### Or: waves that take the shortest path through infinity

*guest post by Tim van Beek*

Water waves can do a lot of things that light waves cannot, like “breaking”:

In mathematical models this difference shows up through the kind of partial differential equation (PDE) that models the waves:

• light waves are modelled by *linear* equations while

• water waves are modelled by *nonlinear* equations.

Physicists like to point out that linear equations model things that do not interact, while nonlinear equations model things that interact with each other. In quantum field theory, people speak of “free fields” versus “interacting fields”.

Some nonlinear PDE that describe fluid flows turn out to also describe geodesics on infinite-dimensional Riemannian manifolds. This fascinating observation is due to the Russian mathematician Vladimir Arnold. In this blog post I would like to talk a little bit about the concepts involved and show you a…

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