Vector Solitons Research Articles

Period-doubling bifurcation and polarization dynamics of vector solitons in a fiber laser

Period-doubling bifurcation and polarization dynamics of vector solitons in a fiber laser

Algorithms for solving the inverse scattering problem for the Manakov model

The paper considers algorithms for solving inverse scattering problems based on the discretization of the Gelfand–Levitan–Marchenko integral equations, associated with the system of nonlinear Schrödinger equations of the Manakov model. The numerical algorithm of the first order approximation for solving the scattering problem is reduced to the inversion of a series of nested block Toeplitz matrices using the Levinson-type bordering method. Increasing the approximation accuracy violates the Toeplitz structure of block matrices. Two algorithms are described that solve this problem for second order accuracy. One algorithm uses a block version of the Levinson bordering algorithm, which recovers the Toeplitz structure of the matrix by moving some terms of the systems of equations to the right-hand side. Another algorithm is based on the Toeplitz decomposition of an almost block-Toeplitz matrix and the Tyrtyshnikov bordering algorithm. The speed and accuracy of calculations using the presented algorithms are compared on an exact solution (the Manakov vector soliton).

Classification of Vector Fields and Soliton Structures on a Tangent Bundle with a Ricci Quarter-Symmetric Metric Connection

Consider $TM$ as the tangent bundle of a (pseudo-)Riemannian manifold $M$, equipped with a Ricci quarter-symmetric metric connection $\overline{\nabla }$. This research article aims to accomplish two primary objectives. Firstly, the paper undertakes the classification of specific types of vector fields, including incompressible vector fields, harmonic vector fields, concurrent vector fields, conformal vector fields, projective vector fields, and $% \widetilde{\varphi }(Ric)$ vector fields, within the framework of $\overline{% \nabla }$ on $T\dot{M}$. Secondly, the paper establishes the necessary and sufficient conditions for the tangent bundle $TM$ to become as a Riemannian soliton and a generalized Ricci-Yamabe soliton with regard to the connection $\overline{\nabla }$.

Some Properties of the Potential Field of an Almost Ricci Soliton

In this article, we are interested in finding necessary and sufficient conditions for a compact almost Ricci soliton to be a trivial Ricci soliton. As a first result, we show that positive Ricci curvature and, for a nonzero constant c, the integral of Ric(cξ,cξ) satisfying a generic inequality on an n-dimensional compact and connected almost Ricci soliton (Mn,g,ξ,σ) are necessary and sufficient conditions for it to be isometric to the n-sphere Sn(c). As another result, we show that, if the affinity tensor of the soliton vector field ξ vanishes and the scalar curvature τ of an n-dimensional compact almost Ricci soliton (Mn,g,ξ,σ) satisfies τnσ−τ≥0, then (Mn,g,ξ,σ) is a trivial Ricci soliton. Finally, on an n-dimensional compact almost Ricci soliton (Mn,g,ξ,σ), we consider the Hodge decomposition ξ=ξ¯+∇h, where divξ¯=0, and we use the bound on the integral of Ricξ¯,ξ¯ and an integral inequality involving the scalar curvature to find another characterization of the n-sphere.

Numerical study of vector solitons with the oscillatory phase backgrounds in the integrable coupled nonlinear Schrödinger equations

In this paper, we numerically investigate vector solitons with oscillatory phase backgrounds in the integrable coupled nonlinear Schrödinger equations, which are widely applied to varieties of physical contexts such as the simultaneous propagation of nonlinear optical pulses and the dynamics of two-components Bose–Einstein condensates. We develop the time-splitting Chebyshev–Galerkin method based on a transformation to accurately compute the vector soliton solutions. Compared to the finite difference method, numerical experiments show that the method with spectral accuracy and high efficiency is necessary for simulating the dynamics evolution of vector solitons. Combined with modulation instability conditions, linear stability analysis and direct numerical simulation, we reveal that the bright-dark and dark-dark solitons with various combinations of parameters under perturbations have qualitative differences. Particularly, vector solitons in unstable background with different wave numbers present distinct dynamics evolutions. The results help us to understand soliton dynamics with oscillatory phase backgrounds and the superposition between nonlinear waves.

Linear-coupling-induced double-period pulsating vector solitons in lasers.

Pulsating solitons is a universal phenomenon originating from the Hopf-type bifurcation in dissipative systems such as lasers and microresonators. Here, we report the vector soliton in a fiber laser pulsating with two periods of different orders of magnitude. The short-period pulsation manifests as the period-tripling facilitated by the linear coupling between orthogonal polarization components, which breaks the self-consistent evolution of the vector soliton over a single round trip (RT). The long-period pulsation arises from the mode competition between the two polarization components mediated by various cavity effects. The interplay between linear coupling and mode competition gives rise to the robust double-period pulsating (DPP) vector soliton. Our results provide a clear physical mechanism for the broadly observed double-period breather, which has a significant value in exploring breathers with complex dynamics and multiple comb spectroscopy.

