Spatial solitons and stability in self-focusing and defocusing Kerr nonlinear media with generalized parity-time-symmetric Scarff-II potentials

Nonlinear light control in optical couplers: Harnessing PPTT-symmetry for enhanced beam propagation

This study explored the evolution of nonlinear eigenmodes in coupled optical systems supported by PT-symmetric Rosen-Morse complex potential, in which one channel is with gain and the other is with loss. We assessed that the threshold potential above which PT-symmetry breakdown occurs is enhanced by coupling constant, by examining low- and high-frequency eigenmodes of ground and first excited states. The stability of eigenmodes was verified by stability analysis using Bogoliubov-de-Gennes (BdG) equations and it was established that even though the Rosen-Morse potential-supported system can create eigenmodes, it cannot support stable soliton solutions for any potential values. The investigation was extended using the modified Rosen-Morse potential that is nearly PT-symmetric and deduced the conditions for better-defined thresholds, improved damping of growth of perturbation which destabilizes eigenmodes, and advanced control mechanisms to manage perturbations and potential interactions. Propagation dynamics of the eigenmodes and power switching between channels have been studied and the controlling mechanism has been discussed to use coupled systems as optical regulators to precisely direct light between multiple paths. We have explored the significance of couplers in signal-processing applications because they control the intensity of various frequency modes. Optical couplers can be used to develop devices that let light travel in one direction while restricting it in the other which find applications in optical sensing.

Soliton and rogue wave excitations in the Chen-Lee-Liu derivative nonlinear Schrödinger equation with two complex PT-symmetric potentials

We demonstrate that fundamental nonlinear localized modes can exist in the Chen-Lee-Liu equation modified by several parity-time (PT) symmetric complex potentials. The explicit formula of analytical solitons is derived from the physically interesting Scarf-II potential, and families of spatial solitons in internal modes are numerically captured under the optical lattice potential. By the spectral analysis of linear stability, we observe that these bright solitons can remain stable across a broad scope of potential parameters, despite the breaking of the corresponding linear PT-symmetric phases. When these bright spatial solitons interact with external incident waves, they can always maintain their original shape, while the external incident wave may remain unchanged or may generate a reflected wave after the interaction. Then, the adiabatic switching of potential parameters is carried out in a way that allows these bright solitons to be excited from one unstable bound state to another alternative stable bound state. Many other intriguing properties associated with these nonlinear localized modes including the lateral power flow are further analyzed meticulously. Various high-order rogue waves induced by modulation instability in these PT-symmetric systems are generated too. These results may be useful to construct novel optical soliton communication schemes or design related optical materials.

Higher-dimensional exceptional points and peakon dynamics triggered by spatially varying Kerr nonlinear media and PT δ(x) potentials

Higher-dimensional PT-symmetric potentials constituted by delta-sign-exponential (DSE) functions are created in order to show that the exceptional points in the non-Hermitian Hamiltonian can be converted to those in the corresponding one-dimensional (1D) geometry, no matter the potentials inside are rotationally symmetric or not. These results are first numerically observed and then are proved by mathematical methods. For spatially varying Kerr nonlinearity, 2D exact peakons are explicitly obtained, giving birth to families of stable square peakons in the rotationally symmetric potentials and rhombic peakons in the nonrotationally symmetric potentials. By adiabatic excitation, different types of 2D peakons can be transformed stably and reciprocally. Under periodic and mixed perturbations, the 2D stable peakons can also travel stably along the spatially moving potential well, which implies that it is feasible to manage the propagation of the light by regulating judiciously the potential well. However, the vast majority of high-order vortex peakons are vulnerable to instability, which is demonstrated by the linear-stability analysis and by direct numerical simulations of propagation of peakon waveforms. In addition, 3D exact and numerical peakon solutions including the rotationally symmetric and the nonrotationally symmetric ones are obtained, and we find that incompletely rotationally symmetric peakons can occur stably in completely rotationally symmetric DSE potentials. The 3D fundamental peakons can propagate stably in a certain range of potential parameters, but their stability may get worse with the loss of rotational symmetry. Exceptional points and exact peakons in n dimensions are also summarized.

