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A connection formula for the Second Painlevé Transcendent

Part of the Lecture Notes in Mathematics book series (LNM,volume 964)

Abstract

We consider a particular case of the Second Painlevé Transcendent

$$y^{''} = xy + 2y^3$$

It is known that if y(x) ∼ kAi(x) as x → +∞, then if 0<k<1,

$$y(x) \sim d\left| x \right|^{ - {\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 4$}}} sin\{ \frac{2}{3}\left| x \right|3/2 - \frac{3}{4}d2\ell n\left| x \right| - c\} asx \to - \infty$$

where d(k) and c(k) are the connection formulae for this nonlinear ordinary differential equation.

The lecture shows that

$$d2(k) = - \pi - 1\ell n(1 - k2)$$

which confirms the numerical estimates of Abtowitz and Segar.

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References

  1. M.J. Ablowitz and H. Segur, "Asymptotic solutions of the Korteweg-de Vries equation", Stud. Appl. Math. 57 pp.13–44 (1977).

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  2. M.J. Ablowitz and H. Segur, "Exact solution of a Painlevé Transcendent", Phys. Rev. Lett. 38 pp.1103–1106 (1977).

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  3. S.P. Hastings and J.B. McLeod, "A Boundary Value Problem Associated with the Second Painlevé Transcendent and the Korteweg-de Vries equation", Arch. Rat. Mech. Anal. 73 pp.31–51 (1980).

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  4. E.L. Ince, "Ordinary Differential Equations", Dover (1944).

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© 1982 Springer-Verlag

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Clarkson, P.A., McLeod, J.B. (1982). A connection formula for the Second Painlevé Transcendent. In: Everitt, W., Sleeman, B. (eds) Ordinary and Partial Differential Equations. Lecture Notes in Mathematics, vol 964. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0064994

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  • DOI: https://doi.org/10.1007/BFb0064994

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11968-5

  • Online ISBN: 978-3-540-39561-4

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