The Gambier Mapping, Revisited

Abstract

We examine critically the Gambier equation and show that it is the generic linearisable equation containing, as reductions, all the second-order equations which are integrable through linearisation. We then introduce the general discrete form of this equation, the Gambier mapping, and present conditions for its integrability. Finally, we obtain the reductions of the Gambier mapping, identify their integrable forms and compute their continuous limits.

11 pages, no figures, to be published in Physica A

References in corpus (2)

• The Gambier Mapping
• Again, Linearizable Mappings

Cited by in corpus (12)

• Generating Finite Dimensional Integrable Nonlinear Dynamical Systems
• Linearisable Mappings and the Low-Growth Criterion
• Dirac--Lie systems and Schwarzian equations
• Singularity confinement and algebraic integrability
• Again, Linearizable Mappings
• A Quasi-Lie Schemes Approach to Second-Order Gambier Equations
• Dynamical Studies of Equations from the Gambier Family
• Singularities and growth of higher order discrete equations
• Discrete systems related to some equations of the Painlevé-Gambier classification
• Detecting discrete integrability: the singularity approach
• Multiplicative equations related to the affine Weyl group E${}_8$
• Schlesinger Transformations for Linearisable Equations