Interaction of Fronts: Wavenumber Domains

Bistable systems allow states that connect the two stable states as x goes from -¥ to ¥. In the simplest situations like in a real Ginzburg-Landau equation for a subcritical pitch-fork bifurcation,

¶tA = ¶x2A+mA+A3-A5,

two fronts of opposite polarity experience an attractive interaction. Thus, the two fronts are given to leading order by A_F(x-x_L(t)) and A_F(x_R(t)-x) , respectively and the distance L = x_R-x_L between the fronts satisfies

d dt L = m1-ge-L/x,

(1)

where m₁ is proportional to the control parameter m and g > 0 . If the interaction is purely attractive, localized solutions given by stationary solutions to (1) are unstable, i.e. if their length is slightly too small they collapse and if their length is slightly too large their length diverges and the nonlinear state expands into the whole system. For initial states corresponding to an array of fronts this implies an annihilation of fronts of opposite polarity as illustrated in the figure below.

The figure shows the temporal evolution of the local wave number in a fourth-order Ginzburg-Landau equation

¶tA = D¶x2A-¶x4A+mA-|A|2A,

which for D < 0 describes systems that exhibit a transition to spatially periodic patterns with two different wavenumbers. The initial state has a homogeneous wavenumber which is unstable. For sufficiently large m this evolution can be described by a fourth-order phase equation (Brand and Deissler, Phys. Rev. Lett. 63 (1989) 508, Riecke, Europhys. Lett. 11 (1990) 213, Raitt and Riecke, Physica D 82 (1995) 79). Such a formation of domains has been observed in experiments by the group of M. Dubois (Hegseth et al., Europhys. Lett. 17 (1992) 413) and is also related to zig-zag structures in two-dimensional structures. The evolution of a periodic structure into an array of domains with different wave numbers can be seen in a movie (500kB). The green line gives the local wave number and the red and yellow line the real and imaginary part of the complex amplitude of the pattern. Note how short domains disappear by an annihilation of the domain walls between high and low wave-number domains.

In more complex systems the interaction between the fronts can be oscillatory in space. Then the fronts can form stable bound states and the coarsening observed in (1) only occurs in an initial transient. A sequence of states that is obtained when D is increased towards 0 from negative values is shown in the next figure. Time progresses from the bottom towards the top. After each of the snapshots the parameter D is changed and the figure above the snapshot shows the ensuing motion of the fronts via the changes in position of the zero-crossings of the wavenumber. In the snapshots the thick solid line indicates the local wavenumber and the thin lines shows the corresponding pattern. The locking of adjacent fronts is clearly seen, for instance, in part f) where two adjacent fronts drift sideways while keeping their (locked) distance fixed. As D is further increased towards 0 localized states of certain lengths disappear (Raitt and Riecke, Phys. Rev. E 55 (1997) 5448).

Experimentally, such stable domains of patterns with different wavenumbers have been observed in convection in narrow channels (Hegseth et al., Europhys. Lett. 17 (1992) 413) and in Faraday waves excited on ferrofluids (Mahr and Rehberg, Phys. Rev. Lett. 80 (1998) 89).