Isospectral integrability analysis of dynamical systems on discrete manifolds

Isospectral Integrability Analysis of Dynamical Systems on Discrete Manifolds

It is shown how functional-analytic gradient-holonomic structures can be used for an isospectral integrability analysis of nonlinear dynamical systems on discrete manifolds. The approach developed is applied to obtain detailed proofs of the integrability of the discrete nonlinear Schrödinger, Ragnisco–Tu and Riemann–Burgers dynamical systems.

On Integrability and Chaos in Discrete Systems

The scalar nonlinear Schrödinger (NLS) equation and a suitable discretization are well known integrable systems which exhibit the phenomena of “effective” chaos. Vector generalizations of both the continuous and discrete system are discussed. Some attention is directed upon the issue of the integrability of a discrete version of the vector NLS equation.

Towers of isospectral manifolds

Given two isospectral not isometric manifolds, we construct a new couple of such manifolds as the total spaces of two Riemannian submersions with totally geodesic fibers isometric to the given ones and of basis any other given manifold. By iteration, we obtain families of isospectral not isometric manifolds.

On the Construction of Isospectral Riemannian Manifolds

Isospectral locally symmetric manifolds

In this article we construct closed, isospectral, non-isometric locally symmetric manifolds. We have three main results. First, we construct arbitrarily large sets of closed, isospectral, non-isometric manifolds. Second, we show the growth of size these sets of isospectral manifolds as a function of volume is super-polynomial. Finally, we construct pairs of infinite towers of finite covers of a...