内容来源：https://www.quantamagazine.org/the-hidden-math-of-ocean-waves-crashes-into-view-20251015/

内容总结：

【本报综合报道】意大利数学家团队近日在流体力学领域取得重大突破，首次从数学上完整证明了海洋波浪不稳定性产生的内在规律。这项研究揭开了困扰学界两个世纪的谜题：为何看似规律的行进波纹会突然瓦解成混乱浪涌。

研究缘起于的里雅斯特港的日常观测。国际高等研究院的阿尔贝托·马斯佩罗教授常从办公室窗口观察亚得里亚海北端的海湾。当著名的"博拉"山风掠过阿尔卑斯山猛吹入港时，原本涌向码头的水流会发生倒灌，但这些逆流波浪从未真正抵达远海——它们在港口出口处逐渐消散，最终归于平静。

这种看似简单的自然现象背后，隐藏着欧拉流体方程的精深数学难题。尽管欧拉在300年前就建立了描述流体运动的基本方程，但即便是最平稳的斯托克斯波列（即连续均匀推进的波浪）也长期无法被精确求解。瑞士洛桑联邦理工学院博士后保罗·文图拉坦言："接触数学前，我以为水波原理早已被彻底掌握，实际上它们充满未解之谜。"

研究团队发现，当斯托克斯波列遭遇特定频率扰动时，会产生被称为"本杰明-菲尔不稳定性"的现象。更令人惊讶的是，美国数学家伯纳德·德康尼克与凯蒂·奥利维拉斯在2011年通过计算机模拟发现，这种不稳定性会随扰动频率升高呈交替出现的"岛屿链"模式——稳定区间与不稳定区间无限交替出现。

马斯佩罗团队联合罗马第三大学的利维娅·科尔西等学者，通过建立特殊矩阵模型，将波动稳定性问题转化为复杂代数计算。在罗格斯大学多伦·泽尔伯格教授协助下，他们最终证实了这种"不稳定岛屿链"确实存在于欧拉方程的解集中，相关论文长达45页的研究过程展现了理论数学与计算科学的深度融合。

该成果不仅解释了为何小型皮划艇的短频扰动可能被波浪吸收，而巨型邮轮的长波扰动会引发链式崩溃，更为预测海啸演变、优化航运安全提供了新数学模型。正如布朗大学数学家沃尔特·施特劳斯评价："这不仅是单项突破，更代表着多方向分析方法的浪潮。"

目前研究团队正致力于将新方法应用于更多流体力学难题。对于的里雅斯特港那些被山风驱散的波浪，马斯佩罗表示："虽尚未证实港口现象与我们的数学模型完全对应，但我宁愿相信它们正遵循着同样的不稳定性规律。"

中文翻译：

海浪背后隐藏的数学原理浮出水面

引言

阿尔贝托·马斯佩罗说，他工作中最好的福利就是窗外的景色。他的办公室位于意大利古老港口城市的里雅斯特的一座小山上，在国际高等研究院内，可以俯瞰亚得里亚海北端的一片广阔海湾。"这非常鼓舞人心，"这位数学家说，"毫无疑问，这是我见过最美的景色。"

意大利人称的里雅斯特为"波拉风之城"，得名于其著名的"波拉"风。这种风从阿尔卑斯山不规则地吹下，席卷整个城市。当波拉风足够强劲时，它会使海浪逆向而行。海浪不再拍打码头，而是流向远离城市的方向，退回开阔的大海。

但它们从未真正到达那里。在这些狂风大作的日子里，马斯佩罗从窗口望去，可以看到退去的海浪在离开港口时慢慢消散，最终让位于平静、静止的海面。

数学家用来研究水和其他流体流动的方程——由莱昂哈德·欧拉在大约300年前首次写下——看起来相当简单。如果你知道每个水滴的位置和速度，并通过假设没有内摩擦（即粘性）来简化数学，那么求解欧拉方程就能让你预测水在任何时间段内的演变。我们在世界海洋中看到的丰富现象——海啸、漩涡、离岸流——都是欧拉方程的解。

