Hamiltonian structure of real Monge-Ampère equations

Details Hamiltonian structure of real Monge-Ampère equations Hamiltonian structure of real Monge-Ampère equations

1998-12-21

arxiv.org/abs/solv-int/9812023v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

published in J. Phys. A 29 (1996) 3257

Scientific paper

10.1088/0305-4470/29/12/029

The real homogeneous Monge-Amp\`{e}re equation in one space and one time dimensions admits infinitely many Hamiltonian operators and is completely integrable by Magri's theorem. This remarkable property holds in arbitrary number of dimensions as well, so that among all integrable nonlinear evolution equations the real homogeneous Monge-Amp\`{e}re equation is distinguished as one that retains its character as an integrable system in multi-dimensions. This property can be traced back to the appearance of arbitrary functions in the Lagrangian formulation of the real homogeneous Monge-Amp\`ere equation which is degenerate and requires use of Dirac's theory of constraints for its Hamiltonian formulation. As in the case of most completely integrable systems the constraints are second class and Dirac brackets directly yield the Hamiltonian operators. The simplest Hamiltonian operator results in the Kac-Moody algebra of vector fields and functions on the unit circle.

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