High-order Perturbation Research Articles

Theoretical studies of the Spin Hamiltonian parameters and local structures for Cu2+ centers in MNB ternary glasses

ABSTRACTThe spin Hamiltonian parameters (g factors g|| and g⊥ and the hyperfine structure constants A|| and A⊥) for the doped Cu2+ ion (in the form of CuO) in ternary glasses (i.e. xMgO·(30-x)Na2O·69B2O3·CuO, with 5 < x < 17 mol%) are theoretically investigated based on the high-order perturbation formulas for a tetragonally elongated octahedral 3d9 complex. In these formulas, the required crystal-field parameters are estimated from the superposition model which enables correlation of the crystal-field parameters and hence the spin Hamiltonian parameters with the tetragonal distortion (characterized by relative tetragonal elongation δ along the C4 axis due to the Jahn–Teller effect) of [CuO6]10− cluster. The concentration dependences of the spin Hamiltonian parameters are illustrated by the approximately linear increases of the cubic field parameter Dq and the covalency factor N as well as the relative elongation δ with increasing the MgO concentration x. Based on the calculation, the [CuO6]10− clusters in the MNB glasses are found to suffer the relative elongations of about δ (≈ 0.125 Å) along the tetragonal axis due to the Jahn–Teller effect. The theoretical results show good agreement with the experimental data. And the improvement is also achieved in present work with respect to the previous theoretical analysis based on the conventional crystal-field model formulas by including the ligand orbital and spin–orbit coupling contributions.

Stationary distribution of a regime-switching predator–prey model with anti-predator behaviour and higher-order perturbations

In this paper, we study a regime-switching predator–prey model with anti-predator behaviour and higher-order perturbations. We obtain the ergodic property by constructing a suitable stochastic Lyapunov function with regime-switching, which provides us a biological perspective of cycling phenomena of a population system, and can better describe the stochastic persistence of a population system in practice. We find that these restrictive assumptions on the functional response are relative weak and valid for many types of response functions.

Perturbative manifolds and the Noether generators of nth-order Poisson equations

A manifold that contains small perturbations will induce a perturbed partial differential equation. The partial differential equation that we select is the Poisson equation – in order to explore the interplay between the geometry of the manifold and the perturbations. Specifically, we show how the problem of symmetry determination, for higher-order perturbations, can be elegantly expressed via geometric conditions.

Element sets for high-order Poincaré mapping of perturbed Keplerian motion

The propagation and Poincar\'e mapping of perturbed Keplerian motion is a key topic in celestial mechanics and astrodynamics, e.g. to study the stability of orbits or design bounded relative trajectories. The high-order transfer map (HOTM) method enables efficient mapping of perturbed Keplerian orbits over many revolutions. For this, the method uses the high-order Taylor expansion of a Poincar\'e or stroboscopic map, which is accurate close to the expansion point. In this paper, we investigate the performance of the HOTM method using different element sets for building the high-order map. The element sets investigated are the classical orbital elements, modified equinoctial elements, Hill variables, cylindrical coordinates and Deprit's ideal elements. The performances of the different coordinate sets are tested by comparing the accuracy and efficiency of mapping low-Earth and highly-elliptical orbits perturbed by $J_2$ with numerical propagation. The accuracy of HOTM depends strongly on the choice of elements and type of orbit. A new set of elements is introduced that enables extremely accurate mapping of the state, even for high eccentricities and higher-order zonal perturbations. Finally, the high-order map is shown to be very useful for the determination and study of fixed points and centre manifolds of Poincar\'e maps.

Energy dependent parity-pair behavior in NO + He collisions.

