Singular asymptotics for the Clarkson-McLeod solutions of the fourth Painlev\'e equation

TL;DR

This paper analyzes the asymptotic behavior of Clarkson-McLeod solutions to the fourth Painlevé equation, revealing their pole structure and extending previous conjectures using advanced mathematical techniques.

Contribution

It provides the first detailed singular asymptotics of these solutions as x approaches negative infinity, confirming and extending Clarkson and McLeod's conjecture.

Findings

Solutions have infinitely many poles on the negative real axis for certain parameters.

Explicit connection formulas between asymptotic regimes are derived.

The asymptotic analysis confirms the pole distribution conjecture.

Abstract

We consider the Clarkson-McLeod solutions of the fourth Painlev\'e equation. This family of solutions behave like $\kappa D_{\alpha-\frac{1}{2}}^2(\sqrt{2}x)$ as $x\rightarrow +\infty$, 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 $\kappa \left( \kappa -\kappa ^*\right )>0$ for some real constant $\kappa ^*$. The connection formulas are also explicitly evaluated. This proves and extends Clarkson and McLeod's conjecture that when the parameter $\kappa >\kappa ^*>0$, the Clarkson-McLeod solutions have infinitely many simple poles on the negative real axis.