Discovery of the exact 3D one-way wave equation

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 computational and topological physics. Here, by borrowing techniques from relativistic 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 the exact one-way wave equation in three dimensions. Surprisingly, we find that this equation necessarily – similarly 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.