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Professor explains neverending wave phenomenon

By Kristen Martin
On November 6, 2011

  • Students listen to a lecture given by Alex Kasman, a math professor at the College of Charleston in South Carolina, on the Soliton Theory Friday in Ross Hall. Spencer Duncan

In 1834, Scotsman John Scott Russell discovered a wave that travelled incessantly in the water.

Russell, a ship designer, was interested in discovering how to recreate this wave, but was ridiculed for his modernized thinking. It was not until the 1960s that a formula was created to track these solitary waves using algebraic geometry to create the Soliton Theory.

Alex Kasman, a math professor at the College of Charleston in South Carolina, visited UNC Friday to lecture on the evolution of the Soliton Theory.

Kasman said a soliton is a permanent, localized disturbance in a non-linear wave and the formula for the theory is a mix between linear and nonlinear wave equations.

"For nonlinear equations, you can't write down a solution for the formula," Kasman said. "That explains why Airy and Stokes said there couldn't be a solitary wave."

Sir George Biddell Airy and Sir George Gabriel Stokes were mathematicians who disproved Russell's theory of the solitary waves, and therefore Russell was only recognized for creating the largest ship used to lay the first trans-Atlantic telegraph cable.

"He died not only with people not believing his theory, but also broke," Kasman said.

In the early 20th century, mathematicians Diederik Kortewig and Gustav de Vries created the KdV equation to model waves on a canal and used algebraic geometry to solve it.

"The solution showed it had solitary waves and they thought it was a rare occurrence," Kasman said.

But these waves were not a rare occurrence, as proven in the 1950s by Enrico Fermi, John Pasta and Stan Ulam.

"They discovered that the wave didn't disappear," Kasman said. "It kept going and going and going, but they kept their findings top secret."

In 1965, Norman Zabusky and Martin David Kruskal, New Jersey mathematicians, discovered an additional wave that collides with another to become the solitary wave.

"The two waves become more like particles and thus become solitons," Kasman explained.

It was not until 1967 when Joseph Langley Burchnall and Theodore William Chaundy used the KdV equation to track these waves.

"They used old algebraic geometry to solve it, where people thought there'd be no application," Kasman said. "They discovered that solitons are everywhere."

Ironically, the original cables laid using Russell's boat did not take advantage of his theory, but now there are fiber optic telegraph cables across the Atlantic Ocean that use solitons to send information back and forth.

"They didn't use Russell's boat but his favorite idea (the Soliton Theory)," Kasman said. "He was a man ahead of his time."

 


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