Colliding molecules behave fundamentally differently at high and low collision energies. At high energies, a collision can be described to a large extent using classical mechanics, and the scattering process can be compared to a billiard-ball-like collision. At low collision energies, the wave character of the collision partners dominates, and only quantum mechanics can predict the outcome of an encounter. It is, however, not so clear how these limits evolve into each other as a function of the collision energy. Here, we investigate and visualize this evolution using a special feature of the differential cross sections for inelastic collisions between NO radicals and He atoms. The so-called "parity-pair" transitions have similar differential cross sections at high collision energies, whereas their cross sections are significantly different in the quantum regime at low energies. These transitions can be used as a probe for the quantum nature of the collision process. The similarity of the parity-pair differential cross sections at high energies could be theoretically explained if the first-order Born approximation were applicable. We found, however, that the anisotropy of the NO-He interaction potential is too strong for the first-order Born approximation to be valid, so higher-order perturbations must be taken into account.

Symmetry breaking during inflation of a toroidal membrane

Membrane structures with symmetry often exhibit geometric instability under finite inflation. We observe and study this phenomenon in the case of a flat toroidal membrane with axisymmetry and reflection symmetry. The membrane is modeled as a Mooney-Rivlin hyperelastic material. The stability of the symmetric inflation path is studied perturbatively which reveals a zone of instability. Using higher order perturbation, the asymmetric shape is subsequently determined. As the inflation proceeds, first the axisymmetry is broken spontaneously through a supercritical pitchfork bifurcation, and is later restored at a reverse subcritical pitchfork bifurcation. The extent of the symmetry breaking zone depends on the material and geometric parameters of the toroidal structure. It is found that a stout torus can completely eliminate the occurrence of such symmetry breaking bifurcations on its inflation path.

Asymmetric regularization of the ground and excited state of the 4He nucleus

We find the threshold structure of the two- and three-nucleon systems, with the deuteron and 3H/3He as the only bound nuclei, sufficient to predict a pair of four-nucleon states: a deeply bound state which is identified with the helium-4 ground state, and a shallow, unstable state at an energy 0.38(25) MeV above the triton-proton threshold which is consistent with data on the first excited state of helium-4. The analysis employs the framework of Pionless EFT at leading order with a generalized regulator prescription which probes renormalization-group invariance of the two states with respect to higher-order perturbations including asymmetrical disturbances of the short-distance structure of the interaction. In addition to this invariance of the bound-state spectrum and the diagonal triton-proton 1S0 phase shifts in the helium-4 channel with respect to the short-distance structure of the nuclear interaction, our multi-channel calculations with a resonating-group method demonstrate the increasing sensitivity of nuclei to the neutron-proton P-wave interaction. We show that two-nucleon phase shifts, the triton channel, and three-nucleon negative-parity channels are less sensitive with respect to enhanced two-nucleon P-wave attraction than the four-nucleon triton-proton 1S0 phase shifts.

Determination of quark masses from nf=4 lattice QCD and the RI-SMOM intermediate scheme

We determine the charm and strange quark masses in the $\overline{\text{MS}}$ scheme, using $n_f=2+1+1$ lattice QCD calculations with highly improved staggered quarks (HISQ) and the RI-SMOM intermediate scheme to connect the bare lattice quark masses to continuum renormalisation schemes. Our study covers analysis of systematic uncertainties from this method, including nonperturbative artefacts and the impact of the non-zero physical sea quark masses. We find $m_c^{\overline{\text{MS}}}(3 \text{GeV}) = 0.9896(61)$ GeV and $m_s^{\overline{\text{MS}}}(3 \text{GeV}) = 0.08536(85)$ GeV, where the uncertainties are dominated by the tuning of the bare lattice quark masses. These results are consistent with, and of similar accuracy to, those using the current-current correlator approach coupled to high-order continuum QCD perturbation theory, implemented in the same quark formalism and on the same gauge field configurations. This provides a strong test of the consistency of methods for determining the quark masses to high precision from lattice QCD. We also give updated lattice QCD world averages for $c$ and $s$ quark masses.

