Waves play a central role in systems ranging from fluids to fiber optics, yet their long-term behavior can be difficult to predict.

Gong Chen, assistant professor in the School of Mathematics, has received a $450,000 National Science Foundation Faculty Early Career Development (CAREER) Award to study the long-term behavior of solitons — coherent, particle-like waves observed across physics and mathematics.

While most waves spread out as they disperse (for example, ripples on water), a soliton is different: it can keep a coherent shape while moving, and in some cases, it can interact with other waves and still emerge recognizable afterward.

“Solitons are important because they appear across many areas of science, including fluid dynamics, optics, plasma physics, field theory, and models from mathematical physics,” says Chen. “From a mathematical point of view, they are a beautiful testing ground for understanding nonlinear behavior.”

Predicting Waves

According to Chen, a guiding idea in the field is that complex nonlinear waves may eventually resolve into a collection of stable solitons alongside dispersive radiation — the more diffuse portion of the wave that spreads out and weakens. Chen’s research focuses on how these waves behave over long periods of time, especially when multiple solitons interact.

“If we start with a complicated nonlinear wave, can we predict what it will look like far in the future?” asks Chen. “I want to understand not just whether a wave is stable, but how it evolves: how stability can fail, how energy is exchanged, and how complicated wave motion eventually organizes itself.” 

His work examines multi-soliton systems and more complex wave structures, including topological solitons, where long-range interactions and internal fluctuations make the mathematics more challenging. 

Chen is developing new mathematical frameworks tailored to these moving and interacting waves, including tools from spectral theory and nonlinear scattering. These approaches allow researchers to analyze wave behavior with new precision in settings where existing methods are limited.

A key part of this work involves nonlinear dispersive equations, which capture the competing effects that shape wave systems. 

“A complicated wave may contain several solitons, some radiation that spreads away, and small oscillations trapped near the solitons,” explains Chen. “Nonlinear dispersive equations allow us to ask precise questions: Which part of the wave persists? Which part disperses? How much energy is released during a collision? Does the system eventually simplify into solitons plus radiation?”

Although the work is theoretical, it strengthens the foundation for models used widely in science and engineering.

“A better theoretical understanding of solitons and dispersive waves improves the reliability of these models,” Chen says. “It helps us know when coherent structures should persist, when they should radiate energy, and when instability or collision effects may change the outcome.”

Beyond Research: Teaching and Impact

Chen plans to use the CAREER Award to integrate research and education. He is organizing a summer school focused on dispersive waves and developing new courses, including a second-level course in partial differential equations that emphasizes connections to physical phenomena.

“The CAREER Award provides the stability and long-term support needed to pursue a coherent research program rather than isolated projects,” says Chen. “For me, the award is especially meaningful because the research and education components are closely connected.”

Chen aims to help students connect mathematical theory with real-world phenomena. 

For Chen, that connection is what makes the field compelling.

“One point I would emphasize is that soliton dynamics is a place where abstract mathematics meets very intuitive physical pictures,” says Chen. “Understanding when that particle-like behavior persists, when it breaks down, and what remains afterward is both mathematically deep and scientifically natural.”

The NSF Faculty Early Career Development Program is a five-year grant designed to help promising researchers establish a foundation for a lifetime of leadership in their field. Known as CAREER awards, the grants are NSF’s most prestigious funding for early-career faculty.