On a Lagrangian Reduction and a Deformation of Completely Integrable Systems

2015/01/31 by Alexis Arnaudon
Mathematics · Physics and Astronomy · #Deformation (meteorology) #Homotopy and Cohomology in Algebraic Topology #Integrable system #Korteweg–de Vries equation #Lagrangian #Nonlinear Waves and Solitons #Norm (philosophy) #Quantum Mechanics and Non-Hermitian Physics #Reduction (mathematics) #Sobolev space #math-ph #math.MP #nlin.SI

paper · pdf · doi:10.1007/s00332-016-9300-2

arxiv created 2016/04/05 · openalex publication_date 2016/04/21 · arxiv updated 2016/05/25 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

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 H¹ 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.

Citations

• The Inverse Scattering Transform‐Fourier Analysis for Nonlinear Problems
• Inverse Scattering Transform for the Camassa-Holm equation
• The Coupled Modified Korteweg-de Vries Equations
• The Camassa-Holm Equation: A Loop Group Approach
• On a novel integrable generalization of the nonlinear Schrödinger equation
• The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations