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By S. P. Novikov

During this publication, Professor Novikov describes fresh advancements in soliton conception and their kinfolk to so-called Poisson geometry. This formalism, that is relating to symplectic geometry, is intensely beneficial for the examine of integrable structures which are defined when it comes to differential equations (ordinary or partial) and quantum box theories. Professor Novikov examines numerous Hamiltonian structures, in the framework of Poisson geometry, to illustrate its energy. This booklet can be of curiosity to mathematicians and physicists.

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For KdV we have N = 2m + 1, m ≥ 1 and and the corresponding metric has signature (m, m + 1). More exactly, the HT systems corresponding to the KdV equation might be described using two different Hamiltonian structures (GZF and LM) —see Lecture 4. After averaging we get two different HTPB’s corresponding to these structures; the restriction of the metric to the variables (U, J) is still the same but they do not give complete systems of coordinates; the action variables are different for GZF and LM.

Benney discussed in 1970 the “m-phase” analogue of G. Whitham’s method (who developed it for m = 1 only) but there was no concrete base for serious development because the family of m-gap solutions was found for the first time in 1974. Serious analogues of Whitham’s theory for the cases m > 1 started in the late seventies only; they are due to H. Flashka, G. Forest, D. McLaughlin, Dobrokhotov and V. Maslov. Suppose now that the parameters (u1 , . . , uN ) are not constants but only “slow functions”: up = up (X, T ), X = εx, T = εt.

The influence of the small viscosity on the situation above has been investigated in 1987 by the author in collaboration with V. Avilov and Krichever (and also by L. Pitaevski and A. Gurevitch —see in [10]). The foundation of the averaging (non-linear WKB) method for the KdVsystem with small dispersion term Ψt = 6ΨΨx + δΨxxx , δ → 0 has been investigated by P. Lax and C. Levermore starting from 1983, and by S. Venakides starting from 1987. Some important asymptotic results were obtained also by R.

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