In 2011, Deconinck and Oliveras simulated various disturbances at increasingly higher frequencies and observed what happened to Stokes waves. As expected, for disturbances above a certain frequency, the waves persisted.
But as the pair continued to augment the frequency, they suddenly began to see destruction again. Initially, Oliveras was concerned that there was an error in the computer program. “Part of me thought, This can’t be true,” she said. “But the more I dug, the more it took.”
In fact, as the disturbance frequency increased, an alternating pattern emerged. First there was a frequency range where the waves became unstable. This was followed by a period of stability, followed by another period of instability, and so on.
Deconinck and Oliveras published their discovery as a counterintuitive assumption: that this archipelago of instability extends endlessly. They called all the unstable intervals “isoles” – Italian for “islands”.
It was strange. The couple had no explanation as to why the instabilities would reoccur, let alone an infinite number of times. They at least wanted proof that their surprising observation was correct.
Photo: Courtesy of Katie Oliveras
For years, no one could make any progress. Then, during a 2019 workshop, Deconinck turned to Maspero and his team. He knew they had extensive experience studying the mathematics of wave phenomena in quantum physics. Perhaps they can find a way to prove that these striking patterns follow from Euler’s equations.
The Italian group immediately got to work. They started with the lowest set of frequencies, which seemed to cause the waves to die. They first used physical techniques to represent each of these low-frequency instabilities as arrays or matrices of 16 numbers. These numbers are encoded as instability increases and distort Stokes waves in time. Mathematicians realized that if one of the numbers in the matrix was always zero, the instability would not augment and the waves would continue to exist. If the number were positive, instability would augment and eventually destroy the waves.
To show that this number was positive for the first batch of instability, mathematicians had to calculate a gigantic sum. It took 45 pages and almost a year of work to solve it. Once they did this, they turned their attention to the infinitely many periods of higher frequency killing interference – isolation.
First, they worked out a general formula – another complicated sum – that would give them the number they needed for each isolate. Then, using a computer program, they solved the formula for the first 21 isoles. (Then the calculations became too intricate for the computer to handle.) All the numbers were positive, as expected, and also seemed to follow a uncomplicated pattern that suggested they would be positive for all the other isolates as well.