On a Lagrangian reduction and a deformation of completely integrable systems

TL;DR

This paper develops a Lagrangian reduction framework for integrable systems on loop groups, introduces a Sobolev norm deformation, and explores the resulting weakly integrable hierarchies, including new and known equations like Camassa-Holm and deformed NLS.

Contribution

It presents a novel Lagrangian reduction approach on loop groups with Sobolev norm deformation, leading to weak integrability and new equations in integrable hierarchies.

Findings

Derived a deformation of integrable hierarchies using Sobolev $H^1$ norm

Identified weak integrability in deformed equations

Obtained new equations including a Camassa-Holm type and deformed NLS

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 $H^1$ 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.