arXiv subjects

Peter A Clarkson

Publications and source records attributed to Peter A Clarkson.

5 recordsLinked to original sources

arXiv2017

Properties of Generalized Freud Polynomials

We consider the semi-classical generalized Freud weight function \[w_λ(x;t) = |x|^{2λ+1}\exp(-x^4 +tx^2),\qquad x\in\mathbb{R},\] with $ λ>-1$ and $t\in\mathbb{R}$ parameters. We analyze the asymptotic behavior of the sequences of monic polynomials that are orthogonal with respect to $w_λ(x;t)$, as well as the asymptotic behavior of the recurrence coefficient, when the degree, or alternatively, the parameter $t$, tend to infinity. We also investigate existence and uniqueness of positive solutions of the nonlinear difference equation satisfied by the recurrence coefficients and prove properties of the zeros of the generalized Freud polynomials.

nlin.SI↗

arXiv2016

On Airy Solutions of the Second Painlevé Equation

In this paper we discuss Airy solutions of the second Painlevé equation (\mbox{\rm P$_{\rm II}$}) and two related equations, the Painlevé XXXIV equation ($\mbox{\rm P}_{34}$) and the Jimbo-Miwa-Okamoto $σ$ form of \mbox{\rm P$_{\rm II}$}\ (\mbox{\rm S$_{\rm II}$}), are discussed. It is shown that solutions which depend only on the Airy function $\mathop{\rm Ai}\nolimits(z)$ have a completely difference structure to those which involve a linear combination of the Airy functions $\mathop{\rm Ai}\nolimits(z)$ and $\mathop{\rm Bi}\nolimits(z)$. For all three equations, the special solutions which depend only on $\mathop{\rm Ai}\nolimits(t)$ are \textit{tronquée} solutions, i.e.\ they have no poles in a sector of the complex plane. Further for both $\mbox{\rm P}_{34}$\ and \mbox{\rm S$_{\rm II}$}, it is shown that amongst these \textit{tronquée} solutions there is a family of solutions which have no poles on the real axis.

nlin.SI↗

arXiv2013

Recurrence coefficients for discrete orthonormal polynomials and the Painlevé equations

We investigate semi-classical generalizations of the Charlier and Meixner polynomials, which are discrete orthogonal polynomials that satisfy three-term recurrence relations. It is shown that the coefficients in these recurrence relations can be expressed in terms of Wronskians of modified Bessel functions and confluent hypergeometric functions, respectively for the generalized Charlier and generalized Meixner polynomials. These Wronskians arise in the description of special function solutions of the third and fifth Painlevé equations.

nlin.SI↗

arXiv2009

Special Polynomials and Exact Solutions of the Dispersive Water Wave and Modified Boussinesq Equations

Exact solutions of the dispersive and modified equations are expressed in terms of special polynomials associated with rational solutions of the fourth Painleve equation, which arises as generalized scaling reductions of these equations. Generalized solutions that involve an infinite sequence of arbitrary constants are also derived which are analogues of generalized rational solutions for the Korteweg-de Vries, Boussinesq and nonlinear Schrodinger equations

nlin.SI↗

arXiv2009

Vortices and Polynomials

The relationship between point vortex dynamics and the properties of polynomials with roots at the vortex positions is discussed. Classical polynomials, such as the Hermite polynomials, have roots that describe the equilibria of identical vortices on the line. Stationary and uniformly translating vortex configurations with vortices of the same strength but positive or negative orientation are given by the zeros of the Adler-Moser polynomials, which arise in the description of rational solutions of the Korteweg-de Vries equation. For quadupole background flow, vortex configurations are given by the zeros of polynomials expressed as wronskians of Hermite polynomials. Further new solutions are found in this case using the special polynomials arising the in the description of rational solutions of the fourth Painleve equation.

nlin.PS↗