On the evolution of scattering data under perturbations of the Toda lattice

TL;DR

This paper investigates how scattering data evolve over time in perturbed Toda lattices, revealing eigenvalue shifts and emergence of new eigenvalues, which influence the formation of solitary waves.

Contribution

It provides an analytical and numerical analysis of scattering data evolution in perturbed Toda lattices, highlighting eigenvalue perturbations and the emergence of new eigenvalues.

Findings

Eigenvalues in scattering data shift to new values under perturbations.

New eigenvalues emerge from the continuous spectrum during evolution.

Solitary waves correspond to perturbed eigenvalues.

Abstract

We present the results of an analytical and numerical study of the long-time behavior for certain Fermi-Pasta-Ulam (FPU) 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 vs. the unperturbed equations. We find that the eigenvalues present initially in the scattering data converge to new, slightly perturbed eigenvalues under the perturbed dynamics of the lattice equation. To these eigenvalues correspond solitary waves that emerge from the solitons in the initial data. We also find that new eigenvalues emerge from the continuous spectrum as the lattice system is let to evolve under the perturbed dynamics.