(PhysOrg.com) -- A coupled line of swinging pendulums apparently has nothing in common with an elastic film that buckles and folds under compression while floating on a liquid, but scientists at the University of Chicago and Tel Aviv University have discovered a deep connection between the two phenomena.
Energy carried in ordinary waves, like those seen on the ocean near a beach, quickly disperses. But the energy in the coupled pendulums and in compressed elastic film concentrates into different kinds of waves, ones with discrete packets of energy called solitons.
Solitons express themselves in other realms as well. In telecommunications, for example, pulsing light travels as solitons through optical fibers. What is special about solitons, and this very special way of localizing energy, is that it does not disperse, said Haim Diamant, associate professor of chemistry at Tel Aviv University. It remains a very well-defined, focused packet.
The connection, which Diamant and UChicagos Thomas Witten report Oct. 14 in the journal Physical Review Letters, is subtle and more easily appreciated in visual form (see graphic). Witten graphed the swing angle of the pendulums as time progresses. Then he made a curve whose angle with the horizontal varies to match the swing angles along the diagonal edge of the graph. The resulting curve takes the same shape as the profile of a folded elastic film floating on a liquid, like the film shown at the bottom of the picture.
Witten and Diamant began collaborating while the latter worked at UChicagos James Franck Institute as a postdoctoral scientist from 1999 to 2002. They have been working together on puzzles emerging from the laboratory of another collaborator, Ka Yee Lee, UChicago professor in chemistry, ever since. Lees research group studies the complex mixture of lipids and proteins that lines the sacs of the lung. These molecular linings fold and unfold as we inhale and exhale; the folding appears important for normal breathing.
Slowly growing energy
The energy applied to the films deformation starts out weak, then grows stronger. Once the wrinkling energy in the film grows stronger, it concentrates itself into a fold shaped like the folding curve in the figure.
Though the fold appears in a specific place on the film, the motion resulting from folding extends over a big region. Usually big things are slow, said Witten, the Homer J. Livingston Professor in Physics. But this is a big thing that is not slow. Its a rapid jerk, and we want to see what enables such rapid, large-scale motion.
Lee and her associates aim to understand breathing mechanics using synthetic films only one layer of molecules thick to simulate the surfactant that lines the microscopic air sacs found in the lungs. Diamant and Witten sought to solve the equation that exactly describes the fold shape of such a film. They knew it would be a difficult task, given that it was a nonlinear equation, one in which simple changes produce complicated effects.
A typical nonlinear problem might absorb decades of work without yielding a solution; this one seemed different. There were strange hints that told us this problem might be solved exactly, Diamant said. Its very rare that a nonlinear problem can be solved exactly.
These hints had appeared in numerical simulations of the folding process generated by Enrique Cerda, a collaborator at the University of Santiago in Chile. It was a miracle that we found an exact solution, but we had a strong feeling that it existed, Diamant said.
Once Diamant and Witten solved the problem, they realized that the solution resembled the sine-Gordon equation, well-known among mathematicians and physicists, which describes how a coupled line of swinging pendulums concentrate their energy.
Their resulting paper lays out the first example the authors know of in which soliton motion of a dynamical system can help scientists understand material deformation. The materials in this instance involve a thin, rigid layer floating on a fluid surface, a structure commonly found in biological tissues and syntheti