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

Abstract

The Clarkson-McLeod solutions of the fourth Painlevé equation behave like $κD_{α-\frac{1}{2}}^2(\sqrt{2}x)$ as $x\rightarrow +\infty$, where $κ$ is some real constant and $D_{α-\frac{1}{2}}(x)$ 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 $σ$-form of this family of solutions are also derived. Furthermore, we find a determinantal representation of the $σ$-form of the Clarkson-McLeod solutions via an integrable operator with the parabolic cylinder kernel.