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We study the Hankel determinant generated by the Laguerre weight with jumpdiscontinuities at $t_k, k=1,\cdots,m$. By employing the ladder operatorapproach to establish Riccati equations, we show that$\sigma_n(t_1,\cdots,t_m)$, the logarithmic derivative of the $n$-dimensionalHankel determinant, satisfies a generalization of the $\sigma$-from ofPainlev\'{e} V equation. Through investigating the Riemann-Hilbert problem forthe associated orthogonal polynomials and via Lax pair, we express $\sigma_n$in terms of solutions of a coupled Painlev\'{e} V system. We also buildrelations between the auxiliary quantities introduced in the above two methods,which provides connections between the Riccati equations and Lax pair. Inaddition, when each $t_k$ tends to the hard edge of the spectrum and $n$ goesto $\infty$, the scaled $\sigma_n$ is shown to satisfy a generalizedPainlev\'{e} III equation.

Laguerre Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlev\'{e} V System