  In this work, we explore the dynamics of time varying photonic media with an
optical Kerr nonlinearity and an associated phase transition. The interplay
between a periodically modulated permittivity and the nonlinearity induces a
continuous transition of electromagnetic waves to a state with broken spatial
and time translation symmetries. This transition gives rise to a lattice-like
wave pattern, in many ways similar to a spatial crystallization in solids.
Symmetry breaking triggers the emergence of soft, Goldstone-like modes, which
propagate as deformations of the lattice structure, as well as massive
Higgs-like modes - spatially uniform oscillations of the field amplitude. We
extend the analysis of the non-equlibrium symmetry breaking to 2+1 dimensional
time varying media and discuss pattern formation as well as the connection to
discrete time crystals.
