Ermakov-Painleve II Symmetry Reduction of a Korteweg Capillarity System

A class of nonlinear Schrodinger 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-Painleve II equation which is linked, in turn, to the integrable Painleve XXXIV equation. A nonlinear Schrodinger encapsulation of a Korteweg-type capillary system is thereby used in the isolation of such a Ermakov-Painleve II reduction valid for a multiparameter class of free energy functions. Iterated application of a Backlund 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 Painleve 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.