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 compare the results of our two papers with the results of the paper Aratyn H., Gomes J.F., Zimerman A.H., Higher order Painleve equations and their symmetries via reductions of a class of integrable models, J. Phys. A: Math. Theor., V. 44} (2011), Art. No. 235202.
Based on the notion of Darboux-KP chain hierarchy and its invariant submanifolds we construct some class of constraints compatible with integrable lattices. Some simple examples are given.
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.
This note is designed to show some classes of differential-difference equations admitting Lax representation which generalize evolutionary equations known in the literature.
We show that by Miura-type transformation the Itoh-Narita-Bogoyavlenskii lattice, for any $ngeq 1$, is related to some differential-difference (modified) equation. We present corresponding integrable hierarchies in its explicit form. We study the elementary Darboux transformation for modified equations.
We observe that Dickeys stabilizing chain can be naturally included into two-dimensional chain of infinitely many copies of equations of KP hierarchy.
We introduce two classes of homogeneous polynomials and show their role in constructing of integrable hierarchies for some integrable lattices.
