Dipole Solitons Research Articles

Solitons in one-dimensional non-Hermitian moiré photonic lattice

We study the excitation and stabilization of gap solitons and dipole solitons in a non-Hermitian moiré photonic lattice with self-focusing effect. The lattice is constituted by superimposition of a parity-time symmetric potential and an anti-parity-time symmetric potential with inconsistent frequencies. We find that the single gap solitons within semi-infinite bandgap can indeed stably persist, and their ss positions within the lattice can be precisely adjusted through controlling the oblique angles and propagation constants of incident solitons. Further study shows that the lattice enables dipole solitons with specific soliton energy to overcome inherent repulsion, facilitating their propagation within the lattice. This work unveils the existence of stable gap solitons supported by non-Hermitian moiré photonic lattices, emphasizing the pivotal roles of propagation constants and incident angles in regulating soliton propagation.

Two-dimensional vortex dipole, tripole, and quadrupole solitons in nonlocal nonlinearity with Gaussian potential well and barrier

Two-dimensional vortex dipole, tripole, and quadrupole solitons in nonlocal nonlinearity with Gaussian potential well and barrier

Dynamics of dipole solitons on a continuous-wave background in an inhomogeneous nonlinear medium with higher-order dispersion

Dynamics of dipole solitons on a continuous-wave background in an inhomogeneous nonlinear medium with higher-order dispersion

Solitons and vortices formation in deformed photonic graphene

We investigate the formation and evolution of two-dimensional solitons and vortices in deformed photonic graphene, encompassing fundamental solitons, multipole solitons, and vortex solitons. Using the plane-wave expansion method, we analyze the bandgap structure of photonic graphene, revealing Dirac cones whose characteristics are influenced by deformation. Soliton generation results from lattice potential deformation, linear energy gap modulation, and nonlinear effects. Specifically, fundamental and dipole solitons, exhibiting out-of-phase behavior, propagate robustly within photonic graphene, maintaining structural integrity over extended distances, while other soliton solutions are unstable. Furthermore, we explore the realization and propagation characteristics of vortex solitons with unit topological charge within photonic graphene.

Three-Wave Mixing of Dipole Solitons in One-Dimensional Quasi-Phase-Matched Nonlinear Crystals

A quasi-phase-matched technique is introduced for soliton transmission in a quadratic [χ (2)] nonlinear crystal to realize the stable transmission of dipole solitons in a one-dimensional space under three-wave mixing. We report four types of solitons as dipole solitons with distances between their bimodal peaks that can be laid out in different stripes. We study three cases of these solitons: spaced three stripes apart, one stripe apart, and confined to the same stripe. For the case of three stripes apart, all four types have stable results, but for the case of one stripe apart, stable solutions can only be found at ω 1 = ω 2, and for the condition of dipole solitons confined to one stripe, stable solutions exist only for Type1 and Type3 at ω 1=ω 2. The stability of the soliton solution is solved and verified using the imaginary time propagation method and real-time transfer propagation, and soliton solutions are shown to exist in the multistability case. In addition, the relations of the transportation characteristics of the dipole soliton and the modulation parameters are numerically investigated. Finally, possible approaches for the experimental realization of the solitons are outlined.

Transformation of rotating dipole and vortex solitons in an anharmonic potential

We investigate the basic properties of dipole and vortex solitons, both quiescent and rotating, supported by the axially symmetric anharmonic potential in a medium with cubic-quintic nonlinearity. The static solitons exhibit nonmonotonous behavior in terms of propagation constant as a function, and feature bistability regions. In the rotating frame, the dipole solitons transform into vortex solitons and the topological charge increases gradually with the growth of rotation frequency. Linear stability analysis and direct simulations reveal that rotating dipole solitons with higher charges are more stable than those with lower charges. Meanwhile, dipole solitons can rotate persistently during propagation and preserve their shape over multiple rotation periods. Our findings provide an alternative way for the transformation to achieve dipole and vortex solitons with higher topological charges.

Fundamental, dipole, and vortex solitons in fractional nonlinear Schrödinger equation with a parity-time-symmetric periodic potential

We investigate the existence and stability of two-dimensional (2D) solitons in the framework of the nonlinear fractional Schrödinger equation (NLFSE) with focusing nonlinearity and a parity-time (PT) symmetric periodic potential. Numerical studies reveal that the fundamental, dipole, and localized vortex solitons exist in a semi-infinite band gap. The Lévy index (α) cannot change the phase transition point of the PT-symmetric periodic potential, but it affects the existence and stability of these solitons significantly. With the growth of the Lévy index, the stability regions become markedly wider for the fundamental, dipole solitons and vortex solitons with the topological charge of 2 (S=2). But for a unit topological charge (S=1), the vortex soliton cannot maintain its stability under the usual diffraction effect (α=2). The result shows that the fractional-order diffraction helps to stabilize the vortex solitons with S=1.