Dynamics of the spatial restricted three-body problem stabilized by Hamiltonian structure-preserving control

The local invariant (stable, unstable, and center) manifolds of a saddle $$\times $$ center $$\times $$ center equilibrium point can be used to construct a three-dimensional Hamiltonian structure-preserving (HSP) controller. The linear stability of the controlled Hamiltonian system is verified when one of the proposed criteria is satisfied. The nonlinear dynamics of a linearly stabilized Hamiltonian system is analyzed by normalizing the Hamiltonian high-order perturbation terms using the Lie series method. The analytical solutions of the invariant tori are then obtained in trigonometric series form. A theorem for the Nekhoroshev stability of the controlled Hamiltonian system is provided. When the constructed HSP controller is applied to a photogravitational restricted three-body problem with oblateness, a hyperbolic artificial equilibrium point at which a criterion is satisfied can be stabilized, and bounded Lissajous trajectories near the equilibrium point are obtained. The allocation law demonstrates that the attitude angles and lightness number of the solar sail can serve as control parameters to provide the required acceleration of the HSP controller.

A first step towards effectively nonperturbative scattering amplitudes in the perturbative regime

We propose an effectively nonperturbative approach to calculating scattering amplitudes in the perturbative regime. We do this in a discretized momentum space by using the QSE method to calculate all the contributions (to all orders in perturbation theory) to the scattering eigenstates that are above a precision cutoff. We then calculate the scattering amplitude by directly taking the inner product between these eigenstates. In the current work we have analyzed this procedure for a λϕ4 theory in one spatial dimension and compared our results with perturbation theory obtaining favorable results suggestive that further research in this direction might be worthwhile. In particular, we show that the efficiency of our method scales much better than second- and higher-order perturbation theory as the momentum lattice spacing decreases and as the eigenstate energy increases.

A high-order perturbation of surfaces method for vector electromagnetic scattering by doubly layered periodic crossed gratings

The accurate simulation of scattering of electromagnetic waves in three dimensions by a diffraction grating is crucial in many applications of engineering and scientific interest. In this contribution we present a novel High-Order Perturbation of Surfaces method for the numerical approximation of vector electromagnetic scattering by a doubly periodic layered medium. For this we restate the governing time harmonic Maxwell equations as vector Helmholtz equations which are coupled by transmission boundary conditions at the layer interface. We then apply the method of Transformed Field Expansions which delivers a Fourier collocation, Legendre–Galerkin, Boundary Perturbation approach to solve the problem in transformed coordinates. A sequence of numerical simulations demonstrate the efficient and robust spectral convergence which can be achieved with the proposed algorithm.

XYZ-SU3 breakings from Laplace sum rules at higher orders

We present new compact integrated expressions of SU3 breaking corrections to QCD spectral functions of heavy–light molecules and four-quark [Formula: see text]-like states at lowest order (LO) of perturbative (PT) QCD and up to [Formula: see text] condensates of the Operator Product Expansion (OPE). Including next-to-next-to-leading order (N2LO) PT corrections in the chiral limit and next-to-leading order (NLO) SU3 PT corrections, which we have estimated by assuming the factorization of the four-quark spectral functions, we improve previous LO results for the [Formula: see text]-like masses and decay constants from QCD spectral sum rules (QSSR). Systematic errors are estimated from a geometric growth of the higher order PT corrections and from some partially known [Formula: see text] nonperturbative contributions. Our optimal results, based on stability criteria, are summarized in Tables 18–21 while the [Formula: see text] and [Formula: see text] channels are compared with some existing LO results in Table 22. One can note that, in most channels, the SU3 corrections on the meson masses are tiny: [Formula: see text] (respectively [Formula: see text]) for the [Formula: see text] (respectively [Formula: see text])-quark channel but can be large for the couplings ([Formula: see text]). Within the lowest dimension currents, most of the [Formula: see text] and [Formula: see text] states are below the physical thresholds while our predictions cannot discriminate a molecule from a four-quark state. A comparison with the masses of some experimental candidates indicates that the [Formula: see text] [Formula: see text] might have a large [Formula: see text] molecule component while an interpretation of the [Formula: see text] candidates as four-quark ground states is not supported by our findings. The [Formula: see text] [Formula: see text] and [Formula: see text] are compatible with the [Formula: see text], [Formula: see text] molecules and/or with the axial-vector [Formula: see text] four-quark ground state. Our results for the [Formula: see text], [Formula: see text] and for different beauty states can be tested in the future data. Finally, we revisit our previous estimates1 for the [Formula: see text] and [Formula: see text] and present new results for the [Formula: see text].

