On Long-Time Asymptotics for the Toda Lattice and Its Hamiltonian Perturbations

This dissertation is devoted to the study of long-time asymptotics for solutions of the Toda lattice and its Hamiltonian perturbations. First, we present the results of an analytical and numerical study of the long-time behavior for certain Fermi-Pasta-Ulam lattices viewed as perturbations of the completely integrable Toda lattice. Our main tools are the direct and inverse scattering transforms for doubly-infinite Jacobi matrices, which are well-known to linearize the Toda flow. We focus in particular on the evolution of the associated scattering data under the perturbed dynamics in comparison to the evolution under the Toda dynamics. Second, we solve the initial value problem for the completely integrable Toda lattice equations numerically by implementing the inverse scattering transform. Our method is based on the nonlinear steepest descent techniques for the Riemann-Hilbert formulation of the inverse scattering transform for Jacobi matrices; and it captures the solutions accurately for arbitrary spatial and temporal parameters without using time-stepping methods. As part of this effort, we introduce contour deformations for the collisionless shock region by constructing the so-called g-function.