TITLE: Are there Soliton Solutions for Maxwell's Equations?
QUESTION [9 upvotes]: 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!

REPLY [4 votes]: 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.