Theophanes E. Raptis, Christos D. Papageorgiou



Beltrami Flows, Non-Diffracting Waves and the Axion Beltrami-Maxwell Postulates

pdf PDF


We present a particular class of solutions in Cartesian, cylindrical and spherical coordinates of the non-dispersive travelling wave variety that propagate an envelope of varying vorticity some of which include topological waves with parallel electric and magnetic components. The significance of these solutions is examined in the recently proposed Axion-Maxwell field theory with potential applications in material science and topological insulators.


Beltrami-Maxwell postulates, Non-diffractive optics, Axions, Symmetry breaking


[1] J. C. Maxwell, “A Dynamical Theory of the Electromagnetic Field.”, Phil. Trans. R. Soc., 155, 459-512 (1865),

[2] J. C. Maxwell, (1873) “A Treatise on Electricity and Magnetism”, Clarendon Press.

[3] H. Marmanis (1996), “Analogy between the Electromagnetic and Hydrodynamic Equations: Applications to Turbulence.”, PhD Thesis, Div. App. Math., Brown University.

[4] P. Holland, “Hydrodynamic Construction of the Electromagnetic Field. ”, Proc. R. Soc., 461(2063) (2005)

[5] T. Kambe, “A new formulation of Compressible Fluids by analogy with Maxwell’s equations.”, Front. Fund. Phys. & Phys. Ed. Res., Springer Proc. Phys. 287-295 (2010).

[6] G. E. Marsh, (1996), “Force-Free Magnetic Fields: Solutions, Topology and Applications”, World Sci.

[7] E. Beltrami, Rendiconti del Reale Instituto Lombardo, Series II, 22 (1889).

[8] A. Lakhtakia, (1994) “Beltrami Fields in Chiral Media”, World Sci.: Series in Contemporary Chemical Physics.

[9] L. Silberstein, “Elektromagnetische Grundgleichungen in bivektorieller Behandlung”, Annalen der Physic,327(3), 579-586 (1907)

[10] A. Aste, “Complex Representation Theory of the Electromagnetic Field”, J. Geom. Symm. Phys., 28, 47-58 (2012)

[11] D. Hestenes, “Reforming the Mathematical Language of Physics”, Am. J. Phys. 71(2), 104-121 (2003), and, “Spacetime Physics with Geometric Algebra.” Am. J. Phys. 71 (7), 691-714 (2003), “Oersted” papers


[12] H. Yabe, Y. Mushiake, “Electromagnetic Fields in Twisted Coordinate Systems.”, Electron. Comm. Jap. (Part I: Communications), 64(12), 43-50 (1981).

[13] C. Chu, T. Ohkawa, “Transverse Electroamgnetic Waves with E||B.”, Phys. Rev. Lett., 48(13), 837-838 (1982) and Phys. Rev. Lett. 58, 424 (1982).

[14] J. E. Gray, “Electromagnetic Waves with E parallel to B.”, J. Phys.A: Math. Gen., 25, 5373- 5376 (1992).

[15] K. Uehara et al., “Non-Transverse Electromagnetic Waves with Parallel Electric and Magnetic Fields.”, J. Phys. Soc. Jap., 58(10), 3570- 3575 (1989) and “Electromagnetic plane Waves with parallel Electric and Magnetic fields in free space.”, Am. J. Phys., 58(4) 394-396 (1989)

[16] T. Nishiyama, “General plane or spherical electromagnetic waves with electric and magnetic fields parallel to each other.”, Wave Motion, 54, 58- 65 (2015).

[17] C. M. Ko, I. G. Jang, “Force-free magnetic fields on curvilinear coordinate surfaces.”, Astrophys. Space Sci., 245, 117-129 (1996).

[18] B. C. Low, “On the hydrodynamic stability of a class of laminated force-free magnetic fields.”, Sol. Phys., 115, 269-276 (1988) and Astrophys. J., 330, 992-996 (1988).

[19] N. Manton, P. Sutcliffe, (2004) “Topological Solitons”, Cambridge Univ. Press.

[20] R. D. Perccei, H. R. Quinn, “CP Conservation in the presence of Pseudoparticles.”, Phys. Rev. Lett. 38(25), 1440-1443 (1977) and “Constraints imposed by CP Conservation in the presence of Pseudoparticles.”, Phys. Rev. D, 16(6), 1791-1797 (1977).

[21] S. Weinberg, “A New Light Boson?”, Phys. Rev. Lett., 40(4) 223–226 (1978).

[22] F. Wilczek, “Problem of Strong P and T invariance in the presence of Instantons.”, Phys. Rev. Lett., 40(5), 279–282 (1978).

[23] C. Abel et al., “Search for Axionlike Dark Matter through Nuclear Spin Precession in Electric and Magnetic Fields.”, Phys. Rev. X, 7, 041034 (2017).

[24] P. Skivie, “Dark Matter Axions.”, Int. J. Mod. Phys. A, 25(02n03), 554-563 (2010).

[25] R. Li et al., “Experimental realization of a topological crystalline insulator in SnTe.”, Nature Phys., 6, 284-288 (2010).

[26] L. Visinelli, “Axion-Electromagnetic Waves.”, Mod. Phys. Lett. A, 28, 1350162 (2013).

[27] S. Chandrashekhar, “On Force-Free Magnetic Fields.” Proc. N.A.S., 42(1), 1-5 (1956) and “On Force-Free Magnetic Fields.”, 44(4), 285-289 (1958)

[28] L. Woltzer, “A Theorem on Force-Free Magnetic Fields.”, Proc. N.A.S., 44(6), 489-491 (1958)

[29] H. E. Moses, “Solution of Maxwell’s Equations in Terms of a Spinor Notation: the Direct and Inverse Problem.”, Phys. Rev. 113(6), 1670-1679 (1958)

[30] J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: Transverse electric mode.”, J. App. Phys. 54, 1179 (1983).

[31] P. Hillion, “More on focus wave modes in Maxwell equations.”, J. App. Phys., 60, 2981 (1986).

[32] J. Lu, J. F. Greenleaf, “Non-diffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realization.”, IEEE Trans. Ultrason. Ferroelec. Freq. Contr. 39(1), 19- 31 (1992) and “Biomedical ultrasound beam forming.”, Ultrasound Med. Biol., 20(5), 403-428 (1994).

[33] H. E. Hernandez-Figueroa, E. Recami, M. Zamboni-Rached, (2013)“Non-Diffracting Waves”, Wiley – Verlag.

[34] W. Pauli, “Relativistic Field Theories of Elementary Particles.”, Rev.Mod. Phys. 13, 213 (1941).

[35] B.R. Martin, G. Shaw, (2008) “Particle Physics”, J. Wiley & Sons.

Cite this paper

Theophanes E. Raptis, Christos D. Papageorgiou. (2017) Beltrami Flows, Non-Diffracting Waves and the Axion Beltrami-Maxwell Postulates . Journal of Electromagnetics, 2, 14-22


Copyright © 2017 Author(s) retain the copyright of this article.
This article is published under the terms of the Creative Commons Attribution License 4.0