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Generalization of the Maxwell equation and relation to the elastic equation
Author(s) -
Soga Hideo
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5624
Subject(s) - mathematics , maxwell's equations , independent equation , mathematical analysis , generalization , eigenvalues and eigenvectors , partial differential equation , maxwell relations , wave equation , differential equation , simultaneous equations , inhomogeneous electromagnetic wave equation , electromagnetic field , physics , quantum mechanics , optical field
In this paper, we propose a hyperbolic system of first‐order pseudo‐differential equations as generalization of the Maxwell equation. We state basic properties of this system corresponding to the ones of the (usual) Maxwell equation and explain that several known generalized Maxwell equations presented by some researchers can be integrated into the system. Namely, their equations can be regarded as our equation in special cases. Their generalized equations admit not only transversal but also longitudinal waves and are examined from the physical viewpoint. Using the present system, from the mathematical viewpoint, we interpret the meaning for presence of the longitudinal wave (with the transversal one) in their generalized equations. This presence means existence of more than one non‐zero characteristic root for the system (ie, non‐zero eigenvalue of the symbol). We prove also that our system becomes a first‐order expression of (generalized) elastic equations. Furthermore, it is shown that introducing the elastic equations implies expressing the generalized Maxwell equations by the potentials.