Stability analysis of nonlinear localized modes in the coupled Gross-Pitaevskii equations with P ⁢ T -symmetric Scarf-II potential

We study the nonlinear localized modes in two-component Bose-Einstein condensates with parity-time-symmetric Scarf-II potential, which can be described by the coupled Gross-Pitaevskii equations. Firstly, we investigate the linear stability of the nonlinear modes in the focusing and defocusing cases, and get the stable and unstable domains of nonlinear localized modes. Then we validate the results by evolving them with 5% perturbations as an initial condition. Finally, we get stable solitons by considering excitations of the soliton via adiabatical change of system parameters. These findings of nonlinear modes can be potentially applied to physical experiments of matter waves in Bose-Einstein condensates.

Self-defocusing nonlinear coupled system with PT-symmetric super-Gaussian potential

The stationary solutions of the coupled nonlinear Schrödinger equation with self-defocusing nonlinearity and super-Gaussian form of parity-time (PT) symmetric potential in an optical system have been analyzed. The stationary eigenmodes of the ground and excited states and the influence of the gain/loss coefficient on the eigenvalue spectra are discussed. The threshold condition of the PT-symmetric phase transition of the high and low-frequency modes has been studied. Also, the variation of the threshold values with the coupling constant and the effect of the nonlinearity on the eigenmodes are analyzed. The stability of the solution is verified using the linear-stability analysis. In addition, the power distribution of the fundamental solutions with the propagation, in the two channels of the system, is analyzed in the PT and broken PT regimes.

Deformation of optical solitons in a variable-coefficient nonlinear Schrödinger equation with three distinct PT-symmetric potentials and modulated nonlinearities

We investigate deformed/controllable characteristics of solitons in inhomogeneous parity-time (PT)-symmetric optical media. To explore this, we consider a variable-coefficient nonlinear Schrödinger equation involving modulated dispersion, nonlinearity, and tapering effect with PT-symmetric potential, which governs the dynamics of optical pulse/beam propagation in longitudinally inhomogeneous media. By incorporating three physically interesting and recently identified forms of PT-symmetric potentials, namely, rational, Jacobian periodic, and harmonic-Gaussian potentials, we construct explicit soliton solutions through similarity transformation. Importantly, we investigate the manipulation dynamics of such optical solitons due to diverse inhomogeneities in the medium by implementing step-like, periodic, and localized barrier/well-type nonlinearity modulations and revealing the underlying phenomena. Also, we corroborate the analytical results with direct numerical simulations. Our theoretical exploration will provide further impetus in engineering optical solitons and their experimental realization in nonlinear optics and other inhomogeneous physical systems.

Spontaneous symmetry breaking and ghost states supported by the fractional PT-symmetric saturable nonlinear Schrödinger equation

We report a novel spontaneous symmetry breaking phenomenon and ghost states existed in the framework of the fractional nonlinear Schrödinger equation with focusing saturable nonlinearity and PT-symmetric potential. The continuous asymmetric soliton branch bifurcates from the fundamental symmetric one as the power exceeds some critical value. Intriguingly, the symmetry of fundamental solitons is broken into two branches of asymmetry solitons (alias ghost states) with complex conjugate propagation constants, which is solely in fractional media. Besides, the dipole and tripole solitons (i.e., first and second excited states) are also studied numerically. Moreover, we analyze the influences of fractional Lévy index ( α) and saturable nonlinear parameters (S) on the symmetry breaking of solitons in detail. The stability of fundamental symmetric soliton, asymmetric, dipole, and tripole solitons is explored via the linear stability analysis and direct propagations. Moreover, we explore the elastic/semi-elastic collision phenomena between symmetric and asymmetric solitons. Meanwhile, we find the stable excitations from the fractional diffraction with saturation nonlinearity to integer-order diffraction with Kerr nonlinearity via the adiabatic excitations of parameters. These results will provide some theoretical basis for the study of spontaneous symmetry breaking phenomena and related physical experiments in the fractional media with PT-symmetric potentials.

