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One very promising approach to solving integrable systems is based on the notion of a dressing chain [395, 438, 448]. This method covers soliton, ﬁnite- gap, rational, and other important solutions [423, 438] within the universal scheme, reducing the starting problem to a solution of closed sets of nonlinear ordinary equations with the bi-Hamiltonian structure [438]. Here we derive the dressing chain equations and study the above classes of solutions. A scheme to reconstruct the potential entering an associated linear prob- lem depends on a class to which solutions of the inverse problem belong [378]. The coeﬃcients (potentials) of the linear equation are elements of some alge- braic structure. Such a structure is generated by transformations that preserve the functional form of the equation. Hence, a general set of potentials splits into subsets invariant with respect to the action of the transformations. These transformations, for example, the Darboux (Schlesinger, Moutard) transfor- mations, are generated by transformations of eigenfunctions φ of a given diﬀerential operator. As a general remark, note that it is possible to approx- imate locally solutions of the linear problem by a sequence of the Moutard and Ribacour transformations [170]. The Darboux transformation (DT) pro- −1 duces
Published: Jan 1, 2007
Keywords: Spectral Problem; Elliptic Function; Spectral Curve; Darboux Transformation; Chain Equation
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