Symmetric 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.

Spontaneous symmetry breaking and vortices in a tri-core nonlinear fractional waveguide

We introduce a waveguiding system composed of three linearly-coupled fractional waveguides, with a triangular (prismatic) transverse structure. It may be realized as a tri-core nonlinear optical fiber with fractional group-velocity dispersion (GVD), or, possibly, as a system of coupled Gross–Pitaevskii equations for a set of three tunnel-coupled cigar-shaped traps filled by a Bose–Einstein condensate of particles moving by Lévy flights. The analysis is focused on the phenomenon of spontaneous symmetry breaking (SSB) between components of triple solitons, and the formation and stability of vortex modes. In the self-focusing regime, we identify symmetric and asymmetric soliton states, whose structure and stability are determined by the Lévy index of the fractional GVD, the inter-core coupling strength, and the total energy, which determines the system’s nonlinearity. Bifurcation diagrams (of the supercritical type) reveal regions where SSB occurs, identifying the respective symmetric and asymmetric ground-state soliton modes. In agreement with the general principle of the SSB theory, the solitons with broken inter-component symmetry prevail with the increase of the energy in the weakly-coupled system. Three-components vortex solitons (which do not feature SSB) are studied too. Because the fractional GVD breaks the system’s Galilean invariance, we also address mobility of the vortex solitons, by applying a boost to them.

Parity–time symmetric solitons of the complex KP equation

Parity–time symmetric solitons of the complex KP equation

Rotationally symmetric transverse magnetic vector wave propagation for nonlinear optics

In this paper the theory and simulation results are presented for 3D cylindrical rotationally symmetric spatial soliton propagation in a nonlinear medium using a modified finite-difference time-domain general vector auxiliary differential equation method for transverse magnetic polarization. The theory of 3D rotationally symmetric spatial solitons is discussed, and compared with two (1 + 1)D, termed “2D” for this paper, hyperbolic secant spatial solitons, with a phase difference of π (antiphase). The simulated behavior of the 3D rotationally symmetric soliton was compared with the interaction of the two antiphase 2D solitons for different source hyperbolic secant separation distances. Lastly, we offer some possible explanations for the simulated soliton behavior.

Multipole solitons and vortex solitons in nonlocal nonlinear media

The nonlinear Schrödinger equation (NLSE) under nonlocal nonlinear media (NNM) is described and the approximate analytical solutions of the vector multipole solitons and vortex optical soliton clusters are obtained via the variational method. The results show that the structure of the optical solitons is determined by modulation depth and topological charge. In the propagation process, the spatial soliton has an observable rotation property. Under certain conditions, the rotating space modulated vortex optical solitons degenerate into circular symmetric vortex optical solitons. The results can be extended to other physical systems.

Periodic transmission theory of circularly symmetric multi-ring solitons in nonlinear Schrödinger equation

In this paper, the evolution characteristics of periodic transmission of circularly symmetric multi-ring solitons in optical nonlocal materials based on nonlinear Schrödinger equation are investigated in detail. The transmission expression of circularly symmetric multi-ring solitons has been derived. It was found that the number and size of rings in these solitons can be controlled by initial parameters. The transmission of circularly symmetric multi-ring solitons is similar to that of high-order temporal solitons in nonlinear fibers, exhibiting periodic variations. When the input energy is a specific value, the statistical width of circularly symmetric multi-ring solitons remains constant during transmission, otherwise it exhibits periodic changes, which can be considered as generalized breathing solitons. The influence of various parameters on the transmission characteristics has been analyzed in detail, and some important transmission characteristics have been intuitively demonstrated through numerical simulation.

Discrete and Semi-Discrete Multidimensional Solitons and Vortices: Established Results and Novel Findings.

This article presents a concise survey of basic discrete and semi-discrete nonlinear models, which produce two- and three-dimensional (2D and 3D) solitons, and a summary of the main theoretical and experimental results obtained for such solitons. The models are based on the discrete nonlinear Schrödinger (DNLS) equations and their generalizations, such as a system of discrete Gross-Pitaevskii (GP) equations with the Lee-Huang-Yang corrections, the 2D Salerno model (SM), DNLS equations with long-range dipole-dipole and quadrupole-quadrupole interactions, a system of coupled discrete equations for the second-harmonic generation with the quadratic (χ(2)) nonlinearity, a 2D DNLS equation with a superlattice modulation opening mini-gaps, a discretized NLS equation with rotation, a DNLS coupler and its PT-symmetric version, a system of DNLS equations for the spin-orbit-coupled (SOC) binary Bose-Einstein condensate, and others. The article presents a review of the basic species of multidimensional discrete modes, including fundamental (zero-vorticity) and vortex solitons, their bound states, gap solitons populating mini-gaps, symmetric and asymmetric solitons in the conservative and PT-symmetric couplers, cuspons in the 2D SM, discrete SOC solitons of the semi-vortex and mixed-mode types, 3D discrete skyrmions, and some others.

