Antisymmetric Solitons Research Articles

Symmetry-breaking bifurcations of pure-quartic solitons in dual-core couplers.

We investigate spontaneous symmetry- and antisymmetry-breaking bifurcations of solitons in a nonlinear dual-core waveguide with the pure-quartic dispersion and Kerr nonlinearity. Symmetric, antisymmetric, and asymmetric pure-quartic solitons (PQSs) are found, and their stability domains are identified. The bifurcations for both the symmetric and antisymmetric PQSs are of the supercritical type (alias phase transitions of the second kind). Direct simulations of the perturbed evolution of PQSs corroborate their stability boundaries predicted by the analysis of small perturbations.

PT-symmetric solitons and parameter discovery in self-defocusing saturable nonlinear Schrödinger equation via LrD-PINN.

We propose a physical information neural network with learning rate decay strategy (LrD-PINN) to predict the dynamics of symmetric, asymmetric, and antisymmetric solitons of the self-defocusing saturable nonlinear Schrödinger equation with the PT-symmetric potential and boost the predicted evolutionary distance by an order of magnitude. Taking symmetric solitons as an example, we explore the advantages of the learning rate decay strategy, analyze the anti-interference performance of the model, and optimize the network structure. In addition, the coefficients of the saturable nonlinearity strength and the modulation strength in the PT-symmetric potential are reconstructed from the dataset of symmetric soliton solutions. The application of more advanced machine learning techniques in the field of nonlinear optics can provide more powerful tools and richer ideas for the study of optical soliton dynamics.

Symmetric and antisymmetric vector solitons for the fractional quadric-cubic coupled nonlinear Schrödinger equation

The fractional quadric-cubic coupled nonlinear Schrödinger equation is concerned, and vector symmetric and antisymmetric soliton solutions are obtained by the square operator method. The relationship between the Lévy index and the amplitudes of vector symmetric and antisymmetric solitons is investigated. Two components of vector symmetric and antisymmetric solitons show a positive and negative trend with the Lévy index, respectively. The stability intervals of these solitons and the propagation constants corresponding to the maximum and minimum instability growth rates are studied. Results indicate that vector symmetric solitons are more stable and have better interference resistance than vector antisymmetric solitons.

Symmetric and antisymmetric solitons in the fractional nonlinear schrödinger equation with saturable nonlinearity and PT-symmetric potential: Stability and dynamics

We report symmetric and antisymmetric solitons in the fractional nonlinear Schrödinger equation with the defocused saturable nonlinearity and the PT-symmetric potential. Both symmetric and antisymmetric solitons can exist and maintain their symmetries as the power of the soliton increases. We find that the strong saturable nonlinearity suppresses the change of the propagation constant as the soliton power increases. The stability of symmetric and antisymmetric solitons is studied by the linear stability analysis and also verified by the direct numerical simulation. The results show that the high soliton power and strong saturable nonlinearity can restrain the instability of symmetric solitons, and make symmetric solitons propagate stably and possess strong robustness. However, antisymmetric solitons become very unstable and possess less robust with the increase of the soliton power.

Symmetric and Anti-Symmetric Solitons of the Fractional Second- and Third-Order Nonlinear Schrödinger EquationSupported by the Zhejiang Provincial Natural Science Foundation of China (Grant No. LR20A050001), the National Natural Science Foundation of China (Grant No. 12075210), and the Scientific Research and Developed Fund of Zhejiang A&F University (Grant No. 2021FR0009).

The fractional second- and third-order nonlinear Schrödinger equation is studied, symmetric and antisymmetric soliton solutions are derived, and the influence of the Lévy index on different solitons is analyzed. The stability and stability interval of solitons are discussed. The anti-interference ability of stable solitons to the small disturbance shows a good robustness.

Existence, symmetry breaking bifurcation and stability of two-dimensional optical solitons supported by fractional diffraction.

