Integrable Lagrangians and modular forms

Abstract

We investigate non-degenerate Lagrangians of the form $$\int f(u_x,u_y,u_t)dxdydt$$ such that the corresponding Euler-Lagrange equations $(f_{u_x}{)}_x+(f_{u_y}{)}_y+(f_{u_t}{)}_t=0$ 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.

17 pages, latex

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