### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 051, 10 pages      arXiv:1705.00518      https://doi.org/10.3842/SIGMA.2017.051
Contribution to the Special Issue on Symmetries and Integrability of Difference Equations

### Self-Dual Systems, their Symmetries and Reductions to the Bogoyavlensky Lattice

Allan P. Fordy a and Pavlos Xenitidis b
a) School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
b) School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7FS, UK

Received May 01, 2017, in final form June 26, 2017; Published online July 06, 2017

Abstract
We recently introduced a class of ${\mathbb{Z}}_N$ graded discrete Lax pairs and studied the associated discrete integrable systems (lattice equations). In particular, we introduced a subclass, which we called ''self-dual''. In this paper we discuss the continuous symmetries of these systems, their reductions and the relation of the latter to the Bogoyavlensky equation.

Key words: discrete integrable system; Lax pair; symmetry; Bogoyavlensky system.

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References

1. Bogoyavlensky O.I., Integrable discretizations of the KdV equation, Phys. Lett. A 134 (1988), 34-38.
2. Fordy A.P., Xenitidis P., $\mathbb{Z}_N$ graded discrete Lax pairs and discrete integrable systems, arXiv:1411.6059.
3. Fordy A.P., Xenitidis P., ${\mathbb Z}_N$ graded discrete Lax pairs and integrable difference equations, J. Phys. A: Math. Theor. 50 (2017), 165205, 30 pages.
4. Marì Beffa G., Wang J.P., Hamiltonian evolutions of twisted polygons in ${\mathbb{RP}}^n$, Nonlinearity 26 (2013), 2515-2551, arXiv:1207.6524.
5. Mikhailov A.V., Xenitidis P., Second order integrability conditions for difference equations: an integrable equation, Lett. Math. Phys. 104 (2014), 431-450, arXiv:1305.4347.
6. Yamilov R., Symmetries as integrability criteria for differential difference equations, J. Phys. A: Math. Gen. 39 (2006), R541-R623.