Solitary wave blocking

We assume the following properties of the unperturbed version of this equation ( :

1.   There exists a stable solution in the form of a solitary wave, , with width , defined, for example, as the separation between points in which some component of v has a chosen value.
2.   This solitary wave can develop from initial conditions in the form of this same wave laterally squeezed, i.e. , k;SPMgt;1, with width, , shorter than normal but longer than some minimal width, ( ), but if , the wave decays.
3.   The typical time for development/establishment of the wave profile is . If, as is often the case, the excitation waves are supported by processes of different time scales, then will be the slowest of them (i.e. the time constant of the limiting stage). The meaning of this parameter will be seen more clearly from its use below.

If but is any positive constant, then assumption 1 implies that (4) has a stationary solitary wave solution , with width . If K(t) is not constant but changes slowly, we may expect that the solution will have the same form as this wave, slowly adjusting its width accordingly. If K(t) changes too rapidly, only then may the wave solution collapse.

With the assumptions listed above, this can be formalised in a phenomenological model. Let us assume, for simplicity, that the dynamics of the wave width is linear with constant relaxation time . As the instantaneous equilibrium state should be changing in accordance with K(t), this gives

 

This equation is in terms of the spatial variable used in (4), which physically corresponds to the width measured in the direction of the flow. The rescaled width , corresponding to real width measured perpendicularly to the wavefront, then obeys

The starting and the final asymptotic value of is . In between, it decreases below but always remains positive. The minimal value of is achieved at which gives

 

The system of equations (8,10) determines, in principle, the critical shear and the corresponding time of the break; however, the exact solution is rather tedious.

To obtain a simple analytical estimate, let us consider the case of practical interest, when the ratio of the activation timescale to the inhibition timescale is

Propagation of excitation waves in this limit is described by the Fife approximation [7]. In particular, the limiting stage of wave formation is establishment of its width, which happens on the time scale of , and minimal possible width from which the wave can recover is of the order of the activation front width, , whereas, in contrast, the normal wave width is, obviously, .

In this limit we should expect that the breakup occurs only if the effective diffusivity changes very quickly compared with . Assuming, in a self-consistent way, that and neglecting the change of L(t) during this time interval, , to leading order we obtain a system of two equations:

  

Using (5) in (12), , then (13) implies , and so the solution is

 

in agreement with the original assumptions[8].

Thus, in this case, there is a critical shear, above which the wave is quenched, and the value of this shear depends on the difference between the time scales of activator and inhibitor, which is a measure of the excitability of the system; the more excitable the system, the bigger the shear needed to destroy the wave.