Ermakov-Painlevé II Symmetry Reduction of a Korteweg Capillarity System

A class of nonlinear Schr\"{o}dinger 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-Painlev\'{e} II equation which is linked, in turn, to the integrable Painlev\'{e} XXXIV equation. A nonlinear Schr\"{o}dinger encapsulation of a Korteweg-type capillary system is thereby used in the isolation of such a Ermakov-Painlev\'{e} II reduction valid for a multi-parameter class of free energy functions. Iterated application of a B\"{a}cklund 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 Painlev\'{e} 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.