Degenerate Vector Research Articles

Response solutions and quasi-periodic degenerate bifurcations for quasi-periodically forced systems**This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11171185, 11571201).

This paper includes two parts. In the first part, we first focus on quasi-periodic time dependent perturbations of one-dimensional quasi-periodically forced systems with degenerate equilibrium. We study the system in two cases, for one of which system admits a response solution under a non-resonant condition on the frequency vector weaker than Brjuno–Rüssmann’s and for another of which system also admits a response solution without any non-resonant conditions. Next, we investigate the existence of response solutions of a quasi-periodic perturbed system with degenerate (including completely degenerate) equilibrium under Brjuno–Rüssmann’s non-resonant condition by using the Herman method. In the second part, we consider, firstly, the quasi-periodic perturbation of a universal unfolding of one-dimensional degenerate vector field . Secondly, we consider the perturbation of a universal unfolding of normal two-dimensional Hamiltonian system with completely degenerate equilibrium. With KAM theory and singularity theory, we show that these two classes of universal unfolding can persist on large Cantor sets under Brjuno–Rüssmann’s non-resonant condition, which implies all the invariant tori in the integrable part and all the bifurcation scenario can survive on large Cantor sets. The result for Hamiltonian system can apply directly to the response context for quasi-periodically forced systems. Our results in this paper can be regarded as an improvement with respect to several results in various literature (Broer et al 2005 Nonlinearity 18 1735–69; Broer et al 2006 J. Differ. Equ. 222 233–62; Wagener 2005 J. Differ. Equ. 216 216–81; Xu 2010 J. Differ. Equ. 250 551–71; Xu and Jiang 2010 Ergod. Theor. Dynam. Syst. 31 599–611; Lu and Xu 2014 Nonlinear Differ. Equ. Appl. 21 361–70).

Theoretical analyses on orbital angular momentum modes in conventional graded-index multimode fibre

Orbital angular momentum (OAM) modes are another mode basis to represent spatial modes. There have been increasing interests in exploiting OAM modes in specialty fibres. In this paper, we present a comprehensive characterisation of OAM modes in conventional graded-index multimode fibre (MMF). 1) We synthesise the circularly polarized OAM modes by properly combining two fold degenerate cylindrical vector modes (eigenmodes) and analyse the total angular momentum, i.e. spin angular momentum and orbital angular momentum. 2) We divide all the OAM modes of the conventional graded-index MMF into 10 OAM mode groups with effective refractive index differences between different mode groups above 10−4 enabling low-level inter-group crosstalk. 3) We study the chromatic dispersion, differential group delay, effective mode area, and nonlinearity for each OAM mode group over a wide wavelength ranging from 1520 to 1630 nm covering the whole C band and L band. 4) We discuss the performance tolerance to fibre ellipticity and bending. 5) We further address the robustness of performance against fibre perturbations including the core size, index contrast and the imperfect index profile of the practically fabricated MMFs. The obtained results may provide theoretical basis for further space-division multiplexing applications employing OAM modes in conventional graded-index MMF.

Multi-Channel Phase Regenerator Based on Polarization-Assisted Phase-Sensitive Amplification

We propose and demonstrate simultaneous phase regeneration of six nonreturn-to-zero binary phase shift keying signals in a single nonlinear medium. The regenerating system, which exhibits binary steplike phase transfer functions, uses four-wave mixing in degenerate dual-pump vector parametric amplifiers implemented in the same nonlinear optical fiber, followed by polarization filtering. Bit-error-ratio measurements confirm optical-signal-to-ratio improvement and negligible crosstalk across all the regenerated channels, which were impaired at the regenerator input by broadband phase noise.

Root Systems In Finite Symplectic Vector Spaces

We study realizations of root systems in possibly degenerate symplectic vector spaces over finite fields, up to symplectic isomorphisms. The main result of this article is the classification of such realizations for the field 𝔽2. Thereby, each root system requires a specific degree of degeneracy of the symplectic vector space. Our main motivation for this article is that for each such realization of a root system we can construct a Nichols algebra over a nonabelian group.

Polarization-Assisted Phase-Sensitive Processor

We propose and experimentally demonstrate a phase-sensitive optical processor, capable of generating two codirectional and π-phase-shifted phase-sensitive amplifiers (PSAs) in a single device. Phase-sensitive operation is obtained by polarization mixing a phase-locked signal/idler pair generated in a degenerate dual-pump vector parametric amplifier based on four-wave mixing in a highly nonlinear fiber. We refer to this configuration as a polarization assisted PSA and demonstrate some of the applications that it may find. First, we experimentally demonstrate the regeneration of a binary phase shift keying signal in a system that requires only a very low nonlinear phase shift of 0.35 rad. Second, we decompose a quadrature phase shift keying (QPSK) signal into its in-phase and quadrature components. While application to a QPSK signal is shown in our demonstration, we demonstrate numerically that any complex modulation format signal can be decomposed using this approach. Finally, we use our processor to regenerate QPSK signals in a single nonlinear device.

