Narrow Results

This paper is mainly a review of the multi-Hamiltonian nature of Toda and generalized Toda lattices corresponding to the classical simple Lie groups but it includes also some new results. The areas investigated include master symmetries, recursion operators, higher Poisson brackets, invariants and group symmetries for the systems. In addition to the positive hierarchy we also consider the negative hierarchy which is crucial in establishing the bi-Hamiltonian structure for each particular simple Lie group. Finally, we include some results on point and Noether symmetries and an interesting connection with the exponents of simple Lie groups. The case of exceptional simple Lie groups is still an open problem.

The purpose of this article is to give a streamlined and self-contained treatment of the long-time asymptotics of the Toda lattice for decaying initial data in the soliton and in the similarity region via the method of nonlinear steepest descent.

Using the tangent bundle conjecture for the Lyapunov exponents, which considers these quantities as a generalization of eigenvalues to nonlinear systems, this work tries to identify zero exponents of a given system, if it exhibits one, via algebraic techniques. The existence of zero Lyapunov exponents for the Toda, Hénon-Heiles and two decoupled simple harmonic oscillator systems are shown.

A direct method for constructing integrable expanding models for lattice soliton hierarchies is developed through enlarging associated Lax pairs. As illustrated by examples, the integrable expanding models for Toda and relativistic Toda lattice hierarchies are investigated.

The numerical behavior of the truncated 3-particle Toda lattice (3pTL) is reviewed and studied in more detail (than in previous papers) and at higher energies (at odd-orders n ≤ 9). We further extended our study to higher truncations at odd-orders, n = 2k + 1, k = 1, …, 9. We have located the majority of the families of periodic orbits along with their main bifurcations. By using: (a) the method of Poincaré surface of section, (b) the maximum Lyapunov characteristic number and (c) the ratio of the families of ordered periodic orbits, we studied the topology of the nine odd-order Hamiltonians with respect to their order of truncation.

We complete the study of the numerical behavior of the truncated 3-particle Toda lattice (3pTL) with even truncations at orders n = 2k, k = 2, …, 10. We use (as in Part I): (a) the method of Poincaré surface of section, (b) the maximum Lyapunov characteristic number and (c) the ratio of the families of ordered periodic orbits. We derived some similarities and quite many differences between the odd and even order expansions.

We study the three-term recurrence coefficients ${\beta }_n,{\gamma }_n,$ of polynomial sequences orthogonal with respect to a perturbed linear functional depending on a variable $z.$ We obtain power series expansions in $z,$ and asymptotic expansions as $n\to \infty .$ We use our results to settle some conjectures proposed by Walter Van Assche and collaborators.

We study the generalized ultradiscrete periodic Toda lattice which has tropical spectral curve. We introduce a tropical analogue of Fay's trisecant identity, and apply it to construct a general solution to .