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 κDα−122(2x) as x→+∞, where κ is some real constant and Dα−12(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→−∞. 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.