Your problem is unclear. The solution of Maxwell's equations is a chronic problems !

Maybe I was not clear because of an error in English. I now recognize that the correct term for what I look for is vortex-like solutions, not vertex-like ones. Sorry, I have now corrected the question.

But why do you say that solutions of MEs is a problem, and that it is chronic? There are an infinity of solutions to MEs and many were well studied. All I ask is: has somebody heard of, or studied, vortex-like solutions?

Dear Stan,

Yes, Maxwell's equations predict vortex solutions, not in vacuum but in different material media. For instance, the vortex in II-superconductors, the Eddy currents in a good conductors or the Hall effect ....The solutions of Maxwell equations in vacuum are well known to be of a wave equation.

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Daniel Baldomir, thanks. I know well about those solutions inside specific material media. I am looking for solutions in vacuum, maybe ABOVE planar structures, but in vacuum. Something analogous to the atmospheric twisters above the Earth surface, or vortices in water bodies (though those extend below the surface). EMF vortices in vacuum above surfaces, or even totally isolated and spacially self-contained. I have looked a lot for any hint of them in scientific literature of all kinds (physics, IEEE, etc) but found nothing. This is my last attempt at finding whether such solutions are really still unexplored. I will publish the stuff anyway, but it is somewhat surprising not to find any reference.

Also, I am testing how people react to the concept. Indeed, it seems that I was immediately misunderstood. It does not surprise me, but it helps me to formulate my thesis better. After all, it is an author's responsability to make himself understood, not vice versa. So, whomever comments on this, I will be in any case grateful for the comments :-)

Dear Stan,

In vacuum Maxwell Electrodynamics has not vortice's solutions: only waves. You need to put some meterial and employ magnetic field as the main field contribution.

Stan Sykora

I also looked but did not find. The suggestion was/is the electric fields are not fields but clouds of vortices. I suggest 2 types of vortices. One like the atmospheric twister, the other like smoke rings. But they must be vortices of something - the ether or plenum in the STOE.

The critical thing by analogy is similar vortices repel and different vortices annihilate each other which could result in a force directing the source (the particle) toward the point of annihilation (between the particles).

Daniel Baldomir , you say " In vacuum Maxwell Electrodynamics has not vortice's solutions". Have you published this result, or otherwise could you give me the reference in which it has been studied and proved? That would be just as good for me as a reference to a paper presenting a vortex solution.

Also, I am not sure why you add "only waves"? According to the wave theories of decades ago, the whole Universe is just waves. Thanks to superposition and interference, waves can be used to build anything, including vortices.

But I am not after any physics, much less any metaphysical speculations. Mine is not a physics question at all. I am after vortex-like math solutions to the PDE's called "Maxwell equations in vacuum" or else a math proof that they can not exist. Possibly literature references, if any exist.

John Hodge, thanks; your comment that you "looked but did not find" supports my feeling that so far MEs were mathematically not studied exhaustively enough; not beyond the obvious solutions. But your subsequent physics speculations do not address my simple question.

I am looking for hard math references before dwelling into it myself.

Dear Stan,

It is not true that you can built all that you want with waves. The wave solution is very different than the vortice-like solution. But it is true that following the fluid model in electrodynamics it was people trying to obtain this kind of solutions with what is known Beltrami fields. If you want to go to this issue I recommend you the following references:

1-D.Reed, Foundational Electrodynamics and Beltrami Vector Fields, in Advanced Electromagnetism: Foundations, Theory, Applications, D. Grimes, T.W. Barrett (eds), World Scientific, Singapore, 1995.

2-Wu J-Z, Ma H-Y and Zhou M-D. Vorticity and Vortex Dynamics. Belin: Springer-Verlag, 2006

3.https://books.google.es/books?isbn=9814501085

Thanks a lot, Daniel Baldomir , I was not aware of the book by Wu.

Btw, I agree with you that not everything can be build with waves, but a hundred years ago many physicists thought it was possible. That is why QM "wavefunctions" are called wavefunctions, even when they regard spacially localized objects.

To better delimit the object of my interest: I have strong empirical evidence that an RF electromagnetic field above a chiral surface plane behaves anomalously, possibly compatibly with the presence of vortex-like formations. Standard simulation programs (like Comsol) do not seem to be able to explain actual experimental results (though, obviously, one only gets from them what one puts in). So sure, I am talking about EM field in vacuum, but bounded by a special type of planar surface. I am well aware that boundary conditions are always an issue. Even for spherical waves, one has boundary conditions - a point-like source, and zero field at infinity. Not to speak about boundary conditions in waveguides, clystrons, etc. My boundary is a chiral plane.

There are certain authors who claim having found this type of solutions also related "electromagnetic scalar waves"...and they were advertised as so far secret and hidden solution of Maxell's equations with very strange and interesting properties (but of course don't exist). Apart from this see the statement of Stan Sykora on thit point. For further details you may consult the homepage of G. Bruhn (Darmstadt university). However there exist scalar waves in acoustics/mechanics and vortices in superconductors (Abrikosov vortices)

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Thank you, Fritz Caspers .

