International Journal of Geometric Methods in Modern Physics

In this paper, we introduce three new transformations and establish connections between moving non-null curves and soliton equations according to Bishop frame in Minkowski 3-space. Later we find formulas for differentials of these three new transformations associated with the nonlinear heat system and repulsive type nonlinear Schrödinger equation.

In this paper, three similarity solutions of a mathematical model for the gravitational waves of the Milne expanding empty spacetime, namely, the Gordon-type equation of the Milne metric will be determined. This equation is a linear second-order partial differential equation, dependent on a functional parameter. Formerly, some Lie reductions of the equation have been found, only for the special cases of its parameter. In this work, by applying a symmetry method for the Gordon-type equation, its new Lie reductions will be found, in the general form of the parameter. Then, its three similarity solutions will be presented.

We study a very general four-dimensional field theory model describing the dynamics of a massless higher spin N symmetric tensor field particle interacting with a geometrical background. This model is invariant under the action of an extended linear diffeomorphism. We investigate the consistency of the equations of motion, and the highest spin degrees of freedom are extracted by means of a set of covariant constraints. Moreover, the highest spin equations of motions (and in general all the highest spin field 1-PI irreducible Green functions) are invariant under a chain of transformations induced by a set of N - 2 Ward operators, while the auxiliary fields equations of motion spoil this symmetry. The first steps to a quantum extension of the model are discussed on the basis of the algebraic field theory. Technical aspects are reported in Appendices, in particular, one of them is devoted to illustrate the spin-2 case.

In the present paper, the equivalence problem for fourth-order differential operators with one variable under general fiber-preserving transformation using the Cartan method of equivalence is applied. Two versions of equivalence problems are considered. First, the direct equivalence problem and second equivalence problem is to determine the sufficient and necessary conditions on two fourth-order differential operators such that there exists a fiber-preserving transformation mapping one to the other according to gauge equivalence.

We consider four-dimensional homogeneous pseudo-Riemannian manifolds with non-trivial isotropy and completely classify the cases giving rise to non-trivial homogeneous Ricci solitons. In particular, we show the existence of non-compact homogeneous (and also invariant) pseudo-Riemannian Ricci solitons which are not isometric to solvmanifolds, and of conformally flat homogeneous pseudo-Riemannian Ricci solitons which are not symmetric.

The construction of a finite-order bicomplex whose morphisms are the horizontal and vertical derivatives of differential forms on finite-order jet prolongations of fibered manifolds over one-dimensional bases is presented. In particular, relationship between the morphisms and classes entering the variational sequence and the associated finite-order bicomplex is studied. Properties of classes entering the infinite-order bicomplex, induced from the finite-order variational sequences by means of an infinite canonical construction, are formulated as a remark, insisting further research.

In this paper, Cayley formula is derived for 4 × 4 semi-skew-symmetric real matrices in . For this purpose, we use the decomposition of a semi-skew-symmetric matrix A = θ₁A₁ + θ₂A₂ by two unique semi-skew-symmetric matrices A₁ and A₂ satisfying the properties and Then, we find Lorentzian rotation matrices with semi-skew-symmetric matrices by Cayley formula. Furthermore, we give a way to find the semi-skew-symmetric matrix A for a given Lorentzian rotation matrix R such that R = Cay(A).

After a quick review of the wild structure of the complex moduli space of Calabi-Yau 3-folds and the role of geometric transitions in this context (the Calabi-Yau web) the concept of deformation equivalence for geometric transitions is introduced to understand the arrows of the Gross–Reid Calabi-Yau web as deformation-equivalence classes of geometric transitions. Then the focus will be on some results and suitable examples to understand under which conditions it is possible to get simple geometric transitions, which are almost the only well-understood geometric transitions both in mathematics and in physics.