Loss of derivatives in the infinite type

We discuss loss of derivatives for degenerate vector fields obtained from infinite type exponentially non-degenerate hypersurfaces of $\C^2$.

Analytic Integrability of Some Examples of Degenerate Planar Vector Fields

This paper is devoted to the classification of analytic integrable cases of two families of degenerate planar vector fields with a monodromic singular point at the origin. This study falls in the still open degenerate center problem. This classification can be done using the formal normal form theory and knowing a suitable normal form of any differential systems associated to each family.

Quadrature demultiplexing using a degenerate vector parametric amplifier.

We report on quadrature demultiplexing of a quadrature phase-shift keying (QPSK) signal into two cross-polarized binary phase-shift keying (BPSK) signals with negligible penalty at bit-error rate (BER) equal to 10(-9). The all-optical quadrature demultiplexing is achieved using a degenerate vector parametric amplifier operating in phase-insensitive mode. We also propose and demonstrate the use of a novel and simple phase-locked loop (PLL) scheme based on detecting the envelope of one of the signals after demultiplexing in order to achieve stable quadrature decomposition.

Optical Phase Quantizer Based on Phase Sensitive Four Wave Mixing at Low Nonlinear Phase Shifts

We propose and demonstrate a new phase sensitive scheme based on four-wave mixing followed by a polarizer to achieve an ideal binary step-like phase transfer function at nonlinear phase shifts as low as 0.3 radians by significantly increasing the parametric de-amplification component. Phase-sensitive operation is obtained by polarization mixing the phase-locked and orthogonally-polarized signal and idler, which is generated in a degenerate dual-pump vector parametric amplifier.

Experimental analysis of degenerate vector phase-sensitive amplification.

We comprehensively investigate a degenerate vector phase-sensitive amplifier (PSA). We determine the gain dependence on the relative phase and polarization angle between the pumps and the degenerate signal wave. The vector PSA is experimentally shown to be sensitive to the pump states of polarization (SOP) due to polarization mode dispersion in the fiber. However, the scheme performance agrees well with theory under specific pump SOPs and we achieve an on-off gain over 10 dB with a small deviation from the theoretically expected results. In comparison to the scalar scheme, the proposed vector scheme has larger tolerance for pump depletion due to four-wave mixing between pumps and generation of higher-order idlers.

Analytic integrability for some degenerate planar vector fields

In this paper we study the analytic integrability of degenerate vector fields of the form (y3+2ax3y+⋯,−x5−3ax2y2+⋯) around the origin. For these vector fields it is proved that integrability does not imply formal orbital equivalence to the Hamiltonian leading part. Moreover, it is shown the existence of a system in this class which has a center but is neither analytically integrable nor formal orbital reversible.

Super-Whittaker vector at c = 3/2

The degenerate Whittaker vector of the superconformal algebra can be represented in terms of Jack superpolynomials. However, in this representation the norm of the Whittaker vector involves a scalar product with respect to which the Jack superpolynomials are not orthogonal. In this note, we point out that this defect can be cured at c = 3/2 by means of a trick specific to the supersymmetric case. At c = 3/2, we thus end up with a closed-form expression for the norm of the degenerate super-Whittaker vector. Granting the super-version of the AGT conjecture, this closed-form expression should be equal to the -symmetric SU(2) pure-gauge instanton partition function—the corresponding equality taking the form of a rather nontrivial combinatorial identity.

The reversibility problem for quasi-homogeneous dynamical systems

In this paper, we obtain necessary and sufficient conditions for the reversibility of a quasi-homogeneous $n$-dimensional system.As a consequence, we get necessary conditions for an arbitrary system to be orbital-reversible.Moreover, we give sufficient conditions for orbital-reversibility in terms of the existence of Lie symmetries of the vector field.The results obtained are conveniently adapted to the case of planar systems, where we give sufficient conditions for a degenerate planar vector field to have a center at the origin.We apply the results to some case studies. Namely, we consider a family of planar vector fields, where we determine centers which arenot orbital-reversible.We also study some tridimensional systems, where nonlinear involutions are determined in the reversible situations.

Theory analysis of mode coupling in tilted long period fiber grating based on the full vector complex coupled mode theory

The mode coupling in the tilted long period fiber grating (TLPG) is analyzed by using the full vector complex coupled mode theory (FV-CCMT). Compared with the non-tilted LPG, new resonance bands corresponding to the LP1m cladding modes are observed in the transmission spectrum of the TLPG. The coupling to LP1m cladding modes is the combination of the coupling to degenerate vector modes TE0m, TM0m and HE2m with different propagation constants. The maximum coupling between LP01 and LP1m modes occurs at a tilt angle of ∼87°. Numerical results show that the state of polarization (SOP) of the input light would induce the shift of the new resonance wavelength. Also, the RI sensitivity of the new resonance wavelength is much higher than that of the normal resonance wavelength corresponding to the LP0m cladding modes. Such TLPG would find application in chemical and biological sensing fields.

