Singular asymptotics for the Clarkson-McLeod solutions of the fourth Painlevé equation

Abstract

We consider the Clarkson-McLeod solutions of the fourth Painlevé equation. This family of solutions behave like $\kappa D_{\alpha -\frac{1}{2}}^2(\sqrt{2}x)$ as $x\to +\infty$, where $\kappa$ is an arbitrary real constant and $D_{\alpha -\frac{1}{2}}(x)$ is the parabolic cylinder function. Using the Deift-Zhou nonlinear steepest descent method, we obtain the singular asymptotics of the solutions as $x\to -\infty$ when $\kappa \left(\kappa -{\kappa }^{\ast }\right)>0$ for some real constant ${\kappa }^{\ast }$. The connection formulas are also explicitly evaluated. This proves and extends Clarkson and McLeod's conjecture that when the parameter $\kappa >{\kappa }^{\ast }>0$, the Clarkson-McLeod solutions have infinitely many simple poles on the negative real axis.

24 pages, 9 figures, 1 table

Cited by in corpus (1)

• Clarkson-McLeod solutions of the fourth Painlevé equation and the parabolic cylinder-kernel determinant