The line rogue wave solutions of the nonlocal Davey-Stewartson I equation with PT symmetry based on the improved physics-informed neural network

In the paper, we employ an improved physics-informed neural network (PINN) algorithm to investigate the data-driven nonlinear wave solutions to the nonlocal Davey-Stewartson (DS) I equation with parity-time (PT) symmetry, including the line breather, kink-shaped and W-shaped line rogue wave solutions. Both the PT symmetry and model are introduced into the loss function to strengthen the physical constraint. In addition, since the nonlocal DS I equation is a high-dimensional coupled system, this leads to an increase in the number of output results. The PT symmetry also needs to be learned that is not given in advance, which increases challenges in computing for multi-output neural networks. To address these problems, our objective is to assign various levels of weight to different items in the loss function. The experimental results show that the improved algorithm has better prediction accuracy to a certain extent compared with the original PINN algorithm. This approach is feasible to investigate complex nonlinear waves in a high-dimensional model with PT symmetry.

Data driven soliton solution of the nonlinear Schrödinger equation with certain P T-symmetric potentials via deep learning

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 the physics informed neural network to solve the considered nonlinear partial differential equation with the above three potentials. We compare the predicted result with the actual result and analyze the ability of deep learning in solving the considered partial differential equation. We check the ability of deep learning in approximating the soliton solution by taking the squared error between real and predicted values. Further, we examine the factors that affect the performance of the considered deep learning method with different activation functions, namely, ReLU, sigmoid, and tanh. We also use a new activation function, namely, sech, which is not used in the field of deep learning, and analyze whether this new activation function is suitable for the prediction of soliton solution of the nonlinear Schrödinger equation for the aforementioned parity time symmetric potentials. In addition to the above, we present how the network's structure and the size of the training data influence the performance of the physics informed neural network. Our results show that the constructed deep learning model successfully approximates the soliton solution of the considered equation with high accuracy.

Stability and modulation of optical peakons in self-focusing/defocusing Kerr nonlinear media with PT-δ-hyperbolic-function potentials

In this paper, we introduce a class of novel PT- δ-hyperbolic-function potentials composed of the Dirac δ(x) and hyperbolic functions, supporting fully real energy spectra in the non-Hermitian Hamiltonian. The threshold curves of PT symmetry breaking are numerically presented. Moreover, in the self-focusing and defocusing Kerr-nonlinear media, the PT-symmetric potentials can also support the stable peakons, keeping the total power and quasi-power conserved. The unstable PT-symmetric peakons can be transformed into other stable peakons by the excitations of potential parameters. Continuous families of additional stable numerical peakons can be produced in internal modes around the exact peakons (even unstable). Further, we find that the stable peakons can always propagate in a robust form, remaining trapped in the slowly moving potential wells, which opens the way for manipulations of optical peakons. Other significant characteristics related to exact peakons, such as the interaction and power flow, are elucidated in detail. These results will be useful in explaining the related physical phenomena and designing the related physical experiments.

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.

Soliton formation and stability under the interplay between parity-time-symmetric generalized Scarf-II potentials and Kerr nonlinearity

We present an alternative type of parity-time (PT)-symmetric generalized Scarf-II potentials, which makes possible for non-Hermitian Hamiltonians in the classical linear Schrödinger system to possess fully real spectra with unique features such as the multiple PT-symmetric breaking behaviors and to support one-dimensional (1D) stable PT-symmetric solitons of power-law waveform, namely power-law solitons, in focusing Kerr-type nonlinear media. Moreover, PT-symmetric high-order solitons are also derived numerically in 1D and 2D settings. Around the exactly obtained nonlinear propagation constants, families of 1D and 2D localized nonlinear modes are also found numerically. The majority of fundamental nonlinear modes can still keep steady in general, whereas the 1D multipeak solitons and 2D vortex solitons are usually susceptible to suffering from instability. Likewise, similar results occur in the defocusing Kerr-nonlinear media. The obtained results will be useful for understanding the complex dynamics of nonlinear waves that form in PT-symmetric nonlinear media in other physical contexts.

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.

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.

