On a class of three-dimensional integrable Lagrangians

Abstract

We characterize 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. 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, $f=u_x^3/3+u_y^2-u_x{u}_t$ 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, $f=u_x{u}_y{u}_t,f=u_x^2{u}_y+u_y{u}_t,andf=u_x^3/3+u_y^2-u_x{u}_t(\mathrm{dKP}).$ There exists a unique integrable quartic Lagrangian, $f=u_x^4+2u_x^2{u}_t-u_x{u}_y-u_t^2.$ 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.

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 sections