10

Some non-linear differential equations (such as Korteweg–de Vries and Kadomtsev–Petviashvili equations) have "solitary waves" solutions (solitons).

Does the set of partial differential equations known as "Maxwell's equations" theoretically admit such kind of solutions?

In that case, should these solutions appear in the form of "stationary shells" of electromagnetic field? By "stationary", I mean do the solutions maintain their shape?

Thanks for your comments!

Community
  • 1
  • What means "solitary wave solution" for you? Something like compact support and "not changing form"? – Severin Schraven Aug 01 '18 at 13:10
  • @SeverinSchraven Yes! Something like this. I would say, the equivalent of the solitons in the (shallow) water. –  Aug 01 '18 at 13:14
  • @andrea.prunotto: Could you please clarify what you mean by "stationary shells"? – Adrian Keister Aug 01 '18 at 13:31
  • @SeverinSchraven With this term I mean "a bubble" of electromagnetic field, sorry if I cannot be more precise, the 3D equivalent of the 1-D soliton in the water (i.e. a wave that does not involve crests and troughs, but is made only of a crest, or a trough). –  Aug 01 '18 at 14:38
  • @AdrianKeister Sorry, the previous comment was for you! –  Aug 01 '18 at 14:48
  • @andrea.prunotto: Ah, thanks! Does my answer answer this part of your question? I mean, a soliton is a solitary wave, as opposed to a standing or continuous wave. But solitons are not stationary in the sense of staying in one location. As everything in sight is an electromagnetic field (that's what the Maxwell Equations describe), a soliton is a form of the EM field. – Adrian Keister Aug 01 '18 at 14:51
  • With stationary I mean that they maintain the shape. And yes, your answer is very interesting. I wonder if the mathematics of this problem can be used to predict the condition of formation of "ball lightnings", an elusive but known phenomenon, whose dimension can range up to meters (and even more). I know a bit of non-linear pde and I studied electromagnetism, but my knowledge is not enough to rigorously address this problem. –  Aug 01 '18 at 14:51
  • @andrea.prunotto: Ah, I see. Well, my answer has answered your shape question, I think. As for ball lightnings, it doesn't sound like those would happen in a fiber cable. In particular, the boundary conditions for a fiber (necessary to get the equations I mentioned in my answer) probably prevent such a phenomenon. – Adrian Keister Aug 01 '18 at 15:31

2 Answers2

15

The answer is yes to both questions. If you cast Maxwell's Equations in cylindrical coordinates for a fiber optic cable, and you take birefringence into account, you get the coupled nonlinear Schrödinger equations. You can then solve those by means of the Inverse Scattering Transform, which takes the original system of nonlinear pde's (nonlinear because of the coordinate system), transforms them into a coupled system of linear ode's (the Manakov system) which are straight-forward to solve, and then, by means of the Gel'fand-Levitan-Marchenko integral equation, you arrive at the soliton solutions of the original pde's. For references, see C. Menyuk, Application of multiple-length-scale methods to the study of optical fiber transmission, Journal of Engineering Mathematics 36: 113-136, 1999, Kluwer Academic Publishers, Netherlands, and my own dissertation, which includes other references of interest. In particular, Shaw's book Mathematical Principles of Optical Fiber Communication has most of these derivations in it.

The resulting soliton solutions behave mostly like waves, but they also interact in a particle-like fashion; for example, in a collision, they can alter each others' phase - a decidedly non-wave-like behavior. Solitons do not stay in one place; in the case above, they would travel down the fiber cable (indeed, solitons are the reason fiber is the backbone of the Internet!), and self-correct their shape as they go. And, as Maxwell's equations are all about electromagnetic fields, the solutions are, indeed, stationary (in your sense) "shells" of electromagnetic fields.

