Abstract
We propose a new type of traveling wave pattern, one that can adapt to the size of physical system in which it is embedded. Such a system arises when the initial state has an instability for a range of wavevectors, k, that extends down to k = 0, connecting at that point to two symmetry modes of the underlying dynamical system. The Min system of proteins in E. coli is such a system with the symmetry emerging from the global conservation of two proteins, MinD and MinE. For this and related systems, traveling waves can adiabatically deform as the system is increased in size without the increase in node number that would be expected for an oscillatory version of a Turing instability containing an allowed wavenumber band with a finite minimum.
Highlights
We propose a new type of traveling wave pattern, one that can adapt to the size of physical system in which it is embedded
One of the ways in which a non-equilibrium system can lead to pattern formation is via a traveling wave bifurcation‘[1]
The uniform state becomes unstable to modes at finite wave vector k and finite frequency ω leading to a variety of phenomena involving nonlinear traveling wave states. This scenario has proven relevant for processes ranging from binary fluid convection [2, 3] to electro-hydrodynamics in liquid crystals [4] to the sloshing of Min proteins in bacteria [5,6,7,8,9,10]
Summary
We propose a new type of traveling wave pattern, one that can adapt to the size of physical system in which it is embedded. We have in general three terms that are important; the velocity term, the diffusion term and the cubic term that occurs (with opposite sign) in both the cd and cde equations.
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