We report the role of $\mathcal{PT}$-symmetry in switching characteristics of a highly nonlinear fiber Bragg grating (FBG) with cubic-quintic-septic nonlinearities. We demonstrate that the device shows novel bi-(multi-) stable states in the broken regime as a direct consequence of the shift in the photonic band gap influenced by both $\mathcal{PT}$-symmetry and higher-order nonlinearities. We also numerically depict that such FBGs provide a productive test bed where the broken $\mathcal{PT}$-symmetric regime can be exploited to set up all-optical applications such as binary switches, multi-level signal processing and optical computing. Unlike optical bistability (OB) in the traditional and unbroken $\mathcal{PT}$-symmetric FBG, it exhibits many peculiar features such as flat-top stable states and ramp like input-output characteristics before the onset of OB phenomenon in the broken regime. The gain/loss parameter plays a dual role in controlling the switching intensities between the stable states which is facilitated by reversing the direction of light incidence. We also find that the gain/loss parameter tailors the formation of gap solitons pertaining to transmission resonances which clearly indicates that it can be employed to set up optical storage devices. Moreover, the interplay between gain/loss and higher order nonlinearities brings notable changes in the nonlinear reflection spectra of the system under constant pump powers. The influence of each control parameters on the switching operation is also presented in a nutshell to validate that FBG offers more degrees of freedom in controlling light with light.
We study collisions of moving nonlinear-Schrödinger solitons with a $\mathcal{PT}$-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 solito…
Families of coupled solitons of $\mathcal{PT}$-symmetric physical models with gain and loss in fractional dimension and in settings with and without cross-interactions modulation (CIM), are reported. Profiles, powers, stability areas, and propagation dynamics of the obtained $\mathcal{PT}$-symmetric coupled solitons are investigated. By comparing the results of the models with and without CIM, we find that the stability area of the model with CIM is much broader than the one without CIM. Remarkably, oscillating $\mathcal{PT}$-symmetric coupled solitons can also exist in the model of CIM with t…
We study spatial and temporal solitons in the $\mathcal{PT}$ symmetric coupler with gain in one waveguide and loss in the other. Stability properties of the high- and low-frequency solitons are found to be completely determined by a single combination of the soliton's amplitude and the gain/loss coefficient of the waveguides. The unstable perturbations of the high-frequency soliton break the symmetry between its active and lossy components which results in a blowup of the soliton or a formation of a long-lived breather state. The unstable perturbations of the low-frequency soliton separate its…
We construct families of one-dimensional (1D) stable solitons in two-component $\mathcal{PT}$-symmetric systems with spin-orbit coupling (SOC) and quintic nonlinearity, which plays the critical role in 1D setups. The system models light propagation in a dual-core waveguide with skewed coupling between the cores. Stability regions for the solitons are identified in the system's parameter space. They include the main semi-infinite gap, and an additional finite $\textit{annex gap}$. Stability boundaries are identified by means of simulations of the perturbed evolution, which agree with results pr…
We investigate the switching dynamics in a $\mathcal{PT}$-symmetric fiber coupler composed of a saturable nonlinear material as the core. In such a saturable nonlinear medium, bistable solitons may evolve due to the balance between dispersion and saturable nonlinearity, which we extend in the context of the $\mathcal{PT}$-symmetric coupler. Our investigations of power-controlled and phase-sensitive switching show richer soliton switching dynamics than the currently existing conventional counterparts, which may lead to ultrafast and efficient all-optical switching dynamics at very low power owi…
Discrete fundamental and dipole solitons are constructed, in an exact analytical form, in an array of linear waveguides with an embedded $\mathcal{PT}$-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 for…
We introduce a 2D network built of $\mathcal{PT}$-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 $\mathcal{PT}$-symmetric dual-core waveguides embedded into a photonic crystal. The system supports $\mathcal{PT}$-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 t…
The effect of competing nonlinearity on beam dynamics in parity-time $(\mathcal{PT})$ symmetric potentials is investigated. By using numerical methods, the existence of gap solitons is demonstrated in the first Bragg gap of optical $\mathcal{PT}$ symmetric lattices with competing nonlinearity. Meanwhile, the stability of such solitons is analyzed through introducing a small perturbation to the solitary solutions. The abrupt annihilation of the solitons during propagation demonstrates the Bragg gap solitons in $\mathcal{PT}$ symmetric potentials are not stable. In comparison with the on-site ga…
We theoretically demonstrate soliton steering in $\mathcal{PT}$-symmetric coupled nonlinear dimers. We show that if the length of the $\mathcal{PT}$-symmetric system is set to $2π$ contrary to the conventional one which operates satisfactorily well only at the half-beat coupling length, the $\mathcal{PT}$ dimer remarkably yields an ideal soliton switch exhibiting almost 99.99% energy efficiency with an ultra-low critical power.
