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

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