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By S. P. Novikov

During this ebook, Professor Novikov describes contemporary advancements in soliton idea and their relatives to so-called Poisson geometry. This formalism, that's concerning symplectic geometry, is very important for the learn of integrable structures which are defined by way of differential equations (ordinary or partial) and quantum box theories. Professor Novikov examines a number of Hamiltonian structures, in the framework of Poisson geometry, to illustrate its strength. This booklet should be of curiosity to mathematicians and physicists.

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Solitons and geometry

During this publication, Professor Novikov describes contemporary advancements in soliton conception and their kinfolk to so-called Poisson geometry. This formalism, that is on the topic of symplectic geometry, is very priceless for the examine of integrable platforms which are defined when it comes to differential equations (ordinary or partial) and quantum box theories.

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The initial phase U0 may be an arbitrary m-vector. The function F should be periodic with period 2π in each of the first m variables: F = F (η1 , . . , ηm , u1 , . . uN ) ηj (η1 , . . , ηm ) ∈ T m . For any (u = const) we get an exact solution of the original system (“mphase solution”). For m = 0 we get constants (provided Ψ = const satisfies the original equation). The most popular case is m = 1 which is sufficiently non-trivial. Many systems have families of periodic solutions as “traveling waves”: x = U η + U0 , Ψ(x − ct), Ψ(η + T ) = Ψ(η).

It is convenient to introduce an algebraic “filtration” with the properties f (Ψ) ≥ 0, f (Ψ(k) ) ≥ k, f (AB) ≥ f (A) + f (B). For the equation we get the decomposition Ψt = K0 (Ψ) + K1 (Ψ)Ψx + . . where the remaining terms have filtration at least 2. Suppose all Ψ = const are exact solutions. In this case K0 (Ψ) = 0. We are coming to the HT system as the first hydrodynamic type approximation to the original system for the class of “very smooth” functions Ψ(X, T ), X = εx, T = εt (slow modulations of the constant solutions): ΨT = K1 (Ψ)ΨX .

We will present here only special results about very special important solutions (see Lecture 5), based on the results of Krichever and Potemin of 1989. The latest results of other authors may be found in the transactions of the Conference dedicated to this subject held in Lyon in July 1991, which are going to be published soon. 37 Lecture 5 Non-linear WKB (Hydrodynamics of Weakly Deformed Solution Lattices). Consider an evolutional system of non-linear differential equations Ψt = K(Ψ, Ψx , Ψxx , .

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