Boundary conditions are
A(x) ~ a0x as x ^ 0, fx(0) = 0, A(x), fx(x) bounded as x
The problem boils down to finding a0 and Q as functions of q so that the solution of the initial value problem to the time-dependent equations is bounded. One such solution is
A(x) = i — ) tanh (x /V2), fx(x) = (1 - tanh(x/V2), n2 + —
as can be verified. These solutions are generated periodically at the origin, alternatively on either side.
The stability of travelling waves and particularly spiral waves can often be quite difficult to demonstrate analytically; the paper by Feroe (1982) on the stability of excitable FHN waves amply illustrates this. However, some stability results can be obtained, without long and complicated analysis, in the case of the wavetrain solutions of the X-w system. In general analytical determination of the stability of spiral waves is still far from complete although numerical evidence, suggests that many are indeed stable. As briefly mentioned above, more recently Yagisita et al. (1998) have investigated spiral waves on a sphere in an excitable reaction diffusion system. They show, among other things, that the spiral tip rotates. They consider the propagation in both a homogeneous and inhomogeneous medium.
Biological waves exist which are solutions of model mechanisms other than reaction diffusion systems. For example, several of the mechanochemical models for generating pattern and form, which we discuss later in Chapter 6, also sustain travelling wave solutions. Waves which lay down a spatial pattern after passage are also of considerable importance as we shall also see in later chapters. In concluding this chapter, perhaps we should reiterate how important wave phenomena are in biology. Although this is clear just from the material in this chapter they are perhaps even more important in tissue communication during the process of embryological development. Generation of steady state spatial pattern and form is a topic of equal importance and will be discussed at length in subsequent chapters.
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