Abstract
Here we shall give a complete list of results pertaining to the Toda model, a fundamental model on the lattice. We will show that the r-matrix approach applies to this case and may be used to prove the complete integrability of the model in the quasi-periodic case. For the rapidly decreasing boundary conditions we will analyze the mapping F from the initial data of the auxiliary linear problem to the transition coefficients and outline a method for solving the inverse problem, i.e. for F−1 constructing. On the basis of the r-matrix approach it will be shown that F is a canonical trans formation to action-angle type variables establishing the complete integrability of the Toda model in the rapidly decreasing case. We shall also define a lattice version of the LL model, the most general integrable lattice system with two-dimensional auxiliary space.
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Faddeev, L.D., Takhtajan, L.A. (2007). Fundamental Models on the Lattice. 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_7
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DOI: https://doi.org/10.1007/978-3-540-69969-9_7
Publisher Name: Springer, Berlin, Heidelberg
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