2007/07/23 by E. V. Ferapontov, Ferapontov, E. V., A. V. Odesskiĭ +2
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Algebraic and Geometric Analysis #Differential Geometry (math.DG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #High Energy Physics - Theory (hep-th) #Nonlinear Waves and Solitons #Number Theory (math.NT) #hep-th #math.AG #math.DG #math.NT #nlin.SI
paper · pdf · doi:10.48550/arxiv.0707.3433
17 pages, latex
openalex publication_date 2007/07/23 · arxiv created 2007/11/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate non-degenerate Lagrangians of the form ∫ f(ux, uy, ut) dx dy dt such that the corresponding Euler-Lagrange equations (fux)x+ (fuy)y+ (fut)t=0 are integrable by the method of hydrodynamic reductions. We demonstrate that the integrability conditions, which constitute an involutive over-determined system of fourth order PDEs for the Lagrangian density f, are invariant under a 20-parameter group of Lie-point symmetries whose action on the moduli space of integrable Lagrangians has an open orbit. The density of the `master-Lagrangian' corresponding to this orbit is shown to be a modular form in three variables defined on a complex hyperbolic ball. We demonstrate how the knowledge of the symmetry group allows one to linearise the integrability conditions.