Caustic formation in DBI models: Wave propagation on planar domain walls

We investigate propagation of on thin planar domain walls effectively described by the scalar Dirac-Born-Infeld model (DBI). We pay a particular attention to the possibility of caustic formation—the process, which may lead to intensive particle emission by domain walls. It is demonstrated that no singularities arise in DBI in two-dimensional (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 partial differential equation 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 caustic 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 nontrivial structure of DBI characteristics beyond the 2D flat spacetime setup uncovered in this work can significantly affect cusp formation.