Discovery of the exact 3D one-way wave equation - K. L. Tsakmakidis; Stefański, T.P. (Nature Communications, 2025)
The standard wave equation describing symmetrical wave propagation in all directions in three dimensions, was discovered by the French scientist d’Alembert, more than 250 years ago. In the 20th century it became important to search for ‘one-way’ versions of this equation in three dimensions– i.e., an equation describing wave propagation in one direction for all angles, and forbiting it in the opposite direction– for a variety of applications in compu tational and topological physics. Here, by borrowing techniques from relati vistic quantum field theory– in particular, from the Dirac equation–,and starting from Engquist and Majda’s seminal, approximative one-way wave equations, we report the discovery of theexactone-waywaveequationin three dimensions. Surprisingly, we find that this equation necessarily– simi larly to the innate emergence of spin in the Dirac equation– has a topological nature, giving rise to strong, spin-orbit coupling and locking, and non vanishing (integer) Chern numbers.