Mathematicians make breakthrough

13 June 2005

Patterns can be found in the most unlikely of places.

You can see it in the alternating sequences of tiny hills and craters that appear when a tray of sand is shaken, and in the spirals that form when water is drained out of a bath. Even in the spots on a leopard and the stripes on a tiger.

Scientists believe that this wonderland of patterns can be explained not just by the actions of genes - in the case of those spots and stripes - and basic scientific principles, but also by mathematical rules. Pattern formation, as it is called, is a field that occupies many a mathematician trying to pin down, in theory anyway, how and why patterns come about.

But even patterns have their quirks. The quirkiest of these are the solitary features known as solitons, which appear entirely divorced from any pattern. For examples, think of the isolated hump that would appear in shaking sand, or even a solitary wave (think tsunami).

In mathematics, solitons have long graduated from theory into the real world, especially in one-dimensional forms (something that has length, but no width). Perhaps the best known of 1-D solitons are the light pulses that travel through the hair-thin optical fibre, used to transmit information and poised to revolutionise communication technology.

"What is new is solitons in 2-D," says Professor Igor Barashenkov, who specialises in the study of pattern formation and solitons, of UCT's Department of Mathematics and Applied Mathematics. The reason for this, notes Barashenkov, is that solitons in two- and three-dimensional forms are prone to "instabilities".

"The problem is that a soliton has so many directions into which it can evolve, or in which it can escape or lose its shape," he says. "So it always finds a way to either collapse or disperse. It simply can not keep its shape."

Several years ago, though, scientists found that a 2-D soliton can be sustained. As long, however, as there is a continuous energy supply to the soliton, balanced by strong dissipation, ie a continuous loss of energy to the surrounding medium, explains Barashenkov.

"It is this self-regulating balance of gain and loss that keeps the soliton intact."

But now Barashenkov and Professor Reinhard Richter of the experimental physics group at Bayreuth University in Germany have re-written the textbooks. In a pioneering experiment involving a magnetic liquid - for their study, kerosene (paraffin) into which a powder of nano-sized rusty-iron particles has been dissolved - that is placed in a stationery magnetic field, they have shown that a soliton (nine, in fact) can keep its shape even when the energy source that first created it, in this case a hand-held electromagnet, is removed.

Clearly visible to the naked eye, the solitons can survive on the surface of the magnetic liquid for days. "They are robust and stable," says Barashenkov.

Barashenkov and Richter's first-of-a-kind findings appeared in a paper, Two-Dimensional Solitons on the Surface of Magnetic Fluids, in a recent edition of Physical Review Letters (PRL), published by the American Physical Society. It was also singled out for lay-coverage in PRL's sister-publication, the online Physical Review Focus (see

The experiment is, of course, of value in the study of magnetic liquids, which boasts an array of industrial and medical uses, such as in drug targeting.

But the findings also pose new challenges for mathematicians interested in the whys and hows of pattern formation, and who now have to produce an iron-clad theory for the laboratory observation. Barashenkov and others believe, for example, that patterns themselves are in fact made up of individual features such as solitons, much the same way that crystal lattices are made up of individual atoms.

"One answer is that the solitons are the basic building blocks out of which patterns are made," says Barashenkov.

That may take some time to prove. For now, though, it is agreed that the scientists' new findings, like their subject matter, certainly stand out.

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