\mbox{for $x \leq x_0$} \\
J_L(x-x_0,J^*) & \mbox{for $x>x_0$}
\end{array}\right. \,\, .$$ Actually the dependence on $J^*$ can easily be obtained by a shift operation if two functions $\tilde J_L(x)$ and $\tilde J_H(x)$ with initial conditions $\tilde J_{L}(0)=J^0_L$ and $\tilde J_{H}(0)=J^0_H$, where $J^0_L$ and $J^0_H$ are arbitrary chosen values in the high- and low density branch of the CDR. If the range in both branches of the CDR includes $J=0$, one can simply choose $J^0_L=J^0_H=0$ [^7]. Since the ODE (\[dJdx\]) is of first order and does not explicitly depend on $x$, the high and low density solutions unambigiously depend on $\rho$ and are monotonic. Thus different local solutions $J_{L,H}$ can only differ by a shift in the variable $x$. An arbitrary solution $J_{L,H}(x-x_0,J^*)$ can be obtained by shifting $\tilde J_{L,H}(x)$ by an amount $\tilde x_{L,H}(J^*)$ so that the value of the shifted function at $x=0$ is equal to $J^*$. The functions $\tilde x_{L,H}(J^*)$ are just the inverse functions of the unique functions $\tilde J_{L,H}(x)$. Then the local solutions at a point with initial condition $J^*$ are given as $$\label{Greensfunc}
J(x-x_0,J^*)=\tilde J(x-x_0-\tilde x(J^*))\,.$$ The functions $\tilde J_{L,H}(x)$ and $\tilde x_{L,H}(J)$ can for example be obtained by numerical solution of (\[dJdx\]) with 