  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 Painlev\'{e} 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
Painlev\'{e} 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 Painlev\'{e}
II system.
