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In 2011, DeConinck and Oliveras simulated different perturbations at higher and higher frequencies and saw what happened to Stokes waves. As they expected, the waves persisted in disturbances above a certain frequency.
But as the couple continued to increase their frequency, they suddenly began to see the devastation again. At first, Oliveras worried there was a bug in the computer program. “Part of me was like, ‘This can’t be true,'” she said. “But the more you dig, the more it lasts.”
In fact, as the frequency of disturbance increased, an alternating pattern emerged. Initially there was a period of hesitation 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 findings as Counterintuitive guess: This archipelago of instability extends to infinity. They called all unstable periods “isole” – the Italian word for “islands.”
It was strange. The couple had no explanation as to why the instability was appearing again, let alone appearing infinitely many times. They at least wanted proof that their startling observation was true.
Photo: Courtesy of Katie Oliveras
For years, no one was able to make any progress. Then, at a 2019 workshop, DeConinck reached out to Maspero and his team. He knew that they had a lot of experience studying the mathematics of wave phenomena in quantum physics. Perhaps they can discover a way to prove that these amazing patterns arise from Euler’s equations.
The Italian group got to work immediately. They started with the lowest set of frequencies that seemed to cause the waves to die. First, they applied techniques from physics to represent each of these low-frequency instabilities as arrays, or matrices, of 16 numbers. These numbers are encrypted How will instability grow? Stokes waves distort with time. Mathematicians have realized that if one of the numbers in the matrix is always zero, the instability will not increase, and the waves will persist. If the number is positive, the instability will grow and eventually destroy the waves.
To show that this number was positive for the first batch of instabilities, mathematicians had to calculate a huge sum. It took 45 pages and nearly a year of work to solve. Once they did, they turned their attention to the many endless intervals of deadly perturbations of high-frequency waves – the isolation.
First, they came up with a general formula — another complex sum — that would give them the number they needed for each isolate. They then used a computer program to solve the formula for the first 21 islands. (After that, the calculations became too complex for the computer to handle.) All the numbers were positive, as expected, and they also seemed to follow a simple pattern that implied they would be positive for all other types of isolation as well.
