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**Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory.**
*(English.
Russian original)*
Zbl 0712.58032

Russ. Math. Surv. 44, No. 6, 35-124 (1989); translation from Usp. Mat. Nauk 44, No. 6(270), 29-98 (1989).

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.

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:

37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

53C20 | Global Riemannian geometry, including pinching |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

35Q51 | Soliton equations |

35G10 | Initial value problems for linear higher-order PDEs |

35K25 | Higher-order parabolic equations |

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76Y05 | Quantum hydrodynamics and relativistic hydrodynamics |