A Variational Approach to Lax Representations

Abstract

It is shown that the Zakharov-Mihailov (ZM) Lagrangian structure for integrable nonlinear equations derived from a general class of Lax pairs possesses a Lagrangian multiform structure. We show that, as a consequence of this multiform structure, we can formulate a variational principle for the Lax pair itself, a problem that to our knowledge was never previously considered. As an example, we present an integrable $N\times N$ matrix system that contains the AKNS hierarchy, and we exhibit the Lagrangian multiform structure of the scalar AKNS hierarchy by presenting the components corresponding to the first three flows of the hierarchy.

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Cited by in corpus (17)

• On the Zakharov-Mikhailov action: $4$d Chern-Simons origin and covariant Poisson algebra of the Lax connection
• Variational symmetries and Lagrangian multiforms
• Lagrangian multiforms for Kadomtsev-Petviashvili (KP) and the Gelfand-Dickey hierarchy
• Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs
• A connection between the classical r-matrix formalism and covariant Hamiltonian field theory
• Lagrangian 3-form structure for the Darboux system and the KP hierarchy
• Classical Yang-Baxter equation, Lagrangian multiforms and ultralocal integrable hierarchies
• Multiform description of the AKNS hierarchy and classical r-matrix
• Hamiltonian multiform description of an integrable hierarchy
• Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems
• Lagrangian multiforms on Lie groups and non-commuting flows
• Semi-discrete Lagrangian 2-forms and the Toda hierarchy
• Hamiltonian structures for integrable hierarchies of Lagrangian PDEs
• Hamiltonian formulation and aspects of integrability of generalised hydrodynamics
• On the geometry of Lagrangian one-forms
• Lagrangian multiform structure of discrete and semi-discrete KP systems
• Lagrangian Multiform for Cyclotomic Gaudin Models