Investigations on a Riemannian Manifold with a Semi- Symmetric Non-Metric Connection and Gradient Solitons

This article carries out the investigation of a three-dimensional Riemannian manifold N3 endowed with a semi-symmetric type non-metric connection. Firstly, we construct a non-trivial example to prove the existence of a semi-symmetric type non-metric connection on N3. It is established that a N3 with the semi-symmetric type non-metric connection, whose metric is a gradient Ricci soliton, is a manifold of constant sectional curvature with respect to the semi-symmetric type non-metric connection. Moreover, we prove that if the Riemannian metric of N3 with the semi-symmetric type non-metric connection is a gradient Yamabe soliton, then either N3 is a manifold of constant scalar curvature or the gradient Yamabe soliton is trivial with respect to the semi-symmetric type non-metric connection. We also characterize the manifold N3 with a semi-symmetric type non-metric connection whose metrics are Einstein solitons and m-quasi Einstein solitons of gradient type, respectively.

Symmetry-breaking bifurcations and excitations of solitons in linearly coupled NLS equations with PT-symmetric potentials

We address symmetry-breaking bifurcations (SBBs) in the ground state (GS) and dipole-mode (DM) solitons of the one-dimensional linearly coupled NLS equations, modelling the propagation of light in a dual-core planar waveguide with the Kerr nonlinearity and two types of P T -symmetric potentials. The P T -symmetric potential is employed to obtain different types of solutions. A supercritical pitchfork bifurcation occurs in families of symmetric solutions of both the GS and DM types. A novel feature of the system is interplay between breakings of the P T and inter-core symmetries. Stability of symmetric GS and DM modes and their asymmetric counterparts, produced by SBBs of both types, is explored via the linear-stability analysis and simulations. It is found that the instability of P T -symmetric solutions takes place prior to the inter-core symmetry breaking. Surprisingly, stable inter-core-symmetric GS solutions may remain stable while the P T symmetry is broken. Fully asymmetric GS and DM solitons are only partially stable. Moreover, we construct symmetric and asymmetric GS solitons under the action of a pure imaginary localized potential, for which the SBB is subcritical. These results exhibit that stable solitons can still be found in dissipative systems. Finally, excitations of symmetric and asymmetric GS solitons are investigated by making the potential’s parameters or the system’s coupling constant functions, showing that GS solitons can be converted from an asymmetric shape onto a symmetric one under certain conditions. These results may pave the way for the study of linear and nonlinear phenomena in a dual-core planar waveguide with P T potential and related experimental designs.

Polar compactons and solitons in a two dimensional optical waveguide: Theory and simulations

The propagation of optical cylindrical compactons and solitons in a two dimensional nonlocal nonlinear waveguide is deeply investigated by carrying a particular emphasis on its nonlinear response function. Considering the weakly nonlocal medium, the two dimensional nonlocal nonlinear Schrödinger equation describing the propagation of signal light is derived. This equation contains the basic nonlinear Schrödinger equation with additional nonlinear terms proportional to the second order space derivative of coordinates in the transverse directions. By seeking the solution in polar form, this equation is separated into real and imaginary parts, followed by the analysis of the equilibrium points and their stability, predicting the type of solutions. These solutions are found both with linear and nonlinear phases through direct integration. In order to seek polar solutions, the nonlocal nonlinear Schrödinger equation is expressed in cylindrical coordinate and the resulting equation, containing additional terms inversely proportional to the radius r is not easily solvable. The type of solutions of this equation is predicted by the investigation of new equilibrium points and their stability. The analytical cylindrical compactons and pulse solitons as solutions of this equation are provided by using the Rayleigh–Ritz variational method, usually used for symmetric radial solitons to approximate pulse solitons in polar coordinate, and the stability of these solutions is checked through numerical investigations.