Spatial solitons and stability in the one-dimensional and the two-dimensional generalized nonlinear Schrödinger equation with fourth-order diffraction and parity-time-symmetric potentials

We investigate the existence and stability of solitons in parity-time (PT)-symmetric optical media characterized by a generic complex hyperbolic refractive index distribution and fourth-order diffraction (FOD). For the linear case, we demonstrate numerically that the FOD parameter can alter the PT-breaking points. For nonlinear cases, the exact analytical expressions of the localized modes are obtained both in one- and two-dimensional nonlinear Schrödinger equations with self-focusing and self-defocusing Kerr nonlinearity. The effect of FOD on the stability structure of these localized modes is discussed with the help of linear stability analysis followed by the direct numerical simulation of the governing equation. Examples of stable and unstable solutions are given. The transverse power flow density associated with these localized modes is also discussed. It is found that the relative strength of the FOD coefficient can utterly change the direction of the power flow, which may be used to control the energy exchange among gain or loss regions.

Effect of PT symmetry on nonlinear waves for three-wave interaction models in the quadratic nonlinear media

We study the three-wave interaction that couples an electromagnetic pump wave to two frequency down-converted daughter waves in a quadratic optical crystal and PT-symmetric potentials. PT symmetric potentials are shown to modulate stably nonlinear modes in two kinds of three-wave interaction models. The first one is a spatially extended three-wave interaction system with odd gain-and-loss distribution in the channel. Modulated by the PT-symmetric single-well or multi-well Scarf-II potentials, the system is numerically shown to possess stable soliton solutions. Via adiabatical change of system parameters, numerical simulations for the excitation and evolution of nonlinear modes are also performed. The second one is a combination of PT-symmetric models which are coupled via three-wave interactions. Families of nonlinear modes are found with some particular choices of parameters. Stable and unstable nonlinear modes are shown in distinct families by means of numerical simulations. These results will be useful to further investigate nonlinear modes in three-wave interaction models.

Random medium model for cusping of plane waves

We introduce a model for a three-dimensional (3D) Schell-type stationary medium whose degree of potential's correlation satisfies the Fractional Multi-Gaussian (FMG) function. Compared with the scattered profile produced by the Gaussian Schell-model (GSM) medium, the Fractional Multi-Gaussian Schell-model (FMGSM) medium gives rise to a sharp concave intensity apex in the scattered field. This implies that the FMGSM medium also accounts for a larger than Gaussian's power in the bucket (PIB) in the forward scattering direction, hence being a better candidate than the GSM medium for generating highly-focused (cusp-like) scattered profiles in the far zone. Compared to other mathematical models for the medium's correlation function which can produce similar cusped scattered profiles the FMG function offers unprecedented tractability being the weighted superposition of Gaussian functions. Our results provide useful applications to energy counter problems and particle manipulation by weakly scattered fields.

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.

Solitons and their stability in the nonlocal nonlinear Schrödinger equation with PT-symmetric potentials

We report localized nonlinear modes of the self-focusing and defocusing nonlocal nonlinear Schrödinger equation with the generalized PT-symmetric Scarf-II, Rosen-Morse, and periodic potentials. Parameter regions are presented for broken and unbroken PT-symmetric phases of linear bounded states and the linear stability of the obtained solitons. Moreover, we numerically explore the dynamical behaviors of solitons and find stable solitons for some given parameters.

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.

Three-component Gross-Pitaevskii equations in the spin-1 Bose-Einstein condensate: Spin-rotation symmetry, matter-wave solutions, and dynamics

We report new matter-wave solutions of the one-dimensional spin-1 Bose-Einstein condensate system by combining global spin-rotation states and similarity transformation. Dynamical behaviors of non-stationary global spin-rotation states derived from the SU(2) spin-rotation symmetry are discussed, which exhibit temporal periodicity. We derive generalized bright-dark mixed solitons and new rogue wave solutions and reveal the relations between Euler angles in spin-rotation symmetry and parameters in ferromagnetic and polar solitons. In the modulated spin-1 Bose-Einstein condensate system, new solutions are derived and graphically illustrated for different types of modulations. Moreover, numerical simulations are performed to investigate the stability of some obtained solutions for chosen parameters.

