Meronlike solutions of conformally covariant coupled nonlinear field equations

We derive a new class of solutions of conformally covariant coupled spinor and scalar equations including a nonlinear spinor self-coupling term, for which the energy-momentum density is nonzero, but the total energy is zero (“meronlike” solutions).     

Gauge and Lorentz covariant quark propagator in an arbitrary gluon field

The quark propagator in the presence of an arbitrary gluon field is calculated gauge and Lorentz covariantly order by order in terms of powers of the gluon field and its derivatives. The result is independent of the path connecting the ends of the propagator, and the leading order result coincides with the exact propagator in the trivial case of a vanishing gluon field. Received: 5 February 2003 / Published online: 23 May 2003     

Approximate analytical solutions of generalized linear and nonlinear reaction-diffusion equations in an infinite domain

This Letter deals with the solution of unified fractional reaction-diffusion systems in an infinite domain. The results are obtained in compact and elegant forms in terms of generalized Mittag-Leffler functions, which are suitable for numerical computation.     

Derivation and uniqueness of vacuum solutions of conformally covariant coupled nonlinear field equations

We derive the solutions of conformally covariant coupled Dirac and scalar fields including a nonlinear fermion self-coupling term for which the conformally covariant (not the canonical, nor the symmetric) energy-momentum tensor θ_μν vanishes. This “vacuum” state is degenerate.     

Conformally covariant field equations I. first-order equations with non-vanishing mass

The connection of conformal fields with the Mackey theory is discussed. The necessary and sufficient conditions for the finite or infinite component field equation $$\left( {L_\mu \partial ^\mu + m} \right)\psi \left( x \right) = 0$$ to be conformally covariant, are derived. The conditions are then explicitly solved under very general assumptions and thus conformally covariant equations of the above type are explicitly found (Theorem 5). The circumstances under which the equation may be obtained from a Lagrangian are discussed.     

Infinite sets of conservation laws for linear and nonlinear field equations

The relation between an infinite set of conservation laws of a linear field equation and the enveloping algebra of the space-time symmetry group is established. It is shown that each symmetric element of the enveloping algebra of the space-time symmetry group of a linear field equation generates a one-parameter group of symmetries of the field equation. The cases of the Maxwell and Dirac equations are studied in detail. Then it is shown that (at least in the sense of a power series in the coupling constant) the conservation laws of the linear case can be deformed to conservation laws of a nonlinear field equation which is obtained from the linear one by adding a nonlinear term invariant under the group of space-time symmetries. As an example, our method is applied to the Korteweg-de Vries equation and to the massless Thirring model.     

A Gauge and Lorentz covariant approximation for the quark propagator in an arbitrary gluon field

We decompose the quark propagator in the presence of an arbitrary gluon field with respect to a set of Dirac matrices. The four-dimensional integrals which arise in first order perturbation theory are rewritten as line-integrals along certain field lines, together with a weighted integration over the various field lines. It is then easy to transform the propagator into a form involving path ordered exponentials. The resulting expression is non-perturbative and has the correct behavior under Lorentz transformations, gauge transformations and charge conjugation. Furthermore it coincides with the exact propagator in first order of the coupling g. No expansion with respect to the inverse quark mass is involved, the expression can even be used for vanishing mass. For large mass the field lines concentrate near the straight line connection and simple results can be obtained immediately. Received: 31 March 2001 / Revised version: 3 May 2001 / Published online: 8 June 2001     

Soliton solutions of infinite component wave equations

Three dimensional moving solitons, including relativistic Fitzgerald contraction of lengths, are explicitly constructed as stationary solutions of infinite component wave equations.     

Analytic solution of gauge field equations for Lorentz gravity

The exterior analytic solution for a static, spherically symmetric system is given by means of a set of gauge field equations from Lorentz gravity in the curvature coordinate. The correction contributed by the gravitational gauge field in the exterior of a static sphere is obtained for the gravity.     

Classification of an infinite series of static solutions to the vacuum Einstein equations

We propose a classification of an infinite series of the static solutions to the vacuum Einstein equation by means of their soliton number and the real and/or complex pole trajectories. We also show that in the 4-soliton solution there appears an intriguing ring solution of which the curvature invariant has a finite limit at the rings.     

Geometrization of linear perturbation theory for diffeomorphism-invariant covariant field equations. II. Basic gauge-invariant variables with applications to de sitter space-time

In a companion paper, a systematic treatment of linearized perturbations and a new geometric definition of gauge-invariant variables, based on the theory of vector bundles and applicable to the case of an arbitrary system of covariant field equations, were carefully presented. One of the purposes of the present paper is to specify a necessary and sufficient condition that a given, finite set of gaugeinvariant variables, denoted collectively by ω and referred to as the complete set of basic variables, can be used to extract the equivalence classes of perturbations from ω in a unique way. The above set is complete because it has the following property: a knowledge of ω is all one needs in the sense that ifx represents an arbitrary point of the “space-time” manifoldX andG denotes any gauge-invariant tensor field onX, then the value ofG atx∈X is uniquely specified by giving the germs of basic gauge-invariant variables atx∈X. Arguments are proposed that ω also has a stronger property which is more immediately useful: anyG is obtainable directly from the basic variables through purely algebraic and differential operations. These results are of practical interest, and one concrete setting where one is led to the explicit definition of ω occurs when considering the infinitesimal perturbation of the metric tensor itself (pure gravity) defined on a fixed background de Sitter space-time and obeying the linearized empty-space Einstein equations with nonnegative cosmological constant Λ; the case Λ=0 corresponds to linear perturbation theory in Minkowski space-time.     

Constructing infinite sequence exact solutions of nonlinear evolution equations

To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding Bäcklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.     

Modified Dirac equation with Lorentz invariance violation and its solutions for particles in an external magnetic field

The second-order modified Dirac equation leading to the modified dispersion relation due to the Lorentz invariance violation corrections is suggested. The equation is formulated in the 16-component first-order form. I have obtained the projection matrix extracting solutions of the equation with definite spin projections which can be considered as the density matrix for pure spin states. Exact solutions of the equation are found for particles in the external constant and uniform magnetic field. The synchrotron radiation radius within the novel modified Dirac equation is estimated.     

Two-soliton solutions of relativistic field equations

Existence of solutions converging fort - to a superposition of two solitons is shown for a class of scalar, relativistic, field equations in two-dimensional space-time.