Adrian Keister
  • 10,272
  • Thanks for the detailed and illuminating answer, Adrian! A further question for you: Do you know if there are macroscopic conditions in which such phenomenon occurs? I mean, a case in which such solitons can have dimensions of meters, rather than the dimensions of the diameter of a fiber cable? –  Aug 01 '18 at 14:40
  • @andrea.prunotto: Thank you for your kind words. An interesting question, that. I don't know the answer; my hunch/intuition would be no, but I don't have much of anything to back that up. – Adrian Keister Aug 01 '18 at 14:46
  • 1
    @andrea.prunotto I googled: High power, pulsed soliton generation at radio and microwave frequencies. – Keith McClary Aug 01 '18 at 17:30
  • 1
    Also https://en.wikipedia.org/wiki/Ball_lightning#Soliton_hypothesis. – Adrian Keister Aug 01 '18 at 17:33
  • @Adrian,Keith Thanks for your comments! –  Aug 01 '18 at 22:18
  • Thanks for the nice answer. May I ask two questions? i) Why is the system nonlinear because of the cylindrical coordinate? I usually thought coordinate transform doesn't make a linear system nonlinear and presume here it is nonlinear only because of birefringence. Do I misunderstand? ii) I was surprised by "solitons are the reason fiber is the backbone of the Internet". Sounds very interesting. Do you mean that we are routinely transmitting solitons as the information carrier in optical fibers? Any reference mentioning this? Thank you! – xiaohuamao Aug 28 '22 at 01:19
  • @xiaohuamao For an answer to i), see the Menyuk paper for the derivation of the nonlinear pde. I am not entirely sure of the reason for going from linear to nonlinear. ii) Yes, the Internet uses solitons to transmit information. I believe Shaw's book (referenced in my answer) talks about this, but I can't confirm as I do not have a copy. – Adrian Keister Aug 28 '22 at 14:24
6

Adrian has given an interesting answer already, but I think it is worth pointing out two key points which were necessary for his soliton situation. Firstly, it was necessary to impose some specific form of initial-boundary data (to constrain the waves to inside the fibre optic cable), and secondly it was necessary to impose physical assumptions on the medium which actually changed the underlying PDE.

If one considers the source-free Maxwells equations in a vacuum, then you know that the electric and magnetic fields $E$ and $B$ satisfy the standard wave equations $\Box E=0$, $\Box B =0$. If you work on the domain $(x,t)\in \mathbb{R}^3 \times [0,\infty)$ of "open space", and if you pose a Cauchy problem for the equations, which means if you specify some initial data (which must of course satisfy divergence free conditions) along the initial surface $t=0$, then it follows from Kirchoff's formula for the solution that the fields $E$ and $B$ have to decay in time. Specifically, one has $\|D^\alpha E(\cdot,t), D^\alpha B(\cdot,t)\|_{L^\infty(\mathbb{R}^3)} \to 0$ as $t\to \infty$ where $D^\alpha$ represents any chosen choice of composition of partial derivatives.

Thus solutions to Maxwell's equations on an unbounded domain must always "scatter at infinity", and you can't hope to find soliton type solutions.

Aerinmund Fagelson
  • 2,744
  • Thanks for your answer! Very interesting, and neat. I wonder if our atmosphere (where these ball-lightnings occur) can be considered a "bound domain", and how. –  Sep 04 '18 at 19:52
  • Interesting thought! I must admit I don't know much about the physics of lightening, but I think the lightening storm would be a much more complicated electrodynamics problem. My answer is really only supposed to apply to electromagnetic waves travelling in a vacuum (such as those induced by the lightening storm which carry the image to your eye, for instance). – Aerinmund Fagelson Sep 04 '18 at 20:05
  • Also you raise a good point that in reality there is no such thing as an unbounded domain, but for initial disturbances that are sufficiently localized with respect to the scale of the problem (i.e. the size of the of the lightening cloud compared with the distance to an astronaut on the moon) the model should be quite good and one would expect the waves to decay as they travel through space (i.e. the lightening storm may appear bright to you, but will look very dim from the perspective of the astronaut). – Aerinmund Fagelson Sep 04 '18 at 20:06
  • Was just looking at some of the videos of this "ball lightning". I do see your point that it is not clear the soliton behaviour there is driven at all by boundedness constraints on the domain. The lightning ball just sits their localized in the middle of a huge cloud! Probably some very complicated nonlinear phenomena going on! Afraid I'm out of my depth :p – Aerinmund Fagelson Sep 04 '18 at 20:38
  • 1
    I share the same feeling! That's way I was trying to address the problem in a theoretical way. But I have studied non-linear equations only for the water (tidal waves, etc.). By the way, there is the theory due to Kapitza, et al (1955), but it does not explain why these monsters (the ball lightnings) are extremely charged! –  Sep 04 '18 at 21:21