Darboux's Theorem, Lie series and the standardization of the Salerno and Ablowitz-Ladik models

TL;DR

This paper revisits Darboux's Theorem to develop a constructive scheme for Hamiltonian models with non-standard symplectic structures, applying it to the Salerno and Ablowitz-Ladik models, and deriving normal forms and dynamic estimates.

Contribution

It introduces a general, constructive approach to Darboux's Theorem for non-standard symplectic Hamiltonian models, with applications to Salerno and Ablowitz-Ladik models.

Findings

Normal form of the Salerno and AL models obtained via Lie-series

Dynamic estimates derived from the dNLS Hamiltonian

Method applicable even when explicit transformations are unknown

Abstract

In the framework of nonlinear Hamiltonian lattices, we revisit the proof of Moser-Darboux's Theorem, in order to present a general scheme for its constructive applicability to Hamiltonian models with non-standard symplectic structures. We take as a guiding example the Salerno and Ablowitz-Ladik (AL) models: we justify the form of a well-known change of coordinates which is adapted to the Gauge symmetry, by showing that it comes out in a natural way within the general strategy outlined in the proof. Moreover, the full or truncated Lie-series technique in the extended phase-space is used to transform the Salerno model, at leading orders in the Darboux coordinates: thus the dNLS Hamiltonian turns out to be a normal form of the Salerno and AL models; as a byproduct we also get estimates of the dynamics of these models by means of dNLS one. We also stress that, once it is cast into the perturbative approach, the method allows to deal with the cases where the explicit trasformation is not known, or even worse it is not writable in terms of elementary functions.