Vector bright soliton behaviors associated with negative coherent coupling

With the introduction of an auxiliary function, a genuine bilinear system (in contrast to the published trilinear forms) is obtained for the two-coupled nonlinear Schrödinger equations with negative coherent coupling in the optical fiber communications. With symbolic computation, degenerate and nondegenerate vector solitons are derived associated with the corresponding phase-parameter constraints. In virtue of asymptotic analysis and graphical simulation, vector solitons of the single-hump, double-hump, or flat-top profiles are displayed, and the collision mechanisms of such vector solitons are revealed as well; namely, the collisions among degenerate solitons and among nondegenerate solitons are both elastic. The only possible inelastic collision, the collision in the degenerate-nondegenerate case, is pointed out, where a degenerate soliton interacts with a nondegenerate one. Results in this paper may be useful for the optical switching with the combined effects of self-phase modulation, cross-phase modulation, and negative coherent coupling.

Reversibility and quasi-homogeneous normal forms of vector fields

This paper uses tools in quasi-homogeneous normal form theory to discuss certain aspects of reversible vector fields around an equilibrium point. Our main result provides an algorithm, via Lie Triangle, that detects the non-reversibility of vector fields. That is, it is possible to decide whether a planar center is not reversible. Some of the theory developed is also applied to get further results on nilpotent and degenerate polynomial vector fields. We find several families of nilpotent centers which are non-reversible.

On the Integrability of Quasihomogeneous Systems and Quasidegenerate Infinity Systems

The integrability of quasihomogeneous systems is considered, and the properties of the first integrals and the inverse integrating factors of such systems are shown. By solving the systems of ordinary differential equations which are established by using the vector fields of the quasihomogeneous systems, one can obtain an inverse integrating factor of the systems. Moreover, the integrability of a class of systems (quasidegenerate infinity systems) which generalize the so-called degenerate infinity vector fields is considered, and a method how to obtain an inverse integrating factor of the systems from the first integrals of the corresponding quasihomogeneous systems is shown.

Analyticity for singular sums of squares of degenerate vector fields

Recently J. J. Kohn (2005) proved C ∞ hypoellipticity for P k = LL + L|z| 2k L = -L*L-(z k L)* -k z L with L = ∂ ∂z + iz∂ ∂t, (the negative of) a singular sum of squares of complex vector fields on the complex Heisenberg group, an operator which exhibits a loss of k - 1 derivatives. Subsequently, M. Derridj and D. S. Tartakoff proved analytic hypoellipticity for this operator using rather different methods going back to earlier methods of Tartakoff. Those methods also provide an alternate proof of the hypoellipticity given by Kohn. In this paper, we consider the equation P m,k = L m L m + L m |z| 2k L m with L m = ∂ ∂z + iz|z| 2m ∂ ∂t, for which the underlying manifold is only of finite type, and prove analytic hypoellipticity using methods of Derridj and Tartakoff. This operator is also subelliptic with large loss of derivatives, but the exact loss plays no role for analytic hypoellipticity. Nonetheless, these methods give a proof of C ∞ hypoellipticity with precise loss as well, which is to appear in a forthcoming paper by A. Bove, M. Derridj, J. J. Kohn and the author.

A preparation theorem for codimension-one foliations

After gluing foliated complex manifolds, we derive a preparation-like theorem for singularities of codimension-one foliations and planar vector fields (in the real or complex setting). Without computation, we retrieve and improve results of Levinson-Moser for functions, Dufour-Zhitomirskii for nondegenerate codimension-one foliations (proving in turn the analyticity), Str?oyzyna- y Zo
ladek for non degenerate planar vector fields and Bruno-?Ecalle for saddle-node foliations in the plane.

Quantum mechanics of phase-sensitive amplification in a fiber

In a recent paper [C.J. McKinstrie, S. Radic, Opt. Express 12 (2004) 4973], four schemes were described, which produce phase-sensitive amplification in fibers: Degenerate scalar and vector four-wave mixing (FWM), and phase conjugation (PC) combined with frequency conversion (FC), in either order. In this paper, the basic quantum mechanics of these schemes are discussed. For each scheme, formulas are derived for the field-quadrature and photon-number variances. The effects of polarization misalignment on vector FWM, and the effects of unbalanced conversion on PC combined with FC, are studied. Moderate amounts of these imperfections do not degrade significantly the performances of the aforementioned schemes.