On the Liouville Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type arising in the Bogolyubov-Whitham averaging theory

Tsarëv, S. P.

doi:10.1070/rm1984v039n06abeh003195

Cited by 50 publications

(94 citation statements)

References 4 publications

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“…Also, it is even not clear whether N-component system (36), (38) is linearly degenerate. It might seem that this system is simple enough for one to be able to verify these properties by a direct computation, using general definitions of linear degeneracy and integrability for hydrodynamic type systems [35], [39,40] (also see the next Section). To our surprise, even the simplest non-trivial case N = 3 turned out to be complicated enough to require computer algebra to get the confirmation of our hypothesis.…”

Section: 'Cold-gas' Hydrodynamic Reductionsmentioning

confidence: 99%

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Kinetic Equation for a Soliton Gas and Its Hydrodynamic Reductions

et al. 2010

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We introduce and study a new class of kinetic equations, which arise in the description of nonequilibrium macroscopic dynamics of soliton gases with elastic collisions between solitons. These equations represent nonlinear integro-differential systems and have a novel structure, which we investigate by studying in detail the class of Ncomponent 'cold-gas' hydrodynamic reductions. We prove that these reductions represent integrable linearly degenerate hydrodynamic type systems for arbitrary N which is a strong evidence in favour of integrability of the full kinetic equation. We derive compact explicit representations for the Riemann invariants and characteristic velocities of the hydrodynamic reductions in terms of the 'cold-gas' component densities and construct a number of exact solutions having special properties (quasi-periodic, self-similar). Hydrodynamic symmetries are then derived and investigated. The obtained results shed the light on the structure of a continuum limit for a large class of integrable systems of hydrodynamic type and are also relevant to the description of turbulent motion in conservative compressible flows.

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Section: 'Cold-gas' Hydrodynamic Reductionsmentioning

confidence: 99%

“…is called semi-Hamiltonian (see [39,40]) if it (i) has N mutually distinct eigenvalues λ = λ i (U) defined by the equation…”

Section: 'Cold-gas' Hydrodynamic Reductionsmentioning

confidence: 99%

Kinetic Equation for a Soliton Gas and Its Hydrodynamic Reductions

et al. 2010

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“…According to the results of Tsarev [27], [28], equations (25) imply that the systems of hydrodynamic type…”

Section: Deformations Of N-orthogonal Coordinate Systems Inducing Resmentioning

confidence: 99%

“…associated with the diagonal metric (24) of constant curvature 1 (these operators are compatible by virtue of (27)). According to the results of Tsarev [27], [28], equations (25) imply that the systems of hydrodynamic type…”

Section: Deformations Of N-orthogonal Coordinate Systems Inducing Resmentioning

confidence: 99%

Compatible Poisson brackets of hydrodynamic type

Ferapontov

2001

J. Phys. A: Math. Gen.

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Some general properties of compatible Poisson brackets of hydrodynamic type are discussed, in particular:-an invariant differential-geometric criterion of the compatibility based on the Nijenhuis tensor; -the Lax pair with a spectral parameter governing compatible Poisson brackets in the diagonalizable case; -the connection of this problem with the class of surfaces in Euclidean space which possess nontrivial deformations preserving the Weingarten operator.

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“…Novikov was proved by S.P. Tsarev ( [37,38]) who suggested a method for solving of diagonal Hamiltonian systems…”

Section: Introductionmentioning

confidence: 99%

On the canonical forms of the multi-dimensional averaged Poisson brackets

Maltsev

2016

Journal of Mathematical Physics

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We consider here special Poisson brackets given by the "averaging" of local multi-dimensional Poisson brackets in the Whitham method. For the brackets of this kind it is natural to ask about their canonical forms, which can be obtained after transformations preserving the "physical meaning" of the field variables. We show here that the averaged bracket can always be written in the canonical form after a transformation of "Hydrodynamic Type" in the case of absence of annihilators of initial bracket. However, in general case the situation is more complicated. As we show here, in more general case the averaged bracket can be transformed to a "pseudocanonical" form under some special ("physical") requirements on the initial bracket.

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On the Liouville Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type arising in the Bogolyubov-Whitham averaging theory

Cited by 50 publications

References 4 publications

Kinetic Equation for a Soliton Gas and Its Hydrodynamic Reductions

Kinetic Equation for a Soliton Gas and Its Hydrodynamic Reductions

Compatible Poisson brackets of hydrodynamic type

On the canonical forms of the multi-dimensional averaged Poisson brackets

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