On a class of three-dimensional integrable Lagrangians

We characterize non-degenerate Lagrangians of the form Z 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. The integrability conditions constitute an over-determined system of fourth order PDEs for the Lagrangian density f, which is in involution. The moduli space of integrable Lagrangians, factorized by the action of a natural equivalence group, is threedimensional. Familiar examples include the dispersionless Kadomtsev-Petviashvili (dKP) and the Boyer-Finley Lagrangians, f = u3 x/3 + u2 y − uxut and f = u2 x + u2 y − 2eut , 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 = uxuyut, f = u2 xuy + uyut and f = u3 x/3 + u2 y − uxut (dKP). There exists a unique integrable quartic Lagrangian, f = u4 x + 2u2 xut − uxuy − u2t . 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’.