Around Chaotic Disturbance and Irregularity for Higher Order Traveling Waves

Many unknown features in the theory of wave motion are still captivating the global scientific community. In this paper, we consider a model of seventh order Korteweg–de Vries (KdV) equation with one perturbation level, expressed with the recently introduced derivative with nonsingular kernel, Caputo-Fabrizio derivative (CFFD). Existence and uniqueness of the solution to the model are established and proven to be continuous. The model is solved numerically, to exhibit the shape of related solitary waves and perform some graphical simulations. As expected, the solitary wave solution to the model without higher order perturbation term is shown via its related homoclinic orbit to lie on a curved surface. Unlike models with conventional derivative (γ=1) where regular behaviors are noticed, the wave motions of models with the nonsingular kernel derivative are characterized by irregular behaviors in the pure factional cases (γ&lt;1). Hence, the regularity of a soliton can be perturbed by this nonsingular kernel derivative, which, combined with the perturbation parameter ζ of the seventh order KdV equation, simply causes more accentuated irregularities (close to chaos) due to small irregular deviations.

Gauge invariant canonical cosmological perturbation theory with geometrical clocks in extended phase-space — A review and applications

The theory of cosmological perturbations is a well-elaborated field and has been successfully applied, e.g. to model the structure formation in our universe and the prediction of the power spectrum of the cosmic microwave background. To deal with the diffeomorphism invariance of general relativity, one generally introduces combinations of the metric and matter perturbations, which are gauge invariant up to the considered order in the perturbations. For linear cosmological perturbations, one works with the so-called Bardeen potentials widely used in this context. However, there exists no common procedure to construct gauge invariant quantities also for higher-order perturbations. Usually, one has to find new gauge invariant quantities independently for each order in perturbation theory. With the relational formalism introduced by Rovelli and further developed by Dittrich and Thiemann, it is in principle possible to calculate manifestly gauge invariant quantities, that is quantities that are gauge invariant up to arbitrary order once one has chosen a set of so-called reference fields, often also called clock fields. This article contains a review of the relational formalism and its application to canonical general relativity following the work of Garcia, Pons, Sundermeyer and Salisbury. As the starting point for our application of this formalism to cosmological perturbation theory, we also review the Hamiltonian formulation of the linearized theory for perturbations around FLRW spacetimes. The main aim of our work will be to identify clock fields in the context of the relational formalism that can be used to reconstruct quantities like the Bardeen potential as well as the Mukhanov–Sasaki variable. This requires a careful analysis of the canonical formulation in the extended ADM-phase-space where lapse and shift are treated as dynamical variables. The actual construction of such observables and further investigations thereof will be carried out in our companion paper.

Two-step deferred correction algorithm for the problem of fluid film squeezed between rotating disks

In this paper, a new collocation method is analyzed for the solution of a system of nonlinear differential equations arisen from modeling the flow of a Newtonian magnetic lubricant film with magnetic induction effects, squeezed between two rotating disks with a time-dependent distance. The source problem is a system of nonlinear partial differential equations which is converted to the system of fourth-order ordinary differential equations by some appropriate transformations. The problem is then transformed to an equivalent second-order system subject to its proper boundary conditions. Eighth-order superconvergent approximations are obtained for the solution based on sextic B-spline. High-order perturbations of the problem are used to find higher orders of convergence. We proceed the superconvergence in two successive steps by perturbing the right-hand side of the problem appropriately. The convergence of the method is analyzed via Green’s function approach. The numerical results verify the theoretical orders of convergence and show the efficiency of the methods as well.