Stable parity-time-symmetric nonlinear modes and excitations in a derivative nonlinear Schrödinger equation

The effect of derivative nonlinearity and parity-time-symmetric (PT-symmetric) potentials on the wave propagation dynamics is explored in the derivative nonlinear Schrödinger equation, where the physically interesting Scarf-II and harmonic-Hermite-Gaussian potentials are chosen. We study numerically the regions of unbroken and broken linear PT-symmetric phases and find some stable bright solitons of this model in a wide range of potential parameters even though the corresponding linear PT-symmetric phases are broken. The semielastic interactions between particular bright solitons and exotic incident waves are illustrated such that we find that particular nonlinear modes almost keep their shapes after interactions even if the exotic incident waves have evidently been changed. Moreover, we exert the adiabatic switching on PT-symmetric potential parameters such that a stable nonlinear mode with the unbroken linear PT-symmetric phase can be excited to another stable nonlinear mode belonging to the broken linear PT-symmetric phase.

Stability, integrability, and nonlinear dynamics of PT-symmetric optical couplers with cubic cross-interactions or cubic-quintic nonlinearities

We explore the parity-time-( PT)-symmetric optical couplers with the cubic both self- and cross-interactions corresponding to self- and cross-phase modulations. When the coefficient of the cubic cross-interaction is chosen as the different values, we find three distinct cases for two branches, including the stable-stable modes (linear unbroken PT-symmetric phase), stable-unstable modes (linear unbroken PT-symmetric phase), as well as unstable-unstable modes (linear broken PT-symmetric phase). Moreover, we find the periodic trajectories for some parameters. Similarly, we also explore the PT-symmetric optical couplers with cubic-quintic self-phase modulations. We numerically give the stable and unstable regions of the cubic-quintic system. Moreover, we also find the periodic trajectories for some parameters in the Stokes domain.

On stable solitons and interactions of the generalized Gross-Pitaevskii equation with PT- and non- PT-symmetric potentials

We report the bright solitons of the generalized Gross-Pitaevskii (GP) equation with some types of physically relevant parity-time- ( PT-) and non- PT-symmetric potentials. We find that the constant momentum coefficient Γ can modulate the linear stability and complicated transverse power-flows (not always from the gain toward loss) of nonlinear modes. However, the varying momentum coefficient Γ(x) can modulate both unbroken linear PT-symmetric phases and stability of nonlinear modes. Particularly, the nonlinearity can excite the unstable linear mode (i.e., broken linear PT-symmetric phase) to stable nonlinear modes. Moreover, we also find stable bright solitons in the presence of non- PT-symmetric harmonic-Gaussian potential. The interactions of two bright solitons are also illustrated in PT-symmetric potentials. Finally, we consider nonlinear modes and transverse power-flows in the three-dimensional (3D) GP equation with the generalized PT-symmetric Scarff-II potential.

Solitonic dynamics and excitations of the nonlinear Schrödinger equation with third-order dispersion in non-Hermitian PT-symmetric potentials

Solitons are of the important significant in many fields of nonlinear science such as nonlinear optics, Bose-Einstein condensates, plamas physics, biology, fluid mechanics, and etc. The stable solitons have been captured not only theoretically and experimentally in both linear and nonlinear Schrödinger (NLS) equations in the presence of non-Hermitian potentials since the concept of the parity-time -symmetry was introduced in 1998. In this paper, we present novel bright solitons of the NLS equation with third-order dispersion in some complex -symmetric potentials (e.g., physically relevant -symmetric Scarff-II-like and harmonic-Gaussian potentials). We find stable nonlinear modes even if the respective linear -symmetric phases are broken. Moreover, we also use the adiabatic changes of the control parameters to excite the initial modes related to exact solitons to reach stable nonlinear modes. The elastic interactions of two solitons are exhibited in the third-order NLS equation with -symmetric potentials. Our results predict the dynamical phenomena of soliton equations in the presence of third-order dispersion and -symmetric potentials arising in nonlinear fiber optics and other physically relevant fields.