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Affine linear and D4 symmetric lattice equations: Symmetry analysis and reductions
, 2008
"... We consider lattice equations on Z² which are autonomous, affine linear and possess the symmetries of the square. Some basic properties of equations of this type are derived, as well as a sufficient linearization condition and a conservation law. A systematic analysis of the Lie point and the genera ..."
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Cited by 15 (3 self)
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We consider lattice equations on Z² which are autonomous, affine linear and possess the symmetries of the square. Some basic properties of equations of this type are derived, as well as a sufficient linearization condition and a conservation law. A systematic analysis of the Lie point and the generalized three and fivepoint symmetries is presented. It leads to the generic form of the symmetry generators of all the equations in this class, which satisfy a certain nondegeneracy condition. Finally, symmetry reductions of certain lattice equations to discrete analogues of the Painlevé equations are considered.
Viallet C., Integrable lattice equations with vertex and bond variables
 J. Phys. A: Math. Theor
"... We present integrable lattice equations on a two dimensional square lattice with coupled vertex and bond variables. In some of the models the vertex dynamics is independent of the evolution of the bond variables, and one can write the equations as nonautonomous “YangBaxter maps”. We also present a ..."
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Cited by 6 (1 self)
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We present integrable lattice equations on a two dimensional square lattice with coupled vertex and bond variables. In some of the models the vertex dynamics is independent of the evolution of the bond variables, and one can write the equations as nonautonomous “YangBaxter maps”. We also present a model in which the vertex and bond variables are fully coupled. Integrability is tested with algebraic entropy as well as multidimensional consistency. 1
Integrable discretisations for a class of nonlinear Schrödinger equations on Grassmann algebras
 Physics Letters A
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INTEGRABLE LATTICE MAPS: Q5, A RATIONAL VERSION OF Q4
, 802
"... We give a rational form of a generic twodimensional “quad ” map, containing the socalled Q4 case [1, 2, 3, 4, 5, 6], but whose coefficients are free. Its integrability is proved using the calculation of algebraic entropy. We first explain the setting, i.e. what are two dimensional lattice maps on ..."
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Cited by 1 (0 self)
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We give a rational form of a generic twodimensional “quad ” map, containing the socalled Q4 case [1, 2, 3, 4, 5, 6], but whose coefficients are free. Its integrability is proved using the calculation of algebraic entropy. We first explain the setting, i.e. what are two dimensional lattice maps on a square
Poisson YangBaxter maps with binomial Lax matrices
 Journal of Mathematical Physics
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Yang–Baxter maps associated to elliptic curves
, 2009
"... We present Yang–Baxter maps associated to elliptic curves. They are related to discrete versions of the KricheverNovikov and the LandauLifshits equations. A lifting of scalar integrable quad–graph equations to two–field equations is also shown. 1 ..."
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We present Yang–Baxter maps associated to elliptic curves. They are related to discrete versions of the KricheverNovikov and the LandauLifshits equations. A lifting of scalar integrable quad–graph equations to two–field equations is also shown. 1
On discrete integrable equations of higher order
, 2013
"... We study 2D discrete integrable equations of order 1 with respect to one independent variable and m with respect to another one. A generalization of the multidimensional consistency property is proposed for this type of equations. The examples are related to the Bäcklund– Darboux transformations f ..."
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We study 2D discrete integrable equations of order 1 with respect to one independent variable and m with respect to another one. A generalization of the multidimensional consistency property is proposed for this type of equations. The examples are related to the Bäcklund– Darboux transformations for the lattice equations of Bogoyavlensky type.