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On the Increasing Tritronquée Solutions of the Painlevé-II Equation
Symmetry, integrability and geometry: methods and applications, Tome 14 (2018) Cet article a éte moissonné depuis la source Math-Net.Ru

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The increasing tritronquée solutions of the Painlevé-II equation with parameter $\alpha$ exhibit square-root asymptotics in the maximally-large sector $|\arg(x)|\tfrac{2}{3}\pi$ and have recently appeared in applications where it is necessary to understand the behavior of these solutions for complex values of $\alpha$. Here these solutions are investigated from the point of view of a Riemann–Hilbert representation related to the Lax pair of Jimbo and Miwa, which naturally arises in the analysis of rogue waves of infinite order. We show that for generic complex $\alpha$, all such solutions are asymptotically pole-free along the bisecting ray of the complementary sector $|\arg(-x)|\tfrac{1}{3}\pi$ that contains the poles far from the origin. This allows the definition of a total integral of the solution along the axis containing the bisecting ray, in which certain algebraic terms are subtracted at infinity and the poles are dealt with in the principal-value sense. We compute the value of this integral for all such solutions. We also prove that if the Painlevé-II parameter $\alpha$ is of the form $\alpha=\pm\tfrac{1}{2}+\mathrm{i} p$, $p\in\mathbb{R}\setminus\{0\}$, one of the increasing tritronquée solutions has no poles or zeros whatsoever along the bisecting axis.
Mots-clés : Painlevé-II equation; tronquée solutions. 
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     author = {Peter D. Miller},
     title = {On the {Increasing} {Tritronqu\'ee} {Solutions} of the {Painlev\'e-II} {Equation}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a124/}
}
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Peter D. Miller. On the Increasing Tritronquée Solutions of the Painlevé-II Equation. Symmetry, integrability and geometry: methods and applications, Tome 14 (2018). http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a124/

[1] Baik J., Buckingham R., DiFranco J., Its A., “Total integrals of global solutions to Painlevé II”, Nonlinearity, 22 (2009), 1021–1061, arXiv: 0810.2586 | DOI | MR | Zbl

[2] Bertola M., Tovbis A., “Universality for the focusing nonlinear {S}chrödinger equation at the gradient catastrophe point: rational breathers and poles of the tritronquée solution to Painlevé I”, Comm. Pure Appl. Math., 66 (2013), 678–752, arXiv: 1004.1828 | DOI | MR | Zbl

[3] Bilman D., Ling L., Miller P. D., Extreme superposition: rogue waves of infinite order and the Painlevé-III hierarchy, arXiv: 1806.00545

[4] Bothner T., “Transition asymptotics for the Painlevé II transcendent”, Duke Math. J., 166 (2017), 205–324, arXiv: 1502.03402 | DOI | MR | Zbl

[5] Bothner T., Miller P. D., Rational solutions of the Painlevé-III equation: large parameter asymptotics, arXiv: 1808.01421

[6] Bothner T., Miller P. D., Sheng Y., “Rational solutions of the Painlevé-III equation”, Stud. Appl. Math., 141 (2018), 626–679, arXiv: 1801.04360 | DOI

[7] Boutroux P., “Recherches sur les transcendantes de {M} {P}ainlevé et l'étude asymptotique des équations différentielles du second ordre”, Ann. Sci. École Norm. Sup., 30 (1913), 255–375 | DOI | MR | Zbl

[8] Boutroux P., “Recherches sur les transcendantes de M Painlevé et l'étude asymptotique des équations différentielles du second ordre (suite)”, Ann. Sci. École Norm. Sup., 31 (1914), 99–159 | DOI | MR | Zbl

[9] Buckingham R. J., Miller P. D., “The sine-Gordon equation in the semiclassical limit: critical behavior near a separatrix”, J. Anal. Math., 118 (2012), 397–492, arXiv: ; Corrigenda, J. Anal. Math., 119 (2013), 403–405 1106.5716 | DOI | MR | Zbl | DOI | MR | Zbl

