Total integrals of Ablowitz-Segur solutions for the inhomogeneous Painlevé II equation

Abstract

In this paper, we establish a formula determining the value of the Cauchy integrals of the real and purely imaginary Ablowitz-Segur solutions for the inhomogeneous second Painlevé equation. Our approach relies on the analysis of the corresponding Riemann-Hilbert problem and the construction of an appropriate parametrix in a neighborhood of the origin. Obtained integral formulas are consistent with already known analogous results for Ablowitz-Segur solutions of homogeneous Painlevé II equation.

The results concerning applications to fluid dynamics are moved to forthcoming publication