Octonionic Lorenz-like condition

In this study, the octonion algebra and its general properties are defined by the Cayley–Dickson’s multiplication rules for octonion units. The field equations, potential equations and Maxwell equations for electromagnetism are investigated with the octonionic equations and these equations can be compared with their vectorial representations. The potential and wave equations for fields with sources are also provided. By using Maxwell equations, a Lorenz-like condition is newly suggested for electromagnetism. The existing equations including the photon mass provide the most acknowledged Lorenz condition for the magnetic monopole and the source.     

Realizations of Affine Weyl Group Symmetries on the Quantum Painlevé Equations by Fractional Calculus

We realize affine Weyl group symmetries on the Schr?dinger equations for the quantum Painlevé equations, by fractional calculus. This realization enables us to construct an infinite number of hypergeometric solutions to the Schr?dinger equations for the quantum Painlevé equations. In other words, since the Schr?dinger equations for the quantum Painlevé equations are equivalent to the Knizhnik?CZamolodchikov equations, we give one method of constructing hypergeometric solutions to the Knizhnik?CZamolodchikov equations.     

Analysis of structure function equations up to the seventh order

We examine balances of structure function equations up to the seventh order N = 7 for longitudinal, mixed and transverse components. Similarly, we examine the traces of the structure function equations, which are of interest because they contain invariant scaling parameters. The trace equations are found to be qualitatively similar to the individual component's equations. In the even-order equations, the source terms proportional to the correlation between velocity increments and the pseudo-dissipation tensor are significant, while for odd N, source terms proportional to the correlation of velocity increments and pressure gradients are dominant. Regarding the component equations, one finds under the inertial range assumptions as many equations as unknown structure functions for even N, i.e. can solve for them as function of the source terms. On the other hand, there are more structure functions than equations for odd N under the inertial range assumptions. Similarly, there are not enough linearly independent equations in the viscous range r → 0 for orders N > 3.     

Generalized linear Schrödinger equations and their Darboux transformations

We construct explicit Darboux transformations of arbitrary order for a class of generalized, linear Schrödinger equations. Our construction contains the well-known Darboux transformations for Schrödinger equations with position-dependent mass, Schrödinger equations coupled to a vector potential and Schrödinger equations for weighted energy.     

A Symmetry Connection Between Hyperbolic and Parabolic Equations

Abstract

We give ansatzes obtained from Lie symmetries of some hyperbolic equations which reduce these equations to the heat or Schrödinger equations. This enables us to construct new solutions of the hyperbolic equations using the Lie and conditional symmetries of the parabolic equations. Moreover, we note that any equation related to such a hyperbolic equation (for example the Dirac equation) also has solutions constructed from the heat and Schrödinger equations.     

Discrete stochastic evolution rules and continuum evolution equations

The continuum evolution equations are derived from updating rules for three classes of stochastic models. The first class corresponds to models whose stochastic continuum equations are of the Langevin type obtained after carrying out a “local average” known as coarse-graining. The second class consists of a hierarchy of continuum equations for the correlations of the dynamical variables obtained after making an average over realizations. This average generates a hierarchy of deterministic partial differential equations except when the dynamical variables do not depend on the values of the neighboring dynamical variables, in which case a hierarchy of ordinary differential equations is obtained. The third class of evolution equations for the correlations of the dynamical variable constitutes another hierarchy after calculating an average over both realizations and all the sites of the lattice. This double average generates a hierarchy of deterministic ordinary differential equations. The second and third classes of equations are truncated using a mean field (m,n)-closure approximation in order to obtain a finite set of equations. Illustrative examples of every class are given.     

Generalized Euler–Poincaré Equations on Lie Groups and Homogeneous Spaces,Orbit Invariants and Applications

We develop the necessary tools, including a notion of logarithmic derivative for curves in homogeneous spaces, for deriving a general class of equations including Euler–Poincaré equations on Lie groups and homogeneous spaces. Orbit invariants play an important role in this context and we use these invariants to prove global existence and uniqueness results for a class of PDE. This class includes Euler–Poincaré equations that have not yet been considered in the literature as well as integrable equations like Camassa–Holm, Degasperis–Procesi, μCH and μDP equations, and the geodesic equations with respect to right-invariant Sobolev metrics on the group of diffeomorphisms of the circle.     

Coupled channel T-operator equations with connected kernels: (I). Three-body problem with pairwise interactions

Previous work on T-operator coupled equations for two-channel systems is generalized and applied to the problem of three bodies interacting via pair potentials. Sets of coupled, integral equations for the two-body arrangement channel T-operators are derived using a channel coupling array W, and the connectedness properties of the kernels of these equations are discussed. It is shown that either disconnected or connected (iterated) kernels can be obtained by various choices of W. One particular realization of the coupled equations is seen to be similar but not identical to the Lovelace form of the Faddeev equations. Since the matrix form of the coupled equations is similar to the one-body Lippmann-Schwinger equation, the introduction of Møller wave operators is straightforward, and these are used to derive coupled integral equations for the channel state vectors.     

