(one maximum!), there are two possible values for the density, $\rho_H$ and $\rho_L$ (with $\rho_H>\rho_L$), leading to two possible local solutions of (\[dJdx\]), a *high density solution* $J_H(x)$ and a *low density solution* $J_L(x)$: $$J^* \begin{array}{l}\nearrow \\ \searrow \end{array}
\begin{array}{l}
\rho_H \longrightarrow J_H(x-x_0,J^*) \\ \\
\rho_L \longrightarrow J_L(x-x_0,J^*)
\end{array}$$ Taking into account shock dynamics, a constraint on the selection of a physical solution is given by the collective velocity $$v_c(x)=J'(\rho(x))$$ where $J(\rho)$ is the current density relation and the prime denotes the derivative with respect to $\rho$ [@kolomeisky_shockdyn]. A solution can only propagate away from the initial point if the direction of $v_c$ is pointing away from it, i.e. left of it only solutions with $v_c<0$ can exist, while right of it solutions must have $v_c>0$. In a system with a single maximum at density $\rho_m$ in the CDR, $\frac{dJ}{d\rho}>0$ for $\rho<\rho_m$ and $\frac{dJ}{d\rho}<0$ for $\rho>\rho_m$, thus left of an initial point, only the high density solution $J_H$ can be realized, while right of it only $J_L$ can physically exist. This principle is displayed in Fig. \[initpoint\_fig\], top. Hence, each initial condition at a point $x_0$ can have its own solutions. We denote these *local solutions* by $$J(x-x_0,J^*)=\left\lbrace\begin{array}{lc}
J_H(x-x_0,J^*) & 