×

One class of relativistically invariant first-order equations. (English. Russian original) Zbl 1458.35133

Differ. Equ. 56, No. 12, 1575-1586 (2020); translation from Differ. Uravn. 56, No. 12, 1621-1633 (2020).
Summary: We introduce a new class of first-order partial differential equations (coined as covariantly equipped systems of equations) invariant with respect to (pseudo)orthogonal changes of Cartesian coordinates in a (pseudo)Euclidean space. We propose a procedure for reducing the Cauchy problem for a system of equations to a Cauchy problem for a covariantly equipped system of equations (the covariant equipping procedure). We prove that the covariant equipping procedure can be applied to the Cauchy problem for Maxwell’s equations.

MSC:

35F50 Systems of nonlinear first-order PDEs
35Q61 Maxwell equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Friedrichs, K. O., Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math., 7, 2, 345-391 (1954) · Zbl 0059.08902 · doi:10.1002/cpa.3160070206
[2] Godunov, S. K.; Romenskii, E. I., Elementy mekhaniki sploshnykh sred i zakony sokhraneniya (Elements of Continuum Mechanics and Conservation Laws) (1998), Novosibirsk: Nauchn. Kniga, Novosibirsk · Zbl 1053.74001
[3] Marchuk, N. G., A generalization of the Yang_Mills equations, Proc. Steklov Inst. Math., 306, 157-177 (2019) · Zbl 1439.35416 · doi:10.1134/S0081543819050158
[4] Dezin, A. A., Invariant differential operators and boundary value problems, Tr. Mat. Inst. Akad. Nauk SSSR, 68, 3-88 (1962)
[5] Kähler, E., Der Innere Differentialkalkul, Randiconti di Mat. (Roma). Ser. 5, 21, 425 (1962) · Zbl 0127.31404
[6] Atiyah, M., Vector fields on manifolds, in Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen, Köln, 1970, p. 200. · Zbl 0193.52303
[7] Ivanenko, D.; Landau, L., Zur Theorie des magnetischen Electrons, Z. Phys., 48, 340 (1928) · JFM 54.0971.03 · doi:10.1007/BF01339119
[8] Benn, I. M.; Tucker, R. W., An Introduction to Spinors and Geometry with Applications to Physics (1987), Bristol: A. Hilger, Bristol · Zbl 0798.53001
[9] Obukhov, Yu. N.; Solodukhin, S. N., Reduction of the Dirac equation and its connection with the Ivanenko-Landau-Kähler equation, Theor. Math. Phys., 94, 2, 198-210 (1993) · Zbl 0796.53077 · doi:10.1007/BF01019331
[10] Marchuk, N. G., Uravneniya teorii polya i algebry Klifforda (Field Theory Equations and Clifford Algebras) (2018), Moscow: URSS, Moscow
[11] Marchuk, N. G., On a method for reducing the wave equation to a system of first-order equations, Dokl. Akad. Nauk SSSR, 276, 6, 1309-1311 (1984)
[12] Godunov, S. K., Uravneniya matematicheskoi fiziki (Equation of Mathematical Physics) (1979), Moscow: Nauka, Moscow
[13] Mizohata, S., Henbibun houteishiki ron (The Theory of Partial Differential Equations) (1965), Tokyo: Iwanami Shoten, Tokyo
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.