# 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"? – Severin Schraven Aug 1 '18 at 13:10
• @SeverinSchraven Yes! Something like this. I would say, the equivalent of the solitons in the (shallow) water. – user559615 Aug 1 '18 at 13:14
• @andrea.prunotto: Could you please clarify what you mean by "stationary shells"? – Adrian Keister Aug 1 '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). – user559615 Aug 1 '18 at 14:38
• @AdrianKeister Sorry, the previous comment was for you! – user559615 Aug 1 '18 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 '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 1 '18 at 14:46
• @andrea.prunotto I googled: High power, pulsed soliton generation at radio and microwave frequencies. – Keith McClary Aug 1 '18 at 17:30
• – Adrian Keister Aug 1 '18 at 17:33
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.