Location of Poles for the Hastings–McLeod Solution to the Second Painlevé Equation

Huang, Min; Xu, Shuai‐Xia; Zhang, Lun

doi:10.1007/s00365-015-9307-1

Cited by 13 publications

(14 citation statements)

References 32 publications

(61 reference statements)

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“…For example, the well-known Hastings-McLeod solution of P II (2) with α = 0 [27], which arises in numerous applications, is a tronquée solution with no poles on the real axis, cf. [28,29].…”

Section: Resultsmentioning

confidence: 99%

“…For example, the well-known Hastings-McLeod solution of P II (1.2) with α = 0 [27], which arises in numerous applications, is a tronquée solution with no poles on the real axis, cf. [28,29,30]. The special solutions q n (z; 0) and p n (z; 0) of P II and P 34 arise in recent study by Clarkson, Loureiro, and Van Assche [24] of the discrete system…”

Section: Discussionmentioning

confidence: 99%

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On Airy Solutions of the Second Painlevé Equation

Clarkson

2016

Studies in Applied Mathematics

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

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Section: Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

On Airy Solutions of the Second Painlevé Equation

Clarkson

2016

Studies in Applied Mathematics

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“…where s := (2m 2 ) 1/3 ln(r/r × ). This equation does admit a particular solution-namely the Hastings-McLeod solution b HM (see[9], up to a change of sign x → −x)-which asymptotically satisfies(46) b HM (s) ∼ thus displaying the correct behavior for r → 0. Asymptotic matching with the bulk solution (39) takes care of itself as r → r + × ; there the Hastings-McLeod solution becomes (47) B HM (r) ∼ 2 ln 1…”

mentioning

confidence: 99%

Optimal Heat Transfer and Optimal Exit Times

Marcotte¹,

Doering²,

Thiffeault³

et al. 2018

SIAM J. Appl. Math.

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A heat exchanger can be modeled as a closed domain containing an incompressible fluid. The moving fluid has a temperature distribution obeying the advection-diffusion equation, with zero temperature boundary conditions at the walls. Starting from a positive initial temperature distribution in the interior, the goal is to flux the heat through the walls as efficiently as possible. Here we consider a distinct but closely related problem, that of the integrated mean exit time of Brownian particles starting inside the domain. Since flows favorable to rapid heat exchange should lower exit times, we minimize a norm of the exit time. This is a time-independent optimization problem that we solve analytically in some limits, and numerically otherwise. We find an (at least locally) optimal velocity field that cools the domain on a mechanical time scale, in the sense that the integrated mean exit time is independent on molecular diffusivity in the limit of large-energy flows.

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“…However, recently, it was shown that all tritronquée solutions of the Painlevé‐I equation are actually analytic down to the origin in the asymptotically pole‐free sector, proving a conjecture of Dubrovin. See for related results on certain solutions of the Painlevé‐II equation . It is not known whether the tritronquée solutions

{\overset{W}{true}}^{\pm} false(ξ; m false)

of the Painlevé‐II equation are exactly pole‐free in the sector

- \frac{2}{3} π < Arg false(ξ false) < \frac{2}{3} π

.…”

Section: Numerical Observations and Formal Scaling Theorymentioning

confidence: 99%

Rational Solutions of the Painlevé‐III Equation

2018

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All of the six Painlevé equations except the first have families of rational solutions, which are frequently important in applications. The third Painlevé equation in generic form depends on two parameters m and n, and it has rational solutions if and only if at least one of the parameters is an integer. We use known algebraic representations of the solutions to study numerically how the distributions of poles and zeros behave as n∈Z increases and how the patterns vary with m∈C. This study suggests that it is reasonable to consider the rational solutions in the limit of large n∈Z with m∈C being an auxiliary parameter. To analyze the rational solutions in this limit, algebraic techniques need to be supplemented by analytical ones, and the main new contribution of this paper is to develop a Riemann–Hilbert representation of the rational solutions of Painlevé‐III that is amenable to asymptotic analysis. Assuming further that m is a half‐integer, we derive from the Riemann–Hilbert representation a finite dimensional Hankel system for the rational solution in which n∈Z appears as an explicit parameter.

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Location of Poles for the Hastings–McLeod Solution to the Second Painlevé Equation

Cited by 13 publications

References 32 publications

On Airy Solutions of the Second Painlevé Equation

On Airy Solutions of the Second Painlevé Equation

Optimal Heat Transfer and Optimal Exit Times

Rational Solutions of the Painlevé‐III Equation

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