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Clarkson-McLeod solutions of the fourth Painlev\'e equation and the parabolic cylinder-kernel determinant
- Source :
- J. Differ. Equ. 352 (2023) 249--307
- Publication Year :
- 2023
-
Abstract
- The Clarkson-McLeod solutions of the fourth Painlev\'e equation behave like $\kappa D_{\alpha-\frac{1}{2}}^2(\sqrt{2}x)$ as $x\rightarrow +\infty$, 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.<br />Comment: 52 pages, 11 figures
- Subjects :
- Mathematical Physics
30E15, 33E17, 34E05, 41A60
Subjects
Details
- Database :
- arXiv
- Journal :
- J. Differ. Equ. 352 (2023) 249--307
- Publication Type :
- Report
- Accession number :
- edsarx.2301.05807
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1016/j.jde.2022.12.027