但这些方程通常无法求解。即使是最简单、最常见的解之一——描述一列平稳滚动的波浪——从欧拉方程中提取出来也是一场数学噩梦。直到大约30年前，我们对这些波浪的了解大部分还仅仅来自现实世界的观察和猜测。在很大程度上，证明似乎如同幻想。

"在开始研究数学之前，我以为水波是已经被充分理解的东西——根本不成问题，"瑞士洛桑联邦理工学院博士后、马斯佩罗以前的研究生保罗·文图拉说。"但实际上，它们就是很奇怪。"

几十年来一直困扰数学家的一个奇怪现象是，即使摩擦力极小，那列平稳滚动的波浪最终仍然会解体并变得不规则。数学家们没有预料到如此不稳定的行为会从这样一个简单的起点产生。他们想要证明这一点——证明不稳定性是欧拉方程自然产生的结果。但他们不知道如何做到。

现在，马斯佩罗和文图拉，连同他们在的里雅斯特的同事马西米利亚诺·贝尔蒂以及罗马第三大学的利维娅·科尔西，终于提出了这样一个证明，精确地指出了这些不稳定性在何时发生，在何时不发生。这一成果仅仅是正在开始改变我们对地球波浪数学理解复兴浪潮中的最新进展。数学家们一直在使用新的计算工具来构建关于波浪行为的猜想。并且，他们现在正在发展复杂的纸笔技术来证明这些猜想。

"这不是某一件特定的事情。这是在多个方向上涌现的全新类型的分析浪潮，"布朗大学的数学家沃尔特·施特劳斯说。"我印象非常深刻。"

缓慢的潮汐

古希腊人常将海浪拍岸那不稳定的节奏比作笑声。考虑到这些波浪一直难以被人类理解，也许他们是对的：海洋一直在嘲笑我们。

即使在17世纪末18世纪初启蒙运动的鼎盛时期，当波浪占据了大量科学讨论时，海洋似乎总是拥有最终决定权。一些科学家已经测量了声波的速度，牛顿和他的批评者就光的波动性陷入了争论。但人类已知的最古老的波浪——水波——仍然是一个数学谜团。

一个多世纪后，这种情况才开始改变。19世纪初，乔治·斯托克斯爵士对海浪产生了浓厚兴趣，起因是他小时候在爱尔兰斯莱戈的家附近游泳时，一个巨浪差点把他卷到海里。1847年，他发表了一篇关于该主题的里程碑式论文。他从欧拉关于无粘性流体的方程入手，并添加了数学条件，即其顶部表面是完全"自由"的——可以呈现任何它喜欢的形状。

"它们看起来并不糟，"施特劳斯在谈到由此产生的方程时说。"但是，只要看一看有微风吹过的湖面。你会看到所有这些复杂的形式，比如白浪和滚动波浪，有些彼此平行，有些则不是。"

这些变化多端的形式，当被理解为欧拉方程的解时，在数学上是截然不同且极其难以处理的。对流体的初始状态做最微小的改变，它都可能以完全不同的方式演变——涟漪和涡流可能变成 rogue waves（畸形波）和海啸。

这些自由的、移动的表面正是斯托克斯想要研究的。但挑战是巨大的。描述 confined within a box（限制在箱子里）或 flowing through a pipe（流过管道）的水的运动已经够难了。但至少，你知道系统边界在哪里——水不能超出那些界限。如果除了重力之外，对水能达到的高度和呈现的形状没有其他限制，数学就变得困难得多。

"如果我早上七点去海滩，海面会非常平静，"科尔西说。"但如果你仔细观察水面，看它如何运动，那简直是一团糟。"

