Boston University, USA

A physicist highlights a three-in-one deal for nonlinear science

As a student of nonlinear phenomena, I am continually amazed by new examples of deterministic chaos, solitary
waves and fractals.

A recent study (R. H. Goodman and R. Haberman Phys. Rev. Lett. 98, 104103; 2007) gave me the rare pleasure of seeing all three of these fundamental manifestations of nonlinearity woven together.

This paper addresses the collisions of solitary waves — localized nonlinear waves that propagate without changing shape and are found in systems ranging from solids to optical fibres.

In the 1980s, with several colleagues, I studied this problem numerically (see, for example, D. K. Campbell and M. Peyrard Physica D 18, 47–53; 1986). We discovered a surprising 'bounce' phenomenon, in which solitary waves would collide, remain trapped for a number (n) of bounces and then escape to infinity. This behaviour occurred only when the waves had specific relative velocities on colliding; these bounce windows were interspersed with regions in which the waves repelled each other immediately.

We developed a heuristic explanation for this behaviour, consistent with the waves behaving like elastic particles that can be deformed, but fell short of developing a full analytical explanation.
Journal club

Goodman and Haberman have now developed an analytical treatment of this effect and have shown, in their words, "that clusters of (n+1)-bounce windows accumulate at the edges of each n-bounce window, repeated at diminishing scales" in an effective fractal structure. This also means that the outcome of a collision is exquisitely sensitive to the initial velocity, a hallmark of deterministic chaos.

If all the above seems dry, take a look at the wonderful graphic (here) from the article, which represents the number of bounces as a function of the collision parameters. The image is certainly worth more than these few hundred words.