Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs

Abstract

We investigate the relation between pluri-Lagrangian hierarchies of $2$-dimensional partial differential equations and their variational symmetries. The aim is to generalize to the case of partial differential equations the recent findings in [Petrera, Suris. J. Nonlinear Math. Phys. 24:sup1, 121--145 (2017)] for ordinary differential equations. We consider hierarchies of $2$-dimensional Lagrangian PDEs (many of which have a natural $(1+1)$-dimensional space-time interpretation) and show that if the flow of each PDE is a variational symmetry of all others, then there exists a pluri-Lagrangian 2-form for the hierarchy. The corresponding multi-time Euler-Lagrange equations coincide with the original system supplied with commuting evolutionary flows induced by the variational symmetries.

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Cited by in corpus (11)

• Variational symmetries and Lagrangian multiforms
• Lagrangian multiforms for Kadomtsev-Petviashvili (KP) and the Gelfand-Dickey 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
• Hamiltonian structures for integrable hierarchies of Lagrangian PDEs
• Semi-discrete Lagrangian 2-forms and the Toda hierarchy
• On the geometry of Lagrangian one-forms
• Lagrangian multiforms and dispersionless integrable systems