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.

中文翻译：

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非齐次 Painlevé II 方程的 Ablowitz-Segur 解的总积分

在本文中，我们建立了一个公式，用于确定非齐次第二 Painlev\'e 方程的实部和纯虚部 Ablowitz-Segur 解的柯西积分值。我们的方法依赖于对相应 Riemann-Hilbert 问题的分析以及在原点邻域中构建适当的参数。获得的积分公式与齐次 Painlev\'e II 方程的 Ablowitz-Segur 解的已知类似结果一致。