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Download file# On a class of three-dimensional integrable Lagrangians

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posted on 22.07.2005, 08:58 by Evgeny FerapontovEvgeny Ferapontov, Karima KhusnutdinovaKarima Khusnutdinova, S.P. TsarevWe 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’.

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