On Hybrid Ermakov-Painlevé Systems. Integrable Reduction

Abstract

Hybrid Ermakov-Painlevé II-IV systems are introduced here in a unified manner. Their admitted Ermakov invariants together with associated canonical Painlevé equations are used to establish integrability properties.

On Modulated NLS-Ermakov Systems

“…Thus, the single component modulated NLS equation (42) is amenable to methods of modern soliton theory such as the inverse scattering procedure as well as admitting geometric properties associated with invariance under a Bäcklund transformation with its concomitant nonlinear superposition principle. Here, the modulated two-component NLS system (4.1) is shown to admit symmetry reduction via (2.2) to an integrable hybrid Ermakov-Painlevé II system as recently set down in [39]. Thus, on introduction of the wave packet representations (2.2) into (4.1), a Ermakov-Painlevé II system results, namely…”

“…Ermakov-Painlevé IV systems of the type (4.30) have been recently introduced in a more general setting in [39]. It is readily shown that, in this case, Σ = q 2 1 + q 2 2 is governed by the canonical integrable Painlevé IV equation…”

Coalescence, deformation and Bäcklund symmetries of Painlevé IV and II equations

“…Hybrid Painlevé equations have been a focus of several papers, e.g. [14,20]. More recently, in reference [3] we introduced the hybrid P III-V model that was obtained as reduction of a class of integrable models known as multi-boson systems [6,7] that generalize the AKNS hierarchy [5].…”