27 research outputs found
On Quadrirational Yang-Baxter Maps
We use the classification of the quadrirational maps given by Adler, Bobenko
and Suris to describe when such maps satisfy the Yang-Baxter relation. We show
that the corresponding maps can be characterized by certain singularity
invariance condition. This leads to some new families of Yang-Baxter maps
corresponding to the geometric symmetries of pencils of quadrics.Comment: Proceedings of the workshop "Geometric Aspects of Discrete and
Ultra-Discrete Integrable Systems" (Glasgow, March-April 2009
Boussinesq-like multi-component lattice equations and multi-dimensional consistency
We consider quasilinear, multi-variable, constant coefficient, lattice
equations defined on the edges of the elementary square of the lattice, modeled
after the lattice modified Boussinesq (lmBSQ) equation, e.g., . These equations are classified into three canonical forms and
the consequences of their multidimensional consistency
(Consistency-Around-the-Cube, CAC) are derived. One of the consequences is a
restriction on form of the equation for the variable, which in turn implies
further consistency conditions, that are solved. As result we obtain a number
of integrable multi-component lattice equations, some generalizing lmBSQ.Comment: 24 page
Infinitely many symmetries and conservation laws for quad-graph equations via the Gardner method
The application of the Gardner method for generation of conservation laws to
all the ABS equations is considered. It is shown that all the necessary
information for the application of the Gardner method, namely B\"acklund
transformations and initial conservation laws, follow from the multidimensional
consistency of ABS equations. We also apply the Gardner method to an asymmetric
equation which is not included in the ABS classification. An analog of the
Gardner method for generation of symmetries is developed and applied to
discrete KdV. It can also be applied to all the other ABS equations
Yang-Baxter maps and symmetries of integrable equations on quad-graphs
A connection between the Yang-Baxter relation for maps and the
multi-dimensional consistency property of integrable equations on quad-graphs
is investigated. The approach is based on the symmetry analysis of the
corresponding equations. It is shown that the Yang-Baxter variables can be
chosen as invariants of the multi-parameter symmetry groups of the equations.
We use the classification results by Adler, Bobenko and Suris to demonstrate
this method. Some new examples of Yang-Baxter maps are derived in this way from
multi-field integrable equations.Comment: 20 pages, 5 figure
Continuous symmetric reductions of the Adler-Bobenko-Suris equations
Continuously symmetric solutions of the Adler-Bobenko-Suris class of discrete
integrable equations are presented. Initially defined by their invariance under
the action of both of the extended three point generalized symmetries admitted
by the corresponding equations, these solutions are shown to be determined by
an integrable system of partial differential equations. The connection of this
system to the Nijhoff-Hone-Joshi "generating partial differential equations" is
established and an auto-Backlund transformation and a Lax pair for it are
constructed. Applied to the H1 and Q1 members of the
Adler-Bobenko-Suris family, the method of continuously symmetric reductions
yields explicit solutions determined by the Painleve trancendents.Comment: 28 pages, submitted to J. Phys. A: Math. Theo
Lagrangian multiform structure for the lattice Gel'fand-Dikii hierarchy
The lattice Gel'fand-Dikii hierarchy was introduced by Nijhoff, Papageorgiou,
Capel and Quispel in 1992 as the family of partial difference equations
generalizing to higher rank the lattice Korteweg-de Vries systems, and includes
in particular the lattice Boussinesq system. We present a Lagrangian for the
generic member of the lattice Gel'fand-Dikii hierarchy, and show that it can be
considered as a Lagrangian 2-form when embedded in a higher dimensional
lattice, obeying a closure relation. Thus the multiform structure proposed in
arXiv:0903.4086v2 [nlin.SI] is extended to a multi-component system.Comment: 12 page
Yang-Baxter maps and multi-field integrable lattice equations
A variety of Yang-Baxter maps are obtained from integrable multi-field
equations on quad-graphs. A systematic framework for investigating this
connection relies on the symmetry groups of the equations. The method is
applied to lattice equations introduced by Adler and Yamilov and which are
related to the nonlinear superposition formulae for the B\"acklund
transformations of the nonlinear Schr\"odinger system and specific
ferromagnetic models.Comment: 16 pages, 4 figures, corrected versio
An algebraic method of classification of S-integrable discrete models
A method of classification of integrable equations on quad-graphs is
discussed based on algebraic ideas. We assign a Lie ring to the equation and
study the function describing the dimensions of linear spaces spanned by
multiple commutators of the ring generators. For the generic case this function
grows exponentially. Examples show that for integrable equations it grows
slower. We propose a classification scheme based on this observation.Comment: 11 pages, workshop "Nonlinear Physics. Theory and Experiment VI",
submitted to TM
Cosymmetries and Nijenhuis recursion operators for difference equations
In this paper we discuss the concept of cosymmetries and co--recursion
operators for difference equations and present a co--recursion operator for the
Viallet equation. We also discover a new type of factorisation for the
recursion operators of difference equations. This factorisation enables us to
give an elegant proof that the recursion operator given in arXiv:1004.5346 is
indeed a recursion operator for the Viallet equation. Moreover, we show that
this operator is Nijenhuis and thus generates infinitely many commuting local
symmetries. This recursion operator and its factorisation into Hamiltonian and
symplectic operators can be applied to Yamilov's discretisation of the
Krichever-Novikov equation
Yang Baxter maps with first degree polynomial 2 by 2 Lax matrices
A family of nonparametric Yang Baxter (YB) maps is constructed by
refactorization of the product of two 2 by 2 matrix polynomials of first
degree. These maps are Poisson with respect to the Sklyanin bracket. For each
Casimir function a parametric Poisson YB map is generated by reduction on the
corresponding level set. By considering a complete set of Casimir functions
symplectic multiparametric YB maps are derived. These maps are quadrirational
with explicit formulae in terms of matrix operations. Their Lax matrices are,
by construction, 2 by 2 first degree polynomial in the spectral parameter and
are classified by Jordan normal form of the leading term. Nonquadrirational
parametric YB maps constructed as limits of the quadrirational ones are
connected to known integrable systems on quad graphs