尽管如此，斯托克斯还是能够提出一个猜想解：水表面有可能形成间隔均匀、朝单一方向传播的波浪。

到了20世纪20年代，数学家们证明了斯托克斯的猜想。此外，他们还发现，如果没有外部干扰，欧拉方程的这些解会永远持续下去：一旦形成，所谓的斯托克斯波将永远欢快地沿着水面传播，其形态保持不变。

但是，如果一艘经过的船只的尾流与波浪的路径交叉呢？波浪是会吸收这种干扰并保持其形态，还是会被永久破坏，转变成完全不同的波浪模式？

几十年来，数学家们假设斯托克斯波是稳定的，意味着任何小的畸变都只会产生最小的影响。毕竟，现实世界充满了这种复杂情况，然而海洋中却遍布斯托克斯波。如果它们在最轻微的扰动下就会解体，那么它们永远无法存活足够长的时间到达岸边。

然而，在1967年，数学家 T. 布鲁克·本杰明决定验证这个基本假设。他让他的学生吉姆·费尔在一个波浪水槽——一种一端装有振荡舵、可以产生斯托克斯波的窄矩形水池——中进行一系列实验。但费尔无法让波浪到达水池的另一端。起初，他以为是实验装置出了问题。但很快，令人惊讶的是，情况变得明朗：这些波浪是不稳定的。

1995年，数学家们最终证明了这种"本杰明-费尔不稳定性"是欧拉方程不可避免的结果。但这项工作让研究人员对这些不稳定性的性质产生了疑问。哪种干扰能摧毁波浪，哪种不能？不稳定性膨胀的速度有多快？太平洋中心的一阵风是否可能引起一列波浪在数周后冲击马里布海滩，还是波浪结构在到达岸边之前就会瓦解？

奇怪的群岛

马斯佩罗从未想过要问为什么离开的里雅斯特海湾的波浪会消失。他的灵感最终来自一台计算机，而不是窗外的景色。

在2019年一个关于波浪数学的研讨会上，他和他的合作者遇到了华盛顿大学的应用数学家伯纳德·德康尼克，后者与西雅图大学的凯蒂·奥利弗拉斯一起，一直在绘制所有可能摧毁斯托克斯波的不同不稳定性图谱。几年前，这两人注意到了一个惊人的模式，并且一直无法停止思考它。

当一列完美的斯托克斯波遇到干扰其形状的扰动时，有时扰动的影响会增长到摧毁整列波浪，有时则几乎不产生干扰。结果取决于扰动的频率——与原始波浪的长度相比，其振荡的程度。皮划艇产生的由短而频繁振荡组成的尾流，将比产生更长、更慢振荡的巨型远洋班轮带来更高频率的冲击。

一般来说，数学家预计波浪更容易从像皮划艇那样的高频干扰中恢复，因为其影响在任何给定时刻都局限于经过波浪的较小区域。另一方面，远洋班轮的尾流可以同时影响整个波浪，永久性地破坏它。本杰明-费尔不稳定性就是由低频干扰引起的。

2011年，德康尼克和奥利弗拉斯模拟了频率越来越高的不同扰动，并观察斯托克斯波发生了什么。正如他们所料，对于高于某个特定频率的扰动，波浪坚持了下来。

但随着两人继续调高频率，他们突然又开始看到破坏的发生。起初，奥利弗拉斯担心是计算机程序有错误。"我的一部分想法是，这不可能是对的，"她说。"但我越是深入研究，这个现象就越是持续出现。"

事实上，随着扰动频率的增加，一种交替的模式出现了。首先是一个频率区间，波浪变得不稳定。接着是一个稳定区间，然后又是一个不稳定区间，如此循环往复。

德康尼克和奥利弗拉斯将他们的发现作为一个违反直觉的猜想发表：这个不稳定性群岛无限延伸下去。他们称所有的不稳定区间为"isole"——意大利语中的"岛屿"。

这很奇怪。两人无法解释为什么不稳定性会再次出现，更不用说无限多次了。他们至少想要一个证明，证实他们惊人的观察是正确的。

多年来，没有人能取得任何进展。然后，在2019年的研讨会上，德康尼克找到了马斯佩罗和他的团队。他知道他们在研究量子物理中类波现象的数学方面经验丰富。也许他们能找到一种方法来证明这些引人注目的模式源于欧拉方程。

