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On a class of three-dimensional integrable Lagrangians. (English) Zbl 1108.37045

Summary: We characterize nondegenerate Lagrangians of the form \[ \int f(u_x, u_y, u_t)\text{d}x\text{d}y\text{d}t \] 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 overdetermined system of fourth-order PDEs for the Lagrangian density \(f\), which is in involution and possesses 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,
\[ f=u_xu_yu_t,\quad f=u^2_xu_y + u_yu_t\;\text{and}\;f= u_{x}^{3}/3+ u_{y}^{2} - u_{x} u_{t}\;\text{(dKP)}. \]
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:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
35Q58 Other completely integrable PDE (MSC2000)

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References:

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