Field theory onR×S 3 topology. IV: Electrodynamics of magnetic moments

The equations of electrodynamics for the interactions between magnetic moments are written on R×S³ topology rather than on Minkowskian space-time manifold of ordinary Maxwell's equations. The new field equations are an extension of the previously obtained Klein-Gordon-type, Schrödinger-type, Weyl-type, and Dirac-type equations. The concept of the magnetic moment in our case takes over that of the charge in ordinary electrodynamics as the fundamental entity. The new equations have R×S³ invariance as compared to the Lorentz invariance of Maxwell's equations. The solutions of the new field equations are given. In this theory the divergence of the electric field vanishes whereas that of the magnetic field does not.Research supported in part by the Colgate Research Council and by the Center for Theoretical Physics, University of Maryland.     

Large Eddy Simulation composition equations for single-phase and two-phase fully multicomponent flows

The Large Eddy Simulation (LES) equations for multicomponent (MC) fuel single-phase (SP) flow and two-phase (TP) flow with phase change are derived from the Direct Numerical Simulation (DNS) equations by filtering the DNS equations using a top-hat filter. Additional to the equations solved for single-component (SC) fuels, composition equations enter the formulation. The species composition is represented through a Probability Distribution Function (PDF), and DNS equations for the PDF moments are solved to find the composition. The TP filtered equations contain three categories of subgrid-scale (SGS) terms: (1) SGS–flux terms, (2) filtered source terms (FSTs) and (3) terms representing the ‘LES assumptions’. For SP flows no FSTs exist. The SGS terms in the LES equations must be either shown negligible or modeled. It is shown that for the composition equations, two equivalent forms of the DNS equations lead to two non-equivalent forms of the LES equations. Criteria are proposed to select the form best suited for LES. These criteria are used in conjunction with evaluations based on a DNS database portraying mixing and phase change, and lead to choosing one of the LES forms which satisfies all criteria. It is shown that the LES assumptions lead to additional SGS terms which require modeling. Further considerations are made for reactive flows.     

Integral equations for N-particle scattering

Integral equations are derived for the N-particle transition operators. The equations couple together only transition operators between two-body channels. The kernel of the equations becomes connected after a single iteration. Transition operators involving channels with three or more particles can be obtained by quadratures from the solution of the equations. It is also shown that the N-particle equations can be reduced to multichannel two-body equations by the use of the quasiparticle method.     

Averaged Lagrangians and the mean effects of fluctuations in ideal fluid dynamics

We begin by placing the generalized Lagrangian mean (GLM) equations for a compressible adiabatic fluid into the Euler–Poincaré (EP) variational framework of fluid dynamics, for an averaged Lagrangian. We then state the EP Averaging Result—that GLM equations arise from GLM Hamilton’s principles in the EP framework. Next, we derive a new set of approximate small-amplitude GLM equations (gℓm equations) at second order in the fluctuating displacement of a Lagrangian trajectory from its mean position. These equations express the linear and nonlinear back-reaction effects on the Eulerian mean fluid quantities by the fluctuating displacements of the Lagrangian trajectories in terms of their Eulerian second moments. The derivation of the gℓm equations uses the linearized relations between Eulerian and Lagrangian fluctuations, in the tradition of Lagrangian stability analysis for fluids. The gℓm derivation also uses the method of averaged Lagrangians, in the tradition of wave, mean flow interaction (WMFI). The gℓm EP motion equations for compressible and incompressible ideal fluids are compared with the Euler-alpha turbulence closure equations. An alpha model is a GLM (or gℓm) fluid theory with a Taylor hypothesis closure (THC). Such closures are based on the linearized fluctuation relations that determine the dynamics of the Lagrangian statistical quantities in the Euler-alpha closure equations. We use the EP Averaging Result to bridge between the GLM equations and the Euler-alpha closure equations. Hence, combining the small-amplitude approximation with THC yields in new turbulence closure equations for compressible fluids in the EP variational framework.     

Self-dual Yang-Mills fields in d=7,8, octonions and ward equations

The Yang-Mills theories in d=7 and d=8 with the arbitrary gauge group G are considered. Generalized self-duality-type relations for gauge fields are reduced to systems of nonlinear differential equations on functions of one variable (Ward equations). Ward equations may be reduced to equations which follow from Yang-Baxter equations. This permits us to obtain new classes of explicit solutions of the Yang-Mills equations in d=7 and d=8.     

From the Becker–Döring to the Lifshitz–Slyozov–Wagner Equations

Connections between two classical models of phase transitions, the Becker–Döring (BD) equations and the Lifshitz–Slyozov–Wagner (LSW) equations, are investigated. Homogeneous coefficients are considered and a scaling of the BD equations is introduced in the spirit of the previous works by Penrose and Collet, Goudon, Poupaud and Vasseur. Convergence of the solutions to these rescaled BD equations towards a solution to the LSW equations is shown. For general coefficients an approach in the spirit of numerical analysis allows to approximate the LSW equations by a sequence of BD equations. A new uniqueness result for the BD equations is also provided.     

