Slow and fast diffusion in ion-exchange

Consider a particular case of (3.1.15) with n = 2, signzj = sign z2. This corresponds to the exchange of two counterions in an ideal ion-exchanger (complete co-ion exclusion). Accordingly, (3.1.15) is rewritten as 

The other peculiarity concerns the beginning of movement of the support s boundary. For an arbitrary initial condition, the former does not start propagating right away rather it takes some finite waiting time to build the boundary concentration gradient that is necessary for the support s propagation to begin. 

In this section we shall concentrate on another somewhat less explored limit case of (3.2.1a). For a — oo, = 1 equation (3.2.1a) yields 

In this section we shall study still another peculiarity of (3.2.3)—the occurrence of uniformly bounded, monotonic travelling waves. These waves, very common in reaction-diffusion (see, e.g., [25]—[27]), seem fairly unexpected in the reactionless diffusion under discussion. Their occurrence here is directly related to the singularity of diffusivity in (3.2.3) and thus can be viewed as the fast diffusional counterpart of the aforementioned peculiarities of the porous medium equation. 

Before we turn to this issue, we would like to substantiate the above discussion of basic features of nonlinear diffusion with some examples based upon the well-known similarity solutions of the Cauchy problems for the relevant diffusion equations. Similarity solutions are particularly instructive because they express the intrinsic symmetry features of the equation [6], [28], [29], Recall that those are the shape-preserving solutions in the sense that they are composed of some function of time only, multiplied by another function of a product of some powers of the time and space coordinates, termed the similarity variable. This latter can usually be constructed from dimensional arguments. Accordingly, a similarity solution may only be available when the Cauchy problem under consideration lacks an explicit length scale. Thus, the two types of initial conditions compatible with the similarity requirement are those corresponding to an instantaneous point source and to a piecewise constant initial profile, respectively, of the form 

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