# Discovery of the exact 3D one-way wave equation

## TL;DR

The paper reports the discovery of a three-dimensional one-way wave equation with topological properties, similar to those found in the Dirac equation.

## Contribution

The discovery of an exact one-way wave equation in 3D with inherent topological characteristics.

## Key findings

- The exact one-way wave equation in three dimensions was derived using techniques from relativistic quantum field theory.
- The equation exhibits topological properties, including strong spin-orbit coupling and non-vanishing Chern numbers.
- This discovery parallels the emergence of spin in the Dirac equation.

## Abstract

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.

The wave equation was first discovered by the French scientist d’Alembert. It has since been intriguing, but so far unsuccessful to answer if ‘one-way’ versions of it might exist. Here, the authors report the discovery of such an equation in three dimensions, showing that it has a topological character.