Frank Merle is used to confronting a messy world. He works on the mathematics of highly nonlinear systems—ones that respond in dramatic, unpredictable ways to even the smallest changes. It’s the same math that explains how, under the right conditions, the atmosphere above a barren plain can produce a roiling tornado.
A linear equation is something like y = 2x, which states that the value of y doubles whenever you double the value of x. But most equations are much more sensitive to changes to their input. A highly nonlinear system is defined by equations that can jump from zero to infinity almost out of nowhere. Sussing out whether a system of equations can exhibit this kind of extreme behavior, called a “singularity” or “blowup,” is a difficult task for mathematicians.
Merle has had enormous success taming these blowups in the equations describing lasers, fluids and quantum mechanics. His trick is to embrace the nonlinear. Whereas most researchers before him treated these phenomena gingerly by making tiny tweaks to a well-behaved, linear world, he has focused them, studying their mathematical consequences directly. “I have a slightly different view of the world,” he says. “I see the world as a more catastrophic place to live.”
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By engaging with the chaos, Merle discovered simplicity. Much of his work focuses on special structures, called “solitons,” that persist amid the mayhem of nonlinear systems. Solitons are able to keep their form and energy while they move about in realms where the gnarliest math reigns like a single rogue wave traversing an entire vast, swirling ocean wholly intact. Merle believes that all nonlinear systems can be dealt with by thinking of them as a bunch of these solitons coming together—chaos belying simplicity.
Today Merle received this year’s Breakthrough Prize in Mathematics for his achievements. The prize comes with a $3-million award. Scientific American spoke with Merle about how he managed to tame some of nature’s most tangled sets of equations.
[An edited transcript of the interview follows.]
What does this prize mean to you?
It came as a shock—it took me some time to recover. It’s a great honor. And it’s exciting, because when I found this new way of seeing these problems, most people were not convinced that I could produce something interesting. Then one problem fell and then another one, so of course now there’s a lot of recognition of all this work.
What was your “new way of seeing problems” in nonlinear dynamics?
I was only concentrating on the nonlinear structure. Most of the work before started from something we understand—linear things—and pushed them slightly into the nonlinear. But my starting point was never the linear structure; it was the nonlinear stuff.
And this led you to put solitons front and center.
Yes, because solitons are a totally nonlinear concept. A soliton is a special solution to nonlinear equations, such as fluid equations, that doesn’t send energy away to infinity—it keeps all its energy contained and keeps the same shape.
When you look at physical quantities in nonlinear systems, they seem to oscillate and change chaotically. But if you look long enough, some emergent structure appears that doesn’t depend that much on how things started. This emerging structure is the soliton. From the mathematical point of view, you don’t initially see why it will appear, yet somehow it does.
Solitons seem much simpler than the crazy, chaotic behavior of nonlinear systems. Yet you believe that the behavior of these systems comes down, somehow, to solitons.
Yes, a family of interacting solitons. This is called the “soliton resolution conjecture.”
It’s been the belief since the 1970s, but people then couldn’t really see the nature of this phenomenon—why exactly it must be true. And mathematically, there’s no way to tackle it, except for a few specific kinds of nonlinear equations.
But the idea is pure beauty. You look at a very complicated situation—your problem is chaotic, with infinitely many parameters—but then, at the end, everything becomes simple, with a finite number of parameters that you can track down and compute.
The equation you discover at the end can be even simpler than you think. There’s a simplicity that’s very hidden, very difficult to see even by experiment, but it appears. There’s a little bit of magic in that.
You used solitons to help study blowup—the phenomenon where nonlinear equations break down and suddenly become infinite. Why does this matter?
For different nonlinear equations, blowup can be either good or not good—either you want blowup, or you don’t. But to know how it works is important either way. In the equation for how focused a laser is, you want blowup because you want to focus your laser as much as possible.
And you proved that the laser equations can blow up under certain conditions. Does that mean the laser actually becomes infinitely focused?
Not really. The mathematical equation says it goes to infinity, but in reality, it doesn’t. It will just become very focused and then stay very focused for a long time.
But the equation is just an approximation. In fact, in all of physics, equations are always approximations. Different physics come out when the laser is very concentrated: sometimes known physics and sometimes completely unknown physics.
You also worked on blowup for fluid equations. How is that different?
In fluid equations, you want to avoid blowups because they’re related to turbulence. But in real life, you have turbulence everywhere, so you need to at least understand it.
I worked on compressible fluids, which are ruled by the Navier-Stokes equation. People already knew that a simplified version of the equation, without any friction, could produce singularities.
But the question was whether having friction could at least slow down the singularity formation or [even] stop it. Our result was to prove that it didn’t stop it—that friction doesn’t stop the blowup.
Isn’t blowup in Navier-Stokes one of the Clay Mathematics Institute’s Millennium Prize Problems? Does that mean that solving it is worth $1 million?
The Clay problem is the same question for incompressible fluids. This was for compressible fluids—the compressibility helps you in some sense. So the Clay problem remains open still.
You also worked on the nonlinear version of the Schrödinger equation governing quantum mechanics. What was the breakthrough there?
You have a linear part of the Schrödinger equation and a nonlinear part. Usually the linear term is the most important, but sometimes—what’s called the “super-critical case”—the nonlinear term can have its own craziness.
Everybody—even myself—thought for a long time that solutions to the Schrödinger equation will never blow up, because any singularity will disperse after some time. For a while, we tried to prove this.
In math, sometimes you almost prove a thing in several different ways, and each time there is some key point missing, something you cannot tame. Maybe you think it’s small.
But after a while, you get this feeling that maybe this is a hint that the opposite might be true. And that small piece turns out to be dramatic, the key element of what becomes your proof of the opposite statement. That’s what happened in this case. So the process of mathematics itself is often nonlinear, too—at least for me.
