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Register a new userOn some class of differential-difference equations admitting Lax representation
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Added by
Andrei Svinin Kirillovich
Publication date
2013
fields
Physics
and research's language is
English
Authors
Andrei K. Svinin
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This note is designed to show some classes of differential-difference equations admitting Lax representation which generalize evolutionary equations known in the literature.
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We consider two infinite classes of ordinary difference equations admitting Lax pair representation. Discrete equations in these classes are parameterized by two integers $kgeq 0$ and $sgeq k+1$. We describe the first integrals for these two classes in terms of special discrete polynomials. We show an equivalence of two difference equations belonged to different classes corresponding to the same pair $(k, s)$. We show that solution spaces $mathcal{N}^k_s$ of different ordinary difference equations with fixed value of $s+k$ are organized in chain of inclusions.
We show some classes of higher order partial difference equations admitting a zero-curvature representation and generalizing lattice potential KdV equation. We construct integrable hierarchies which, as we suppose, yield generalized symmetries for obtained class of partial difference equations. As a byproduct we also derive non-evolutionary differential-difference equations with their Lax pair representation which may be of potential interest.
We introduce two classes of discrete polynomials and construct discrete equations admitting a Lax representation in terms of these polynomials. Also we give an approach which allows to construct lattice integrable hierarchies in its explicit form and show some examples.
In this paper we show that an arbitrary solution of one ordinary difference equation is also a solution for infinite class of difference equations. We also provide an example of such a solution that is related to sequence generated by second-order linear recurrent relations.
We introduce two classes of homogeneous polynomials and show their role in constructing of integrable hierarchies for some integrable lattices.
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