Hydrodynamics of Weakly Deformed Soliton Lattices. Differential Geometry and Hamiltonian Theory Hydrodynamics of Weakly Deformed Soliton Lattices. Differential Geometry and Hamiltonian Theory

CONTENTS Introduction 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