A solvable model for symmetry-breaking phase transitions

Analytically solvable models are benchmarks in studies of phase transitions and pattern-forming bifurcations. Such models are known for phase transitions of the second kind in uniform media, but not for localized states (solitons), as integrable equations which produce solitons do not admit intrinsic transitions in them. We introduce a solvable model for symmetry-breaking phase transitions of both the first and second kinds (alias sub- and supercritical bifurcations) for solitons pinned to a combined linear-nonlinear double-well potential, represented by a symmetric pair of delta-functions. Both self-focusing and defocusing signs of the nonlinearity are considered. In the former case, exact solutions are produced for symmetric and asymmetric solitons. The solutions explicitly demonstrate a switch between the symmetry-breaking transitions of the first and second kinds (i.e., sub- and supercritical bifurcations, respectively). In the self-defocusing model, the solution demonstrates the transition of the second kind which breaks antisymmetry of the first excited state.

Pair of optical solitons in parallel planar waveguides coupled by linear photonic crystal

We propose a simple optical model, which consists of two Kerr nonlinear parallel planar waveguides sandwiching a one-dimensional photonic crystal for demonstrating the phenomenon of spontaneous symmetry breaking in optics. We study the shape and stability of soliton pairs propagating in two planar waveguides and the spontaneous symmetry breaking of the power distribution of solitons in the two waveguides. The results of numerical simulation show that there is supercritical type of spontaneous symmetry breaking in this system. More optical power P and smaller refractive index of photonic crystals [Formula: see text] make soliton solutions more prone to spontaneous symmetry breaking bifurcation. Asymmetric soliton solutions and symmetric soliton solutions correspond to different dispersion relations. Under the given total power P, for the symmetric soliton solutions, the propagation constant k increases linearly with [Formula: see text], and for asymmetric soliton solutions, the propagation constant k decreases nonlinearly with [Formula: see text]. The minimum point of [Formula: see text] is just the demarkation between the symmetric soliton solutions and the asymmetric soliton solutions. If [Formula: see text] is fixed, the minimum point of the Hamiltonian of the system, [Formula: see text], is also the critical point of the two different kinds of soliton solutions. We also study stable and unstable distributions of two kinds of soliton solutions.

One-dimensional Townes solitons in dual-core systems with localized coupling.

The recent creation of Townes solitons (TSs) in binary Bose-Einstein condensates and experimental demonstration of spontaneous symmetry breaking (SSB) in solitons propagating in dual-core optical fibers has drawn renewed interest in the TS and SSB phenomenology in these and other settings. In particular, stabilization of TSs, which are always unstable in free space, is a relevant problem with various ramifications. We introduce a system which admits exact solutions for both TSs and SSB of solitons. It is based on a dual-core waveguide with quintic self-focusing and fused (localized) coupling between the cores. The respective system of coupled nonlinear Schrödinger equationsgives rise to exact solutions for full families of symmetric and asymmetric solitons, which are produced by the supercritical SSB bifurcation (i.e., the symmetry-breaking phase transition of the second kind). Stability boundaries of asymmetric solitons are identified by dint of numerical methods. Unstable solitons spontaneously transform into robust moderately asymmetric breathers or strongly asymmetric states with small intrinsic oscillations. The setup can be used in the design of photonic devices operating in coupling and switching regimes.

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.

Investigating Symmetric Soliton Solutions for the Fractional Coupled Konno–Onno System Using Improved Versions of a Novel Analytical Technique

The present research investigates symmetric soliton solutions for the Fractional Coupled Konno–Onno System (FCKOS) by using two improved versions of an Extended Direct Algebraic Method (EDAM) i.e., modified EDAM (mEDAM) and r+mEDAM. By obtaining precise analytical solutions, this research explores the characteristics and behaviours of symmetric solitons in FCKOS. Further, the amplitude, shape and propagation behaviour of some solitons are visualized by means of a 3D graph. This investigation fosters a more thorough comprehension of non-linear wave phenomena in considered systems and offers helpful insights towards soliton behavior in it. The outcomes reveal that the recommended techniques are successful in constructing symmetric soliton solutions for complex models like the FCKOS.

Spontaneous symmetry breaking and ghost states in two-dimensional fractional nonlinear media with non-Hermitian potential

