53 research outputs found
Integrability and Symmetries of Difference Equations: the Adler-Bobenko-Suris Case
We consider the partial difference equations of the Adler-Bobenko-Suris
classification, which are characterized as multidimensionally consistent. The
latter property leads naturally to the construction of auto-B{\"a}cklund
transformations and Lax pairs for all the equations in this class. Their
symmetry analysis is presented and infinite hierarchies of generalized
symmetries are explicitly constructed.Comment: 16 pages, for the proceedings of the 4th Workshop "Group Analysis of
Differential Equations and Integrable Systems", Cyprus, 200
Darboux transformations with tetrahedral reduction group and related integrable systems
In this paper, we derive new two-component integrable differential difference and partial difference systems by applying a Lax-Darboux scheme to an operator formed from a
Symmetry algebra of discrete KdV equations and corresponding differential-difference equations of Volterra type
A sequence of canonical conservation laws for all the Adler-Bobenko-Suris
equations is derived and is employed in the construction of a hierarchy of
master symmetries for equations H1-H3, Q1-Q3. For the discrete potential and
Schwarzian KdV equations it is shown that their local generalized symmetries
and non-local master symmetries in each lattice direction form centerless
Virasoro type algebras. In particular, for the discrete potential KdV, the
structure of its symmetry algebra is explicitly given. Interpreting the
hierarchies of symmetries of equations H1-H3, Q1-Q3 as differential-difference
equations of Yamilov's discretization of Krichever-Novikov equation,
corresponding hierarchies of isospectral and non-isospectral zero curvature
representations are derived for all of them.Comment: 22 page
Symmetries of ℤN graded discrete integrable systems
We recently introduced a class of ℤN graded discrete Lax pairs and studied the associated discrete integrable systems (lattice equations). We discuss differential–difference equations which then we interpret as symmetries of the discrete systems. In particular, we present nonlocal symmetries which are associated with the 2D Toda lattice
ZN graded discrete Lax pairs and Yang–Baxter maps
We recently introduced a class of ZNZN graded discrete Lax pairs and studied the associated discrete integrable systems (lattice equations). In this paper, we introduce the corresponding Yang–Baxter maps. Many well-known examples belong to this scheme for N=2, so, for N?3, our systems may be regarded as generalizations of these. In particular, for each N we introduce a class of multi-component Yang–Baxter maps, which include HBIII (of Papageorgiou et al. 2010 SIGMA 6, 003 (9 p). (doi:10.3842/SIGMA.2010.033)), when N=2, and that associated with the discrete modified Boussinesq equation, for N=3. For N?5 we introduce a new family of Yang–Baxter maps, which have no lower dimensional analogue. We also present new multi-component versions of the Yang–Baxter maps FIV and FV (given in the classification of Adler et al. 2004 Commun. Anal. Geom. 12, 967–1007. (doi:10.4310/CAG.2004.v12.n5.a1))
On the Lagrangian structure of 3D consistent systems of asymmetric quad-equations
Recently, the first-named author gave a classification of 3D consistent
6-tuples of quad-equations with the tetrahedron property; several novel
asymmetric 6-tuples have been found. Due to 3D consistency, these 6-tuples can
be extended to discrete integrable systems on Z^m. We establish Lagrangian
structures and flip-invariance of the action functional for the class of
discrete integrable systems involving equations for which some of the
biquadratics are non-degenerate and some are degenerate. This class covers,
among others, some of the above mentioned novel systems.Comment: 21 pp, pdfLaTe
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
?_? graded discrete Lax pairs and integrable difference equations
We introduce a class of Z_N graded discrete Lax pairs, with NĂ—N matrices, linear in the
spectral parameter. We give a classification scheme for such Lax pairs and the associated
discrete integrable systems. We present two potential forms and completely classify the
generic case. Many well known examples belong to our scheme for N = 2, so many of our
systems may be regarded as generalisations of these. Even at N = 3, several new integrable
systems arise. Many of our equations are mutually compatible, so can be used together to
form “coloured” lattices
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