547,735 research outputs found

    Functional representation of the Ablowitz-Ladik hierarchy

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    The Ablowitz-Ladik hierarchy (ALH) is considered in the framework of the inverse scattering approach. After establishing the structure of solutions of the auxiliary linear problems, the ALH, which has been originally introduced as an infinite system of difference-differential equations is presented as a finite system of difference-functional equations. The representation obtained, when rewritten in terms of Hirota's bilinear formalism, is used to demonstrate relations between the ALH and some other integrable systems, the Kadomtsev-Petviashvili hierarchy in particular.Comment: 15 pages, LaTe

    Factorization method for second order functional equations

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    We apply general difference calculus in order to obtain solutions to the functional equations of the second order. We show that factorization method can be successfully applied to the functional case. This method is equivariant under the change of variables. Some examples of applications are presented.Comment: 22 pages, examples and new section added, several correction

    Elliptic Dunkl operators, root systems, and functional equations

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    We consider generalizations of Dunkl's differential-difference operators associated with groups generated by reflections. The commutativity condition is equivalent to certain functional equations. These equations are solved in many cases. In particular, solutions associated with elliptic curves are constructed. In the An1A_{n-1} case, we discuss the relation with elliptic Calogero-Moser integrable nn-body problems, and discuss the quantization (qq-analogue) of our construction.Comment: 30 page