The interaction of fractional diffraction and parity-time ({{{{{mathcal{PT}}}}}}) symmetric would bring some unique properties to certain physical system. Here we report a spontaneous symmetry breaking (SSB) phenomenon and ghost states of solitons supported by the two-dimensional (2D) fractional nonlinear Schrödinger (FNLS) equation with focusing and defocusing Kerr media under 2D non-Hermitian {{{{{mathcal{PT}}}}}}-symmetric potential. The continuous asymmetric soliton bifurcates out in a SSB way. For the focusing regime, as the power exceeds a critical value, the asymmetric soliton bifurcates from fundamental symmetric solitons, and when it turns to the defocusing regime, the symmetry breaking phenomenon is found in the branch of the first excited state. The symmetric solitons (including fundamental one and first excited state) are destabilized at this point. Intriguingly, the branches of asymmetry solitons are existing with complex conjugate propagation constants (alias ghost states), which is solely in fractional media. Moreover, we investigate the dependence of fractional Lévy index (α) on the symmetry breaking of solitons in detail. And the stabilities of fundamental symmetric soliton, asymmetric solitons, as well as the first excited states are explored. Moreover, The management of optical field propagation is achieved by modulation of the external potential. Meanwhile, we find the stable excitations of solitons via adiabatic excitations of system parameters. These results will provide some theoretical basis for the study of SSB phenomena and related physical experiments in the 2D fractional media with {{{{{mathcal{PT}}}}}}-symmetric potentials.

Spherically Symmetric Non-Topological Solitons and Boson Stars in Friedberg-Lee-Sirlin Model

We study spherically symmetric non-topological solitons and boson stars in Friedberg-Lee-Sirlin model in 3+1 dimensional spacetime. The non-topological soliton field configuration consists of a massless complex scalar field coupled to a real scalar field with symmetry breaking potential. The complex scalar field becomes massive when the real scalar field goes to one of the degenerate vacuums. Their existence is supported by the conserved charge and the mass-energy only on certain frequency range. When gravity is taken into account, they form configuration called boson stars. The gravitational interaction influences the quantity of the conserved charge and the mass-energy of the system. We present the dependence of symmetry breaking energy scale on the solution of boson stars.

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.

Dynamical Structures of Multi-Solitons and Interaction of Solitons to the Higher-Order KdV-5 Equation

In this study, we build multi-wave solutions of the KdV-5 model through Hirota’s bilinear method. Taking complex conjugate values of the free parameters, various colliding exact solutions in the form of rogue wave, symmetric bell soliton and rogue waves form; breather waves, the interaction of a bell and rogue wave, and two colliding rogue wave solutions are constructed. To explore the characteristics of the breather waves, localized in any direction, the higher-order KdV-5 model, which describes the promulgation of weakly nonlinear elongated waves in a narrow channel, and ion-acoustic, and acoustic emission in harmonic crystals symmetrically is analyzed. With the appropriate parameters that affect and manage phase shifts, transmission routes, as well as energies of waves, a mixed solution relating to hyperbolic and sinusoidal expression are derived and illustrated by figures. All the single and multi-soliton appeared symmetric about an axis of the wave propagation. The analyzed outcomes are functional in achieving an understanding of the nonlinear situations in the mentioned fields.

Deformation conjecture: deforming lower dimensional integrable systems to higher dimensional ones by using conservation laws

Utilizing some conservation laws of (1+1)-dimensional integrable local evolution systems, it is conjectured that higher dimensional integrable equations may be regularly constructed by a deformation algorithm. The algorithm can be applied to Lax pairs and higher order flows. In other words, if the original lower dimensional model is Lax integrable (possesses Lax pairs) and symmetry integrable (possesses infinitely many higher order symmetries and/or infinitely many conservation laws), then the deformed higher order systems are also Lax integrable and symmetry integrable. For concreteness, the deformation algorithm is applied to the usual (1 + 1)-dimensional Korteweg-de Vries (KdV) equation and the (1 + 1)-dimensional Ablowitz-Kaup-Newell-Segur (AKNS) system (including nonlinear Schrödinger (NLS) equation as a special example). It is interesting that the deformed (3+1)-dimensional KdV equation is also an extension of the (1 + 1)-dimensional Harry-Dym (HD) type equations which are reciprocal links of the (1+1)-dimensional KdV equation. The Lax pairs of the (3 + 1)-dimensional KdV-HD system and the (2 + 1)-dimensional AKNS system are explicitly given. The higher order symmetries, i.e., the whole (3 + 1)-dimensional KdV-HD hierarchy, are also explicitly obtained via the deformation algorithm. The single soliton solution of the (3 + 1)-dimensional KdV-HD equation is implicitly given. Because of the effects of the deformation, the symmetric soliton shape of the usual KdV equation is no longer conserved and deformed to be asymmetric and/or multi-valued. The deformation conjecture holds for all the known (1 +1)-dimensional integrable local evolution systems that have been checked, and we have not yet found any counter-example so far. The introduction of a large number of (D + 1)-dimensional integrable systems of this paper explores a serious challenge to all mathematicians and theoretical physicists because the traditional methods are no longer directly valid to solve these integrable equations.