Hard to shock DBI: wave propagation on planar domain walls

We investigate propagation of generic waves on thin planar domain walls effectively described by the scalar DBI model. We pay a particular attention to the possibility of caustic (shock) formation - the process, which may lead to intensive particle emission by domain walls. It is demonstrated that no singularities arise in DBI in 2D flat spacetime in the hyperbolic case, if one starts from smooth initial conditions. Technically, this happens because the same family characteristics of the relevant PDE remain parallel at all the times, albeit not being straight lines generically. Crucially, characteristic curves cease to be parallel beyond the simplified setup of DBI in 2D flat spacetime. In particular, this is shown to be the case in $D>2$ for spherical waves, in an expanding Universe, and in the case of a minimal deformation of DBI necessary for avoiding the domain wall problem in cosmology. However, we prove that DBI remains shock free in the hyperbolic case in all these physically relevant situations. This strongly suggests that caustics can form on planar domain walls only due to the loss of hyperbolicity, and they have a cusp profile. We demonstrate, how the non-trivial structure of DBI characteristics beyond the 2D flat spacetime setup uncovered in this work can significantly affect cusp formation.