这个意大利小组立即投入工作。他们从似乎会导致波浪消失的最低一组频率开始。首先，他们应用物理学中的技术，将这些低频不稳定性中的每一个表示为16个数字的阵列或矩阵。这些数字编码了不稳定性将如何随时间增长并扭曲斯托克斯波。数学家们意识到，如果矩阵中的某个数字始终为零，不稳定性就不会增长，波浪就会继续存在。如果这个数字是正数，不稳定性就会增长并最终摧毁波浪。

为了证明第一批不稳定性的这个数字是正数，数学家们必须计算一个巨大的求和式。他们花了45页纸和近一年的工作才解决它。一旦完成，他们便将注意力转向无限多个高频杀波扰动的区间——即那些"岛屿"。

首先，他们找到了一个通用公式——另一个复杂的求和式——可以为他们提供每个"岛屿"所需的数字。然后他们使用一个计算机程序计算了前21个"岛屿"的公式解。（在那之后，计算变得太复杂，计算机无法处理。）这些数字都是正数，正如预期的那样——而且它们似乎遵循一个简单的模式，暗示所有其他"岛屿"对应的数字也将是正数。

但模式不是证明，马斯佩罗和他的同事们不确定如何继续。于是他们转向全球的计算机专家社区寻求帮助。

堤坝决口

马斯佩罗一直在数学文献中搜寻任何可能对他有帮助的东西。他认为问题在于，他需要以某种方式简化他必须进行的计算。他找到了一本书，其中罗格斯大学的数学家多伦·蔡尔伯格概述了在计算机上执行困难代数计算的算法方法。在无法将这些方法应用于自己情况后，马斯佩罗直接联系了蔡尔伯格。

"我们最近遇到了一些无法解决的组合问题，"他给蔡尔伯格的邮件开头写道。"我们想知道您是否能帮助我们。"

蔡尔伯格对此很感兴趣。"这个问题正好对我的胃口，"他说。经过一番努力，他让他称之为 Shalosh B. Ekhad 的计算机（该计算机作为合著者出现在他所有的论文中）计算了前2000个"岛屿"的求和，验证了输出结果都是正数，并且符合意大利团队确定的模式。然后，他召集了他的计算机代数爱好者网络来帮忙，并提出以能证明该模式永远持续下去的人的名义，向"整数序列在线百科全书"捐赠100美元。

2024年2月，蔡尔伯格付了款。在与两位经常合作的伙伴进行了漫长的邮件交流后，他带着一个完整的证明回来了，证明这些求和永远不会等于零。

德康尼克和奥利弗拉斯是对的：他们的"岛屿"是真实存在的。这一结果意味着，数学家们现在终于精确地知道了哪些类型的扰动会杀死斯托克斯波，哪些不会——这是他们两个世纪以来一直希望理解的事情。

"这感觉就像是，天啊，太感谢了，"奥利弗拉斯说。

这也给数学家们留下了更多的工作。为什么波浪会以这种交替的模式生与死？"好吧，那些'岛屿'是真实的，"她说。"现在我们必须重视它们。"

这一成果仅仅是近期旨在阐明水波数学的一系列论文中的最新进展。数学家们正在结合计算和理论技术的进步，以更好地理解欧拉方程的解，使他们能够证明越来越多关于波浪行为的猜想。马斯佩罗和他的同事们希望，他们的方法现在可以用来解决该领域的其他问题。

至于马斯佩罗办公室窗外被波拉风吹动的波浪，以及它们最终衰退为平静水面——目前，他还不能肯定他的团队的数学是否能解释这一确切现象。"我不知道是否有联系，"他说。"但我愿意相信是同样的不稳定性在起作用。"