Envelope vector solitons in nonlinear flexible mechanical metamaterials

In this paper, we employ a combination of analytical and numerical techniques to investigate the dynamics of lattice envelope vector soliton solutions propagating within a one-dimensional chain of flexible mechanical metamaterial. To model the system, we formulate discrete equations that describe the longitudinal and rotational displacements of each individual rigid unit mass using a lump element approach. By applying the multiple-scales method in the context of a semi-discrete approximation, we derive an effective nonlinear Schrödinger equation that characterizes the evolution of rotational and slowly varying envelope waves from the aforementioned discrete motion equations. We thus show that this flexible mechanical metamaterial chain supports envelope vector solitons where the rotational component has the form of either a bright or a dark soliton. In addition, due to nonlinear coupling, the longitudinal displacement displays kink-like profiles thus forming the 2-components vector soliton. These findings, which include specific vector envelope solutions, enrich our knowledge on the nonlinear wave solutions supported by flexible mechanical metamaterials and open new possibilities for the control of nonlinear waves and vibrations.

Ricci Soliton Vector Fields of a Sub-Class of Perfect Fluid Bianchi Type-I Spacetimes in f(T) Theory of Gravity

Ricci Soliton Vector Fields of a Sub-Class of Perfect Fluid Bianchi Type-I Spacetimes in f(T) Theory of Gravity

Energy-conserving SAV-Hermite–Galerkin spectral scheme with time adaptive method for coupled nonlinear Klein–Gordon system in multi-dimensional unbounded domains

In this article, we construct an efficient numerical scheme to solve the coupled nonlinear Klein–Gordon system in a multi-dimensional unbounded domain Rd(d=1,2,3). We first use the SAV method to equivalently deform the original system, then we treat the nonlinear part of the system explicitly in temporal discretization. Thanks to the SAV method, we ensure the unconditional energy conservation of the discrete system, as a cost, we only need to solve two completely linearized decoupled systems. To avoid the errors caused by domain truncation and the construction of artificial boundary conditions, we apply the Hermite–Galerkin spectral method with scaling factor for spatial discretization. In order to save computational costs, we utilize two time adaptive strategies to adjust the time step size, through which we can improve computational efficiency without sacrificing accuracy. Finally, we presented numerical experiments to demonstrate the accuracy, efficiency, and energy conservation properties of our algorithm, and applied it to simulate the interactions of two-dimensional and three-dimensional vector solitons.

On finite time Type I singularities of the Kähler–Ricci flow on compact Kähler surfaces

We show that the underlying complex manifold of a complete non-compact two-dimen­sional shrinking gradient Kähler–Ricci soliton (M,g,X) with soliton metric g with bounded scalar curvature \operatorname{R}_{g} whose soliton vector field X has an integral curve along which \operatorname{R}_{g}\not\to0 is biholomorphic to either \mathbb{C}\times\mathbb{P}^{1} or to the blowup of this manifold at one point, and that the soliton metric g is toric. We also identify the corresponding soliton vector field X in each case. Given these possibilities, we then prove a strong form of the Feldman–Ilmanen–Knopf conjecture for finite time Type I singularities of the Kähler–Ricci flow on compact Kähler surfaces, leading to a classification of the bubbles of such singularities in this dimension.

Vector multipole solitons of fractional-order coupled saturable nonlinear Schrödinger equation

Three kinds of vector multipole solitons of fractional coupled saturable nonlinear Schrödinger equation are reported, including fractional dipole-dipole, dipole-tripole and tripole-dipole vector soliton solutions. Firstly, their existence domains, which are modulated by potential function parameters, are constructed in a certain interval. Secondly, the stable regions of three kinds of vector multipole solitons, which are modulated by the soliton power of each component, are found. The properties of solitons are explored through these existence and stability domains. Finally, the stability of three kinds of fractional vector multipole solitons is verified by the numerical evolution. Compared with the integer-order results, there are differences in the existence and stable regions of soliton solutions, and the Lévy index affects the existence and stability of vector multipole solitons.

Coupled circularly polarized electromagnetic soliton states in magnetized plasmas

The interaction between two co-propagating electromagnetic pulses in a magnetized plasma is considered, from first principles, relying on a fluid-Maxwell model. Two circularly polarized wavepackets by same group velocities are considered, characterized by opposite circular polarization, to be identified as left-hand- or right hand circularly polarized (i.e. LCP or RCP, respectively). A multiscale perturbative technique is adopted, leading to a pair of coupled nonlinear Schrödinger-type (NLS) equations for the modulated amplitudes of the respective vector potentials associated with the two pulses. Systematic analysis reveals the existence, in certain frequency bands, of three different types of vector soliton modes: an LCP-bright/RCP-bright coupled soliton pair state, an LCP-bright/RCP-dark soliton pair, and an LCP-dark/RCP-bright soliton pair. The value of the magnetic field plays a critical role since it determines the type of vector solitons that may occur in certain frequency bands and, on the other hand, it affects the width of those frequency bands that are characterized by a specific type of vector soliton (type). The magnetic field (strength) thus arises as an order parameter, affecting the existence conditions of each type of solution (in the form of an envelope soliton pair). An exhaustive parametric investigation is presented in terms of frequency bands and in a wide range of magnetic field (strength) values, leading to results that may be applicable in beam-plasma interaction scenarios as well as in space plasmas and in the ionosphere.