The scalar wave equation in Kasner spacetime is solved, first for a particular choice of Kasner parameters, by relating the integrand in the wave packet to the Bessel functions. An alternative integral representation is also displayed, which relies upon the method of integration in the complex domain for the solution of hyperbolic equations with variable coefficients. In order to study the propagation of wave fronts, we integrate the equations of bicharacteristics which are null geodesics, and we are able to express them, for the first time in the literature, with the help of elliptic integrals for another choice of Kasner parameters. For generic values of the three Kasner parameters, the solution of the Cauchy problem is built through a pair of integral operators, where the amplitude and phase functions in the integrand solve a coupled system of partial differential equations. The first is the so-called transport equation, whereas the second is a nonlinear equation that reduces to the Eikonal equation if the amplitude is a slowly varying function. Remarkably, the analysis of such a coupled system is proved to be equivalent to building first an auxiliary covariant vector having vanishing divergence, while all nonlinearities are mapped into solving a covariant generalization of the Ermakov–Pinney equation for the amplitude function. Last, from a linear set of equations for the gradient of the phase one recovers the phase itself. This is the parametrix construction that relies upon Fourier–Maslov integral operators, but with a novel perspective on the nonlinearities in the dispersion relation. Furthermore, the Adomian method for nonlinear partial differential equations is applied to generate a recursive scheme for the evaluation of the amplitude function in the parametrix. The resulting formulas can be used to build self-dual solutions to the field equations of noncommutative gravity, as has been shown in the recent literature.

A geometric setup for constrained variational calculus is presented. The analysis deals with the study of the extremals of an action functional defined on piecewise differentiable curves, subject to differentiable, non-holonomic constraints. Special attention is paid to the tensorial aspects of the theory. As far as the kinematical foundations are concerned, a fully covariant scheme is developed through the introduction of the concept of infinitesimal control. The standard classification of the extremals into normal and abnormal ones is discussed, pointing out the existence of an algebraic algorithm assigning to each admissible curve a corresponding abnormality index, related to the co-rank of a suitable linear map. Attention is then shifted to the study of the first variation of the action functional. The analysis includes a revisitation of Pontryagin's equations and of the Lagrange multipliers method, as well as a reformulation of Pontryagin's algorithm in Hamiltonian terms. The analysis is completed by a general result, concerning the existence of finite deformations with fixed endpoints.

We study the confinement of charged Dirac particles in a plane embedded in 3 + 1 space–time due to the presence of a constant magnetic field that is tilted on the given plane. We focus on the nature of the solutions of the Dirac equation and on how they depend on the choice of vector potential that gives rise to the magnetic field. In particular, we select a "Landau gauge" such that the momentum is conserved along the direction of the vector potential yielding spinor wave functions, which are localized in the plane containing the magnetic field and normal to the vector potential. These wave functions are expressed in terms of the Hermite polynomials. We point out the relevance of these findings to the relativistic quantum Hall effect and compare with the results obtained for a constant magnetic field normal to the plane in 2 + 1 dimensions.

In the traditional Kaluza–Klein theory, the cylinder condition and the constancy of the extra-dimensional radius (scalar field) imply that time-like geodesics on the five-dimensional bundle project to solutions of the Lorentz force equation on spacetime. This property is lost for nonconstant scalar fields, in fact there appears new terms that have been interpreted mainly as new forces or as due to a variable inertial mass and/or charge. Here we prove that the additional terms can be removed if we assume that charged particles are coupled with the same spacetime conformal structure of neutral particles but through a different conformal factor. As a consequence, in Kaluza–Klein theory the proper time of the charged particle might depend on the charge-to-mass ratio and the scalar field. Then we show that the compatibility between the equation of the projected geodesic and the classical limit of the Klein–Gordon equation fixes unambiguously the conformal factor of the coupling metric solving the conformal ambiguity problem of Kaluza–Klein theories. We confirm this result by explicitly constructing the projection of the Klein–Gordon equation and by showing that each Fourier mode, even for a variable scalar field, satisfies the Klein–Gordon equation on the base.

The Kastler–Kalau–Walze theorem, announced by A. Connes, shows that the Wodzicki residue of the inverse square of the Dirac operator is proportional to the Einstein–Hilbert action of general relativity. In this paper, we prove a Kastler–Kalau–Walze type theorem for five-dimensional manifolds with boundary.

We consider the Noether Symmetry Approach for a cosmological model derived from a tachyon scalar field T with a Dirac–Born–Infeld Lagrangian and a potential V(T). Furthermore, we assume a coupled canonical scalar field ϕ with an arbitrary interaction potential B(T, ϕ). Exact solutions are derived consistent with the accelerated behavior of cosmic fluid.