initial conditions $J^0_{L,H}$.

![\[initpoint\_fig\] Top: Local solutions in the vicinity of a point with an initial condition $J^*$. Due to the non-unique inversion of the current-density relation, there are two possible solutions. Since for a physical solution the direction of the collective velocity must point away from this position, only solutions with *maximal current* are realized.\
Bottom: Intersection point of local solutions of the density profile. The constraint that only upward shocks can exist implies that only solutions with *minimal current* are physically realized. ](initpoint.eps "fig:"){width="0.7\columnwidth"}\
![\[initpoint\_fig\] Top: Local solutions in the vicinity of a point with an initial condition $J^*$. Due to the non-unique inversion of the current-density relation, there are two possible solutions. Since for a physical solution the direction of the collective velocity must point away from this position, only solutions with *maximal current* are realized.\
Bottom: Intersection point of local solutions of the density profile. The constraint that only upward shocks can exist implies that only solutions with *minimal current* are physically realized. ](intersec.eps "fig:"){width="0.7\columnwidth"}

Selection of the global current profile {#global_sec}
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The physically realized *global* current profile in the steady state is also determined by shock dynamics [@popkov1; @kolomeisky_shockdyn]. Shocks manifest themselves as discontinuities 