Two-dimensional vortex dipole solitons in nonlocal nonlinearity with PT-symmetric Scarff-II potential.

We investigate the dynamics and stability of two-dimensional (2D) vortex dipole solitons in nonlocal nonlinearity with PT-symmetric Scarff-II potential. We analyze the solitons with single charge and higher-order charge using analytical and numerical methods. By the variational approach, we can obtain analytical solutions for the model. It is found that the nonlocality degree affects the evolution of the beams. We discover that the vortex dipole solitons will undergo stable deformation rather than maintaining their basic profile when the nonlocality is strong. Moreover, the stability of the vortex dipole solitons depends on the potential depth and there exists a threshold, below which the beams can keep their shapes and propagate stably whether the nonlocality is weak, intermediate, or strong. Numerical simulations are consistent with the analytical results.

Wave-speed management of dipole, bright and W-shaped solitons in optical metamaterials

Wave-speed management of soliton pulses in a nonlinear metamaterial exhibiting a rich variety of physical effects that are important in a wide range of practical applications, is studied both theoretically and numerically. Ultrashort electromagnetic pulse transmission in such inhomogeneous system is described by a generalized nonlinear Schrödinger equation with space-modulated higher-order dispersive and nonlinear effects of different nature. We present the discovery of three types of periodic wave solutions that are composed by the product of Jacobi elliptic functions in the presence of all physical processes. Envelope solitons of the dipole, bright and W-shaped types are also identified, thus illustrating the potentially rich set of localized pulses in the system. We develop an effective similarity transformation method to investigate the soliton dynamics in the presence of the inhomogeneities of media. The application of developed method to control the wave speed of the presented solitons is discussed. The results show that the wave speed of dipole, bright and W-shaped solitons can be effectively controlled through spatial modulation of the metamaterial parameters. In particular, the soliton pulses can be decelerated and accelerated by suitable variations of the distributed dispersion parameters.

Physics-informed Neural Network method for predicting soliton dynamics supported by complex PT-symmetric potentials

Abstract We examine the deep learning technique referred to as the physics-informed neural network method for approximating non-linear Schrödinger equation under considered parity time symmetric potentials and obtaining multifarious soliton solutions. For the first time, neural networks founded principally physical information are adopted to figure out the solution the examined non-linear partial differential equation and generate six different types of soliton solutions, which are basic, dipole, tripole, quadruple, pentapole and sextupole solitons we consider. We make comparisons between the predicted and actual soliton solutions to see whether deep learning is capable of seeking the solution the partial differential equation described before. We may assess whether physics-informed neural network is capable of effectively providing approximate soliton solutions through the evaluation of squared error between the predicted and numerical results. Besides, we also scrutinize how different activation mechanisms and network architectures impact the capability of selected deep learning technique works. Through the findings we can prove that the neural networks model we established can be utilized to accurately and effectively approximate non-linear Schrödinger equation under consideration and predict the dynamics of soliton solution.

Physics-Informed Neural Network Method for Predicting Soliton Dynamics Supported by Complex Parity-Time Symmetric Potentials

We examine the deep learning technique referred to as the physics-informed neural network method for approximating the nonlinear Schrödinger equation under considered parity-time symmetric potentials and for obtaining multifarious soliton solutions. Neural networks to found principally physical information are adopted to figure out the solution to the examined nonlinear partial differential equation and to generate six different types of soliton solutions, which are basic, dipole, tripole, quadruple, pentapole, and sextupole solitons we consider. We make comparisons between the predicted and actual soliton solutions to see whether deep learning is capable of seeking the solution to the partial differential equation described before. We may assess whether physics-informed neural network is capable of effectively providing approximate soliton solutions through the evaluation of squared error between the predicted and numerical results. Moreover, we scrutinize how different activation mechanisms and network architectures impact the capability of selected deep learning technique works. Through the findings we can prove that the neural networks model we established can be utilized to accurately and effectively approximate the nonlinear Schrödinger equation under consideration and to predict the dynamics of soliton solution.

