Dynamics of generalizations of the Toda lattice.

Abstract

The tridiagonal Toda lattice equation is a fundamental example of an explicitly solvable differential equation. It has considerable other structure as well: it can be interpreted Lie algebraicly, it is completely integrable, and it is gradient on the level sets of its integrals. In addition, its qualitative behavior can be compactly described in term of flows on polytopes. In this thesis, we analyze and compare the long term behavior of several generalizations of the Toda lattice. In particular, for two of the generalizations (the block double bracket equation and block asymmetric Toda equations), we analyze the structure of their equilibrium manifolds. We use this analysis to show that the differential equations exhibit sorting behavior similar to, but different from, the sorting behavior of the original tridiagonal Toda lattice. We also show that the block double bracket equation is gradient with respect to a normal metric. In addition, a connection is made between the matrix Riccati equation and the block asymmetric Toda equation in a low dimension. This connection allows us to determine conditions for blow up in the low dimension for some special cases. We simulate some solutions to the block double bracket equation, the block asymmetric Toda equation, and the symmetric non-Abelian Toda equation. Finally, we analyze the tridiagonalization of the full Toda equation and deduce its gradient behavior from the tridiagonalization.