Title: Ermakov-Painlevé II Symmetry Reduction of a Korteweg Capillarity System,
Abstract: A class of nonlinear Schrödinger equations involving a triad of power law
terms together with a de Broglie-Bohm potential is shown to admit symmetry
reduction to a hybrid Ermakov-Painlevé II equation which is linked, in turn,
to the integrable Painlevé XXXIV equation. A nonlinear Schrödinger
encapsulation of a Korteweg-type capillary system is thereby used in the
isolation of such a Ermakov-Painlevé II reduction valid for a multi-parameter
class of free energy functions. Iterated application of a Bäcklund
transformation then allows the construction of novel classes of exact solutions
of the nonlinear capillarity system in terms of Yablonskii-Vorob'ev polynomials
or classical Airy functions. A Painlevé XXXIV equation is derived for the
density in the capillarity system and seen to correspond to the symmetry
reduction of its Bernoulli integral of motion.