Static vector solitons in a topological mechanical lattice

Topological solitons, renowned for their stability and particle-like collision behaviors, have sparked interest in developing macroscopic-scale information processing devices. However, the exploration of interactions between multiple topological solitons in mechanical systems remains elusive. In this study, we construct a topological mechanical lattice supporting static vector solitons that represent quantized degrees of freedom and can freely propagate across the system. Drawing inspiration from coupled double atomic chains with sublattice symmetry breaking, we design a mechanical analogue featuring topologically protected boundary modes and induce independent modes to finite motions along branched motion pathways. Through a continuum theory, we describe the evolution of boundary modes with vector solitons composed of superposed kink solutions, identifying them as minimum energy pathways on the rugged effective potential surface with multiple degenerate ground states. Our results reveal the connection between transformable topological lattices and multistable systems, providing insight into nonlinear topological mechanics.

Coupled nonlinear Schrödinger system: role of four-wave mixing effect on nondegenerate vector solitons

Coupled nonlinear Schrödinger system: role of four-wave mixing effect on nondegenerate vector solitons

Wave pulses’ physical properties in birefringent optical fibres containing two vector solitons with coupled fractional LPD equation with Kerr’s law nonlinearity

Wave pulses’ physical properties in birefringent optical fibres containing two vector solitons with coupled fractional LPD equation with Kerr’s law nonlinearity

Coherent manipulation of vectorial soliton beam in sodium like atomic medium

A four-level, sodium like atomic medium, driven by a probe and control fields, is used to modify the paraxial vectorial light beam in the medium. A uniform packet of vector arrows is controlled, whose shape is stable and undistorted during propagation, known as vectorial soliton. To the best of our knowledge, significant control over vectorial soliton is reported for the first time in the present work. Optical vectorial solitonic beams are controlled and manipulated with the parameters of the coupled fields with the system. The vectorial soliton is a strong, undistorted function of detunings, Rabi frequencies, and collective phase of the deriving fields, respectively. The modified vectorial solitons may have useful applications in telecommunication technology for delivering data bit streams over long distances and data processing.

Buildup and synchronization regimes of a vector pure-quartic soliton molecule in a fiber laser cavity.

Pure-quartic solitons (PQSs) have recently received increasing attention due to their energy-width scaling over the traditional soliton, which has expanded our understanding of soliton dynamics with high-order dispersion in nonlinear systems. Here, we numerically reveal the asynchronization and synchronization processes of the sub-pulse within the vector PQS molecule in a mode-locked fiber laser by solving the coupled Ginzburg-Landau equations. During the establishment of a vector PQS molecule, the repulsion, attraction, and finally stabilization processes have been observed. Specifically, sub-pulse disappearance, regeneration, and finally synchronization with the other pulses are also investigated. Our analysis of the pulse energy, time interval, and relative phase evolution dynamics with the round trip indicates that the asynchronization and synchronization within the vector PQS molecule associate tightly with the gain competition and the cross-phase modulation. Our findings provide insights into the internal mutual dynamics within the vector soliton molecule and offer guidance for the applications of PQS.

Symmetry broken vectorial Kerr frequency combs from Fabry-Pérot resonators

Spontaneous symmetry breaking of a pair of vector temporal cavity solitons has been established as a paradigm to modulate optical frequency combs, and finds many applications in metrology, frequency standards, communications, and photonic devices. While this phenomenon has successfully been observed in Kerr ring resonators, the counterpart exploiting linear Fabry-Pérot cavities is still unexplored. Here, we consider field polarization properties and describe a vector comb generation through the spontaneous symmetry breaking of temporal cavity solitons within coherently driven, passive, Fabry-Pérot cavities with Kerr nonlinearity. Global coupling effects due to the interactions of counter-propagating light restrict the maximum number of soliton pairs within the cavity - even down to a single soliton pair - and force long range polarization conformity in trains of vector solitons.

Nonlinear tunnelling of 3D partially nonlocal nonautonomous nondegenerate vector solitons in a linear external potential

Nonlinear tunnelling of 3D partially nonlocal nonautonomous nondegenerate vector solitons in a linear external potential

Conformal Ricci almost solitons with certain soliton vector fields

Conformal Ricci almost solitons with certain soliton vector fields