Total integrals of Ablowitz-Segur solutions for the inhomogeneous Painlev\'e II equation

TL;DR

This paper derives explicit formulas for the total integrals of Ablowitz-Segur solutions to the inhomogeneous Painlevé II equation using Riemann-Hilbert analysis, extending known results from the homogeneous case.

Contribution

It provides the first explicit integral formulas for inhomogeneous Ablowitz-Segur solutions, expanding the understanding of their integral properties.

Findings

Derived formulas for Cauchy integrals of solutions

Confirmed consistency with homogeneous Painlevé II results

Enhanced analytical tools for inhomogeneous Painlevé equations

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 Painlev\'e 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 Painlev\'e II equation.