Conditionally integrable PT -symmetric hierarchies and near-Lax pairs of the KdV and KPII equations

Abstract

Earlier work is generalized to obtain novel conditionally-integrable hierarchies of various PT -symmetric, nonlinear partial differential equations. The Painlevé Test yields the possible integrable cases. These are labeled by the integer n, the order of the dominant pole in the Laurent expansion for the solution. For the PT -symmetric Korteweg–de Vries (KdV) equation which has been considered earlier, further analysis using the extended homogeneous balance technique gives the near-Lax Pair. Next, a PT -Symmetric hierarchy of (2 + 1) Kadomtsev-Petviashvili equations is considered. The Painlevé Test and Invariant Painlevé analysis in (2 + 1) dimensions yield special solutions and Bäcklund Transformations for the integrable cases.