# Are there Soliton Solutions for Maxwell's Equations?

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?

• What means "solitary wave solution" for you? Something like compact support and "not changing form"? Aug 1, 2018 at 13:10
• @SeverinSchraven Yes! Something like this. I would say, the equivalent of the solitons in the (shallow) water.
– user559615
Aug 1, 2018 at 13:14
• @andrea.prunotto: Could you please clarify what you mean by "stationary shells"? Aug 1, 2018 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).
– user559615
Aug 1, 2018 at 14:38
• @AdrianKeister Sorry, the previous comment was for you!
– user559615
Aug 1, 2018 at 14:48

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.

• 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?
– user559615
Aug 1, 2018 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. Aug 1, 2018 at 14:46
• @andrea.prunotto I googled: High power, pulsed soliton generation at radio and microwave frequencies. Aug 1, 2018 at 17:30
• Aug 1, 2018 at 17:33
– user559615
Aug 1, 2018 at 22:18

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.

• 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.
– user559615
Sep 4, 2018 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). Sep 4, 2018 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). Sep 4, 2018 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 Sep 4, 2018 at 20:38
• 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!
– user559615
Sep 4, 2018 at 21:21