Ermakov-Painlevé II Symmetry Reduction of a Korteweg Capillarity System

Abstract

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 multiparameter 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.

Moving boundary problems for a canonical member of the WKI inverse scattering scheme: conjugation of a reciprocal and Möbius transformation

“…Hybrid Ermakov-Painlevé systems have been derived via symmetry reduction of solitonic models in nonlinear elastodynamics [82], Korteweg capillarity theory [83] and cold plasma physics [84]. Two-point Dirichlet boundary value problems for a particular Ermakov-Painlevé II reduction arising out of a Nernst-Planck three-ion electrodiffusion have been treated in [85].…”

Cyclic Maya diagrams and rational solutions of higher order Painlevé systems

“…The special polynomials associated with rational solutions of the Painlevé equations arise in several applications: 1.the Yablonskii‐Vorob'ev polynomials arise in the transition behavior for the semiclassical sine‐Gordon equation, in boundary value problems, in moving boundary problems, and in symmetry reductions of a Korteweg capillarity system and cold plasma physics; 2.the generalized Hermite polynomials arise as multiple integrals in random matrix theory, in supersymmetric quantum mechanics, in the description of vortex dynamics with quadrupole background flow, and as coefficients of recurrence relations for orthogonal polynomials; …”

“…1. the Yablonskii-Vorob'ev polynomials arise in the transition behavior for the semiclassical sine-Gordon equation, 19 in boundary value problems, 20 in moving boundary problems, [21][22][23] and in symmetry reductions of a Korteweg capillarity system 24 and cold plasma physics; 25 2. the generalized Hermite polynomials arise as multiple integrals in random matrix theory, 26 in supersymmetric quantum mechanics, [27][28][29][30] in the description of vortex dynamics with quadrupole background flow, 31 and as coefficients of recurrence relations for orthogonal polynomials 32-34 ; 3. the generalized Okamoto polynomials arise in supersymmetric quantum mechanics 29 and generate rational-oscillatory solutions of the de-focusing nonlinear Schrödinger equation 35 ; and 4. the Umemura polynomials arise as multivortex solutions of the complex sine-Gordon equation, 36,37 and in multiple-input multiple-output wireless communication systems. 38 A very successful approach in the study of rational solutions to Painlevé equations has been through the geometric methods developed by the Japanese school, most notably by Noumi and Yamada.…”

“…(i) the Yablonskii-Vorob'ev polynomials arise in the transition behaviour for the semi-classical sine-Gordon equation [14], in boundary value problems [8], in moving boundary problems [65][66][67], and in symmetry reductions of a Korteweg capillarity system [68] and cold plasma physics [69]; (ii) the generalized Hermite polynomials arise as multiple integrals in random matrix theory [31],…”

An constructive proof for the Umemura polynomials for the third Painlevé equation

“…• the Yablonskii-Vorob'ev polynomials arise in the transition behaviour for the semi-classical sine-Gordon equation [13]; in boundary boundary problems [3]; in moving boundary problems [79,80,81]; and in Ermakov-Painlevé symmetry reductions of a Korteweg capillarity system [82] and cold plasma Physics [83];…”