Recherches sur les transcendantes de M. Painlevé et l'étude asymptotique des équations différentielles du second ordre

Boutroux, Pierre

doi:10.24033/asens.661

Cited by 174 publications

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“…More detailed asymptotic expansions were noted already in [4]. However, (1.2) suffices for the present work.…”

mentioning

confidence: 94%

“…All together, we can thus combine four different types of solutions with the same width of the smooth band. Tronquée solutions with the right pole field extending far along the negative real axis were analyzed already in [4]. There does not appear to be any restrictions on where bands of this type can be placed in the left half-plane.…”

Section: 4mentioning

confidence: 99%

“…Introduction. The six Painlevé transcendents were discovered about 100 years ago [4], [13], [23], [24], and have since received extensive attention. They satisfy nonlinear second order ODEs, cannot be reduced to other types special functions, and obey the Painlevé property of all its solutions being free from movable branch points (however, the solutions may have movable poles or movable isolated essential singularities) [18], [21].…”

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confidence: 99%

“…u(z) is real-valued along the real axis), the solutions can form pole patterns similar to the very schematic illustrations in Figure 1.1. Further analytic motivations for these sectors, including their relations to the tronquée (truncated) cases and the regularities within the pole fields, go back to [4], and are also summarized in Section 12.3 of [16]. Within smooth areas, the left-hand side (LHS) of (1.1) becomes negligible for increasing |z|, and the two terms in the RHS will balance, leading to the estimate u(z) = ± − z 6 + o (1) .…”

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confidence: 99%

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A numerical methodology for the Painlevé equations

Fornberg

Weideman

2011

Journal of Computational Physics

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Abstract. The six Painlevé transcendents P I -P V I have both applications and analytic properties that make them stand out from most other classes of special functions. Although they have been the subject of extensive theoretical investigations for about a century, they still have a reputation for being numerically challenging. In particular, their extensive pole fields in the complex plane have often been perceived as 'numerical mine fields'. In the present work, we note that the Painlevé property in fact provides the opportunity for very fast and accurate numerical solutions throughout such fields. When combining a Taylor/Padé-based ODE initial value solver for the pole fields with a boundary value solver for smooth regions, numerical solutions become available across the full complex plane. We focus here on the numerical methodology, and illustrate it for the P I equation. In later studies, we will concentrate on mathematical aspects of both the P I and the higher Painlevé transcendents.Key words. Painlevé transcendents, P I equation, Taylor series method, Padé approximation, Chebyshev collocation method.1. Introduction. The six Painlevé transcendents were discovered about 100 years ago [4], [13], [23], [24], and have since received extensive attention. They satisfy nonlinear second order ODEs, cannot be reduced to other types special functions, and obey the Painlevé property of all its solutions being free from movable branch points (however, the solutions may have movable poles or movable isolated essential singularities) [18], [21]. They arise in many branches of physics, including classical and quantum spin models, scattering theory, self-similar solutions to nonlinear dispersive wave PDEs, string theory and random matrix theory. For a summary of their applications, with extensive references, we refer to Sections 32.13 -32.16 in the NIST Handbook of Mathematical Functions [21].The attention that the Painlevé functions have received over the last century has primarily been limited to their analytic properties and applications. The numerical computation of these functions, by contrast, has received comparatively little attention. With the exceptions of [8], [20], [22] (discussed later), numerical studies have so far mostly focused on pole-free intervals along the real line [17], [21], [25] or on a polefree region in the complex plane [11]. A possible explanation for this state of affairs is that the large pole fields in the complex plane associated with the Painlevé ODEs may be thought of as a challenge to standard numerical methods for the integration of ODEs.We argue in the present work that these pole fields provide, in fact, especially advantageous integration opportunities. A 'pole friendly' algorithm based on Padé approximation is devised that is capable of integrating accurately and stably through pole fields for distances in the 10 5 − 10 6 range. This allows numerical studies not only of 'near-fields', but also of transition processes towards 'far fields', for which an extensive body of asymptotic...

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“…More detailed asymptotic expansions were noted already in [4]. However, (1.2) suffices for the present work.…”

mentioning

confidence: 94%

Section: 4mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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A numerical methodology for the Painlevé equations

Fornberg

Weideman

2011

Journal of Computational Physics

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“…The only equations that have been studied with reasonable completeness are certain special equations whose general solution is meromorphic: Painleve equations (see [1] and [4]). The properties of meromorphic solutions of Painleve equations are investigated in detail in the classical paper [73]. We consider a very simple class of higher-order equations, for which a complete investigation of its meromorphic solutions is possible in certain cases.…”

Section: M)mentioning

confidence: 99%

Meromorphic solutions of algebraic differential equations

Erëmenko¹

1982

Russ. Math. Surv.

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Universality for the Focusing Nonlinear Schrödinger Equation at the Gradient Catastrophe Point: Rational Breathers and Poles of the Tritronquée Solution to Painlevé I

Bertola

Tovbis

2013

Comm Pure Appl Math

167

319

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The semiclassical (zero‐dispersion) limit of solutions $q=q(x,t,\epsilon)$ to the one‐dimensional focusing nonlinear Schrödinger equation (NLS) is studied in a scaling neighborhood D of a point of gradient catastrophe ($x_0,t_0$). We consider a class of solutions, specified in the text, that decay as $|x| \rightarrow \infty$. The neighborhood D contains the region of modulated plane wave (with rapid phase oscillations), as well as the region of fast‐amplitude oscillations (spikes). In this paper we establish the following universal behaviors of the NLS solutions q near the point of gradient catastrophe: (i) each spike has height $3|q{_0}(x_0,t_0)|$ and uniform shape of the rational breather solution to the NLS, scaled to the size ${\cal O}(\epsilon)$; (ii) the location of the spikes is determined by the poles of the tritronquée solution of the Painlevé I (P1) equation through an explicit map between D and a region of the Painlevé independent variable; (iii) if $(x,t)\in D$ but lies away from the spikes, the asymptotics of the NLS solution $q(x,t, \epsilon)$ is given by the plane wave approximation $q_0(x,t, \epsilon)$, with the correction term being expressed in terms of the tritronquée solution of P1. The relation with the conjecture of Dubrovin, Grava, and Klein about the behavior of solutions to the focusing NLS near a point of gradient catastrophe is discussed. We conjecture that the P1 hierarchy occurs at higher degenerate catastrophe points and that the amplitudes of the spikes are odd multiples of the amplitude at the corresponding catastrophe point. Our technique is based on the nonlinear steepest‐descent method for matrix Riemann‐Hilbert problems and discrete Schlesinger isomonodromic transformations. © 2013 Wiley Periodicals, Inc.

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Recherches sur les transcendantes de M. Painlevé et l'étude asymptotique des équations différentielles du second ordre

Cited by 174 publications

References 0 publications

A numerical methodology for the Painlevé equations

A numerical methodology for the Painlevé equations

Meromorphic solutions of algebraic differential equations

Universality for the Focusing Nonlinear Schrödinger Equation at the Gradient Catastrophe Point: Rational Breathers and Poles of the Tritronquée Solution to Painlevé I

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