Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 0712.58032
Dubrovin, B.A.; Novikov, S.P.
Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory. (English. Russian original)
[J] Russ. Math. Surv. 44, No.6, 35-124 (1989); translation from Usp. Mat. Nauk 44, No.6(270), 29-98 (1989). ISSN 0036-0279

The authors review results due to the authors and to their colleagues in Moscow concerning the general theory of Hamiltonian systems of hydrodynamic type and the hydrodynamics of weakly deformed soliton lattices that were obtained in 1982-1988. \par A short introduction - without details - to soliton lattices and the Whitham equation is given. \par Contents: Introduction \par Chapter I. Hamiltonian theory of systems of hydrodynamic type \par {\S}1. General properties of Poisson brackets \par {\S}2. Hamiltonian formalism of systems of hydrodynamic type and Riemannian geometry \par {\S}3. Generalizations: differential-geometric Poisson brackets of higher orders, differential-geometric Poisson brackets on a lattice, and the Yang-Baxter equation \par {\S}4. Riemann invariants and the Hamiltonian formalism of diagonal systems of hydrodynamic type. Novikov's conjecture. Tsarev's theorem. The generalized hodograph method\par Chapter II. Equations of hydrodynamics of soliton lattices \par {\S}5. The Bogolyubov-Whitham averaging method for field-theoretic systems and soliton lattices. The results of Whitham and Hayes for Lagrangian systems \par {\S}6. The Whitham equations of hydrodynamics of weakly deformed soliton lattices for Hamiltonian field-theoretic systems. The principle of conservation of the Hamiltonian structure under averaging \par {\S}7. Modulations of soliton lattices of completely integrable evolutionary systems. Krichever's method. The analytic solution of the Gurevich-Pitaevskij problem on the dispersive analogue of a shock wave. \par {\S}8. Evolution of the oscillatory zone in the KdV theory. Multi-valued functions in the hydrodynamics of soliton lattices. Numerical studies \par {\S}9. Influence of small viscosity on the evolution of the oscillatory zone.\par References.

MSC 2000:: *37J35 Completely integrable systems, etc.
37K10 Completely integrable systems etc.
53C20 Riemannian manifolds (global)
58-02 Research monographs (global analysis)
35Q51 Solitons
35G10 Initial value problems for linear higher-order PDE
35K25 Higher order parabolic equations, general
76-02 Research monographs (fluid mechanics)
76Y05 Nonclassical hydrodynamics

Keywords: survey; bibliography; Hamiltonian systems of hydrodynamic type; soliton lattices; Whitham equation; Poisson brackets; completely integrable evolutionary systems

Cited in: Zbl 1078.14049 Zbl 0853.58054 Zbl 0807.58016

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]