Sampling from Gaussian Copula with a conditional distribution approach

Question

In the paper "Cheng and al. (2007)" on pages 193-194, the authors propose an algorithm that generates variables with a given copula function $C$ being the joint distribution. This procedure is detailed in the book An Introduction to Copulas (1999) , by Roger B. Nelsen, on page 35-37.

The algorithm is the following

1. Generate two independent uniform (0,1) variates $u$ and $t$;

2. Set $v=c_{u}^{(-1)}(t)$, where $c_u^{(-1)}$ denotes a quas-inverse of $c_u(v)=P[V \leq v |U=u]=\frac{\partial C(u,v)}{\partial u}$;

3. The desired pair $(x,y)$ is given by $x=F_{1}^{-1}(u)$ and $y=F_{2}^{-1}(v)$, where $F_1$ and $F_2$ are the marginal distributions.

Now imagine, as suggested later in the study mentioned above, that the copula function is the Gaussian copula given by

\begin{equation} C_\rho(u,v)= \int_{\infty}^{\Phi^{-1}(u)}\int_{\infty}^{\Phi^{-1}(v)} \frac{1}{2\pi\sqrt{(1-\rho^2)}}\exp\left[\frac{-(s^2-2\rho st+t^2)}{2(1-\rho^2)}\right] \,dsdt, \end{equation}

and both marginal distributions $F_1$ and $F_2$ are Laplace distribution functions given by

\begin{align} F(x) &= \int_{-\infty}^x \!\!f(u)\,\mathrm{d}u = \begin{cases} \frac12 \exp \left( \frac{x-\mu}{b} \right) & \mbox{if }x < \mu \\ 1-\frac12 \exp \left( -\frac{x-\mu}{b} \right) & \mbox{if }x \geq \mu \end{cases} \\ &=\tfrac{1}{2} + \tfrac{1}{2} \text{sgn}(x-\mu) \left(1-\exp \left(-\frac{|x-\mu|}{b} \right ) \right ). \end{align}

After a few searches, it appears that the natural approach with the Gaussian Copula is to simulate from the multivariate standard normal distribution with an appropriate correlation matrix and not using the conditional distribution approach suggested by the algorithm described above.

Since this method is used in the paper I am trying to reproduce, I would like to use the algorithm given above with the Gaussian copula and with the Laplace distribution as marginal distribution. Therefore, I wanted to know

1. if it is possible,

2. Since I am having troubles trying it and understanding it, could someone describe all the steps for deriving the partial derivative of the Gaussian Copula and then inverting it.

Thank you for your help.