Solitons in Space–Time Crystals: Selection, Holonomy and Twin Channels

Media modulated in space and time carry known solitons: the momentum-gap pair of a photonic time crystal, Bragg gap solitons, event solitons and the phase-locked solitons of a parametric pump. This article asks what the symmetry of the modulation does to them. The harmonic content selects the family. A uniform harmonic gives counter-propagating pairs, a static one Bragg solitons, both together event solitons, and travelling harmonics one-way, phase-locked solitons. The glide time reversal forbids every static odd harmonic, so a glide-invariant medium has no Bragg and no event soliton at the first zone edge. Its channel is a parametrically driven nonlinear Schrödinger equation, and its locked soliton exists in a full Maxwell model of a dispersive medium. A locked soliton lives in the half-integer time sector and has two branches. Around a closed path it returns as (-1)^(l+tau+N) times itself, where l counts the pump winding, tau a Möbius twist and N the cells at the zone edge. A smooth slip of the pump phase flips its branch; a sharp one destroys it. A winding pump phase is an exact boost: it drives the soliton, strengthens it, and makes it orbit a pump vortex. On a locked background the walls have the parity of the same sum. The two partners of the forced doublet are the forward and the backward channel. Their solitons annihilate on collision, and a ring keeps the majority direction. In a projective medium each pump channel locks solitons in its own window. The results are measured on model equations, a resonator ring and a one-dimensional field model, not on a fabricated device.