The physics of solid materials is enormously rich territory. In materials such as high-temperature superconductors and other "exotic" materials, its usually what the electrons do in the material that is most interesting. The atoms that make up the crystal lattice aren't quite so interesting. In fact, physicists have had a good basic picture of the vibrations that take place within a (periodic) crystal lattice for roughly a century, since the work of Brillouin, Debye and Einstein. It's all quite easily understood in terms of extended periodic waves of vibration, or phonons -- or so it seemed until recently. Now it appears that the standard picture may have left a whole lot out -- especially exotic waves that act like particles and zip around according to rules all their own.
They're called "discrete breathers" and they are quintessentially nonlinear forms that seem to live inside lots of ordinary materials. Pour energy (with radio waves, lasers, etc) into liquid Helium, or into a magnetic material, and you often find that its not just extended phonons that get stirred up -- rather the energy ends up moving about in a handful of localized pulses that persist for very long times. Researchers increasingly think that understanding the origins and habits of these breathers will be crucial to understanding materials when pushed out of equilbrium, and may also be important even in equilibrium.
There 's currently a lot of work going on to explore these breathers, both in real materials, and also in fascinating "artificial" materials such as mesoscopic arrays of cantilevers etched into semiconductors (which mimic the workings of an atomic lattice). In experiments, researchers can stir up these exotic objects, and watch them move, learning the laws by which they interact. This article in New Scientist looks at how these excitations work, why they were overlooked for so long, what they might mean for physics in the future, and how they might change our most fundamental view of how ordinary matter works.
To learn more about discrete breathers, I strongly recommend looking at the work of Al Sievers of Cornell University and also of Sergej Flach of the Max Planck Institute in Dresden. In particular, for those who know a little physics and mathematics, this article in Physics Today offers a very nice description of the basic ideas.