We consider the Hankel determinant generated by the Gaussian weight with two jump discontinuities. Utilizing the results of [C. Min and Y. Chen, Math. Meth. Appl. Sci. {\bf 42} (2019), 301--321] where a second order PDE was deduced for the log derivative of the Hankel determinant by using the ladder operators adapted to orthogonal polynomials, we derive the coupled Painleve IV system which was established in [X. Wu and S. Xu, arXiv: 2002.11240v2] by a study of the Riemann-Hilbert problem for orthogonal polynomials. Under double scaling, we show that, as $n\rightarrow\infty$, the log derivative of the Hankel determinant in the scaled variables tends to the Hamiltonian of a coupled Painleve II system and it satisfies a second order PDE. In addition, we obtain the asymptotics for the recurrence coefficients of orthogonal polynomials, which are connected with the solutions of the coupled Painleve II system.

中文翻译：

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具有两个跳跃不连续性、偏微分方程以及耦合的 Painlevé II 和 IV 系统的高斯酉系综

我们考虑由具有两个跳跃不连续性的高斯权重生成的 Hankel 行列式。利用 [C. Min 和 Y. Chen，数学。冰毒。应用程序 科学。{\bf 42} (2019), 301--321] 其中通过使用适用于正交多项式的阶梯算子推导出 Hankel 行列式的对数导数的二阶偏微分方程，我们推导出耦合的 Painleve IV 系统，该系统在[X。Wu and S. Xu, arXiv: 2002.11240v2] 通过研究正交多项式的 Riemann-Hilbert 问题。在双标度下，我们表明，作为 $n\rightarrow\infty$，标度变量中 Hankel 行列式的对数导数趋向于耦合 Painleve II 系统的哈密顿量，并且它满足二阶偏微分方程。此外，我们获得了正交多项式的递推系数的渐近线，