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  • Open Access

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

Liang Zhang1, Zhiwei Fan2,*, and Yiming Pan1,†

  • 1School of Physical Science and Technology and Center for Transformative Science, ShanghaiTech University, Shanghai 200031, China
  • 2School of Engineering, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom

  • *Contact author: zhiwei.fan@newcastle.ac.uk
  • †Contact author: yiming.pan@shanghaitech.edu.cn
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Phys. Rev. Research 8, 023107 – Published 1 May, 2026

DOI: https://doi.org/10.1103/q226-dkk4

Abstract

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.

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Phys. Rev. Research 8, 023107– Published 1 May, 2026
  • Received 25 March 2025
  • Accepted 4 March 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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