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A. K. Svinin

Publications and source records attributed to A. K. Svinin.

10 recordsLinked to original sources
arXiv2005

Integrable Mappings Related to the Extended Discrete KP Hierarchy

We investigate self-similar solutions of the extended discrete KP hierarchy. It is shown that corresponding ansatzes lead to purely discrete equations with dependence on some number of parameters together with equations governing deformations with respect to these parameters. Some examples are provided. In particular, it is shown that the well known discrete first Painleve equation (dPI) and its hierarchy arises as self-similar reduction of Volterra lattice hierarchy which in turn can be treated as a reduction of the extended discrete KP hierarchy. It is written down equations which naturally generalize dPI. It is shown that theses discrete systems describe Bäcklund transformations of Noumi-Yamada systems of type $A_{2(n-1)^{(1)}}$. We also consider Miura transformations relating different infinite- and finite-field integrable mappings. Simplest example of this kind of Miura transformations is given.

nlin.SI↗
arXiv2004

Extended discrete KP hierarchy and its reductions from a geometric viewpoint

We interpret the recently suggested extended discrete KP (Toda lattice) hierarchy from a geometrical point of view. We show that the latter corresponds to the union of invariant submanifolds $S_0^n$ of the system which is a chain of infinitely many copies of Darboux-KP hierarchy, while the intersections $S_0^n\cap S_{l-1}^{ln-r}$ yields a number of reductions to $l$-field lattices.

nlin.SI↗
arXiv2004

Invariant Submanifolds of Darboux-KP Chain and Extension of the Discete KP Hierarchy

Invariant submanifolds of the so-called Darboux-KP chain are investigated. It is shown that restriction of dynamics on some class of invariant submanifolds yields the extension of the discrete KP hierarchy, while the intersections leads to Lax pairs for a broad class of differential-difference systems with finite number of fields. Some attention is given to investigation of self-similar reductions. It is shown that self-similar ansatzes lead to purely discrete equations with dependence on some number of parameters together with equations governing deformations with respect to these parameters. Some examples are provided. In particular it is shown that well known first discrete Painleve equation (dPI) corresponds to Volterra lattice hierarchy. It is written down equations which naturally generalize dPI in the sense that they have first Painleve transcedent in continuous limit.

nlin.SI↗
arXiv2001

A class of integrable lattices and KP hierarchy

We introduce a class of integrable $l$-field first-order lattices together with corresponding Lax equations. These lattices may be represented as consistency condition for auxiliary linear systems defined on sequences of formal dressing operators. This construction provides simple way to build lattice Miura transformations between one-field lattice and $l$-field ($l\ge 2$) ones. We show that the lattices pertained to above class is in some sense compatible with KP flows and define the chains of constrained KP Lax operators.

nlin.SI↗
arXiv2001

n-th discrete KP hierarchy

We report an infinite class of discrete hierarchies which naturally generalize familiar discrete KP one.

nlin.SI↗