Covariant Field Equations in Supergravity

Covariance is a useful property for handling supergravity theories. In this paper, we prove a covariance property of supergravity field equations: under reasonable conditions, field equations of supergravity are covariant modulo other field equations. We prove that for any supergravity there exist such covariant equations of motion, other than the regular equations of motion, that are equivalent to the latter. The relations that we find between field equations and their covariant form can be used to obtain multiplets of field equations. In practice, the covariant field equations are easily found by simply covariantizing the ordinary field equations.     

整数自旋粒子的方程和波函数

从高自旋态的Bargmann-Wigner方程出发,建立了整数自旋粒子的运动方程,通过求解方程得到了一套整数自旋粒子波函数,并建立了等效Largrange形式.     

A simple and direct method for generating travelling wave solutions for nonlinear equations

We propose a simple and direct method for generating travelling wave solutions for nonlinear integrable equations. We illustrate how nontrivial solutions for the KdV, the mKdV and the Boussinesq equations can be obtained from simple solutions of linear equations. We describe how using this method, a soliton solution of the KdV equation can yield soliton solutions for the mKdV as well as the Boussinesq equations. Similarly, starting with cnoidal solutions of the KdV equation, we can obtain the corresponding solutions for the mKdV as well as the Boussinesq equations. Simple solutions of linear equations can also lead to cnoidal solutions of nonlinear systems. Finally, we propose and solve some new families of KdV equations and show how soliton solutions are also obtained for the higher order equations of the KdV hierarchy using this method.     

Discretizing constant curvature surfaces via loop group factorizations: The discrete sine- and sinh-Gordon equations

The sine- and sinh-Gordon equations are the harmonic map equations for maps of the (Lorentz) plane into the 2-sphere. Geometrically they correspond to the integrability equations for surfaces of constant Gauss and constant mean curvature. There is a well-known dressing action of a loop group on the space of harmonic maps. By discretizing the vacuum solutions we obtain via the dressing action completely integrable discretizations (in both variables) of the sine- and sinh-Gordon equations. For the sine-Gordon equation we get Hirota's discretization. Since we work in a geometric context we also obtain discrete models for harmonic maps into the 2-sphere and discrete models of constant Gauss and mean curvature surfaces.     

Dynamically orthogonal field equations for continuous stochastic dynamical systems

In this work we derive an exact, closed set of evolution equations for general continuous stochastic fields described by a Stochastic Partial Differential Equation (SPDE). By hypothesizing a decomposition of the solution field into a mean and stochastic dynamical component, we derive a system of field equations consisting of a Partial Differential Equation (PDE) for the mean field, a family of PDEs for the orthonormal basis that describe the stochastic subspace where the stochasticity ‘lives’ as well as a system of Stochastic Differential Equations that defines how the stochasticity evolves in the time varying stochastic subspace. These new evolution equations are derived directly from the original SPDE, using nothing more than a dynamically orthogonal condition on the representation of the solution. If additional restrictions are assumed on the form of the representation, we recover both the Proper Orthogonal Decomposition equations and the generalized Polynomial Chaos equations. We apply this novel methodology to two cases of two-dimensional viscous fluid flows described by the Navier–Stokes equations and we compare our results with Monte Carlo simulations.     

A variational principle for symplectic connections

We introduce a variational principle for symplectic connections and study the corresponding field equations. For two-dimensional compact symplectic manifolds we determine all solutions of the field equations. For two-dimensional non-compact simply connected symplectic manifolds we give an essentially exhaustive list of solutions of the field equations. Finally we indicate how to construct from solutions of the field equations on (M, ω) solutions of the field equations on the cotangent bundle to M with its standard symplectic structure.     

