Mathematical advance adds to knowledge describing waves

February 25, 2016 // By Jean-Pierre Joosting
University at Buffalo mathematician Gino Biondini and postdoctoral researcher Dionyssios Mantzavinos have published a new paper that advances the art of describing a wave. The findings, published in Physical Review Letters, pertain to wave forms ranging from light waves in optical fibers to water waves in the sea.

The study explores what happens when a regular wave pattern has small irregularities, a question that scientists have been trying to answer for the last 50 years.

Researchers have long known that in many cases such minor imperfections grow and eventually completely distort the original wave as it travels over long distances, a phenomenon known as "modulational instability." But the UB team has added to this story by showing, mathematically, that many different kinds of disturbances evolve to produce wave forms belonging to a single class, denoted by their identical asymptotic state.

Biondini, a professor of mathematics in the UB College of Arts and Sciences and an adjunct faculty member in the UB physics department, says the first great success in using math to represent waves came in the 1700s. The so-called wave equation, used to describe the propagation of waves such as light, sound and water waves, was discovered by Jean le Rond d'Alembert in the middle of that century. But the model has limitations.

"The wave equation is a great first approximation, but it breaks down when the waves are very large — or, in technical parlance — 'nonlinear,'" Biondini said. "So, for example, in optical fibers, the wave equation is great for moderate distances, but if you send a laser pulse (which is an electromagnetic wave) through an optical fiber across the ocean or the continental U.S., the wave equation is not a good approximation anymore. "Similarly, when a water wave whitecaps and overturns, the wave equation is not a good description of the physics anymore."

Over the next 250 years, scientists and mathematicians continued to develop new and better ways to describe waves. One of the models that researchers derived in the middle of the 20th century is the nonlinear Schrödinger equation, which helps to characterize wave trains in a variety of physical contexts, including in nonlinear optics and in deep water.