@article {43966,
	title = {Discovery of the exact 3D one-way wave equation},
	journal = {Nature Communications},
	volume = {16},
	year = {2025},
	pages = {5719},
	abstract = {The standard wave equation describing symmetrical wave propagation in all directions in three dimensions, was discovered by the French scientist d{\textquoteright}Alembert, more than 250 years ago. In the 20th century it became important to search for {\textquoteleft}one-way{\textquoteright} versions of this equation in three dimensions{\textendash} i.e., an equation describing wave propagation in one direction for all angles, and forbiting it in the opposite direction{\textendash} for a variety of applications in compu tational and topological physics. Here, by borrowing techniques from relati vistic quantum field theory{\textendash} in particular, from the Dirac equation{\textendash},and starting from Engquist and Majda{\textquoteright}s seminal, approximative one-way wave equations, we report the discovery of theexactone-waywaveequationin three dimensions. Surprisingly, we find that this equation necessarily{\textendash} simi larly to the innate emergence of spin in the Dirac equation{\textendash} has a topological nature, giving rise to strong, spin-orbit coupling and locking, and non vanishing (integer) Chern numbers.},
	url = {https://www.nature.com/articles/s41467-025-61220-3.epdf?sharing_token=mKXkdwtUt5QIrc23HQipodRgN0jAjWel9jnR3ZoTv0OGnBKC95ILaWLutYy4W8vCwI2KQK3-EIOqrkFImheUKEpufMw1QzW4ILyuojznhEhKDvyl2t20qzlJjVNm6M12zSkVM5Eqc76i2CSCUJ46ncFU3oBG-XndlLLGeU49rsc\%3D},
	author = {K. L. Tsakmakidis and Stefa{\'n}ski, T.P.}
}