硅二极管对高功率微波的非线性响应计算

 考虑二极管内部载流子所满足的非线性、耦合、刚性方程组中可能考虑的因素,利用自得研制的半导体器件模拟程序mPND1D, 对硅二极管在高功率微波激励下的非线性特性进行了数值计算,结果显示出硅二极管对徽波信号响应的非线性。     

Necessary and sufficient conditions for the existence of a Lagrangian in field theory. III. Generalized analytic representations of tensorial field equations

In this paper we first study the equivalence transformations of class C², regular, tensorial, quasi-linear systems of field equations which (a) preserve the continuity, regularity, and quasi-linear structure of the systems; and (b) occur within a fixed system of Minkowski coordinates and field components. We identify, among the transformations of this class, those which either induce or preserve a self-adjoint structure of the field equations and we term them genotopic and isotopic transformations, respectively. We then give the necessary and sufficient conditions for an equivalence transformation of the above type to be either genotopic or isotopic. By using this methodology, we then extend the theorem on the necessary and sufficient condition for the existence of ordered direct analytic representations introduced in the preceding paper to the case of ordered indirect analytic representations in terms of the conventional Lagrange equations; we introduce a method for the construction of a Lagrangian, when it exists, in this broader context; and we explore some implications of the underlying methodology for the problem of the structure of the Lagrangian capable of representing interactions within the framework of the indirect analytic representations. Some of the several aspects which demand an inspection prior to the use of this analytic approach in actual models are pointed out. In particular, we indicate a possible deep impact in the symmetries and conservation laws of the system generated by the use of the concept of indirect analytic representation. As a preparatory step prior to the analysis of these problems, we study some methodological aspects which underlie the generalized Lagrange equations postulated in the first paper of this series for the case when they are regular, namely, when they are simple equivalence transformations of the conventional Lagrange equations. We first introduce a generalization of the action principle capable of inducing the generalized as well as the conventional equations. In this way we establish that the former equations are “bona fide” analytic equations. Finally, as our most general analytic framework for the case of unconstrained field equations, we work out the necessary and sufficient condition for the existence of ordered direct analytic representations of quasi-linear systems in terms of the generalized analytic equations and study their relationship to the conventional representations.     

The ultraviolet behaviour of integrable quantum field theories,affine Toda field theory

We investigate the thermodynamic Bethe ansatz (TBA) equations for a system of particles which dynamically interacts via the scattering matrix of affine Toda field theory and whose statistical interaction is of a general Haldane type. Up to the first leading order, we provide general approximated analytical expressions for the solutions of these equations from which we derive general formulae for the ultraviolet scaling functions for theories in which the underlying Lie algebra is simply laced. For several explicit models we compare the quality of the approximated analytical solutions against the numerical solutions. We address the question of existence and uniqueness of the solutions of the TBA equations, derive precise error estimates and determine the rate of convergence for the applied numerical procedure. A general expression for the Fourier transformed kernels of the TBA equations allows one to derive the related Y-systems and a reformulation of the equations into a universal form.     

Transport equations for chiral fermions to order ? and electroweak baryogenesis: Part I

This is the first in a series of two papers. In this first part, we use the Schwinger-Keldysh formalism to derive semiclassical Boltzmann transport equations, accurate to order ?, for massive chiral fermions, scalar particles, and for the corresponding CP-conjugate states. Our considerations include complex mass terms and mixing fermion and scalar fields, such that CP-violation is naturally included, rendering the equations particularly suitable for studies of baryogenesis at a first order electroweak phase transition. We provide a quantitative criterion in which case the reduction to the diagonal kinetic equations in the mass eigenbasis is justified, leading to a quasiparticle picture even in the case of mixing scalar or fermionic particles. Within the approximations we make, it is possible to first study the Boltzmann equations without the collision term. In a second paper [Ann. Phys. xxx (2004) xxx] we discuss the collision terms and reduce the Boltzmann equations to fluid equations.     

THE HOLONOMY FUNCTIONAL FORMULATION OF YANG-MILLS FIELD EQUATIONS

Starting from the integral formalism of gauge fields,we reformulate the equa-tions of motion for gauge fields——sourceless Yang-Mills equations——as functional-differential equations of the holonomy functional (i.e.,the phase factors of closedloops).From these equations we derive a set of infinite number of conservationlaws in the functional-differential from for the gauge theories in a two-dimensionalspace-time.     

