the inner regions. Hence, we restrict the value $\kappa\in \mathbb{R}_0^{+}$ and refer the reader to the end of \[anasolIP\] and \[secgenrestrict\] for additional mathematical limitations required for the solvability of the system. The constant $S_{\infty}>0$ is a free parameter. Without loss of generality we set[^6] $$S_{\infty}=1\label{eq:sbnd}$$ for both the implosion as well as the explosion problem. This BC is the only condition for fixing the density solution because we either need to know the inner BC ($S_0$) or the outer BC ($S_\infty$) together with the jump position $\xi_0$. The jump position is given by the RHC which we will discuss in \[secRHC\].

We derive the inner BC for the velocity distribution () by a linearization of $$\frac{\mathrm{d}u}{\mathrm{d}\xi}\approx\frac{\kappa\xi-u\alpha}{\xi} \mbox{,\ \ for\ }\quad 0<\xi\ll1\quad\mbox{and}\quad 0<\left|u\right|\ll1\label{eq:appr_u}$$ and find the approximate solution $$u\left(\xi^{\ll1}\right)=u_0=\frac{\kappa\xi}{\alpha+1}+c_{0}\xi^{-\alpha}.\label{eq:solv_ibnd}$$ The constant $c_0$ has to be zero, because the fluid must halt at the wall or the center (see \[fig:Figtube\]a, b). To satisfy we need the inner starting position $\xi\neq0$ and set $\left.\xi\right|_0=10^{-3}$, being valid for both, the IP and EP.

The outer velocity BCs are fixed in the same manner, whereas we define the dependent variable $\eta\doteq\xi^{-1}$ such that $\eta\rightarrow 0 \mbox{\ for\ } \xi\rightarrow \infty$. This enables us to 