Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Relativistic Toda-type systems
arXiv:1408.2405 · doi:10.1088/1751-8113/48/8/085203
Abstract
We establish the pluri-Lagrangian structure for families of Bäcklund transformations of relativistic Toda-type systems. The key idea is a novel embedding of these discrete-time (one-dimensional) systems into certain two-dimensional pluri-Lagrangian lattice systems. This embedding allows us to identify the corner equations (which are the main building blocks of the multi-time Euler-Lagrange equations) with local superposition formulae for Bäcklund transformations. These superposition formulae, in turn, are key ingredients necessary to understand and to prove commutativity of the multi-valued Bäcklund transformations. Furthermore, we discover a two-dimensional generalization of the spectrality property known for families of Bäcklund transformations. This result produces a family of local conservations laws for two-dimensional pluri-Lagrangian lattice systems, with densities being derivatives of the discrete 2-form with respect to the Bäcklund (spectral) parameter. Thus, a relation of the pluri-Lagrangian structure with more traditional integrability notions is established.
25 pages