Clarkson-McLeod solutions of the fourth Painlevé equation and the parabolic cylinder-kernel determinant. (English) Zbl 1509.30027
Summary: The Clarkson-McLeod solutions of the fourth Painlevé equation behave like \(\kappa D_{\alpha - \frac{1}{2}}^2 (\sqrt{2} x)\) as \(x \to +\infty\), where \(\kappa\) is some real constant and \(D_{\alpha - \frac{1}{2}} (x)\) is the parabolic cylinder function. Using the Deift-Zhou nonlinear steepest descent method, we derive the asymptotic behaviors for this class of solutions as \(x \to - \infty\). This completes a proof of Clarkson and McLeod’s conjecture on the asymptotics of this family of solutions. The total integrals of the Clarkson-McLeod solutions and the asymptotic approximations of the \(\sigma\)-form of this family of solutions are also derived. Furthermore, we find a determinantal representation of the \(\sigma\)-form of the Clarkson-McLeod solutions via an integrable operator with the parabolic cylinder kernel.
MSC:
| 30E25 | Boundary value problems in the complex plane |
| 30E15 | Asymptotic representations in the complex plane |
| 33E17 | Painlevé-type functions |
| 41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
Software:
DLMFReferences:
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