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 κDα−122(2x) as x→+∞, where κ is some real constant and Dα−12(x) parabolic cylinder function. Using Deift-Zhou nonlinear steepest descent method, we derive asymptotic behaviors for this class x→−∞. This completes a proof Clarkson McLeod's conjecture on asymptotics family solutions. total integrals approximations σ-form are also derived. Furthermore, find determinantal representation via an integrable operator with kernel.