Event soliton formation in mixed energy-momentum gaps of nonlinear spacetime crystals

We report the formation of a soliton—termed “event soliton”—in nonlinear photonic spacetime crystals (STCs). In these media, simultaneous spatiotemporal periodic modulation of the dielectric constant generates mixed frequency (ω) and wave vector (k) gaps. Under Kerr nonlinearity, the event solitons emerge as fully localized entities in both spacetime and energy-momentum domains, providing a tangible demonstration of the concept of “event” in relativity. The ωk-gap mixture arises from the coexistence and competition between time-reflected and Bragg-reflected waves due to the spatiotemporal modulation. We propose a partial differential equation to capture various spatiotemporal patterns and present numerical simulations to validate our theoretical predictions, reflecting a three-way balance among k-gap opening, ω-gap opening, and nonlinearity. Our work opens avenues for fundamental studies and fosters experimental prospects for implementing spacetime crystals and pattern formation in time-varying photonics.