英文来源：

The Hidden Math of Ocean Waves Crashes Into View
Introduction
The best perk of Alberto Maspero’s job, he says, is the view from his window. Situated on a hill above the ancient port city of Trieste, Italy, his office at the International School for Advanced Studies overlooks a broad bay at the northern tip of the Adriatic Sea. “It’s very inspiring,” the mathematician said. “For sure the most beautiful view I’ve ever had.”
Italians call Trieste la città della bora, after its famed “bora” wind, which blows erratically down off the Alps and over the city. When the bora is strong enough, it drives the waves into reverse. Instead of breaking against the docks, they stream away from the city, back toward the open sea.
But they never actually get there. Watching from his window on these gusty days, Maspero can see the retreating waves slowly disperse as they exit the port, eventually giving way to a calm, still surface.
The equations that mathematicians use to study the flow of water and other fluids — which Leonhard Euler first wrote down nearly 300 years ago — look simple enough. If you know the location and velocity of each droplet of water, and simplify the math by assuming there’s no internal friction, or viscosity, then solving Euler’s equations will allow you to predict how the water will evolve over any time period. The rich menagerie of phenomena we see in the world’s oceans — tsunamis, whirlpools, riptides — are all solutions to Euler’s equations.
But the equations are usually impossible to solve. Even one of the simplest and most common kinds of solutions — one that describes a steady train of gently rolling waves — is a mathematical nightmare to extract from Euler’s equations. Until about 30 years ago, the bulk of what we knew about these waves came only from a mix of real-world observations and guesswork. For the most part, proofs seemed like a fantasy.
“Before starting math, I thought water waves were something very understood — not a problem at all,” said Paolo Ventura, a postdoctoral fellow at the Swiss Federal Institute of Technology Lausanne and Maspero’s former graduate student. “But in reality, they are just strange.”
One strange phenomenon that has perplexed mathematicians for decades is that, even when friction is minimal, that steady train of gently rolling waves still eventually falls apart and becomes irregular. Mathematicians hadn’t expected to see such unstable behavior emerge from such a simple starting point. They wanted to prove it — to show that instabilities arise naturally from the Euler equations. But they couldn’t figure out how to do it.
Now Maspero and Ventura, along with their Trieste colleague Massimiliano Berti and Livia Corsi of Roma Tre University, have finally presented such a proof, showing exactly when these instabilities occur and when they don’t. The result is just the latest in a renaissance that’s starting to transform our mathematical understanding of Earth’s waves. Mathematicians have been using new computational tools to formulate conjectures about how waves behave. And they’re now developing sophisticated pen-and-paper techniques to prove those conjectures.
“It’s not one particular thing. It’s a whole wave of new types of analysis in multiple directions,” said Walter Strauss, a mathematician at Brown University. “I’m very impressed.”
A Slow Tide
The ancient Greeks often compared the unsteady beat of waves against the shore to laughter. Considering how those waves have eluded human understanding, perhaps they were right: The ocean has been laughing at us all along.
Even at the height of the Enlightenment in the late 17th and early 18th centuries, when waves took up much of the scientific discourse, the ocean always seemed to have the last word. A number of scientists had measured the speed of sound waves, and Newton and his detractors were locked in a conflict over the wavelike nature of light. But the oldest waves known to humans remained a mathematical enigma.
It would take more than a century for this to start to change. In the early 1800s, Sir George Stokes became fascinated with ocean waves when, as a boy, he was swimming near his home in Sligo, Ireland, and an enormous wave almost dragged him out to sea. In 1847, he published a monumental treatise on the topic. He started with Euler’s equations for a fluid with no viscosity and added the mathematical condition that its top surface be totally “free” — allowed to take any shape it pleased.
“They don’t look bad,” Strauss said of the resulting equations. “But just take a look at a lake with a little wind on it. You get all these complicated forms, like whitecaps and rolling waves, some parallel to each other, some not.”
Each of these varied forms, when understood as a solution to Euler’s equations, is mathematically distinct and terribly unwieldy. Make the tiniest change to the fluid’s initial state, and it might evolve in a vastly different way — bumps and eddies can become rogue waves and tsunamis.
