On a class of three-dimensional integrable Lagrangians

We characterize non-degenerate Lagrangians of the form $\int f(u_x, u_y, u_t) dx dy dt$ 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. The integrability conditions constitute an over-determined system of fourth order PDEs for the Lagrangian density $f$, which is in involution and possess interesting differential-geometric properties. The moduli space of integrable Lagrangians, factorized by the action of a natural equivalence group, is three-dimensional. Familiar examples include the dispersionless Kadomtsev-Petviashvili (dKP) and the Boyer-Finley Lagrangians, $f=u_x^3/3+u_y^2-u_xu_t$ and $f=u_x^2+u_y^2-2e^{u_t}$, respectively. A complete description of integrable cubic and quartic Lagrangians is obtained. Up to the equivalence transformations, the list of integrable cubic Lagrangians reduces to three examples, $f=u_xu_yu_t, f=u_x^2u_y+u_yu_t, and f=u_x^3/3+u_y^2-u_xu_t ({\rm dKP}).$ There exists a unique integrable quartic Lagrangian, $f=u_x^4+2u_x^2u_t-u_xu_y-u_t^2.$ We conjecture that these examples exhaust the list of integrable polynomial Lagrangians which are essentially three-dimensional (it was verified that there exist no polynomial integrable Lagrangians of degree five). We prove that the Euler-Lagrange equations are integrable by the method of hydrodynamic reductions if and only if they possess a scalar pseudopotential playing the role of a dispersionless `Lax pair'. MSC: 35Q58, 37K05, 37K10, 37K25. Keywords: Multi-dimensional Dispersionless Integrable Systems, Hydrodynamic Reductions, Pseudopotentials.Comment: 12 pages A4 format, standard Latex 2e. In the file progs.tar we include the programs needed for computations performed in the paper. Read 1-README first. The new version includes two new section

Staeckel systems generating coupled KdV hierarchies and their finite-gap and rational solutions

We show how to generate coupled KdV hierarchies from Staeckel separable systems of Benenti type. We further show that solutions of these Staeckel systems generate a large class of finite-gap and rational solutions of cKdV hierarchies. Most of these solutions are new.Comment: 15 page

From St\"{a}ckel systems to integrable hierarchies of PDE's: Benenti class of separation relations

We propose a general scheme of constructing of soliton hierarchies from finite dimensional St\"{a}ckel systems and related separation relations. In particular, we concentrate on the simplest class of separation relations, called Benenti class, i.e. certain St\"{a}ckel systems with quadratic in momenta integrals of motion.Comment: 24 page