On the structure of Hamiltonian operators in the field theory

Astashov, A. M.; Vinogradov, A. M.

doi:10.1016/0393-0440(86)90022-7

Cited by 25 publications

(29 citation statements)

References 6 publications

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“…Since the generalized function on the right-hand side of (32) has finite order and support on the diagonal χ = y = z, (32) is equivalent to a finite system of quadratic relations involving the coefficients B{-and their derivatives with respect to χ and u s< ". We shall not give here the explicit form of this system (see below for systems of hydrodynamic type and also [11], [ 12], [65]). Let us observe that a sufficient collection of relations is obtained if the Jacobi identity is verified only for linear functionals of the form…”

Section: Chapter I Hamiltonian Theory Of Systems Of Hydrodynamic Typementioning

confidence: 99%

Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory

Dubrovin¹,

Novikov²

1989

Russ. Math. Surv.

405

601

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CONTENTSIntroduction 35 Chapter I. Hamiltonian theory of systems of hydrodynamic type 45 § 1. General properties of Poisson brackets 45 §2. Hamiltonian formalism of systems of hydrodynamic type and 55 Riemannian geometry §3. Generalizations: differential-geometric Poisson brackets of higher orders, 66 differential-geometric Poisson brackets on a lattice, and the Yang-Baxter equation §4. Riemann invariants and the Hamiltonian formalism of diagonal systems 71 of hydrodynamic type. Novikov's conjecture. Tsarev's theorem. The generalized hodograph method Chapter II. Equations of hydrodynamics of soliton lattices 78 §5. The Bogolyubov-Whitham averaging method for field-theoretic systems 78 and soliton lattices. The results of Whitham and Hayes for Lagrangian systems §6. The Whitham equations of hydrodynamics of weakly deformed soliton 84 lattices for Hamiltonian field-theoretic systems. The principle of conservation of the Hamiltonian structure under averaging §7. Modulations of soliton lattices of completely integrable evolutionary 96 systems. Krichever's method. The analytic solution of the Gurevich-Pitaevskii problem on the dispersive analogue of a shock wave. §8. Evolution of the oscillatory zone in the KdV theory. Multi-valued 105 functions in the hydrodynamics of soliton lattices. Numerical studies §9. Influence of small viscosity on the evolution of the oscillatory zone 113 References 118 -Q -. Ρ l/\ u $ I \ " u which resemble the equations of hydrodynamics of a compressible fluid. These are Riemann type equations or "systems of hydrodynamic type" in our terminology. We shall call equations (4) the Whitham equations, or equations of hydrodynamics of weakly deformed soliton lattices. Sometimes they are called equations of slow modulation of parameters. Later, these problems were studied by Luke [89], Maslov [43], Ablowitz and Benney [63], Hayes [80], Whitham [58], and Gurevich and Pitaevskii [14], [15]. The aspects discussed include the sufficiency of equations (4) for the construction of asymptotic solutions in the case m = 1, their explicit form in some important particular cases, and generalizations to the multiphase case m > 1 (although at that time finite-zone solutions were not yet known and the discussion was not sufficiently explicit). Applications to physical problems in dispersive hydrodynamics were found in [14], [15]. Equations (4) for non-degenerate Lagrangian systems (all in the case m = 1) were derived in [93].The theory of multi-phase systems began to develop rapidly only after the formulation in 1974-75 of the above-mentioned theory of finite-zone (algebraic-geometric) solutions of integrable soliton systems, which actually made it possible to consider multi-phase analogues of Whitham's equations (4) in the case m > 1 (see the papers [ 19], [73]). In [73], [75], [76] Flaschka, McLaughlin and others derived equations (4) from the theory of Riemann surfaces used in the construction of finite-zone solutions and obtained a number of useful generalizations of Whitham's results for the case m > 1 whic...

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Section: Chapter I Hamiltonian Theory Of Systems Of Hydrodynamic Typementioning

confidence: 99%

Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory

Dubrovin¹,

Novikov²

1989

Russ. Math. Surv.

405

601

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“…Normal forms for Hamiltonian operators of order ≤5 and a ''variational'' analog of the Darboux Lemma were presented in [99,100]. In [101], one can find normal forms for operators of order ≤ 11 and some classification results for operators of higher order.…”

Section: Remark 12mentioning

confidence: 98%

“…Consider an example. (99) be the Burgers equation (its Lie algebra of symmetries was fully described in [70]). Direct computations show that (99) does not possess symmetries of the form ϕ = ϕ(x, t, u), but if one extends the setting by a new (nonlocal) variable w such that…”

Section: Nonlocal Symmetriesmentioning

confidence: 99%

Geometry of jet spaces and integrable systems

Krasil’shchik

Verbovetsky

2011

Journal of Geometry and Physics

106

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An overview of some recent results on the geometry of partial differential equations in application to integrable systems is given. Lagrangian and Hamiltonian formalism both in the free case (on the space of infinite jets) and with constraints (on a PDE) are discussed. Analogs of tangent and cotangent bundles to a differential equation are introduced and the variational Schouten bracket is defined. General theoretical constructions are illustrated by a series of examples.Comment: 54 pages; v2-v6 : minor correction

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“…[5,6,8,13] and references therein. It is thus no wonder that the study and, in particular, the classification of Hamiltonian operators is a subject of ongoing interest, see for instance [1,3,7,6,11,12] Although it is well known [15,16] for these systems [16].…”

Section: Introductionmentioning

confidence: 99%

The Darboux coordinates for a new family of Hamiltonian operators and linearization of associated evolution equations

Vodová¹

2011

Nonlinearity

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u is the dependent variable and D is the total derivative with respect to the independent variable. We present a differential substitution that reduces any linear combination of these operators to an operator with constant coefficients and linearizes any evolution equation which is bi-Hamiltonian with respect to a pair of any nontrivial linear combinations of the operators H (N,0) . We also give the Darboux coordinates for H (N,0) for any odd N 3.

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On the structure of Hamiltonian operators in the field theory

Cited by 25 publications

References 6 publications

Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory

Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory

Geometry of jet spaces and integrable systems

The Darboux coordinates for a new family of Hamiltonian operators and linearization of associated evolution equations

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