We study existence, bifurcation and stability of two-dimensional optical solitons in the framework of fractional nonlinear Schrödinger equation, characterized by its Lévy index, with self-focusing and self-defocusing saturable nonlinearities. We demonstrate that the fractional diffraction system with different Lévy indexes, combined with saturable nonlinearity, supports two-dimensional symmetric, antisymmetric and asymmetric solitons, where the asymmetric solitons emerge by way of symmetry breaking bifurcation. Different scenarios of bifurcations emerge with the change of stability: the branches of asymmetric solitons split off the branches of unstable symmetric solitons with the increase of soliton power and form a supercritical type bifurcation for self-focusing saturable nonlinearity; the branches of asymmetric solitons bifurcates from the branches of unstable antisymmetric solitons for self-defocusing saturable nonlinearity, featuring a convex shape of the bifurcation loops: an antisymmetric soliton loses its stability via a supercritical bifurcation, which is followed by a reverse bifurcation that restores the stability of the symmetric soliton. Furthermore, we found a scheme of restoration or destruction the symmetry of the antisymmetric solitons by controlling the fractional diffraction in the case of self-defocusing saturable nonlinearity.

Symmetry breaking in a two-component system with repulsive interactions and linear coupling

We extend the well-known theoretical treatment of the spontaneous symmetry breaking (SSB) in two-component systems combining linear coupling and self-attractive nonlinearity to a system in which the linear coupling competes with repulsive interactions. First, we address one- and two-dimensional (1D and 2D) ground-state (GS) solutions and 2D vortex states with topological charges S=1 and 2, maintained by a confining harmonic-oscillator (HO) potential. The system can be implemented in BEC and optics. By means of the Thomas-Fermi approximation and numerical solution of the underlying coupled Gross-Pitaevskii equations, we demonstrate that SSB takes place, in the GSs and vortices alike, when the cross-component repulsion is stronger than the self-repulsion in each component. The SSB transition is categorized as a supercritical bifurcation, which gives rise to states featuring broken symmetry in an inner area, and intact symmetry in a surrounding layer. Unlike stable GSs and vortices with S=1, the states with S=2 are unstable against splitting. We also address SSB for 1D gap solitons in the system including a lattice potential. In this case, SSB takes place under the opposite condition, i.e., the cross-component repulsion must be weaker than the self-repulsion, and SSB is exhibited by antisymmetric solitons.

Spontaneous symmetry breaking in purely nonlinear fractional systems.

Spontaneous symmetry breaking, a spontaneous course of breaking the spatial symmetry (parity) of the system, is known to exist in many branches of physics, including condensed-matter physics, high-energy physics, nonlinear optics, and Bose-Einstein condensates. In recent years, the spontaneous symmetry breaking of solitons in nonlinear wave systems is broadly studied; understanding such a phenomenon in nonlinear fractional quantum mechanics with space fractional derivatives (the purely nonlinear fractional systems whose fundamental properties are governed by a nonlinear fractional Schrödinger equation), however, remains pending. Here, we survey symmetry breaking of solitons in fractional systems (with the fractional diffraction order being formulated by the Lévy index α) of a nonlinear double-well structure and find several kinds of soliton families in the forms of symmetric and anti-symmetric soliton states as well as asymmetric states. Linear stability and dynamical properties of these soliton states are explored relying on linear-stability analysis and direct perturbed simulations, with which the existence and stability regions of all the soliton families in the respective physical parameter space are identified.

Symmetry breaking of spatial Kerr solitons in fractional dimension

We study symmetry breaking of solitons in the framework of a nonlinear fractional Schr\"{o}dinger equation (NLFSE), characterized by its L\'{e}vy index, with cubic nonlinearity and a symmetric double-well potential. Asymmetric, symmetric, and antisymmetric soliton solutions are found, with stable asymmetric soliton solutions emerging from unstable symmetric and antisymmetric ones by way of symmetry-breaking bifurcations. Two different bifurcation scenarios are possible. First, symmetric soliton solutions undergo a symmetry-breaking bifurcation of the pitchfork type, which gives rise to a branch of asymmetric solitons, under the action of the self-focusing nonlinearity. Second, a family of asymmetric solutions branches off from antisymmetric states in the case of self-defocusing nonlinearity through a bifurcation of an inverted-pitchfork type. Systematic numerical analysis demonstrates that increase of the L\'{e}vy index leads to shrinkage or expansion of the symmetry-breaking region, depending on parameters of the double-well potential. Stability of the soliton solutions is explored following the variation of the L\'{e}vy index, and the results are confirmed by direct numerical simulations.