Solutions of Yang's EuclideanR-gauge equations and self-duality

Under some assumptions and transformations of variables, Yang's equations forR-gauge fields on Euclidean space lead to conformally invariant equations permitting one to obtain infinitely many other solutions from any solution of these conformally invariant equations. These conformally invariant equations closely resemble the mathematically interesting generalized Lund-Regge equations. Some exact solutions of these conformally in variant equations are obtained. Except for some singular situations, these solutions are self-dual.     

Anisotropic solutions of the Einstein-Boltzmann equations: I. General formalism

Given a choice of a timelike vector field, a particle distribution function in a general curved space-time can be analysed into spherical harmonics; the Liouville and Boltzmann equations can then be written as a set of equations relating these spherical harmonic components. We obtain these equations and the resulting equations for the spherical harmonic moments of the distribution function. An orthonormal tetrad formalism is used as an aid in our calculations; the set of moment equations used can be completed by giving Einstein's field equations as equations for the rotation coefficients of this tetrad. We discuss time and space reversal symmetry properties of the Boltzmann equation, but leave applications of the set of equations obtained to further papers.     

A connection between the Einstein and Yang-Mills equations

It is our purpose here to show an unusual relationship between the Einstein equations and the Yang-Mills equations. We give a correspondence between solutions of the self-dual Einstein vacuum equations and the self-dual Yang-Mills equations with a special choice of gauge group. The extension of the argument to the full Yang-Mills equations yields Einstein's unifield equations. We try to incorporate the full Einstein vacuum equations, but the approach is incomplete. We first consider Yang-Mills theory for an arbitrary Lie-algebra with the condition that the connection 1-form and curvature are constant on Minkowski space. This leads to a set of algebraic equations on the connection components. We then specialize the Lie-algebra to be the (infinite dimensional) Lie-algebra of a group of diffeomorphisms of some manifold. The algebraic equations then become differential equations for four vector fields on the manifold on which the diffeomorphisms act. In the self-dual case, if we choose the connection components from the Lie-algebra of the volume preserving 4-dimensional diffeomorphism group, the resulting equations are the same as those obtained by Ashtekar, Jacobsen and Smolin, in their remarkable simplification of the self-dual Einstein vacuum equations. (An alternative derivation of the same equations begins with the self-dual Yang-Mills connection now depending only on the time, then choosing the Lie algebra as that of the volume preserving 3-dimensional diffeomorphisms.) When the reduced full Yang-Mills equations are used in the same context, we get Einstein's equations for his unified theory based on absolute parallelism. To incorporate the full Einsteinvacuum equations we use as the Lie group the semi-direct product of the diffeomorphism group of a 4-dimensional manifold with the group of frame rotations of anSO(1, 3) bundle over the 4-manifold. This last approach, however, yields equations more general than the vacuum equations.Andrew Mellon Postdoctoral fellow and Fulbright ScholarSupported in part by NSF grant no. PHY 80023     

Coupled fractional nonlinear differential equations and exact Jacobian elliptic solutions for exciton–phonon dynamics

An improved quantum model for exciton–phonon dynamics in an α-helix is investigated taking into account the interspine coupling and the influence of power-law long-range exciton–exciton interactions. Having constructed the model Hamiltonian, we derive the lattice equations and employ the Fourier transforms to go in continuum space showing that the long-range interactions (LRI) lead to a nonlocal integral term in the equations of motion. Indeed, the non-locality originating from the LRI results in the dynamic equations with space derivatives of fractional order. New theoretical frameworks are derived, such that: fractional generalization of coupled Zakharov equations, coupled nonlinear fractional Schrödinger equations, coupled fractional Ginzburg–Landau equations, coupled Hilbert–Zakharov equations, coupled nonlinear Hilbert–Ginzburg–Landau equations, coupled nonlinear Schrödinger equations and coupled nonlinear Hilbert–Schrödinger equations. Through the F-expansion method, we derive a set of exact Jacobian solutions of coupled nonlinear Schrödinger equations. These solutions include Jacobian periodic solutions as well as bright and dark soliton which are important in the process of energy transport in the molecule. We also discuss of the impact of LRI on the energy transport in the molecule.     

The conservation of source strength in linear field theories and the superpotentials

In linear field theories for vector potentialsAi and tensor potentialsgik=gki, the Maxwell and the linearized Einstein equations are the only field equations from which true conservation laws result for each gauge of the field equations.     

Gauge-invariant perturbations in anisotropie homogeneous cosmological models

In a recent paper K. Tomita and M. Den found a set of coupled differential equations for spatially flat, anisotropic homogeneous,N- dimensional cosmological models. Some particular exact solutions of those differential equations for a few specific equations of state were obtained by D. Lorentz-Petzold. In the present work we solve those differential equations completely.