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On a Class of Three-Dimensional Integrable Lagrangians

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Abstract

We characterize non-degenerate Lagrangians of the form

such that the corresponding Euler-Lagrange equations 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 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 , 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,

There exists a unique integrable quartic Lagrangian,

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'.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK

    E.V. Ferapontov & K.R. Khusnutdinova

  2. Krasnoyarsk State Pedagogical University, 89 Lebedevoi, 660049, Krasnoyrsk, Russia

    S.P. Tsarev

Authors
  1. E.V. Ferapontov
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  2. K.R. Khusnutdinova
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  3. S.P. Tsarev
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Corresponding author

Correspondence to E.V. Ferapontov.

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Communicated by G.W. Gibbons

SPT acknowledges partial financial support from the RFBR grant 04-01-00130.

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Ferapontov, E., Khusnutdinova, K. & Tsarev, S. On a Class of Three-Dimensional Integrable Lagrangians. Commun. Math. Phys. 261, 225–243 (2006). https://doi.org/10.1007/s00220-005-1415-5

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