On high-order perturbation expansion for the study of long–short wave interactions

In high-order analysis and simulation of long–short surface wave interaction using mode decomposition, ‘divergent’ terms of the form $k_{S}a_{L}=O(\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D716})\gg 1$ appear in the high-order expansions, where $k_{L,S}$, $a_{L,S}$ are respectively the long, short modal wavenumbers and amplitudes, with $\unicode[STIX]{x1D6FE}\equiv k_{S}/k_{L}\gg 1$ and $k_{L}a_{L}\sim k_{S}a_{S}=O(\unicode[STIX]{x1D716})$ finite. We address the effect of these terms on the numerical scheme, showing numerical cancellation at all orders $m$; but increasing ill-conditioning of the numerics with $\unicode[STIX]{x1D6FE}$ and $m$, which we quantify. In the context of mode decomposition, we show theoretical exact cancellation of the divergent terms up to $m=3$, extending the existing result of Brueckner &amp; West (J. Fluid Mech., vol. 196, 1988, pp. 585–592) and supporting the conjecture that this is obtained for all orders $m$. We show the latter by developing a theoretical proof for any $m$ using a Dirichlet–Neumann operator and mathematical induction. The implication of the theoretical proof on the numerical simulation of long–short wave interaction is discussed.

A new high-order numerical method for solving singular two-point boundary value problems

Recently, Goh et al. (2012) [ 28 ] proposed a numerical technique based on quartic B-spline collocation for solving a class of singular boundary value problems (SBVP) with Neumann and Dirichlet boundary conditions (BC). This method is only fourth-order accurate. In this paper, we propose an optimal numerical technique for solving a more general class of nonlinear SBVP subject to Neumann and Robin BC. The method is based on high order perturbation of the problem under consideration. The convergence of the proposed method is analyzed. To demonstrate the applicability and efficiency of the method, we consider four numerical examples, three of which arise in various physical models in applied science and engineering. A comparison with other available numerical solutions has been carried out to justify the advantage of the proposed technique. Numerical result reveals that the proposed method is sixth order convergent, which in turn is two orders of magnitude larger than in Goh et al. (2012) [ 28 ].

Research of the spin-Hamiltonian parameters for the tetragonal Co2+ tetrahedral centers in Y3Al5O12 and Y3Ga5O12 crystals

In this article, the calculations with the high-order perturbation formulas founded on the two-spin-orbit-parameter (of dn ion and that of ligand) model are performed for the spin-Hamiltonian parameters (zero-field splitting D and g factors g//, g⊥) of the tetragonal Co2+ tetrahedral centers in Y3Al5O12 and Y3Ga5O12 crystals. The calculated results by using only one adjustable parameter are comparable with the experimental values. The impurity-caused angular distortions in the Co2+ centers in both crystals are also evaluated. The results are discussed.

Nonlinear phenomena in general relativity

The perturbation theory plays an important role in studying structure formation in cosmology and post-Newtonian physics, but not all phenomena can be described by the linear perturbation theory. Thus, It is necessary to study exact solutions or higher order perturbations. Specifically, we study black hole (apparent) horizons and the cosmological event horizon formation in the perturbation theory. We emphasize that in the perturbative regime of the gravitational potential these horizons cannot form in the lower order. Studying the infinite plane metric, we show that to capture the cosmological constant effect we need at least a second order expansion.

Stable, high-order computation of impedance-impedance operators for three-dimensional layered medium simulations.

The faithful modelling of the propagation of linear waves in a layered, periodic structure is of paramount importance in many branches of the applied sciences. In this paper, we present a novel numerical algorithm for the simulation of such problems which is free of the artificial singularities present in related approaches. We advocate for a surface integral formulation which is phrased in terms of impedance-impedance operators that are immune to the Dirichlet eigenvalues which plague the Dirichlet-Neumann operators that appear in classical formulations. We demonstrate a high-order spectral algorithm to simulate these latter operators based upon a high-order perturbation of surfaces methodology which is rapid, robust and highly accurate. We demonstrate the validity and utility of our approach with a sequence of numerical simulations.