zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory. (English) Zbl 0712.58032

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.

A short introduction - without details - to soliton lattices and the Whitham equation is given.

Contents: Introduction

Chapter I. Hamiltonian theory of systems of hydrodynamic type

§1. General properties of Poisson brackets

§2. Hamiltonian formalism of systems of hydrodynamic type and Riemannian geometry

§3. Generalizations: differential-geometric Poisson brackets of higher orders, differential-geometric Poisson brackets on a lattice, and the Yang-Baxter equation

§4. Riemann invariants and the Hamiltonian formalism of diagonal systems of hydrodynamic type. Novikov’s conjecture. Tsarev’s theorem. The generalized hodograph method

Chapter II. Equations of hydrodynamics of soliton lattices

§5. The Bogolyubov-Whitham averaging method for field-theoretic systems and soliton lattices. The results of Whitham and Hayes for Lagrangian systems

§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

§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.

§8. Evolution of the oscillatory zone in the KdV theory. Multi-valued functions in the hydrodynamics of soliton lattices. Numerical studies

§9. Influence of small viscosity on the evolution of the oscillatory zone.

References.


MSC:
37J35Completely integrable systems, topological structure of phase space, integration methods
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
53C20Global Riemannian geometry, including pinching
58-02Research monographs (global analysis)
35Q51Soliton-like equations
35G10Initial value problems for linear higher-order PDE
35K25Higher order parabolic equations, general
76-02Research monographs (fluid mechanics)
76Y05Quantum hydrodynamics; relativistic hydrodynamics