Ermakov-Painlevé II Symmetry Reduction of a Korteweg Capillarity System

Abstract

A class of nonlinear Schr\"odinger 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\"odinger 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\"acklund 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.

Locations

• Symmetry Integrability and Geometry Methods and Applications - View - PDF
• arXiv (Cornell University) - View - PDF
• The scientific electronic library of periodicals of the National Academy of Sciences of Ukraine (National Academy of Sciences of Ukraine) - View - PDF
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