We study the dynamics of two-component vector solitons, namely, bright-bright (BB) solitons interacting with parity-time-($\mathcal{PT}$) symmetric potentials. We employ direct numerical simulations to demonstrate the unidirectional segregation of the BB soliton. Using a modified perturbed dynamical variational Lagrangian approximation, we develop an analytical model to verify the results obtained from the numerical calculations. Simplified variational equations of motion suggest that the splitting of BB solitons can be explained by considering the effective force between the two components.
We construct families of discrete solitons (DSs) in an array of self-defocusing waveguides with an embedded $\mathcal{PT}$ (parity-time)-symmetric dimer, which is represented by a pair of waveguides carrying mutually balanced gain and loss. Four types of states attached to the embedded defect are found, namely, staggered and unstaggered bright localized modes and gray or anti-gray DSs. Their existence and stability regions expand with the increase of the strength of the coupling between the dimer-forming sites. The existence of the gray and staggered bright DSs is qualitatively explained by di…
We introduce one- and two-dimensional (1D and 2D) models of parity-time ($% \mathcal{PT}$) -symmetric couplers with the mutually balanced linear gain and loss applied to the two cores, and cubic-quintic (CQ) nonlinearity acting in each one. The 2D and 1D models may be realized in dual-core optical waveguides, in the spatiotemporal and spatial domains, respectively. Stationary solutions for $\mathcal{PT}$-symmetric solitons in these systems reduce to their counterparts in the usual coupler. The most essential problem is the stability of the solitons, which become unstable against symmetry break…
We investigate the existence of stable soliton solution in a system described by complex Ginzburg-Landau (CGL) equation with near parity reflection - time reversal ($\mathcal{PT}$) symmetric Rosen-Morse potential. In this study, the stability of solution is examined by numerical analysis to show that solitons are stable for some parameter ranges. The dynamical properties such as evolution and transverse energy flow for both self-focusing and self-defocusing nonlinear mode are also analyzed. The obtained results are useful for experimental designs and applications in related fields.
We demonstrate analytically and numerically the existence of exact solitons in the form of double-kink and fractional-transform, supported by a symmetric transversally modulated defocusing nonlinearity acting as a psuedopotential combined with an antisymmetric gain-loss profile. We explicate here that the $\mathcal{PT}$ symmetry is never broken in the dynamical system under study, even in the absence of any symmetric modulation of linear refractive-index, and the transversally modulated defocusing nonlinearity comes in the way as a requirement to establish the ensuing $\mathcal{PT}$ symmetry.
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 ($\mathcal{PT}$) symmetric. These models describe triple-core nonlinear optical waveguides, with balanced gain and losses in the $\mathcal{PT}$-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 …
We have considered cubic nonlinear Schrödinger equation along with supersymmetric $\mathcal{PT}$ like potential and obtained exact stationary solutions in terms of bright and brigh-dark interacting solitons. The $\mathcal{PT}$ broken and $\mathcal{PT}$ unbroken regions are demonstrated also depicted. Connected and disconnected stable regions of bright and dark solitons are examined incorporating linear stability analysis validated by direct numerical simulations. Moreover, the strength of stability has been illustrated through excitations of bright and dark solitons.
