Integrability in disconnecting air bubbles

This is a rather neat bit of theory. It's motivated by the following paradox: integrable systems remember everything about their initial conditions, whereas the behavior of systems at "finite-time singularities" (the dynamical-systems equivalent of phase transitions; cf. Chapter 10 of Nigel Goldenfeld's book) exhibits "universality," which implies insensitivity to initial conditions. What happens to integrable systems with finite-time singularities, e.g. a disconnecting air bubble? It turns out that, while some properties of the bubble exhibit universal behavior, others are sensitive to the initial conditions.

Memory-encoding vibrations in a disconnecting air bubble
Laura E. Schmidt, Nathan C. Keim, Wendy W. Zhang & Sidney R. Nagel
Nature Phys. 5, 343 - 346 (2009)

Many nonlinear processes, such as the propagation of waves over an ocean or the transmission of light pulses down an optical fibre1, are integrable in the sense that the dynamics has as many conserved quantities as there are independent variables. The result is a time evolution that retains a complete memory of the initial state. In contrast, the nonlinear dynamics near a finite-time singularity, in which physical quantities such as pressure or velocity diverge at a point in time, is believed to evolve towards a universal form, one independent of the initial state2. The break-up of a water drop in air3 or a viscous liquid inside an immiscible oil4, 5 are processes that conform to this second scenario. These opposing scenarios collide in the nonlinearity produced by the formation of a finite-time singularity that is also integrable. We demonstrate here that the result is a novel dynamics with a dual character.