17 research outputs found
Integrable and superintegrable systems associated with multi-sums of products
We construct and study certain Liouville integrable, superintegrable, and
non-commutative integrable systems, which are associated with multi-sums of
products.Comment: 26 pages, submitted to Proceedings of the royal society
Darboux transformations, finite reduction groups and related Yang-Baxter maps
In this paper we construct Yang-Baxter (YB) maps using Darboux matrices which
are invariant under the action of finite reduction groups. We present
6-dimensional YB maps corresponding to Darboux transformations for the
Nonlinear Schr\"odinger (NLS) equation and the derivative Nonlinear
Schr\"odinger (DNLS) equation. These YB maps can be restricted to
dimensional YB maps on invariant leaves. The former are completely
integrable and they also have applications to a recent theory of maps
preserving functions with symmetries \cite{Allan-Pavlos}. We give a
dimensional YB-map corresponding to the Darboux transformation for a
deformation of the DNLS equation. We also consider vector generalisations of
the YB maps corresponding to the NLS and DNLS equation.Comment: 18 pages, revised version. The format of the paper has changed, we
added one sectio
Grassmann extensions of Yang-Baxter maps
In this paper we show that there are explicit Yang–Baxter (YB) maps with Darboux–Lax representation between Grassman extensions of algebraic varieties. Motivated by some recent results on noncommutative extensions of Darboux transformations, we first derive a Darboux matrix associated with the Grassmann-extended derivative nonlinear Schrödinger (DNLS) equation, and then we deduce novel endomorphisms of Grassmann varieties, which possess the YB property. In particular, we present ten-dimensional maps which can be restricted to eight-dimensional YB maps on invariant leaves, related to the Grassmann-extended NLS and DNLS equations. We consider their vector generalisations
Integrable Euler top and nonholonomic Chaplygin ball
We discuss the Poisson structures, Lax matrices, -matrices, bi-hamiltonian
structures, the variables of separation and other attributes of the modern
theory of dynamical systems in application to the integrable Euler top and to
the nonholonomic Chaplygin ball.Comment: 25 pages, LaTeX with AMS fonts, final versio
Liouville integrability and superintegrability of a generalized Lotka–Volterra system and its Kahan discretization
We prove the Liouville and superintegrability of a generalized Lotka–Volterra system and its Kahan discretization
Entwining Yang–Baxter maps related to NLS type equations
We construct birational maps that satisfy the parametric set-theoretical Yang–Baxter equation and its entwining generalisation. For this purpose, we employ Darboux transformations related to integrable nonlinear Schrödinger type equations and study the refactorisation problems of the product of their associated Darboux matrices. Additionally, we study various algebraic properties of the derived maps, such as invariants and associated symplectic or Poisson structures, and we prove their complete integrability in the Liouville sense
Some integrable maps and their Hirota bilinear forms
We introduce a two-parameter family of birational maps, which reduces to a family previously found by Demskoi, Tran, van der Kamp and Quispel (DTKQ) when one of the parameters is set to zero. The study of the singularity confinement pattern for these maps leads to the introduction of a tau function satisfying a homogeneous recurrence which has the Laurent property, and the tropical (or ultradiscrete) analogue of this homogeneous recurrence confirms the quadratic degree growth found empirically by Demskoi et al. We prove that the tau function also satisfies two different bilinear equations, each of which is a reduction of the Hirota-Miwa equation (also known as the discrete KP equation, or the octahedron recurrence). Furthermore, these bilinear equations are related to reductions of particular two-dimensional integrable lattice equations, of discrete KdV or discrete Toda type. These connections, as well as the cluster algebra structure of the bilinear equations, allow a direct construction of Poisson brackets, Lax pairs and first integrals for the birational maps. As a consequence of the latter results, we show how each member of the family can be lifted to a system that is integrable in the Liouville sense, clarifying observations made previously in the original DTKQ case
Darboux transformation for the vector sine-Gordon equation and integrable equations on a sphere
We propose a method for construction of Darboux transformations, which is a new development of the dressing method for Lax operators invariant under a reduction group. We apply the method to the vector sine-Gordon equation and derive its Bäcklund transformations. We show that there is a new Lax operator canonically associated with our Darboux transformation resulting an evolutionary differential-difference system on a sphere. The latter is a generalised symmetry for the chain of Bäcklund transformations. Using the re-factorisation approach and the Bianchi permutability of the Darboux transformations we derive new vector Yang-Baxter map and integrable discrete vector sine-Gordon equation on a sphere