Miura type transformations and homogeneous spaces

Abstract

We relate Miura type transformations (MTs) over an evolution system to its zero-curvature representations with values in Lie algebras g. We prove that certain homogeneous spaces of g produce MTs and show how to distinguish these spaces. For a scalar translation-invariant evolution equation this allows to classify all MTs in terms of homogeneous spaces of the Wahlquist-Estabrook algebra of the equation. For other evolution systems this allows to construct some MTs. As an example, we study MTs over the KdV equation, a 5th order equation of Harry-Dym type, and the coupled KdV-mKdV system of Kersten and Krasilshchik.

17 pages; v3, v2: minor improvements

References in corpus (2)

• Coverings and fundamental algebras for partial differential equations
• Integrability of Kersten-Krasil'shchik coupled KdV-mKdV equations: singularity analysis and Lax pair

Cited by in corpus (5)

• Miura-type transformations for lattice equations and Lie group actions associated with Darboux-Lax representations
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Generalized Landau-Lifshitz systems and Lie algebras associated with higher genus curves
• Simplifications of Lax pairs for differential-difference equations by gauge transformations and (doubly) modified integrable equations
• On continual classes of evolution equations