On the discrete and continuous Miura Chain associated with the Sixth Painlevé Equation
arXiv:solv-int/9906006 · doi:10.1016/S0375-9601(99)00764-1
Abstract
A Miura chain is a (closed) sequence of differential (or difference) equations that are related by Miura or Bäcklund transformations. We describe such a chain for the sixth Painlevé equation (\pvi), containing, apart from \pvi itself, a Schwarzian version as well as a second-order second-degree ordinary differential equation (ODE). As a byproduct we derive an auto-Bäcklund transformation, relating two copies of \pvi with different parameters. We also establish the analogous ordinary difference equations in the discrete counterpart of the chain. Such difference equations govern iterations of solutions of \pvi under Bäcklund transformations. Both discrete and continuous equations constitute a larger system which include partial difference equations, differential-difference equations and partial differential equations, all associated with the lattice Korteweg-de Vries equation subject to similarity constraints.
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References in corpus (2)
Cited by in corpus (6)
- On a Schwarzian PDE associated with the KdV Hierarchy
- First degree birational transformations of the Painlevé equations and their contiguity relations
- New contiguity relation of the sixth Painlevé equation from a truncation
- Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation
- Discrete Painlevé equations and their Lax pairs as reductions of integrable lattice equations
- On the Schlesinger transformations of the sixth Painlevé equation