Propagation of coupled quartic and dipole multi-solitons in optical fibers medium with higher-order dispersions

We present the discovery of two types of multiple-hump soliton modes in a highly dispersive optical fiber with a Kerr nonlinearity. We show that multi-hump optical solitons of quartic or dipole types are possible in the fiber system in the presence of higher-order dispersion. Such nonlinear wave packets are very well described by an extended nonlinear Schrödinger equation involving both cubic and quartic dispersion terms. It is found that the third- and fourth-order dispersion effects in the fiber material may lead to the coupling of quartic or dipole solitons into double-, triple-, and multi-humped solitons. We provide the initial conditions for the formation of coupled multi-hump quartic and dipole solitons in the fiber. Numerical results illustrate that propagating multi-quartic and multi-dipole solitons in highly dispersive optical fibers coincide with a high accuracy to our analytical multi-soliton solutions. It is important for applications that described multiple-hump soliton modes are stable to small noise perturbation that was confirmed by numerical simulations. These numerical results confirm that the newly found multi-soliton pulses can be potentially utilized for transmission in optical fibers medium with higher-order dispersions.

The transport of dipole solitons in a one-dimensional nonlinear photonic crystal

Quadratic χ(2) nonlinearity crystal provide an ideal stage for the researches of the transport of optical solitons. However, the transport of dipole solitons in a homogeneous χ(2) nonlinear medium is unstable. To solve this problem, a one dimensional nonlinearity photonic crystal is designed from a χ(2) nonlinearity crystal by quasi-phase matching technique. The duty cycle of the positive polarized in the transverse direction of the crystal is periodic modulated, so as to realize a stable transport of dipole solitons in it. Besides, relations of the transportation characteristics of the dipole soliton and the modulation parameters are researched in this letter numerically. Moreover, mobility and collision of these dipole soliton through the crystals are discussed through the paper. Some key parameters of the related experiment are also estimated.

Two-dimensional fractional [formula omitted]-symmetric cubic–quintic NLS equation: Double-loop symmetry breaking bifurcations, ghost states and dynamics

In this paper, we address the double-loop symmetry breaking bifurcations of optical vortex solitons in the two-dimensional (2D) fractional cubic–quintic nonlinear Schrödinger (FCQNLS) equation with an imprinted partially parity-time (PPT) symmetric complex potential. The continuous branches of asymmetric solitons bifurcate from the fundamental one as the power exceeds some critical value. It is found that the symmetry breaking point is exactly the point at which the fundamental symmetric soliton is destabilized. Intriguingly, the branches of asymmetry solitons (alias ghost states) are existing with complex conjugate propagation constants, which is solely in fractional media. Moreover, the branch of asymmetric solitons crosses the branch of fundamental symmetric ones when it reaches the second critical value, and eventually merges back into it at the third critical power point. The imaginary part of the propagation constant also exhibits a circular shape. And the stability of the fundamental symmetric soliton is restored. In addition, the stability of two branches of asymmetric solitons differs from each other. The reason generating this phenomenon is analyzed in detail. Besides, the branch of dipole solitons for the first excited state is also discussed numerically, which is always stable in the process of symmetry breaking. The stability and dynamics of symmetric (fundamental and the first excited ones) and asymmetric solitons are explored via the linear stability analysis, direct simulation, modulation as well as adiabatic excitation. These results will provide some theoretical basis for the study of spontaneous symmetry breaking phenomena and related experiments in fractional media with PPT-symmetric potentials.

Three-dimensional vortex dipole solitons in self-gravitating systems.

We derive the nonlinear equationsgoverning the dynamics of three-dimensional (3D) disturbances in a nonuniform rotating self-gravitating fluid under the assumption that the characteristic frequencies of disturbances are small compared to the rotation frequency. Analytical solutions of these equations are found in the form of the 3D vortex dipole solitons. The method for obtaining these solutions is based on the well-known Larichev-Reznik procedure for finding two-dimensional nonlinear dipole vortex solutions in the physics of atmospheres of rotating planets. In addition to the basic 3D x-antisymmetric part (carrier), the solution may also contain radially symmetric (monopole) or/and antisymmetric along the rotation axis (z-axis) parts with arbitrary amplitudes, but these superimposed parts cannot exist without the basic part. The 3D vortex soliton without the superimposed parts is extremely stable. It moves without distortion and retains its shape even in the presence of an initial noise disturbance. The solitons with parts that are radially symmetric or/and z antisymmetric turn out to be unstable, although, at sufficiently small amplitudes of these superimposed parts, the soliton retains its shape for a very long time.