[10] Buckingham R. J., Miller P. D., “Large-degree asymptotics of rational Painlevé-II functions: noncritical behaviour”, Nonlinearity, 27 (2014), 2489–2578, arXiv: 1310.2276 | DOI | MR

[11] Buckingham R. J., Miller P. D., “Large-degree asymptotics of rational Painlevé-II functions: critical behaviour”, Nonlinearity, 28 (2015), 1539–1596, arXiv: 1406.0826 | DOI | MR | Zbl

[12] Claeys T., Grava T., “Painlevé II asymptotics near the leading edge of the oscillatory zone for the Korteweg–de Vries equation in the small-dispersion limit”, Comm. Pure Appl. Math., 63 (2010), 203–232, arXiv: 0812.4142 | DOI | MR | Zbl

[13] Clarkson P. A., “On Airy solutions of the second Painlevé equation”, Stud. Appl. Math., 137 (2016), 93–109, arXiv: 1510.08326 | DOI | MR | Zbl

[14] Deift P. A., Zhou X., “Asymptotics for the Painlevé II equation”, Comm. Pure Appl. Math., 48 (1995), 277–337 | DOI | MR | Zbl

[15] Dubrovin B., Grava T., Klein C., “On universality of critical behavior in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the tritronquée solution to the Painlevé-I equation”, J. Nonlinear Sci., 19 (2009), 57–94, arXiv: 0704.0501 | DOI | MR | Zbl

[16] Flaschka H., Newell A. C., “Monodromy- and spectrum-preserving deformations. I”, Comm. Math. Phys., 76 (1980), 65–116 | DOI | MR | Zbl

[17] Fokas A. S., Its A. R., Kapaev A. A., Novokshenov V. Yu., Painlevé transcendents: the Riemann–Hilbert approach, Mathematical Surveys and Monographs, 128, Amer. Math. Soc., Providence, RI, 2006 | DOI | MR

[18] Gambier B., “Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est a points critiques fixes”, Acta Math., 33 (1910), 1–55 | DOI | MR

[19] Jimbo M., Miwa T., “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II”, Phys. D, 2 (1981), 407–448 | DOI | MR | Zbl

[20] Joshi N., Mazzocco M., “Existence and uniqueness of tri-tronquée solutions of the second Painlevé hierarchy”, Nonlinearity, 16 (2003), 427–439, arXiv: math.CA/0212117 | DOI | MR | Zbl

[21] Kapaev A. A., “Asymptotic expressions for the second Painlevé functions”, Theoret. and Math. Phys., 77 (1988), 1227–1234 | DOI | MR | Zbl

[22] Kapaev A. A., “Global asymptotics of the second Painlevé transcendent”, Phys. Lett. A, 167 (1992), 356–362 | DOI | MR

[23] Kapaev A. A., “Scaling limits in the second Painlevé transcendent”, J. Math. Sci., 83 (1997), 38–61 | DOI | MR

[24] Lu B.-Y., Miller P. D., Universality near the gradient catastrophe point in the semiclassical sine-Gordon equation, in preparation

[25] Novokshenov V. Yu., “Tronquée solutions of the Painlevé II equation”, Theoret. and Math. Phys., 172 (2012), 1136–1146 | DOI | MR | Zbl

[26] Olver F. W. J., Olde Daalhuis A. B., Lozier D. W., Schneider B. I., Boisvert R. F., Clark C. W., Miller B. R., Saunders B. V. (eds), NIST digital library of mathematical functions, Release 1.0.17, , 2017 http://dlmf.nist.gov/

[27] Olver S., RHPackage, Version 0.4, , 2011 https://github.com/dlfivefifty/RHPackage

[28] Tracy C. A., Widom H., “Level-spacing distributions and the Airy kernel”, Comm. Math. Phys., 159 (1994), 151–174, arXiv: hep-th/9211141 | DOI | MR | Zbl

[29] Zhou X., “The Riemann–Hilbert problem and inverse scattering”, SIAM J. Math. Anal., 20 (1989), 966–986 | DOI | MR | Zbl