# How mathematicians study wave equations

## Gigliola Staffilani has been studying wave equations representing physical phenomena since graduate school Ocean waves are one of the many natural phenomena that can be characterized by a wave equation. Courtesy of MIT News

Waves are everywhere, from tsunamis to earthquakes to light. In fact, if you are reading this article aloud, you are producing waves. Even particles can be modeled by waves. Most waves are governed by a mathematical equation known as the wave equation. Gigliola Staffilani, professor of mathematics, has been studying wave equations representing physical phenomena since graduate school.

“When I was an undergraduate in Università di Bologna in Italy, I had to take basic classes in different fields like algebra, analysis, and geometry. I really did not like algebra because it was too rigid — things are either equal or not. In contrast, analysis is mostly inequalities. Algebraists might disagree with my view, but in analysis there is room to solve an easier problem with less strict inequality before solving a bigger problem,” said Staffilani. As a graduate student at the University of Chicago, Staffilani had to choose between two research topics for her PhD thesis. One was in geometry, focusing on elliptic partial differential equations (PDEs). The fundamentals of PDEs had been thoroughly researched at the time, but there were higher level problems left to be solved. The other project was in analysis, focusing on nonlinear wave equations. Her advisor, who was an expert in elliptic PDEs, had started working on it from analysis point of view. “At that time, I was mostly concerned about finishing my PhD thesis, so I picked the second option which had more problems. I didn’t care if that was going anywhere as a field and didn’t have an end goal for myself becoming an expert in that field. I was lucky that it turned out to be a fundamental field in analysis.”

An example of a nonlinear wave equation that Staffilani worked on is one that governs the Bose-Einstein Condensate. The Bose-Einstein Condensate is a state achieved when a dilute gas made up of particles called bosons is cooled down near to absolute zero. As the temperature is lowered, the particles start to interact with each other as a result of quantum effects. In particular, bosons interfere with each other like light waves from different sources. Considering the quantum effects, it is better to model particles as waves. Mathematically, these waves are represented as functions that depend on time and space. The wave functions that models bosons are governed by a nonlinear system of equations called the Gross-Pitaevskii hierarchy. Even though we know the governing equation, we can’t get an explicit solution to the equation. This means we don’t know how the state of particles evolves with time. So, we can’t make a mathematical prediction of the behavior of bosons. However, there are numerical, experimental, and analytical ways to predict what a solution must look like. Based on the implicit properties of the solutions, the Bose-Einstein Condensate can be modeled precisely.

Most of the time, Staffilani uses a mathematical tool called harmonic analysis to deduce the implicit properties of the solutions. Using harmonic analysis, she decomposes the wave functions into their constituent parts that are better understood. For instance, functions can be decomposed into sines and cosines (also called the Fourier transform of the function). Based on the properties of these familiar functions, she deduces the properties of bigger and more complicated functions. Staffilani tries to understand periodic solutions to the wave equation that governs the Bose-Einstein Condensate. Mathematically, the nonlinear aspect makes it harder to study periodicity. The decomposed parts of the solutions — sines and cosines if we are using Fourier Transform — appear as products. Generally, it is harder to keep track of the period of functions in product form. However, we can attribute the periodicity to the constraint imposed by the physical boundary on the system of particles. We can think of a periodic function as some wave in a box that hits the wall and comes back. If we could understand the properties of an implicit periodic solution to the wave equation, we could understand more about the time evolution of the Bose-Einstein Condensate.

Staffilani acknowledged that a mathematical problem sometimes requires an interdisciplinary approach. “Best breakthroughs are done by people who bring ideas from different fields into the one they think they are expert on.” In the past ten years, her research has been focused on using probability in solving nonlinear wave equations arising in physics. When Staffilani tried to answer about the existence and the uniqueness of a solution to the wave equation using the tools in the analysis only, she encountered some counterexamples. “Are the counterexamples manufactured by the fact that we are approaching the problem in a certain theory trying to use certain types of tools or are they really due to intrinsic problems in physics?” Probability helps in making generic claims about the solutions to the wave functions. For instance, we could say that for “almost all” initial states of the physical system, the wave function representing the system must evolve in a particular way. The counterexamples that show up in mathematics could be thought of as “isolated phenomena that don’t really represent physical phenomena.” Using probabilistic tools, one might also predict phenomena that could exist with a very low probability of occurrence.

Recently, Staffilani has been trying to understand the origin of mathematical structures in solutions to the wave equation governing the Bose-Einstein Condensate. The solutions of Gross-Pitaevskii hierarchy that govern the Bose-Einstein Condensate have been found to be the product of solutions to the Schrödinger equation, which is a widely studied equation in quantum mechanics. Interestingly, the solutions are integrable. Integrability can partially be explained by “saying that there are infinitely many conservation laws like the law of conservation of energy,” but its origin is unclear from a mathematical standpoint. With her collaborators, Staffilani wonders what is in the Bose-Einstein Condensate and the Gross-Pitaevskii hierarchy that results in the integrability of solutions to the Schrödinger equation. “I am in the middle of finishing a 100 page paper describing exactly these types of questions.”