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

@article{Rogers2017ErmakovPainlevIS, title={Ermakov-Painlev{\'e} II Symmetry Reduction of a Korteweg Capillarity System}, author={Colin Rogers and Peter A. Clarkson}, journal={Symmetry Integrability and Geometry-methods and Applications}, year={2017}, volume={13}, pages={018} }

A class of nonlinear Schr\"{o}dinger 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\"{o}dinger 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… Expand

#### 9 Citations

Rational solutions of higher order Painlev\'{e} systems I

- Mathematics, Physics
- 2018

This is the first paper of a series whose aim is to reach a complete classification and an explicit representation of rational solutions to the higher order generalizations of $\textrm{PIV}$ and… Expand

Cyclic Maya diagrams and rational solutions of higher order Painlevé systems

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- 2018

This paper focuses on the construction of rational solutions for the $A_{2n}$ Painlev\'e system, also called the Noumi-Yamada system, which are considered the higher order generalizations of PIV. In… Expand

Reciprocal gausson phenomena in a Korteweg capillarity system

- Physics
- Meccanica
- 2019

In previous work in the literature, a kinetic derivation of a logarithmic nonlinear Schrodinger equation incorporating a de Broglie–Bohm term has been obtained in a capillarity context. Here,… Expand

On a Neumann boundary value problem for Ermakov–Painlevé III

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- Electronic Journal of Qualitative Theory of Differential Equations
- 2019

A Neumann-type boundary value problem is investigated for a hybrid Ermakov–Painlevé equation. Existence properties are established and a sequence of approximate solutions is investigated. In an… Expand

On modulated coupled systems. Canonical reduction via reciprocal transformations

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- 2020

Abstract It is shown how classes of modulated coupled systems of sine-Gordon, Demoulin and Manakov-type may be reduced to their unmodulated counterparts via the application of a novel reciprocal… Expand

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

- Mathematics
- 2016

We are concerned with the Umemura polynomials associated with the third Painlev\'e equation. We extend Taneda's method, which was developed for the Yablonskii-Vorob'ev polynomials associated with the… Expand

On modulated multi-component NLS systems: Ermakov invariants and integrable symmetry reduction

- Physics
- Ricerche di Matematica
- 2018

A multi-component, modulated NLS system is presented which admits symmetry reduction to a nonlinear subsystem that is shown to be integrable by application of its admitted Ermakov invariants and a… Expand

On Modulated NLS-Ermakov Systems

- Mathematics
- 2017

Spatial modulated coupled nonlinear Schrödinger systems with symmetry reduction to integrable Ermakov and Ermakov-Painlevé subsystems are investigated.

Electronic Journal of Qualitative Theory of Differential Equations

- 2019

We study some properties of the range of the relativistic pendulum operator P , that is, the set of possible continuous T-periodic forcing terms p for which the equation Px = p admits a T-periodic… Expand

#### References

SHOWING 1-10 OF 151 REFERENCES

The classical Korteweg capillarity system: geometry and invariant transformations

- Mathematics
- 2014

A class of invariant transformations is presented for the classical Korteweg capillarity system. The invariance is an extension of a kind originally introduced in an anisentropic gasdynamics context.… Expand

On generalized Loewner systems: Novel integrable equations in 2+1 dimensions

- Mathematics
- 1993

A reinterpretation and generalization of a class of infinitesimal Backlund transformations originally introduced in a gas‐dynamics context by Loewner in 1952 leads to a linear representation for a… Expand

Application of Uniform Asymptotics to the Second Painlevé Transcendent

- Mathematics, Physics
- 1996

Abstract. In this work we propose a new method for investigating connection problems for the class of nonlinear second‐order differential equations known as the Painlevé equations. Such problems can… Expand

Painlevé equations: nonlinear special functions

- Mathematics
- 2003

The six Painleve equations (PI-PVI) were first discovered about a hundred years ago by Painleve and his colleagues in an investigation of nonlinear second-order ordinary differential equations.… Expand

Large-degree asymptotics of rational Painlevé-II functions: Noncritical behaviour

- Mathematics, Physics
- 2014

This paper is a continuation of our analysis, begun in Buckingham and Miller (2014 Nonlinearity 27 2489–577), of the rational solutions of the inhomogeneous Painleve-II equation and associated… Expand

Backlund flux quantization in a model of electrodiffusion based on Painlevé II

- Physics, Mathematics
- 2012

A previously established model of steady one-dimensional two-ion electrodiffusion across a liquid junction is reconsidered. It involves three coupled first-order nonlinear ordinary differential… Expand

Large-degree asymptotics of rational Painlevé-II functions: critical behaviour

- Mathematics
- 2014

This paper is a continuation of our analysis, begun in Buckingham and Miller (2014 Nonlinearity 27 2489–577), of the rational solutions of the inhomogeneous Painleve-II equation and associated… Expand

Resonance NLS Solitons as Black Holes in Madelung Fluid

- Physics, Mathematics
- 1998

Envelope solitons of the Nonlinear Schrodinger equation (NLS) under quantum potential's influence are studied. Corresponding problem is found to be integrable for an arbitrary strength, s ≠ 1, of the… Expand

Monodromy- and spectrum-preserving deformations I

- Mathematics
- 1980

A method for solving certain nonlinear ordinary and partial differential equations is developed. The central idea is to study monodromy preserving deformations of linear ordinary differential… Expand

A boundary value problem associated with the second painlevé transcendent and the Korteweg-de Vries equation

- Mathematics
- 1980

AbstractThe differential equation considered is
$$y'' - xy = y|y|^\alpha $$
. For general positive α this equation arises in plasma physics, in work of De Boer & Ludford. For α=2, it yields… Expand