduced nonlinear ODE model is derived for the amplitude of localized solutions near phase transition. Above phase transition, this ODE model predicts a family of stable solitons not bifurcating from linear (infinitesimal) modes under a certain sign of nonlinearity. In addition, it predicts periodicallyoscillating nonlinear modes away from solitons. Under the opposite sign of nonlinearity, it predicts unbounded growth of solutions. Below phase transition, solution dynamics is predicted as well. All analytical results are compared to direct computations of the full system and good agreement is observed.

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