On a Lagrangian reduction and a deformation of completely integrable systems
Exactly Solvable and Integrable Systems
2016-05-25 v4 Mathematical Physics
math.MP
Abstract
We develop a theory of Lagrangian reduction on loop groups for completely integrable systems after having exchanged the role of the space and time variables in the multi-time interpretation of integrable hierarchies. We then insert the Sobolev norm in the Lagrangian and derive a deformation of the corresponding hierarchies. The integrability of the deformed equations is altered and a notion of weak integrability is introduced. We implement this scheme in the AKNS and SO(3) hierarchies and obtain known and new equations. Among them we found two important equations, the Camassa-Holm equation, viewed as a deformation of the KdV equation, and a deformation of the NLS equation.
Keywords
Cite
@article{arxiv.1501.02745,
title = {On a Lagrangian reduction and a deformation of completely integrable systems},
author = {Alexis Arnaudon},
journal= {arXiv preprint arXiv:1501.02745},
year = {2016}
}