Gerald Teschl, Universität Wien (Koautoren: W. Bulla, F. Gesztesy, and H. Holden)
The Toda equation describes a linear chain of particles (i.e., a one-dimensional crystal) which interact via exponential nearest neighbor forces. It is connected to many applications in solid state physics, for example, Peierl's model, Langmuir oscillations in plasmas, or solitons in conducting polymers. Mathematically it can be viewed as the prototypical example of a completely integrable wave equation.
I am going to show how the Toda equation can be solved using the theory of hyperelliptic Riemann surfaces and how explicit solutions in terms of theta functions can be found. Along the way we will encounter terms like Lax pairs, solitons, and hierarchies of integrable equations, which you might have heard elsewhere and always wanted to know more about.
|||G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Amer. Math. Soc, Rhode Island, 2000.|