We construct soliton solution of a variable coefficients nonlinear Schrödinger equation in the presence of parity reflection - time reversal $(\mathcal{PT})-$ symmetric Rosen-Morse potential using similarity transformation technique. We transform the variable coefficients nonlinear Schrödinger equation into the nonlinear Schrödinger equation with $\mathcal{PT}-$symmetric potential with certain integrability conditions. We investigate in-detail the features of the obtained soliton solutions with two different forms of dispersion parameters. Further, we analyze the nonlinear tunneling effect of …
We introduce a one-dimensional system combining the $\mathcal{PT}$-symmetric complex periodic potential and the $χ^{(2)}$ (second-harmonic-generating) nonlinearity. The imaginary part of the potential, which represents spatially separated and mutually balanced gain and loss, affects only the fundamental-frequency (FF) wave, while the real potential acts on the second-harmonic (SH) component too. Soliton modes are constructed, and their stability is investigated (by means of the linearization and direct simulations) in semi-infinite and finite gaps in the corresponding spectrum, starting from t…
Although the spinor field in (1+1) dimensions has the right structure to model a dispersive bimodal system with gain and loss, the plain addition of gain to one component of the field and loss to the other one results in an unstable dispersion relation. In this paper, we advocate a different recipe for the $\mathcal{PT}$-symmetric extension of spinor models --- the recipe that does not produce instability of the linear Dirac equation. Having exemplified the physical origins of the $\mathcal P$- and $\mathcal T$-breaking terms, we consider the extensions of three U(1)-invariant spinor models wi…
We investigate the physics informed neural network method, a deep learning approach, to approximate soliton solution of the nonlinear Schrödinger equation with parity time symmetric potentials. We consider three different parity time symmetric potentials, namely Gaussian, periodic and Rosen-Morse potentials. We use physics informed neural network to solve the considered nonlinear partial differential equation with the above three potentials. We compare the predicted result with actual result and analyze the ability of deep learning in solving the considered partial differential equation. We ch…
Nonlinear wave propagation in parity-time ($\mathcal{PT}$) 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 fro…
We consider light pulses in a circular array of $2N$ coupled nonlinear optical waveguides. The waveguides are either hermitian or alternate gain and loss in a $\mathcal{PT}$-symmetric fashion. Simple patterns in the array include a ring of $2N$ pulses travelling abreast, and a breather -- a string of pulses where all even and all odd waveguides flash in turn. In addition, the structure displays solitons gyrating around the necklace by switching from one waveguide to the next. Some of the gyrating solitons are stable while other ones are weakly unstable and evolve into gyrating multiflash strin…
We introduce a ladder-shaped chain with each rung carrying a $\mathcal{PT}$ -symmetric gain-loss dipole. The polarity of the dipoles is staggered along the chain, meaning that a rung bearing gain-loss is followed by one bearing loss-gain. This renders the system $\mathcal{PT}$-symmetric in both horizontal and vertical directions. The system is governed by a pair of linearly coupled discrete nonlinear Schrödinger (DNLS) equations with self-focusing or defocusing cubic onsite nonlinearity. Starting from the analytically tractable anti-continuum limit of uncoupled rungs and using the Newton's met…
Key Laboratory for Physical Electronics and Devices, Ministry of Education, School of Electronic Science and Engineering, Xi’an Jiaotong University（西安交通大学电子科学与工程学院）；Institute of Spectroscopy, Russian Academy of Sciences（俄罗斯科学院光谱研究所）；School of Physics, Chengdu University of Technology（成都理工大学物理学院）；School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院）；State Key Laboratory of Human-Machine Hybrid Augmented Intelligence, Institute of Artificial Intelligence and Robotics, Xi’an Jiaotong University（西安交通大学人工智能与机器人研究院人机混合增强智能全国重点实验室）
We consider an array of dual-core waveguides, which represent an optical realization of a chain of dimers, with an active (gain-loss) coupling between the cores, opposite signs of discrete diffraction in the parallel arrays, and a phase-velocity mismatch between them (which is necessary for the stability of the system). The array provides an optical emulation of the charge-parity ($\mathcal{CP}$) symmetry. The addition of the intracore cubic nonlinearity gives rise to several species of fundamental discrete solitons, which exist in continuous families, although the system is non-Hermitian. The…
We consider parity-time ($\mathcal{PT}$) symmetric arrays formed by $N$ optical waveguides with gain and $N$ waveguides with loss. When the gain-loss coefficient exceeds a critical value $γ_c$, the $\mathcal{PT}$-symmetry becomes spontaneously broken. We calculate $γ_c(N)$ and prove that $γ_c \to 0$ as $N \to \infty$. In the symmetric phase, the periodic array is shown to support $2N$ solitons with different frequencies and polarisations.
