arXiv subjects
Publications and source records attributed to Andrei K. Svinin.
We present and investigate a new infinite family of homogeneous equations which possess the Laurent property. The first representative in this family is the well-known Somos-5 recurrence.
In this paper we analyze recent work \cite{Hone1} by Hone, Roberts and Vanhaecke, where the so-called Volterra map was introduced via the Lax equation that looks similar to the Lax representation for the Mumford's system \cite{Vanhaecke}. This map turns out to be birational and a corresponding dynamical system on an affine space $M_g$ of dimension $3g+1$ was associated with it. This mapping is related to some discrete equation of the order $2g+1$ associated with the Stieltjes continued fraction expansion of a certain function on a hyperelliptic (elliptic) curve of genus $g\geq 1$. The authors of the paper provides examples of this equation for the simplest cases $g=1$ and $g=2$, but for higher values of $g$, corresponding equation turns out to be too cumbersome to write them out. We present an approach in which the mentioned $(2g+1)$-order equation can be written out for all values of $g\geq 1$ in a compact form. This equation is not new and can be found, for example, in \cite{Svinin3}. An essential point in our framework is the use of special class of discrete polynomials which as shown to be closely related to the Stieltjes continued fraction. On the one hand, this allows us to generalize some of the results of the work \cite{Hone1}. On the other hand, many things in this approach can be presented in a more compact and unified form. Ultimately, we believe that this allows us to give a new perspective on this topic.
The main object of study in this paper is the well-known Somos-4 recurrence. We prove a theorem that any sequence generated by this equation also satisfies Gale-Robinson one. The corresponding identity is written in terms of its companion elliptic sequence. An example of such relationship is provided by the second-order linear sequence which, as we prove using Wajda's identity, satisfies the Somos-4 recurrence with suitable coefficients. Also, we construct a class of solutions to Volterra lattice equation closely related to the second-order linear sequence.
In this paper we show that an arbitrary solution of one ordinary difference equation is also a solution for a hierarchy of integrable difference equations. We also provide an example of such a solution that is related to sequence generated by a second-order linear recursion with 2-periodic coefficients.
Sequences of Genocchi numbers of the first and second kind are considered. For these numbers, an approach based on their representation using sequences of polynomials is developed. Based on this approach, for these numbers some identities generalizing the known identities are constructed.
We give an expression of polynomials for higher sums of powers of integers via the higher order Bernoulli numbers.
Some class of sums which naturally include the sums of powers of integers is considered. A number of conjectures concerning a representation of these sums is made.
We consider Tuenter polynomials as linear combinations of descending factorials and show that coefficients of these linear combinations are expressed via a Catalan triangle of numbers. We also describe a triangle of coefficients in terms of some polynomials.
We consider two infinite classes of ordinary difference equations admitting Lax pair representation. Discrete equations in these classes are parameterized by two integers $k\geq 0$ and $s\geq k+1$. 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.
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.
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 $n\geq 1$, 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 introduce two classes of homogeneous polynomials and show their role in constructing of integrable hierarchies for some integrable lattices.
We observe that Dickey's stabilizing chain can be naturally included into two-dimensional chain of infinitely many copies of equations of KP hierarchy.
We show a broad class of constraints compatible with Itoh-Narita-Bogoyavlenskii lattice hierarchy. All these constraints can be written in the form of discrete conservation law $I_{i+1}=I_i$ with appropriate homogeneous polynomial discrete function $I=I[a]$.
We consider the extended discrete KP hierarchy and show that similarity reduction of its subhierarchies lead to purely discrete equations with dependence on some number of parameters together with equations governing deformations with respect to these parameters. It is written down discrete equations which naturally generalize the first discrete Painlevé equation $\mathrm{dP}_{\rm I}$ in a sense that autonomous version of these equations admit the limit to the first Painlevé equation. It is shown that each of these equations describes Bäcklund transformations of Veselov-Shabat periodic dressing lattices with odd period known also as Noumi-Yamada systems of type $A_{2(n-1)}^{(1)}$.