Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs (Q2040981)

The authors use a novel property to recognize solutions of a pluri-Lagrangian problem. The aim is to investigate the relation between pluri-Lagrangian hierarchies of \(2\)-dimensional partial differential equations and their variational symmetries. They generalize the theory from ordinary differential equations, developed in [\textit{M. Petrera} and \textit{Y. B. Suris}, J. Nonlinear Math. Phys. 24, No. Suppl. 1, 121--145 (2017; Zbl 1421.70031)], to partial differential equations. They 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. This paper is organized as follows. Section 1 is an introduction to the subject and summarizes the main results. In Section 2 the authors give a short overview of Lagrangian field theory, recalling some classical notions and definitions. In particular they provide a formulation of the celebrated Noether theorem, which establishes the relation between conservation laws and variational symmetries. In Section 3 they review the notion of continuous \(2\)-dimensional pluri-Lagrangian systems. Section 4 is devoted to new results. Here the authors prove that from a family of variational symmetries one can construct a pluri-Lagrangian structure. In Section 5 they discuss concrete examples (i.e., the potential KdV hierarchy, the nonlinear Schrödinger hierarchy, the Sine-Gordon equation and the modified KdV hierarchy) which illustrate the theoretical results obtained in Section 4. Finally, Section 6 sums up the paper with some conclusions.