Clarkson-McLeod solutions of the fourth Painlev\'e equation and the parabolic cylinder-kernel determinant

TL;DR

This paper analyzes Clarkson-McLeod solutions of the fourth Painlevé equation, deriving their asymptotics, total integrals, and a determinantal representation, thus confirming a conjecture and connecting solutions to parabolic cylinder functions.

Contribution

It provides the first rigorous asymptotic analysis of Clarkson-McLeod solutions as x approaches negative infinity and establishes a determinantal formula linking these solutions to parabolic cylinder kernels.

Findings

Asymptotic behaviors as x→−∞ are derived using the Deift-Zhou method.

Total integrals of the solutions are computed.

A determinantal representation via an integrable operator is established.

Abstract

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