A simple dynamical systems perspective on the interaction of deep-water waves

What is it about?

The very simplest mathematical solutions describing waves on the surface of water are sine waves. These arise out of a simplification of water wave theory that assumes that waves are not too steep, and the equations can be treated as linear. Indeed, some of the waves we see in the ocean look at first glance like sine waves, or superpositions of sine waves. Nevertheless, attempts to generate these sine waves in the laboratory - to conduct experiments on their properties - repeatedly ran into difficulties. In particular, even with a wavemaker oscillating periodically in a prescribed manner, the result in the wave flume was often something very different than expected - and much messier! The reason for this became clear only during the 1950s and 1960s: like a house of cards or a ball on top of a hill, some waves are unstable, and reconfigure themselves when even the slightest disturbance occurs. That's the situation people encountered in the lab: water doesn't like to be in the shape of a sine wave, and energy is redistributed so that a single sine wave becomes a combination of different waves due to this "modulation instability". The word "modulation" here refers to the small disturbance, which can come from almost anywhere. Describing this situation mathematically is quite tricky, and usually requires partial differential equations like the nonlinear Schrödinger equation. Moreoever, while it's usually possible to describe this instability mathematically for a short, initial time (using something called "linear stability theory") it's very hard to follow the longer-time evolution, which usually requires numerical computation. In this paper, we show that you can describe these wave instabilities exactly using a very simple system of ordinary differential equations. These allow you to visualise what happens at all times in the future by looking at phase portraits, which doesn't require any computation.

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Photo by Austin Neill on Unsplash

Photo by Austin Neill on Unsplash

Why is it important?

Looking at the (modulation) instability of water waves through the phase plane actually allows you to capture new, never-before-seen solutions. Because these are very sensitive to even the smallest errors it's essentially impossible to capture them by purely numerical methods, but they pop out of the phase-plane description. This includes new "breather" solutions that turn one wave into two waves (or vice versa), as well as "steady" solutions where waves are mathematically in resonance but don't exchange any energy.