Spatial solitons in a nonlocal fused coupler

We study soliton-modes of optical pulses and their stabilities in a nonlocal nonlinear fused dual-core planar waveguide. Taking the different parameters, we find the symmetric, antisymmetric and asymmetric bright soliton pairs and dipole soliton pairs. During the long-distance propagation, the amplitudes of some solitons adding the white noise exhibit periodic oscillations and the solitons except asymmetric soliton can propagate steadily. We also find that the phases and profiles of solitons in the nonlocal coupler are determined by the propagation constant, the amplitude of the optical field and the region of the fused segment.

Soliton solutions in nonlocal nonlinear coupler

We study the existence of soliton pairs in nonlocal nonlinear coupler and the effects of nonlocal nonlinearity on the stability and types of soliton pairs. Bright soliton, dipole soliton, periodic soliton, double soliton and dark soliton are found, and many kinds of antisymmetric, symmetric and asymmetric solitons can be obtained by adjusting the nonlinear parameters or coupling coefficients. In addition, the stability of the exact bright solitary wave solutions and dipole-mode solitary wave solutions with the white noise perturbation are also investigated numerically. It is investigated that the energy flow and the Hamiltonian, and the energy flow are related to the nonlocal parameter. Moreover, We also obtain some Airy-like solitons by numerical method.

Controllable behaviors of nonautonomous solitons on background of continuous wave and cnoidal wave in $$\mathcal {PT}$$ PT -symmetric dimer with inhomogeneous effect

We obtain exact \(\mathcal {PT}\)-symmetric and \(\mathcal {PT}\)-antisymmetric nonautonomous soliton solutions on background waves. These solutions indicate that dispersion and nonlinear coefficients influence form factors of nonautonomous solitons such as amplitude, width and center; however, linear coupling coefficient and gain/loss parameter only influence phase of solitons. Based on these solutions, the controllable behaviors such as postpone, sustainment and restraint on continuous wave background in an exponential decreasing dispersion system are discussed. Moreover, the propagation behaviors of solitons on the cnoidal wave background in different dispersion systems are also studied.

Vortex soliton in (2+1)-dimensional $$\mathcal {PT}$$ PT -symmetric nonlinear couplers with gain and loss

We study the coupled nonlinear Schrodinger equation in the (2+1)-dimensional inhomogeneous \(\mathcal {PT}\)-symmetric nonlinear couplers and obtain \(\mathcal {PT}\)-symmetric and \(\mathcal {PT}\)-antisymmetric vortex soliton solutions. The dynamical behaviors of the completely localized structures (vortex solitons) are analytically and numerically investigated in a diffraction decreasing system with exponential profile. Numerical results indicate that one vortex soliton with different topological charges can stably propagate a long distance. The space between two humps and the modulation depth of vortex solitons add when the topological charge increases. However, the change tendency of the amplitude and width of vortex solitons is same with the increase in topological charge.

Energy transport in the three coupled α-polypeptide chains of collagen molecule with long-range interactions effect.

The dynamics of three coupled α-polypeptide chains of a collagen molecule is investigated with the influence of power-law long-range exciton-exciton interactions. The continuum limit of the discrete equations reveal that the collagen dynamics is governed by a set of three coupled nonlinear Schrödinger equations, whose dispersive coefficient depends on the LRI parameter r. We construct the analytic symmetric and asymmetric (antisymmetric) soliton solutions, which match with the structural features of collagen related with the acupuncture channels. These solutions are used as initial conditions for the numerical simulations of the discrete equations, which reveal a coherent transport of energy in the molecule for r > 3. The results also indicate that the width of the solitons is a decreasing function of r, which help to stabilize the solitons propagating in the molecule. To confirm further the efficiency of energy transport in the molecule, the modulational instability of the system is performed and the numerical simulations show that the energy can flow from one polypeptide chain to another in the form of nonlinear waves.

