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

The Clarkson-McLeod solutions of the fourth Painlevé equation behave like $\kappa D_{\alpha -\frac{1}{2}}^2(\sqrt{2}x)$ as $x\to +\mathrm{\infty }$, 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 $x\to -\mathrm{\infty }$. 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.