One can find MANY (claimed) vortex solutions for electromagnetic fields and waves but they are usually wrong...however putting finger on the mistake in the derivation is not always trivial..

https://www2.mathematik.tu-darmstadt.de/~bruhn/Meyliana.html

The aerodynamics research also comes close. But close is not good enough.

So, I suggest you solve the vortex issues by analogy or similarity to aerodynamics observations as I did.

When you say vortex-like, do you mean with zero curl except for paths that go round a central filament?

If you do not require that exact definition, I think you will find that the electric and magnetic fields in any finite width beam are vortex-like because they have closed paths.

Looking at it another way, every specific changing line of magnetic flux is surrounded by its own electric field vortex, and each changing electric field line is surrounded by its own magnetic field vortex. That is what Maxwell's curl equations say, I think. If that is true then in general the solutions to Maxwell's equations are vortex-like.

Well, I do not mean trivial solutions. If you take the hydrodynamic equations, for example, they describe all kinds of waves, but that is elementary stuff. A water vortex, be it one in your bathtub or an oceanic malstroem, is something quite different. I suspect that Maxwell equations solutions include something just as different from the usual EM waves as an oceanic vortex is from "normal" water waves. An EM vortex would have to be a near (and hence non-radiative) solution to ME's involving both the magnetic and the electric field, both vortex like, contained mostly in a limited area of space and tapering off to zero anywhere outside that area.

I also suspect that only a small portion of possible qualitatively distinct solutions to MEs have been studied so far. I went through hundreds of texts and they mostly just parrot the previous ones, showing that "if we ASSUME a spherical wave in the dual vector EM field - or a planar one - than we can twiddle its parameters to make it match MEs". Ok, big deal. That is what Maxwell himself said anyway, so it is useless to repeat it. And once the repetitions are removed, there is precious little that remains. Except in the applied areas of wavegiudes, coax cable, clystrons, and maybe antennas (some of which are strongly chiral). But even there, all the resonance modes start from a-priori qualified guesses about how the solutions might look. Yet any mathematician must feel that PDE's on a pair of coupled vector fields must have an infinity of solution types. Ideally, there should exist a math thorem saying that only such and such types of solutions (and, in vacuum, all their linear combinations, of course) may exist.

For there to be a magnetic field vortex there must be a filament of changing electric field or of current through the centre of it - that is the only way the curl can be non-zero, which is required at the centre of a vortex. For an electric field vortex there must be a changing magnetic field filament through the centre of it, so there is no steady state solution. For any finite electric field vortex the centre of the vortex has to have finite width, or the magnetic field intensity would have to be infinite. In free space the magnetic field cannot fall to zero in a finite distance, so the core of the vortex has tapered edges . The electric field is the integral of infinitesimal vortices from changing infinitesimal magnetic field threads. Not sure if this leads anywhere, but it does rule some things out.

Maybe, I am not sure. Filaments can twist in many ways. As I see it now, I should ask some math professionals. Evidently there are no obvious literature references that I missed. Which is what I wanted to know, so thanks you all :-)

There has been a lot of discussion about 20 years ago and earlier on "alternative" electrodynamics including authors like K. Meyl, Myron Evans etc . In this context electromagnetic vortices play a big role as well as electromagnetic scalar waves.

All this has calmed down by now and you can find the detailed derivations and where exactly all (?) the mistakes (published in this context) came in, via the link to the webpage by Gerhard Bruhn (Darmstadt University). In short ..there are no new electrodynamics in the frame of the validity of Maxwells equations and no hidden or so far unknown solutions. Of course we have QED and QCD as a valid extension. One possible exception may be the modified Maxwells equations taking the so far hypothetical axion-electromagnetic coupling into account.

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Sorry, I cannot offer a math solution but only a thought on how to generate em waves which are propagating on a plane surface and whose energy flow (not the E and H fields) has a vortex-like component.

Janice Turner nee Hendry succeeded in simulating a source whose field splits into a radiation part and a surface wave propagating on a corrugated surface, like a Zenneck wave:

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.651.4359&rep=rep1&type=pdf

http://www.ee.ucl.ac.uk/lcs/previous/LCS2011/LCS1137.pdf

I guess if the corrugations in the plane surface were not concentric but would form a spiral or several spirals the surface wave would propagate spirally, too, because it would be directed everywhere (nearly) perpendicular to the corrugations.

Mathematically, the question arises if radial Zenneck waves which are solutions of the Maxwell equations can be "twisted" somehow into spiral Zenneck waves.

Anyway these days everything can nicely be simulated numerically (e.g. CST ,Ansoft or similar) with great precision and with given and well defined boundary and initial conditions the solutions are unambiguous. And before claiming having found an electromagnetic vortex and should be very clear about its precise definition. BTW a nice " riddle" in this context is the setup of a charged capacitor in the field of strong permanent (horseshoe)magnet..(to be found in one of the Feynman books)..E and H are orthogonal and thus there is a BIG flux (yes also at DC) of Poynting vector ..can come into the ballpark of Megawatts..