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

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.