The Toda Lattice as a Soliton Gas

A basic tenet of integrable systems is that, under sufficiently irregular initial data, they can be thought of as dense collections of many solitons, or “soliton gases.” In this talk we focus on the Toda lattice, which is an archetypal example of an integrable Hamiltonian dynamical system. We explain how the system, under certain random initial data, can be interpreted through solitons, and provide a framework for studying how these solitons asymptotically evolve in time. The arguments use ideas from random matrix theory, particularly the analysis of Lyapunov exponents governing the decay rates of eigenvectors of random tridiagonal matrices.