These free, moving surfaces were what Stokes wanted to study. But the challenge was immense. Describing the motion of water confined within a box, or flowing through a pipe, is hard enough. But then, at least, you know where the system’s edges lie — no water can extend beyond those boundaries. If there’s no restriction other than the force of gravity on how high the water can reach and what shape it can take, the math becomes far more difficult.
“If I go to the beach at seven in the morning, it’s going to be very calm,” Corsi said. “But if you really look at the surface, how it moves, it’s a mess.”
Still, Stokes was able to conjecture one solution: that it’s possible for the surface of the water to form evenly spaced waves that travel in a single direction.
In the 1920s, mathematicians proved Stokes’ conjecture. Furthermore, they found that if there are no external disturbances, these solutions to the Euler equations persist forever: Once they form, so-called Stokes waves will continue cruising gaily along the water’s surface for all time, their form unchanged.
But what if the wake of a passing boat crosses the waves’ path? Will the waves absorb this disturbance and maintain their form, or will they be disrupted permanently, transforming into an entirely different pattern of waves?
For decades, mathematicians assumed that Stokes waves are stable, meaning that any small distortion will have a minimal effect. After all, the real world is full of such complications, yet the seas are rife with Stokes waves. If they fell apart at the tiniest poke, they’d never survive long enough to make it to shore.
Still, in 1967, the mathematician T. Brooke Benjamin decided to verify this basic assumption. He had his student Jim Feir perform a series of experiments in a wave tank — a narrow rectangular pool with an oscillating rudder at one end that could produce Stokes waves. But Feir couldn’t get the waves to reach the other end of the pool. At first, he thought there was a problem with the experimental setup. But soon it became apparent that the waves were, surprisingly, unstable.
In 1995, mathematicians finally proved that such “Benjamin-Feir instabilities” are an inevitable consequence of the Euler equations. But the work left researchers wondering about the nature of these instabilities. Which kinds of disturbances can kill waves, and which can’t? How rapidly do the instabilities balloon? Could a gust of wind at the center of the Pacific cause a train of waves to strike Malibu Beach weeks later, or would the formation break down before reaching the shore?
Strange Archipelagos
Maspero had never thought to wonder why the waves exiting Trieste’s bay were dying. His inspiration ultimately came from a computer, not the scene outside his window.
At a 2019 workshop on the mathematics of waves, he and his collaborators met Bernard Deconinck, an applied mathematician at the University of Washington who, along with Katie Oliveras of Seattle University, had been mapping all the different instabilities that could destroy Stokes waves. A few years earlier, the pair had noticed an astonishing pattern, and they hadn’t been able to stop thinking about it.
When a perfect train of Stokes waves encounters a disturbance that distorts the waves’ shape, sometimes the effects of the disturbance grow to destroy the entire train, and sometimes they barely interfere. The outcome depends on the frequency of the disturbance — how much it oscillates compared to the length of the original wave. A kayak, which produces a wake that consists of short, frequent oscillations, will deliver a higher-frequency impact than a massive ocean liner, which produces longer and slower oscillations.
In general, mathematicians expect waves to recover more easily from higher-frequency disruptions like the kayak’s, because their impacts are limited to a smaller region of a passing wave at any given moment. The wake of the ocean liner, on the other hand, can affect the entire wave at once, permanently disrupting it. Benjamin-Feir instabilities are caused by low-frequency disruptions.
In 2011, Deconinck and Oliveras simulated different disturbances with higher and higher frequencies and watched what happened to the Stokes waves. As they expected, for disturbances above a certain frequency, the waves persevered.
But as the pair continued to dial up the frequency, they suddenly began to see destruction again. At first, Oliveras worried that there was a bug in the computer program. “Part of me was like, this can’t be right,” she said. “But the more I dug, the more it persisted.”
In fact, as the frequency of the disturbance increased, an alternating pattern emerged. First there was an interval of frequencies where the waves became unstable. This was followed by an interval of stability, which was followed by yet another interval of instability, and so on.
