Pinning model inhomogeneous

Of course the next step is to introduce inhomogeneities in the model, for example in terms of two different energies Eat a.nd Egc- But this corresponds to considering an inhomogeneous pinning model (and we will come back to this at length). 

We introduce inhomogeneities in the polymer chain, or at the interface or defect line, by assigning to each monomer, or to each interface site, a charge, that is a real number that will determine quantitatively the interaction of the chain with the surrounding environment. For example, an inhomogeneous pinning model is the following ... 

While a quantitative analysis of the periodic pinning model of Section 1.6.1 is not immediate, on a qualitative level the mechanism of the transition is not new with respect to the corresponding homogeneous model. Now instead we are going to present a case in which the inhomogeneous character of the charge distribution is at the base of the localization mechanism. 

In this chapter the aim is above all to stress the basic steps that allow to reduce the weakly inhomogeneous case to the homogeneous one and convey the idea that in this class of models one gets as far as in the homogeneous pinning model treated in Chapter 2. This reduction is not without a price, since in reality formulas become substantially more complex in the details. However it should be stressed from now that new phenomena appear in this set up and one of our purposes is to point out in an informal way the phenomenological richness of periodic models. 

Recently, the breakup of free spiral waves has been observed in the numerical simulations of reaction-diffusion models [42-46]. This effect is principally of the same nature as breaking (or spontaneous depinning) of a pinned spiral wave. The main question is here what induces breaking of the wave front at a certain distance from the free tip. In some of the stimulations (e.g. [42-45]), the breakup is preceded by the onset of meandering which might actually produce the inhomogeneities of the residual inhibitor concentration that force the wave to break. 

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