dorsal/arxiv
View SchemaHamiltonian structure of real Monge-Amp\`ere equations
| Authors | Y. Nutku |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9812023 |
| URL | https://arxiv.org/abs/solv-int/9812023 |
| DOI | 10.1088/0305-4470/29/12/029 |
Abstract
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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"abstract": "The real homogeneous Monge-Amp\\`{e}re equation in one space and one time\ndimensions admits infinitely many Hamiltonian operators and is completely\nintegrable by Magri\u0027s theorem. This remarkable property holds in arbitrary\nnumber of dimensions as well, so that among all integrable nonlinear evolution\nequations the real homogeneous Monge-Amp\\`{e}re equation is distinguished as\none that retains its character as an integrable system in multi-dimensions.\nThis property can be traced back to the appearance of arbitrary functions in\nthe Lagrangian formulation of the real homogeneous Monge-Amp\\`ere equation\nwhich is degenerate and requires use of Dirac\u0027s theory of constraints for its\nHamiltonian formulation. As in the case of most completely integrable systems\nthe constraints are second class and Dirac brackets directly yield the\nHamiltonian operators. The simplest Hamiltonian operator results in the\nKac-Moody algebra of vector fields and functions on the unit circle.",
"arxiv_id": "solv-int/9812023",
"authors": [
"Y. Nutku"
],
"categories": [
"solv-int",
"nlin.SI"
],
"doi": "10.1088/0305-4470/29/12/029",
"title": "Hamiltonian structure of real Monge-Amp\\`ere equations",
"url": "https://arxiv.org/abs/solv-int/9812023"
},
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