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arXiv · 1906.04535

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

Matteo Petrera·Mats Vermeeren
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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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Matteo Petrera, Mats Vermeeren. 2019-06-11. Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs. https://doi.org/10.1007/s40879-020-00436-7

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