On a Lagrangian reduction and a deformation of completely integrable systems

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.

References in corpus (6)

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• Inverse Scattering Transform for the Camassa-Holm equation
• On asymptotically equivalent shallow water wave equations
• On a novel integrable generalization of the nonlinear Schrödinger equation
• A multisymplectic approach to defects in integrable classical field theory
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Cited by in corpus (4)

• Asymptotic Expansions and Solitons of the Camassa-Holm Nonlinear Schrodinger Equation
• Quasi-integrable KdV models, towers of infinite number of anomalous charges and soliton collisions
• $G$-Strands on symmetric spaces
• Solitary Waves of the Two-Dimensional Camassa-Holm--Nonlinear Schrödinger Equation