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.
\Bibitem{RogCla17}
\by Colin~Rogers, Peter~A.~Clarkson
\paper Ermakov--Painlev\'{e}~II Symmetry Reduction of a Korteweg Capillarity System
\jour SIGMA
\yr 2017
\vol 13
\papernumber 018
\totalpages 20
\mathnet{http://mi.mathnet.ru/sigma1218}
\crossref{https://doi.org/10.3842/SIGMA.2017.018}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85016604463}
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This publication is cited in the following 8 articles:
Pablo Amster, Colin Rogers, “On a Dirichlet boundary value problem for an Ermakov–Painlevé I equation. A Hamiltonian EPI system”, Electron. J. Qual. Theory Differ. Equ., 2023, no. 23, 1
Colin Rogers, “Moving boundary problems for a canonical member of the WKI inverse scattering scheme: conjugation of a reciprocal and Möbius transformation”, Phys. Scr., 97:9 (2022), 095207
Clarkson P.A. Gomez-Ullate D. Grandati Y. Milson R., “Cyclic Maya Diagrams and Rational Solutions of Higher Order Painleve Systems”, Stud. Appl. Math., 144:3 (2020), 357–385
C. Rogers, W. K. Schief, B. Malomed, “On modulated coupled systems. Canonical reduction via reciprocal transformations”, Commun. Nonlinear Sci. Numer. Simul., 83 (2020), 105091
C. Rogers, “Reciprocal Gausson phenomena in a Korteweg capillarity system”, Meccanica, 54:10 (2019), 1515–1523
P. Amster, C. Rogers, “On a Neumann boundary value problem for Ermakov-Painleve III”, Electron. J. Qual. Theory Differ., 2019, no. 69, 1–10
C. Rogers, “On modulated multi-component nls systems: Ermakov invariants and integrable symmetry reduction”, Ric. Mat., 68:2 (2019), 615–627
Rogers C., Chow K., “On Modulated NLS-Ermakov Systems”, J. Nonlinear Math. Phys., 24:1, SI (2017), 61–74
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