Abstract. These lecture notes are devoted to the theory of equations of associativity describing geometry of moduli spaces of 2D topological eld theories.

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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...

The basic content of this survey is an exposition of a recently developed method of constructing a broad class of periodic and almost-periodic solutions of non-linear equations of mathematical physics to which (in the rapidly decreasing case) the method of the inverse scattering problem is applicable. These solutions are such that the spectrum of their associated linear differential operators has a finite-zone structure. The set of linear operators with a given finite-zone spectrum is the Jacobian variety of a Riemann surface, which is determined by the structure of the spectrum. We give an explicit solution of the corresponding non-linear equations in the language of the theory of Abelian functions.

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