When Linear Stability Does Not Exclude Nonlinear Instability

We describe a mechanism that results in the nonlinear instability of stationary states even in the case where the stationary states are linearly stable. This instability is due to the nonlinearity-induced coupling of the linearization's internal modes of negative energy with the continuous spectrum. In a broad class of nonlinear Schrödinger equations considered, the presence of such internal modes guarantees the nonlinear instability of the stationary states in the evolution dynamics. To corroborate this idea, we explore three prototypical case examples: (a)an antisymmetric soliton in a double-well potential, (b)a twisted localized mode in a one-dimensional lattice with cubic nonlinearity, and (c)a discrete vortex in a two-dimensional saturable lattice. In all cases, we observe a weak nonlinear instability, despite the linear stability of the respective states.

Discrete solitons and vortices on two-dimensional lattices of PT-symmetric couplers.

We introduce a 2D network built of 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 PT-symmetric dual-core waveguides embedded into a photonic crystal. The system supports 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 through the computation of stability eigenvalues, and verified by direct simulations. Symmetric FSs represent the system's ground state, being stable at lowest values of the power, while anti-symmetric FSs and OnVs are stable at higher powers. Symmetric OnVs, which are also stable at lower powers, are remarkably robust modes: on the contrary to other PT-symmetric states, unstable OnVs do not blow up, but spontaneously rebuild themselves into stable FSs.

Spatial control of the competition between self-focusing and self-defocusing nonlinearities in one- and two-dimensional systems

We introduce a system with competing self-focusing (SF) and self-defocusing (SDF) terms, which have the same scaling dimension. In the one-dimensional (1D) system, this setting is provided by a combination of the SF cubic term multiplied by the delta-function, $\delta (x)$, and a spatially uniform SDF quintic term. This system gives rise to the most general family of 1D-Townes solitons, the entire family being unstable. However, it is completely stabilized by a finite-width regularization of the $\delta $-function. The results are produced by means of numerical and analytical methods. We also consider the system with a symmetric pair of regularized $\delta $-functions, which gives rise to a wealth of symmetric, antisymmetric, and asymmetric solitons, linked by a bifurcation loop, that accounts for the breaking and restoration of the symmetry. Soliton families in 2D versions of both the single- and double-delta-functional systems are also studied. The 1D and 2D settings may be realized for spatial solitons in optics, and in Bose-Einstein condensates.

Fundamental and multipole solitons supported by one-dimensional multilayer photonic crystals potentials

The existence and stability of solitons in one-dimensional multilayer photonic crystals potentials are reported. For all of the solitons, there exist cutoff points of the propagation constant below which the solitons vanish in the semi-infinite gap. The fundamental solitons are stable in the whole range where solitons exist. The antisymmetric dipole solitons can be stable when the propagation constant closes to the cutoff point. The range of stability for symmetric tripole solitons is changed with modulation depth and width of the multilayer photonic crystals potentials. The power of solitons increases with increasing of propagation constant and modulation width or decreasing of modulation depth of the potentials.

PT-symmetric coupler with a coupling defect: soliton interaction with exceptional point.

We study the interaction of a soliton in a parity-time (PT) symmetric coupler which has local perturbation of the coupling constant. This defect does not change the PT-symmetry of the system, but locally can achieve the exceptional point. We found that the symmetric solitons after interaction with the defect either transform into breathers or blow up. The dynamics of antisymmetric solitons are more complex, showing domains of successive broadening of the beam and of the beam splitting in two outward propagating solitons, in addition to the single breather generation and blowup. All the effects are preserved when the coupling strength in the center of the defect deviates from the exceptional point. If the coupling is strong enough, the only observable outcome of the soliton-defect interaction is the generation of the breather.

Antisymmetry breaking of discrete dipole gap solitons induced by a phase-slip defect

In this paper, we study the antisymmetry breaking of discrete dipole soliton induced by a phase-slip one-dimensional discrete lattice, which contains on-site self-repulsive nonlinearity. This system can be realized in waveguide arrays system in optics or Bose–Einstein condensate in optical lattice. Different from the symmetry breaking occurring in the ground-state, antisymmetry breaking occurs in the first excited state, which contains antisymmetry. For this system, stable antisymmetric dipole soliton and anti-asymmetric are found, the symmetry transition between them is supercritical type. It is found that, increasing the total norm of the soliton or decreasing the coupled strength of the defect waveguide can lead to the antisymmetry breaking. Such kind of symmetry breaking can lead to the change of the tendency of some characters of the soliton.