### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 018, 20 pages      arXiv:1701.03238      https://doi.org/10.3842/SIGMA.2017.018

### Ermakov-Painlevé II Symmetry Reduction of a Korteweg Capillarity System

Colin Rogers a and Peter A. Clarkson b
a) Australian Research Council Centre of Excellence for Mathematics & Statistics of Complex Systems, School of Mathematics, The University of New South Wales, Sydney, NSW2052, Australia
b) School of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury, CT2 7FS, UK

Received January 13, 2017, in final form March 15, 2017; Published online March 22, 2017

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.

Key words: Ermakov-Painlevé II equation; Painlevé capillarity; Korteweg-type capillary system; Bäcklund transformation.

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