In 2011, Deconinck and Oliveras experimented with various disturbances at increasing frequencies to see how they affected the Stokes waves. Initially, they found that the waves remained stable with disturbances above a certain frequency. However, when they increased the frequency further, they observed that destruction of the waves began to occur again. Oliveras was initially concerned that there might be an error in the computer program. “Part of me was like, this can’t be right,” she recalled. “But the more I dug, the more it persisted.” They noticed that as the frequency increased, a repeating pattern of instability and stability emerged. There was a range of frequencies where the waves became unstable, followed by periods of stability, then more instability, and so on.
Deconinck and Oliveras shared their surprising findings as a conjecture: that this series of instabilities extends infinitely. They named the unstable intervals “isole”—the Italian word for “islands.” The phenomenon was puzzling, as they had no explanation for the recurring instabilities or why they appeared infinitely. They were determined to find a proof to confirm their unexpected observation.
Years went by without progress until a workshop in 2019, where Deconinck approached Maspero and his team, who had considerable experience in studying wave-like phenomena in quantum physics. He hoped they could help prove that these intriguing patterns originated from the Euler equations. The Italian team jumped right in, beginning with the lowest frequency sets that caused wave decay. They applied physics techniques to represent each low-frequency instability as arrays, or matrices, of 16 numbers, which described how the instability would develop and affect the Stokes waves over time.
The mathematicians realized that if any number in the matrix was zero, the instability wouldn’t grow, allowing the waves to persist. If the number was positive, the instability would escalate, ultimately destroying the waves. “Part of me was like, this can’t be right. But the more I dug, the more it persisted,” said Katie Oliveras from Seattle University.
To demonstrate that the number was positive for the initial set of instabilities, the team had to calculate a massive sum, which took 45 pages and nearly a year to work out. Once that was done, they shifted their focus to the infinitely many higher-frequency disturbances—the isole. They established a general formula to determine the necessary number for each isola. They even wrote a computer program to compute the values for the first 21 isole. Beyond that, the calculations became too complex for the computer. The results were all positive, as they had anticipated, and they appeared to follow a simple pattern, suggesting that they would remain positive for all subsequent isole.
