been able to consider initial data which is pure shock (piecewise constant). This present paper extends the ideas in the theory of a-contraction (in particular as developed in [@MR3519973]).

![In this paper, we study the stability of solutions $u$ (to ) which are $L^2$ perturbations of a piecewise-smooth solution $\bar{u}$, as shown in this schematic. The nonlinearity in the solution $\bar{u}$ causes significant technical challenges not present in the piecewise-constant case (for the piecewise-constant case, see [@Leger2011; @MR3519973]).[]{data-label="figure_diagram"}](art/move_entire_solution_systems){width="\textwidth"}

The point of the present article is this: As discussed for the case of nonlocal scalar balance laws in [@scalar_move_entire_solution], when studying the stability up to a translation in space of solutions piecewise-constant in space, we can view the shift function which is doing the translation as simply determining at which points do we want to see the left hand state of our solution, and at which points do we want to see the right hand state of our solution. However, for piecewise-*smooth* data, the shift function cannot be viewed like this. Instead, the shift function is viewed as artificially translating in space our solution. If the solution is non-constant away from the discontinuity, this artificial translation creates a linear term in 