Classical R-matrices on Poisson algebras and related dispersionless systems

Błaszak, Maciej

doi:10.1016/s0375-9601(02)00421-8

Cited by 76 publications

(77 citation statements)

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“…Another deformation scheme, based on the R-matrix approach, was proposed in [3], see also [34,35]. The specialization of the chain (4) corresponding to a = 0, b = −1, u…”

Section: Introductionmentioning

confidence: 99%

Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor

2007

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The integrability of an m-component system of hydrodynamic type, u t = V (u)u x , by the generalized hodograph method requires the diagonalizability of the m × m matrix V (u). This condition is known to be equivalent to the vanishing of the corresponding Haantjes tensor. We generalize this approach to hydrodynamic chains -infinite-component systems of hydrodynamic type for which the ∞×∞ matrix V (u) is 'sufficiently sparse'. For such systems the Haantjes tensor is well-defined, and the calculation of its components involves finite summations only. We illustrate our approach by classifying broad classes of conservative and Hamiltonian hydrodynamic chains with the zero Haantjes tensor. We prove that the vanishing of the Haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi-Hamiltonian hydrodynamic reductions, thus providing an easy-to-verify necessary condition for the integrability.MSC: 35L40, 35L65, 37K10.

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“…Another deformation scheme, based on the R-matrix approach, was proposed in [3], see also [34,35]. The specialization of the chain (4) corresponding to a = 0, b = −1, u…”

Section: Introductionmentioning

confidence: 99%

Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor

2007

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“…However, Lax function being infinite formal Laurent series leads to the construction of dispersionless infinite-field Benney moment like equations. Such systems for Laurent series at ∞ have been considered earlier in [18]. The original Benney moment equation can be obtained for k = r = 0.…”

Section: Commentsmentioning

confidence: 99%

Meromorphic Lax representations of (1+1)-dimensional multi-Hamiltonian dispersionless systems

Szablikowski

Błaszak

2006

Journal of Mathematical Physics

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Rational Lax hierarchies introduced by Krichever are generalized. A systematic construction of infinite multi-Hamiltonian hierarchies and related conserved quantities is presented. The method is based on the classical R-matrix approach applied to Poisson algebras. A proof, that Poisson operators constructed near different points of Laurent expansion of Lax functions are equal, is given. All results are illustrated by several examples.

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“…(ab, c) A = (a, bc) A , ∀a, b, c ∈ A. Assume R ∈ End(A) is a classical Rmatrix, then for each integer n −1, the formula 11) where H, F are smooth functions on A, defines a Poisson structure on A. Moreover, all {·, ·} n are compatible.…”

Section: Theorem 23 [9]mentioning

confidence: 99%