It is known that multidimensional complex potentials obeying $\mathcal{PT}$-symmetry may possess all real spectra and continuous families of solitons. Recently it was shown that for multi-dimensional systems these features can persist when the parity symmetry condition is relaxed so that the potential is invariant under reflection in only a single spatial direction. We examine the existence, stability and dynamical properties of localized modes within the cubic nonlinear Schrödinger equation in such a scenario of partially $\mathcal{PT}$-symmetric potential.
We investigate the existence and stability of dissipative soliton solution in a system described by complex Ginzburg-Landau (CGL) equation with asymmetric complex potential, which is obtained from original parity reflection - time reversal ($\mathcal{PT}$) symmetric Rosen-Morse potential. In this study, stability of solution is examined by numerical analysis to show that solitons are stable for some parameter ranges for both self-focusing and self-defocusing nonlinear modes. Dynamical properties such as evolution and transverse energy flow for both modes are also analyzed. Obtained results are…
We study the integrable nonlocal nonlinear Schrödinger equation proposed by Ablowitz and Musslimani, that is considered as a particular example of equations with parity-time ($\mathcal{PT}$) symmetric self-induced potential. We consider dynamics (including collisions) of moving solitons. Analytically we develop a collective coordinate approach based on variational methods and examine its applicability in the system. We show numerically that a single moving soliton can pass the origin and decay or be trapped at the origin and blows up at a finite time. Using a standard soliton ansatz, the varia…
The achievable data rates of current fiber-optic wavelength-division-multiplexing (WDM) systems are limited by nonlinear interactions between different subchannels. Recently, it was thus proposed to replace the conventional Fourier transform in WDM systems with an appropriately defined nonlinear Fourier transform (NFT). The computational complexity of NFTs is a topic of current research. In this paper, a fast inverse NFT algorithm for the important special case of multi-solitonic signals is presented. The algorithm requires only $\mathcal{O}(D\log^{2}D)$ floating point operations to compute $D…
Since the parity-time-(PT-) symmetric quantum mechanics was put forward, fundamental properties of some linear and nonlinear models with PT-symmetric potentials have been investigated. However, previous studies of PT-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 $\mathcal{PT}$-symmetric Scarf-II potentials. The broken linear PT symmetry phase may enjoy a restoration with the growing diffraction parameter. Continuous families of one- and…
In the present paper we consider an optical system with a $χ^{(2)}$-type nonlinearity and unspecified $\mathcal{PT}$-symmetric potential functions. Considering this as an inverse problem and positing a family of exact solutions in terms of cnoidal functions, we solve for the resulting potential functions in a way that ensures the potentials obey the requirements of $\mathcal{PT}$-symmetry. We then focus on case examples of soliton and periodic solutions for which we present a stability analysis as a function of their amplitude parameters. Finally, we numerically explore the nonlinear dynamics …
We consider a dipolar Bose-Einstein condensate trapped in an array of two-well systems with an arbitrary orientations of the dipoles relative to the system's axis. The system can be built as a chain of local traps sliced into two parallel lattices by a repelling laser sheet. It is modelled by a pair of coupled discrete Gross-Pitaevskii equations, with dipole-dipole self- and cross-interactions. When the dipoles are not polarized perpendicular or parallel to the lattice, the cross-interaction is asymmetric, replacing the familiar symmetric two-component discrete solitons by two new species of c…
We study fundamental rogue-wave solutions of the focusing nonlinear Schrödinger equation in the limit that the order of the rogue wave is large and the independent variables $(x,t)$ are proportional to the order (the far-field limit). We first formulate a Riemann-Hilbert representation of these solutions that allows the order to vary continuously rather than by integer increments. The intermediate solutions in this continuous family include also soliton solutions for zero boundary conditions spectrally encoded by a single complex-conjugate pair of poles of arbitrary order, as well as other sol…
