«4 + 6»-Decomposition of ten-dimensional octonionic spinor wave equations and derivation of four-dimensional massless Dirac equations

1990 ◽  
Vol 105 (11) ◽  
pp. 1203-1214 ◽  
Author(s):  
H. Tachibana ◽  
K. Imaeda
Keyword(s):  
Wave Equations ◽  
Dirac Equations ◽  
Spinor Wave

scholarly journals Relativistic wave equations with fractional derivatives and pseudodifferential operators

2002 ◽  
Vol 2 (4) ◽  
pp. 163-197 ◽  
Author(s):  
Petr Závada
Keyword(s):  
Fractional Derivatives ◽  
Wave Equations ◽  
Pseudodifferential Operators ◽  
Relativistic Wave Equations ◽  
Green Functions ◽  
Relativistic Wave ◽  
Fractional Powers ◽  
Dirac Equations ◽  
Klein Gordon ◽  
Symmetry Transformations

We study the class of the free relativistic covariant equations generated by the fractional powers of the d′Alembertian operator(□1/n). The equations corresponding ton=1and2(Klein-Gordon and Dirac equations) are local in their nature, but the multicomponent equations for arbitraryn>2are nonlocal. We show the representation of the generalized algebra of Pauli and Dirac matrices and how these matrices are related to the algebra ofSU (n)group. The corresponding representations of the Poincaré group and further symmetry transformations on the obtained equations are discussed. The construction of the related Green functions is suggested.

The wave equation for the Lanczos potential. I

1994 ◽  
Vol 447 (1931) ◽  
pp. 557-575 ◽  
Keyword(s):  
Wave Equation ◽  
Degrees Of Freedom ◽  
Wave Equations ◽  
Conformal Curvature ◽  
Lanczos Potential ◽  
Conformal Curvature Tensor ◽  
Radiation Theory ◽  
Non Local ◽  
Spinor Wave ◽  
Gauge Conditions

The non-local part of the gravitational field in general relativity is described by the 10 component conformal curvature tensor C abcd of Weyl. For this field Lanczos found a tensor potential L abc with 16 independent components. We can make L abc have only 10 effective degrees of freedom by imposing the 6 gauge conditions L ab s :s = 0. Both fields C abcd , L abc satisfy wave equations. The wave equation satisfied by C abcd is nonlinear, even in vacuo . However, a linear spinor wave equation for the Lanczos potential has been found by Illge but no correct tensor wave equation for L abc has yet been published. Here, we derive a correct tensor wave equation for L abc and when it is simplified with the aid of some four­-dimensional identities it is equivalent to Illge’s wave equation. We also show that the nonlinear spinor wave equation of Penrose for the Weyl field can be derived from Illge’s spinor wave equation. A set of analogues of well-known results of classical electromagnetic radiation theory can now be given. We indicate how a Green’s function approach to gravitational radiation could be based on our tensor wave equation, when a global study of space-time is attempted.

COMPLEXIFIED THEORY OF POSITIVE-FREQUENCY MAXWELL-DIRAC FIELDS

1993 ◽  
Vol 08 (21) ◽  
pp. 3697-3719 ◽  
Author(s):  
J.G. CARDOSO
Keyword(s):  
Wave Equations ◽  
Lagrangian Density ◽  
Variational Principles ◽  
Equations Of Motion ◽  
Dirac Equations ◽  
Spinor Form ◽  
Equivalent Wave ◽  
Symmetric Operators ◽  
Positive Frequency ◽  
Component Spinor

We present a method whereby the equations of motion yielding the explicit two-component spinor form of the complete Maxwell-Dirac theory in complex Minkowski space may be directly derived from two variational principles. One of these dynamical statements gives rise to the first half of the electromagnetic theory which particularly appears as the equations of motion involving a slightly modified version of the free part of the conventional Maxwell Lagrangian density. The other principle actually involves a holomorphic two-spinor expression for the full Maxwell-Dirac Lagrangian density which leads to the second half along with the Dirac equations that carry the relevant covariant derivative operator. It is shown explicitly how the Bianchi identities of the complete theory can be established in a manifestly covariant way. A system of essentially equivalent wave equations for positive-frequency photons and electrons is then exhibited. In particular, we make use of certain skew-symmetric operators to obtain an extended form of the Feynman-Gell-Mann equations.

scholarly journals A new construction for spinor wave equations

1998 ◽  
Vol 3 (4) ◽  
pp. 517-534
Author(s):  
S. de Toro Arias ◽  
C. Vanneste
Keyword(s):  
Wave Equations ◽  
New Construction ◽  
Spinor Wave

Wave equations in momentum space

1950 ◽  
Vol 204 (1077) ◽  
pp. 145-169 ◽  
Keyword(s):  
Integral Equations ◽  
Momentum Space ◽  
Wave Equations ◽  
Yukawa Potential ◽  
Dirac Equations ◽  
General Integral ◽  
Space Potential ◽  
Method Of Solution ◽  
The Fourier Transform ◽  
General Method

The non-relativistic and relativistic wave equations of a particle are solved in momentum space for different kinds of fields. It is assumed that the potential, when defined in co-ordinate space, is central; but the method applies also to more general forms of momentum space potentials. First, the general integral equations which correspond, in momentum space, to Schrödinger and Dirac equations are derived, and the angular variables separated. Then, a general method of solution of these integral equations is given, which lies mainly on a transformation into a four-dimensional hyperspace. These results are (relativistically and non-relativistically) applied rigorously to the Coulombian field (hydrogen atom) and to the scalar Yukawa potential. A more general case is further investigated. In this, the momentum space potential can be expanded in a series of powers of the modulus of the momentum (this last application involves a cut-off procedure). A mathematical appendix deals with some properties of the hyperspherical harmonics (especially the integral equation of these functions) and with the Fourier transform of some kinds of central potentials.

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

On the Exact Soliton Solutions of Some Nonlinear PDE by Tan-Cot Method

2018 ◽  
Vol 5 (1) ◽  
pp. 31-36
Author(s):  
Md Monirul Islam ◽  
Muztuba Ahbab ◽  
Md Robiul Islam ◽  
Md Humayun Kabir
Keyword(s):  
Solitary Wave ◽  
Acoustic Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Rectangular Channel ◽  
Soliton Solutions ◽  
Boussinesq Equations ◽  
Travelling Wave ◽  
Ion Acoustic Waves ◽  
Analytical System

For many solitary wave applications, various approximate models have been proposed. Certainly, the most famous solitary wave equations are the K-dV, BBM and Boussinesq equations. The K-dV equation was originally derived to describe shallow water waves in a rectangular channel. Surprisingly, the equation also models ion-acoustic waves and magneto-hydrodynamic waves in plasmas, waves in elastic rods, equatorial planetary waves, acoustic waves on a crystal lattice, and more. If we describe all of the above situation, we must be needed a solution function of their governing equations. The Tan-cot method is applied to obtain exact travelling wave solutions to the generalized Korteweg-de Vries (gK-dV) equation and generalized Benjamin-Bona- Mahony (BBM) equation which are important equations to evaluate wide variety of physical applications. In this paper we described the soliton behavior of gK-dV and BBM equations by analytical system especially using Tan-cot method and shown in graphically. GUB JOURNAL OF SCIENCE AND ENGINEERING, Vol 5(1), Dec 2018 P 31-36

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