Dynamics of Slowly Phase-Modulated Periodic Waves

4 Dynamics of Slowly Phase-Modulated Periodic Waves 

In excitable reaction-diffusion systems, pulses can travel as a periodic wave train. In oscillatory reaction-diffusion systems, too, the existence of plane wave solutions has been theoretically established (Kopell and Howard, 1973 a). In this section we will be concerned with such periodic waves in one space dimension, particularly when the local wavenumber slowly and slightly varies with x. For these systems, the analogy to systems of weakly coupled oscillators might look even weaker. Actually, however, there exists a rather strong formalistic similarity between the two. 

Here again, x and are treated mathematically as independent variables. Correspondingly, the spatial differentiation appearing in (4.3.1) has to be transformed as 

The terms of 0(e) and O(e ) have to be treated as a perturbation. By the term unperturbed system , we therefore mean a strictly periodic system, i.e., system (4.3.1) subject to the periodic condition 

Usually, we have a family of such periodic waves with various T (Rinzel and Keller, 1973). Here we consider the problem for a given value of /. The linearization of the unperturbed system about Xq leads to 

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