arXiv subjects

Pavlos Kassotakis

Publications and source records attributed to Pavlos Kassotakis.

25 records · Page 2Linked to original sources

arXiv2017

Difference systems in bond and face variables and non-potential versions of discrete integrable systems

Integrable discrete scalar equations defined on a~two or a three dimensional lattice can be rewritten as difference systems in bond variables or in face variables respectively. Both the difference systems in bond variables and the difference systems in face variables can be regarded as vector versions of the original equations. As a result, we link some of the discrete equations by difference substitutions and reveal the non-potential versions of some consistent-around-the-cube equations. We obtain higher-point configurations, including pairs of compatible six~points equations on the ${\mathbb Z}^2$ lattice together with associated seven points equations. Also we obtain a variety of compatible ten point equations together with associated ten and twelve point equations on the ${\mathbb Z}^3$ lattice. Finally, we present integrable multiquadratic quad relations.

nlin.SI↗

arXiv2017

Integrable deformations of the Bogoyavlenskij-Itoh Lotka-Volterra systems

We construct a family of integrable deformations of the Bogoyavlenskij-Itoh systems and construct a Lax operator with spectral parameter for it. Our approach is based on the construction of a family of compatible Poisson structures for the undeformed system, whose Casimirs are shown to yield a generating function for the integrals in involution of the deformed systems. We show how these deformations are related to the Veselov-Shabat systems.

math.DS↗

arXiv2016

Integrable reductions of the Bogoyavlenskij-Itoh Lotka-Volterra systems

Given a constant skew-symmetric matrix A, it is a difficult open problem whether the associated Lotka-Volterra system is integrable or not. We solve this problem in the special case when A is a Toepliz matrix where all off-diagonal entries are plus or minus one. In this case, the associated Lotka-Volterra system turns out to be a reduction of Liouville integrable systems, whose integrability was shown by Bogoyavlenskij and Itoh. We prove that the reduced systems are also Liouville integrable and that they are also non-commutative integrable by constructing a set of independent first integrals, having the required involutive properties (with respect to the Poisson bracket). These first integrals fall into two categories. One set consists of polynomial functions which can be obtained by a matricial reformulation of Itoh's combinatorial description. The other set consists of rational functions which are obtained through a Poisson map from the first integrals of some recently discovered superintegrable Lotka-Volterra systems. The fact that these polynomial and rational first integrals, combined, have the required properties for Liouville and non-commutative integrability is quite remarkable; the quite technical proof of functional independence of the first integrals is given in detail.

math-ph↗

arXiv2015

$2^n-$rational maps

We present a natural extension of the notion of nondegenerate rational maps (quadrirational maps) to arbitrary dimensions. We refer to these maps as $2^n-$rational maps. In this note we construct a rich family of $2^n-$rational maps. These maps by construction are involutions and highly symmetric in the sense that the maps and their companion maps have the same functional form.

nlin.SI↗

arXiv2013

Integrable Maps which Preserve Functions with Symmetries

We consider maps which preserve functions which are built out of the invariants of some simple vector fields. We give a reduction procedure, which can be used to derive commuting maps of the plane, which preserve the same symplectic form and first integral. We show how our method can be applied to some maps which have recently appeared in the context of Yang-Baxter maps.

math-ph↗

arXiv2011

Families of Integrable Equations

We present a method to obtain families of lattice equations. Specifically we focus on two of such families, which include 3-parameters and their members are connected through Bäcklund transformations. At least one of the members of each family is integrable, hence the whole family inherits some integrability properties.

nlin.SI↗

arXiv2011

On non-multiaffine consistent-around-the-cube lattice equations

We show that integrable involutive maps, due to the fact they admit three integrals in separated form, can give rise to equations, which are consistent around the cube and which are not in the multiaffine form assumed in papers [1, 2]. Lattice models, which are discussed here, are related to the lattice potential KdV equation by nonlocal transformations (discrete quadratures).

nlin.SI↗