On the compatible weakly nonlocal Poisson brackets of hydrodynamic type
We consider the pairs of general weakly nonlocal Poisson brackets of hydrodynamic type (Ferapontov brackets) and the corresponding integrable hierarchies. We show that, under the requirement of the nondegeneracy of the corresponding first pseudo-Riemannian metric g(0) νμ and also some nondegeneracy...
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| Format: | Article |
| Language: | English |
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Wiley
2002-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171202202069 |
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| _version_ | 1832561962246995968 |
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| author | Andrei Ya. Maltsev |
| author_facet | Andrei Ya. Maltsev |
| author_sort | Andrei Ya. Maltsev |
| collection | DOAJ |
| description | We consider the pairs of general weakly nonlocal Poisson brackets
of hydrodynamic type (Ferapontov brackets) and the corresponding integrable hierarchies. We show that, under the requirement of the nondegeneracy of the corresponding first pseudo-Riemannian metric g(0) νμ and also some nondegeneracy requirement for the nonlocal part, it is possible to introduce a canonical set of integrable hierarchies based on the Casimirs, momentum functional and some canonical Hamiltonian functions. We prove also that all the higher positive Hamiltonian operators and the negative symplectic forms have the weakly nonlocal form in this case. The same result is also true for negative Hamiltonian operators and positive symplectic structures in the case when both pseudo-Riemannian metrics g(0) νμ and g(1) νμ are nondegenerate. |
| format | Article |
| id | doaj-art-1decfee2b05a4bc2a885580a840dac3f |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2002-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-1decfee2b05a4bc2a885580a840dac3f2025-02-03T01:23:39ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252002-01-01321058761410.1155/S0161171202202069On the compatible weakly nonlocal Poisson brackets of hydrodynamic typeAndrei Ya. Maltsev0L. D. Landau Institute for Theoretical Physics, Kosygina Street 2, Moscow 117940, RussiaWe consider the pairs of general weakly nonlocal Poisson brackets of hydrodynamic type (Ferapontov brackets) and the corresponding integrable hierarchies. We show that, under the requirement of the nondegeneracy of the corresponding first pseudo-Riemannian metric g(0) νμ and also some nondegeneracy requirement for the nonlocal part, it is possible to introduce a canonical set of integrable hierarchies based on the Casimirs, momentum functional and some canonical Hamiltonian functions. We prove also that all the higher positive Hamiltonian operators and the negative symplectic forms have the weakly nonlocal form in this case. The same result is also true for negative Hamiltonian operators and positive symplectic structures in the case when both pseudo-Riemannian metrics g(0) νμ and g(1) νμ are nondegenerate.http://dx.doi.org/10.1155/S0161171202202069 |
| spellingShingle | Andrei Ya. Maltsev On the compatible weakly nonlocal Poisson brackets of hydrodynamic type International Journal of Mathematics and Mathematical Sciences |
| title | On the compatible weakly nonlocal Poisson brackets of hydrodynamic type |
| title_full | On the compatible weakly nonlocal Poisson brackets of hydrodynamic type |
| title_fullStr | On the compatible weakly nonlocal Poisson brackets of hydrodynamic type |
| title_full_unstemmed | On the compatible weakly nonlocal Poisson brackets of hydrodynamic type |
| title_short | On the compatible weakly nonlocal Poisson brackets of hydrodynamic type |
| title_sort | on the compatible weakly nonlocal poisson brackets of hydrodynamic type |
| url | http://dx.doi.org/10.1155/S0161171202202069 |
| work_keys_str_mv | AT andreiyamaltsev onthecompatibleweaklynonlocalpoissonbracketsofhydrodynamictype |