Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation

Abstract

We identify a periodic reduction of the non-autonomous lattice potential Korteweg-de Vries equation with the additive discrete Painlevé equation with $E_6^{(1)}$ symmetry. We present a description of a set of symmetries of the reduced equations and their relations to the symmetries of the discrete Painlevé equation. Finally, we exploit the simple symmetric form of the reduced equations to find rational and hypergeometric solutions of this discrete Painlevé equation.

References in corpus (8)

• Lax forms of the $q$-Painlevé equations
• Non-commutative lattice modified Gel'fand-Dikii systems
• Discrete Schlesinger Transformations, their Hamiltonian Formulation, and Difference Painlevé Equations
• Construction of a Lax Pair for the $E_6^{(1)}$ $q$-Painlevé System
• Lax formalism for q-Painleve equations with affine Weyl group symmetry of type E^{(1)}_n
• Integrability of reductions of the discrete KdV and potential KdV equations
• The Lattice Structure of Connection Preserving Deformations for q-Painlevé Equations I
• A systematic approach to reductions of type-Q ABS equations

Cited by in corpus (6)

• Commutation Relations and Discrete Garnier Systems
• Geometric Reductions of ABS equations on an $n$-cube to discrete Painlevé systems
• Lattice equations arising from discrete Painlevé systems (I): $(A_2+A_1{)}^{(1)}$ and $(A_1+A_1^{\prime }{)}^{(1)}$ cases
• Lax pairs of discrete Painlevé equations: $(A_2+A_1{)}^{(1)}$ case
• Lattice equations arising from discrete Painlevé systems. II. $A_4^{(1)}$ case
• Direct Connection between the R_{II} Chain and the Nonautonomous Discrete Modified KdV Lattice