Complex PT-symmetric nonlinear Schrödinger equation and Burgers equation

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.

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.

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.

𝒫𝒯-symmetric and antisymmetric nonlinear states in a split potential box

We introduce a one-dimensional [Formula: see text]-symmetric system, which includes the cubic self-focusing, a double-well potential in the form of an infinitely deep potential box split in the middle by a delta-functional barrier of an effective height ε, and constant linear gain and loss, γ, in each half-box. The system may be readily realized in microwave photonics. Using numerical methods, we construct [Formula: see text]-symmetric and antisymmetric modes, which represent, respectively, the system's ground state and first excited state, and identify their stability. Their instability mainly leads to blowup, except for the case of ε=0, when an unstable symmetric mode transforms into a weakly oscillating breather, and an unstable antisymmetric mode relaxes into a stable symmetric one. At ε>0, the stability area is much larger for the [Formula: see text]-antisymmetric state than for its symmetric counterpart. The stability areas shrink with increase of the total power, P In the linear limit, which corresponds to [Formula: see text], the stability boundary is found in an analytical form. The stability area of the antisymmetric state originally expands with the growth of γ, and then disappears at a critical value of γThis article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)'.

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.

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.

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.

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.

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.

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

We present a unified theoretical study of the bright solitons governed by self-focusing and defocusing nonlinear Schrödinger (NLS) equations with generalized parity-time- (PT) symmetric Scarff-II potentials. Particularly, a PT-symmetric k-wave-number Scarff-II potential and a multiwell Scarff-II potential are considered, respectively. For the k-wave-number Scarff-II potential, the parameter space can be divided into different regions, corresponding to unbroken and broken PT symmetry and the bright solitons for self-focusing and defocusing Kerr nonlinearities. For the multiwell Scarff-II potential the bright solitons can be obtained by using a periodically space-modulated Kerr nonlinearity. The linear stability of bright solitons with PT-symmetric k-wave-number and multiwell Scarff-II potentials is analyzed in detail using numerical simulations. Stable and unstable bright solitons are found in both regions of unbroken and broken PT symmetry due to the existence of the nonlinearity. Furthermore, the bright solitons in three-dimensional self-focusing and defocusing NLS equations with a generalized PT-symmetric Scarff-II potential are explored. This may have potential applications in the field of optical information transmission and processing based on optical solitons in nonlinear dissipative but PT-symmetric systems.