Nonlocal soliton in non-parity-time-symmetric coupler

Parity-time (PT) symmetric is not a necessary condition for achieving a real spectrum and some studies about realizing real spectra in non-PT-symmetric systems with arbitrary gain–loss profiles have been presented recently. By tuning the free parameters in non-PT-symmetric potentials, phase transition could also be induced. Above phase transition point, discrete complex eigenvalues bifurcate out from continuous real eigenvalues in the interior of the continuous spectrum. In this work, we investgate the existence and stability of solitons in nonlocal nonlinear couplers with non-PT-symmetric complex potentials both below and above phase transition. There are several discrete eigenvalues in the linear spectra of the non-PT-symmetric system used here. With the square-operator iteration method, we find that different continuous families of solitions can bifurcate from different discrete linear eigenvalues. Moreover, linear-stability analysis collaborated with direct numerical propagation simulations demonstrates that the nonlocal solitions can be stable in a range of parameter values. we first address the cases below the phase transition. To be specific, when we fix the coupling coefficient and vary the degree of nonlocality, it’s found that fundamental solitons, dipole solitons, tripolar solitons, quadrupole solitons bifurcate from the largest,the second-largest, the third-largest and the fifth-largest discrete eigenvalue, respectively. These nonlocal solitons are all stable in the low power region. With an increase of the degree of nonlocality, the stability region shrinks for the fundamental solitons while it widens for the dipole and multiplole solitons. At the same time, the power of all the stable solitons increases with the increase of the degree of nonlocality. By varying the coupling coefficient, the arrangement of soliton families emerging in the discrete interval of the linear spectrum can be changed. For example, the dipole solitons bifurcate from the third-or fourth-largest discrete eigenvalue while the tripolar solitons bifurcate from the fifth largest discrete eigenvalue. Above phase transition,the fundamental solitons are unstable in the low and high power region but are stable in the moderate power region. The stability region shrinks with the increasing degree of nonlocality. We also find the family of dipole solitons bifurcates from the second-largest discrete eigenvalue, but all the dipole solitons are unstable. In addition, we find that the eigenvalues in linear-stability spectra of solitons emerge as conjugation pairs.

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.

Scalar vortex solitons and vector dipole solitons in whispering gallery mode optical microresonators

Governed by the coupled variable-coefficient Lugiato-Lefever equation, the soliton characteristics of whispering gallery mode microresonators in a coupled system are studied. Based on analytical solutions of the relevant models in the absence of pump, the dynamic evolution of vortex solitons is discussed in the periodic system, and the periodic phase change of vortex solitons and the stability of dipole solitons are discussed under the condition of the external excitation. In addition, the influence of the topological charge on the energy distribution of two-component and total light intensity for dipole solitons is investigated. The vortex solitons show great anti-interference performance after the addition of the external excitation, and dipole solitons can maintain the stability with periodical behaviors during a long evolution process. With the addition of the topological charge, the branch of dipole solitons for the single component and the number of spiral layers for vortex solitons increase.

Bright solitons in fractional coupler with spatially periodical modulated nonlinearity

We study bright solitons in the fractional coupler with a spatially periodical modulated nonlinearity. The results show that the linear coupling constant κ, Lévy index α, chemical potential μ and nonlinear intensity g have a significant influence on the amplitude, width and stability of fundamental solitons, dipole solitons and tripole solitons. We investigate the stability of bright solitons by linear stability analysis and the real-time evolution method. It is found that the bright solitons tend to be stable with the increase of linear coupling constant, Lévy index, chemical potential, and nonlinear intensity in the corresponding parameter interval. The results also show that the distance between adjacent peaks is four times of the nonlinear lattice period for dipole solitons and six times for tripole solitons. We also find that the number of soliton peaks is smaller, and the stable region is wider in the fractional coupler with the spatially periodical modulated nonlinear lattice.

Novel hybrid solitary waves and shrunken-periodic solutions, solitary Moiré pattern and conserved vectors of the (4 + 1)-Fokas equation

In this paper, we present a comprehensive study on the [Formula: see text]-Fokas equation that includes new solutions and conservation laws of the equation. Specifically, for the first time, new shrunken-periodic and fascinating interactions on the bright soliton background have been investigated. Furthermore, a bright, dark, dipole soliton, tripole soliton, multipole soliton, periodic, doubly-periodic, damped-periodic, kink, lump, rogue wave, breather and their interactions have been attained and graphically illustrated. In addition, a significant interaction between four parabolic-periodic solitary waves is first detected and emphasized by harnessing 3D and contour plots. On the other hand, the latter has been compared to an experimentally-induced Moiré effect from a monitor screen camera capture. These outstanding results are achieved using the Lie symmetry approach. The multiple reduction process of the [Formula: see text]-Fokas equation through the Lie algebra spanned elements, and the corresponding optimal system guarantees 23 solutions in general forms. To determine the physical meaning of the latter, an appropriate ansatz for the arbitrary functions and parameters has been proposed and listed in the corresponding tables, which leads to the various types of solitary wave solutions. Finally, the conserved vectors for the underlying equation are derived using the multiplier method.