I'm studying mathematical physics. What is that? It is the intersection of math and physics. This means two things. One is that I'm much more concerned about physical reality than some mathematicians are, and I'm also much more concerned about proofs than most physicists are. So nobody likes me! (That's a joke...)
Suppose you take a fiber optic cable. This is a piece of glass finely drawn into a wire; not a very flexible wire, I might add. The big deal about fiber optic cables is that you can send an enormous amount of information down one. You can send a lot more information down one of those things than you can down an electrical wire. So you've got a fiber optic cable. Suppose you send one single light wave down the cable. Such a wave is called a soliton. The great thing about solitons is that their shape is self-correcting. With an electrical wire, you have to have what's called a repeater to send information down it, because the signal loses its shape over distance and becomes meaningless to the recipient. It's just like the game of gossip, where by the time the message reaches the end of the line, it has changed considerably. A repeater takes the old signal, and re-transmits that signal down the wire so the recipient gets a better signal. With fiber optic cables, you don't need a repeater for miles of cable. This has distinct advantages.
Fiber optic cables have to be very small. Why, might you ask? Because there are fundamental limitations on the size of the cable if you want to send solitons down the cable.
If you send a soliton down a fiber optic cable, that soliton's behavior will be governed by a differential equation, what's called the non-linear Schrodinger equation. We do have exact solutions to this equation, but as it turns out, we can get more information about the soliton if we perform what's called the inverse scattering transform. This is somewhat like a Laplace Transform, only more complicated. The result of applying this transform is a multi-dimensional linear system of differential equations, which is much nicer for analysis. Now this whole process can either take birefringence into account, or not. Birefringence is the phenomenon of differently polarized light waves propagating down the cable at different velocities. If you do not take birefringence into account, then the result of the inverse scattering transform is a two-dimensional system, called the Zacharov-Shabat system. If you do take birefringence into account, then you get a three-dimensional system, called the Manakov system. Right now, I'm working on analyzing the Manakov system.
So there's my research. Any questions? :-)