Deconinck and Oliveras published their finding as a counterintuitive conjecture: that this archipelago of instabilities stretches off to infinity. They called all the unstable intervals “isole” — the Italian word for “islands.”
It was strange. The pair had no explanation for why instabilities would appear again, let alone infinitely many times. They at least wanted a proof that their startling observation was correct.
For years, no one could make any progress. Then, at the 2019 workshop, Deconinck approached Maspero and his team. He knew they had a lot of experience studying the math of wavelike phenomena in quantum physics. Perhaps they could figure out a way to prove that these striking patterns arise from the Euler equations.
The Italian group got to work immediately. They started with the lowest set of frequencies that seemed to cause 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 encoded how the instability would grow and distort the Stokes waves over time. The mathematicians realized that if one of the numbers in the matrix was always zero, the instability would not grow, and the waves would live on. If the number was positive, the instability would grow and eventually destroy the waves.
To show that this number was positive for the first batch of instabilities, the mathematicians had to compute a gigantic sum. It took 45 pages and nearly a year of work to solve it. Once they’d done so, they turned their attention to the infinitely many intervals of higher-frequency wave-killing disturbances — the isole.
First, they figured out a general formula — another complicated sum — that would give them the number they needed for each isola. Then they used a computer program to solve the formula for the first 21 isole. (After that, the calculations got too complicated for the computer to handle.) The numbers were all positive, as expected — and they also seemed to follow a simple pattern that implied they would be positive for all the other isole as well.
But a pattern isn’t a proof, and Maspero and his colleagues weren’t sure how to proceed. So they turned to a global community of computer experts for help.
The Levee Breaks
Maspero had been scouring the mathematical literature for anything that could help him. The problem, he decided, was that he needed to somehow simplify the calculations he had to make. He found a book in which Doron Zeilberger, a mathematician at Rutgers University, outlined algorithmic approaches to performing difficult algebraic calculations on a computer. Unable to adapt them to his case, Maspero reached out to Zeilberger directly.
“We have recently encountered certain combinatorial problems that we cannot solve,” his email to Zeilberger began. “We wonder if you can help us.”
Zeilberger was intrigued. “The question was exactly my cup of tea,” he said. With some work, he was able to get his computer, which he calls Shalosh B. Ekhad (and which appears as a co-author on all his papers), to compute sums for the first 2,000 isole, verifying that the outputs were all positive and that they conformed to the pattern the Italian team had identified. Then he called on his network of computer-algebra enthusiasts to help, offering to make a $100 donation to the On-Line Encyclopedia of Integer Sequences in the name of whoever could establish that the pattern persisted forever.
In February 2024, Zeilberger paid up. After a lengthy email exchange with two of his frequent collaborators, he came back with a complete proof that the sums would never equal zero.
Deconinck and Oliveras had been right: Their isole were real. The result means that mathematicians now finally know precisely which types of disturbances will kill a Stokes wave and which will not — something they have hoped to understand for two centuries.
“It’s just like, holy crap, thank you,” Oliveras said.
It also leaves mathematicians with more work to do. Why do waves live and die in this alternating pattern? “OK, those isole were real,” she said. “Now we have to pay attention to them.”
The result is just the latest in a recent spate of papers that aim to illuminate the mathematics of water waves. Mathematicians are combining advances in computational and theoretical techniques to better understand solutions to the Euler equations, allowing them to prove more and more conjectures about how waves behave. Maspero and his colleagues hope that their methods can now be used to solve other problems in this area.
As for the bora-blown waves outside Maspero’s office window, and their eventual decline into flat water — at the moment, he can’t say for sure whether his team’s math explains this precise phenomenon. “I don’t know if there is a connection,” he said. “But I love to think it’s the same instabilities.”