Affine linear and D4 symmetric lattice equations : symmetry analysis and reductions

Abstract

We consider lattice equations on ${\mathds{Z}}^2$ 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 five-point symmetries is presented. It leads to the generic form of the symmetry generators of all the equations in this class, which satisfy a certain non-degeneracy condition. Finally, symmetry reductions of certain lattice equations to discrete analogues of the Painlevé equations are considered.

arXiv comment31 pages

References in corpus (7)

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• Algebraic entropy
• Continuous Symmetries of Difference Equations
• Yang-Baxter maps and symmetries of integrable equations on quad-graphs
• Lie point symmetries of difference equations and lattices
• Factorisation of Macdonald polynomials
• Yang-Baxter maps and multi-field integrable lattice equations

Cited by in corpus (15)

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• The Generalized Symmetry Method for Discrete Equations
• On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations
• Second order integrability conditions for difference equations. An integrable equation
• Symmetries and integrability of discrete equations defined on a black-white lattice
• Cosymmetries and Nijenhuis recursion operators for difference equations
• Symmetry algebra of discrete KdV equations and corresponding differential-difference equations of Volterra type
• Infinitely many symmetries and conservation laws for quad-graph equations via the Gardner method
• Peculiar symmetry structure of some known discrete nonautonomous equations
• A new recursion operator for the Viallet equation
• Continuous symmetric reductions of the Adler-Bobenko-Suris equations
• A systematic approach to reductions of type-Q ABS equations
• A non autonomous generalization of the $Q_{\text{V}}$ equation