Event Soliton Formation in Mixed Energy-Momentum Gaps of Nonlinear Spacetime Crystals

We report the formation of a novel soliton, termed event soliton, in nonlinear photonic spacetime crystals (STCs). In these media, simultaneous spatiotemporal periodic modulation of the dielectric constant generates mixed frequency (${\omega}$) and wavevector (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 ${\omega}$k-gap mixture arises from the coexistence and competition between time reflected and Bragg reflected waves due to the spatiotemporal modulation. We propose a new 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, ${\omega}$-gap opening, and nonlinearity. Our work opens avenues for fundamental studies and fosters experimental prospects for implementing spacetime crystals in both time-varying photonics and periodically driven condensed matter systems.