Hamiltonian structures for systems of hyperbolic conservation laws

Olver, Peter J.; Nutku, Y.

doi:10.1063/1.527909

Cited by 109 publications

(123 citation statements)

References 16 publications

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“…Brunelli and Das (1997) have obtained a Lax representation of the system (1.2) for γ ∈ N * . Olver and Nutku (1988) have exhibited the Hamiltonian structure of a family of hydrodynamic type, including (1.2), but the value γ = −1 appears as a singular case. Statistical investigations have been pursued recently by Passot and Vázquez-Semadeni (1998), with emphasis on the symmetric role of the case γ = 1.…”

Section: The Thomas-mhd Modelmentioning

confidence: 99%

Meromorphy and topology of localized solutions in the Thomas–MHD model

Fournier

Galtier

2001

J. Plasma Phys.

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Abstract. The one-dimensional MHD system first introduced by J.H. Thomas [Phys. Fluids 11, 1245(1968] as a model of the dynamo effect is thoroughly studied in the limit of large magnetic Prandtl number. The focus is on two types of localized solutions involving shocks (antishocks) and hollow (bump) waves. Numerical simulations suggest phenomenological rules concerning their generation, stability and basin of attraction. Their topology, amplitude and thickness are compared favourably with those of the meromorphic travelling waves, which are obtained exactly, and respectively those of asymptotic descriptions involving rational or degenerate elliptic functions. The meromorphy bars the existence of certain configurations, while others are explained by assuming imaginary residues. These explanations are tested using the numerical amplitude and phase of the Fourier transforms as probes of the analyticity properties. Theoretically, the proof of the partial integrability backs up the role ascribed to meromorphy. Practically, predictions are derived for MHD plasmas.

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Section: The Thomas-mhd Modelmentioning

confidence: 99%

Meromorphy and topology of localized solutions in the Thomas–MHD model

Fournier

Galtier

2001

J. Plasma Phys.

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“…which is trivially related to the rst two was obtained earlier [8] . These operators are compatible.…”

Section: The Equation Tt = E X Xxmentioning

confidence: 76%

“…but for a generic , J 2 is a new nontrivial Hamilton Finally [14]; [8] , we have the fourth Hamiltonian operator…”

Section: Gas Dynamics Hierarchymentioning

confidence: 99%

“…The derivation of the second Hamiltonian operator and proof of Jacobi's identities for multi-component equations of hydrodynamic type consist of a generalization of the processes described in detail in refs. [7] and [8]. We shall assume familiarity with these papers and present only the new results unless the generalization is not entirely straight-forward.…”

Section: Introductionmentioning

confidence: 99%

“…The Hamiltonian structure of the equations of gas dynamics [6]; [7]; [8] which in particular include the shallow water equations [9]; [10] , the Poisson [7] and BornInfeld [11] equations were discussed earlier. In this paper we shall conclude the discussion of the Hamiltonian structure of these 2-component systems by presenting two further examples.…”

Section: Introductionmentioning

confidence: 99%

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Multi-Hamiltonian structure of equations of hydrodynamic type

Gümral

Nutku

1990

Journal of Mathematical Physics

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We complete the discussion of the Hamiltonian structure of 2-component equations of hydrodynamic type by presenting the Hamiltonian operators for Euler's equation governing the motion of plane sound waves of nite amplitude and another quasi-linear second order wave equation. There exists a doubly innite family of conserved Hamiltonians for the equations of gas dynamics which degenerate into one, namely the Benney sequence, for shallow water waves. We present further in nite sequences of conserved quantities for these equations. In the case of multi-component equations of hydrodynamic type, we show that Kodama's generalization of the shallow water equations admits bi-Hamiltonian structure.

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On Hamiltonian perturbations of hyperbolic systems of conservation laws I: Quasi‐Triviality of bi‐Hamiltonian perturbations

Dubrovin

Liu

Zhang

2005

Comm Pure Appl Math

127

239

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We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one-dimensional systems of hyperbolic PDEsUnder certain genericity assumptions it is proved that any bi-Hamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives. The main tool is in constructing the so-called quasiMiura transformation of jet coordinates, eliminating an arbitrary deformation of a semisimple bi-Hamiltonian structure of hydrodynamic type (the quasi-triviality theorem). We also describe, following [35], the invariants of such bi-Hamiltonian structures with respect to the group of Miura-type transformations depending polynomially on the derivatives.

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Hamiltonian structures for systems of hyperbolic conservation laws

Cited by 109 publications

References 16 publications

Meromorphy and topology of localized solutions in the Thomas–MHD model

Meromorphy and topology of localized solutions in the Thomas–MHD model

Multi-Hamiltonian structure of equations of hydrodynamic type

On Hamiltonian perturbations of hyperbolic systems of conservation laws I: Quasi‐Triviality of bi‐Hamiltonian perturbations

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