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Hamiltonian Methods in the Theory of Solitons pp 356–470Cite as

Fundamental Continuous Models

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Part of the book series: Springer Series in Soviet Mathematics ((CLASSICS))

Abstract

We shall give a complete list of results pertaining to two fundamental continuous models, the HM and SG models. For the rapidly decreasing boundary conditions we shall analyze the mapping F from the initial data of the auxiliary linear problem to the transition coefficients and the discrete spectrum, and show how to solve the inverse problem, i. e. how to construct the mapping F−1. We shall see that these models allow an r-matrix approach, which will enable us to show that F is a canonical transformation to variables of action-angle type. It will thus be proved that the HM and SG models are completely integrable Hamiltonian systems. We shall also present a Hamiltonian interpretation of the change to light-cone coordinates in the SG model. To conclude this chapter, we shall explain that in some sense the LL model is the most universal integrable system with two-dimensional auxiliary space.

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  1. Steklov Mathematical Institute, Fontanka 27, 191011, Leningrad, USSR

    Ludwig D. Faddeev & Leon A. Takhtajan

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  1. Ludwig D. Faddeev
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  2. Leon A. Takhtajan
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Faddeev, L.D., Takhtajan, L.A. (2007). Fundamental Continuous Models. In: Hamiltonian Methods in the Theory of Solitons. Springer Series in Soviet Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69969-9_6

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