Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Relativistic Toda-type systems

• Published in: Journal of physics. A, Mathematical and theoretical Vol. 48; no. 8; pp. 85203 - 28
• Main Authors: Boll, Raphael, Petrera, Matteo, Suris, Yuri B
• Format: Journal Article
• Language: English
• Published: IOP Publishing 27.02.2015
• Subjects: Density; Derivatives; Euler-Lagrange equations; integrable systems; Lagrangian mechanics; Lattices; Mathematical analysis; relativistic Toda systems; Spectra; Transformations; Transformations (mathematics); Two dimensional
• DOI: 10.1088/1751-8113/48/8/085203
• ISSN: 1751-8113

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 (2D) 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 2D generalization of the spectrality property known for families of Bäcklund transformations. This result produces a family of local conservations laws for 2D 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.