Field Equations due to a Constant Modification in General Relativity

In this article, by adding a constant to Einstein-Hilbert action, we derive field equations for a non-vacuum space. Also we derive a general solution for these field equations, considering a de Sitter like initial geometric constraint. It is shown that how this additional constant can affect usual gravitational field equations, which are derived from general relativity.     

Brans-Dicke cosmological exact solution in a radiation-filled Robertson-Walker universe

In a recent paper Singh and Deo obtained the field equations in Brans-Dicke theory for a radiation-filled universe with Robertson-Walker metric and solved the equations for a particular case. Here we obtain the complete set of solutions of these equations.     

Equations Holding in Hilbert Lattices

We produce and study several sequences of equations, in the language of orthomodular lattices, which hold in the ortholattice of closed subspaces of any classical Hilbert space, but not in all orthomodular lattices. Most of these equations hold in any orthomodular lattice admitting a strong set of states whose values are in a real Hilbert space. For some of these equations, we give conditions under which they hold in the ortholattice of closed subspaces of a generalised Hilbert space. These conditions are relative to the dimension of the Hilbert space and to the characteristic of its division ring of scalars. In some cases, we show that these equations cannot be deduced from the already known equations, and we study their mutual independence. To conclude, we suggest a new method for obtaining such equations, using the tensorial product. PACS numbers: 02.10, 03.65, 03.67     

Evolution equations of the expected phase space ion densities and correlation functions in a D+T plasma

Summary In this paper we formualte a master equation approach describing a D+T thermonuclear plasma in a lumped phase space. From the first moments of this master equation and performing the pass to the continuous limit the evolution equations for the expected phase space ion densities emerge. Also we have obtained the evolution equations of the equal time correlation and covariance functions. Finally we have deduced the hydrodynamic equations that arise from a master equation approach.     

Posteriori Error Estimation for an Interior Penalty Discontinuous Galerkin Method for Maxwell's Equations in Cold Plasma

In this paper, we develop a residual-based a posteriori error estimator for the time-dependent Maxwell's equations in the cold plasma. Here we consider a semi-discrete interior penalty discontinuous Galerkin (DG) method for solving the governing equations. We provide both the upper bound and lower bound analysis for the error estimator. This is the first posteriori error analysis carried out for the Maxwell's equations in dispersive media.     

Generalized Klein–Gordon field equations with octonion space-time (OST) algebra

The concept of space-time representation is redefined using the octonion space-time (OST) algebra. In this study, describing the properties of octonions and their possible connection with Euclidean space-times, the internal and external space-time events are represented within the OST algebra. Keeping in mind the octonionic dual-Euclidean space-times, we express the homogeneous field equations which leads to the symmetrical nature of internal and external space-times. We derive the generalized Proca–Maxwell equations for massive-dyons in the case of the OST algebra. Accordingly, we have obtained a new set of octonionic Klein–Gordon potential (KGP) and Klein–Gordon field (KGF) equations for massive dyons from the generalized Proca–Maxwell equations. This formalism demonstrates that the octonionic KGP and KGF equations can be expressed in a single equation and it is equivalent to energy-momentum relation for dyons. As such, we have made an attempt to write the conservation of Noetherian current from the octonionic Klein–Gordon equations.     

Interactions of an elastic solid with a viscous fluid: Eigenmode analysis

In this paper we study the interaction of a viscous fluid with an elastic solid. Of particular interest are the eigenmodes of the coupled system. Starting from the Navier-Stokes equations for the fluid and the linear elasticity equations for the solid, we derive the linear equations governing the motion of the system. It is shown how a variational formulation of the problem may be obtained by re-scaling the displacement unknowns. The finite-element technique is then used to discretize the equations. The resulting quadratic eigenvalue problem is solved by means of an inverse iteration procedure.     

广义(3+1)维Zakharov-Kuznetsov方程的对称约化、精确解和守恒律

运用李群分析,得到了广义(3+1)维Zakharov-Kuznetsov(ZK)方程的对称及约化方程,结合齐次平衡原理,试探函数法和指数函数法得到了该方程的群不变解和新精确解,包括冲击波解、孤立波解等.进一步给出了广义(3+1)维ZK方程的伴随方程和守恒律.