Numerical studies of the fourth Painlevé equation

Bassom, Andrew P.; Clarkson, Peter A.; Hicks, A. C.

doi:10.1093/imamat/50.2.167

Cited by 31 publications

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“…When ν is a non-negative integer it has been proved that (1.6) admits exact solutions which decay exponentially as |ξ | → ∞ [18][19][20], which are nonlinear analogues of the bound-state solutions for the linear harmonic oscillator. Clarkson and Cosgrove [21] have shown that a symmetry reduction exists from the derivative nonlinear Schrödinger equation which reduces it to the nonlinear harmonic oscillator (1.6).…”

Section: (K; Z)s(z) (13)mentioning

confidence: 99%

A New Polarimeter for Dilute Aqueous Solutions: The Improved Null-Method Compensation for the Optical Rotatory Power by the Faraday Effect of a Sample

Mitsui

Kinugawa

Sakurai

1999

Jpn. J. Appl. Phys.

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In this paper we study the fourth Painlevé equation and how the concept of isomonodromy may be used to elucidate properties of its solutions. This work is based on a Lax pair which is derived from an inverse scattering formalism for a derivative nonlinear Schrödinger system, which in turn possesses a symmetry reduction that reduces it to the fourth Painlevé equation. It is shown how the monodromy data of our Lax pair can be explicitly computed in a number of cases and the relationships between special solutions of the monodromy equations and particular integrals of the fourth Painlevé equation are discussed. We use a gauge transformation technique to derive Bäcklund transformations from our Lax pair and generalize the findings to examine particular solutions and Bäcklund transformations of a related nonlinear harmonic oscillator equation.

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Section: (K; Z)s(z) (13)mentioning

confidence: 99%

A New Polarimeter for Dilute Aqueous Solutions: The Improved Null-Method Compensation for the Optical Rotatory Power by the Faraday Effect of a Sample

Mitsui

Kinugawa

Sakurai

1999

Jpn. J. Appl. Phys.

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“…What we have developed here is the discrete analogue of the simplest solution in the complementary error function hierarchy; furthermore, this is the discrete analogue of the most elementary of the "bound-state" solutions of PIV derived in [24] (see also [25]), which themselves form a special case complementary error function hierarchy. This technique can be adapted to find discrete counterparts to further exact solutions within the complementary error function hierarchy.…”

Section: Discrete Complementary Error Function Solutionsmentioning

confidence: 99%

New exact solutions of the discrete fourth Painlevé equation

Bassom

1994

Physics Letters A

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“…The fourth Painlevé equation has been studied from various perspectives: see, e.g., [1,4,7,8,10,11,13,14,16,17]. However, the study of asymptotic behaviors in the limit |x| → ∞ for x ∈ C appears to be incomplete in the literature.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Behavior of the Fourth Painlevé Transcendents in the Space of Initial Values

Joshi

Radnović

2016

Constr Approx

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We study the asymptotic behavior of solutions of the fourth Painlevé equation as the independent variable goes to infinity in its space of (complex) initial values, which is a generalization of phase space described by Okamoto. We show that the limit set of each solution is compact and connected and, moreover, that any solution that is not rational has an infinite number of poles and infinite number of zeros.

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Numerical studies of the fourth Painlevé equation

Cited by 31 publications

References 0 publications

A New Polarimeter for Dilute Aqueous Solutions: The Improved Null-Method Compensation for the Optical Rotatory Power by the Faraday Effect of a Sample

A New Polarimeter for Dilute Aqueous Solutions: The Improved Null-Method Compensation for the Optical Rotatory Power by the Faraday Effect of a Sample

New exact solutions of the discrete fourth Painlevé equation

Asymptotic Behavior of the Fourth Painlevé Transcendents in the Space of Initial Values

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