Geodesic
Higher-order dispersive deformations of multidimensional Poisson brackets of hydrodynamic type
Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 2, pp. 214-237

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The theory of multidimensional Poisson vertex algebras provides a completely algebraic formalism for studying the Hamiltonian structure of partial differential equations for any number of dependent and independent variables. We compute the cohomology of the Poisson vertex algebras associated with two-dimensional, two-component Poisson brackets of hydrodynamic type at the third differential degree. This allows obtaining their corresponding Poisson–Lichnerowicz cohomology, which is the main building block of the theory of their deformations. Such a cohomology is trivial neither in the second group, corresponding to the existence of a class of nonequivalent infinitesimal deformations, nor in the third group, corresponding to the obstructions to extending such deformations.
Keywords: Hamiltonian operator
Mots-clés : hydrodynamic Poisson bracket, Poisson vertex algebra, Poisson cohomology. 
@article{TMF_2018_196_2_a2,
     author = {M. Casati},
     title = {Higher-order dispersive deformations of multidimensional {Poisson} brackets of hydrodynamic type},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {214--237},
     publisher = {mathdoc},
     volume = {196},
     number = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2018_196_2_a2/}
}
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M. Casati. Higher-order dispersive deformations of multidimensional Poisson brackets of hydrodynamic type. Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 2, pp. 214-237. http://geodesic.mathdoc.fr/item/TMF_2018_196_2_a2/