Asymptotic front formation in reactive ion-exchange

As illustrated by (3.2.11), for m 2 the first derivative of concentration at the boundary of support is discontinuous that is, a weak shock is formed at the zero concentration front. This stands in accord with the classical Rankine-Hugoniot condition that prescribes for any moving interface Xi(t) 

Equation (3.3.1) implies that with a boundary of the support moving at a finite speed, the derivative at the boundary is finite, discontinuous for m = 2, and blows up one-sidedly for m 2. 

2 Asymptotic Ll stability of these waves has been announced recently by Tak4i [37]. 

It was observed in the previous section that a certain limit case of non-reactive binary ion-exchange is described by the porous medium equation with m = 2 in other words, a weak shock is to be expected at the boundary of the support. Recall that this shock results from a specific interplay of ion migration in a self-consistent electric field with diffusion. Another source of shocks (weak or even strong in the sense to be elaborated upon below) may be fast reactions of ion binding by the ion-exchanger. 

In this section we address formation of concentration shocks in reactive ion-exchange as an asymptotic phenomenon. The prototypical case of local reaction equilibrium of Langmuir type will be treated in detail, following [1], [51], For a treatment of the effects of deviation from local equilibrium the reader is referred to [51]. The methodological point of this section consists of presentation of a somewhat unconventional asymptotic procedure well suited for singular perturbation problems with a nonlinear degeneration at higher-order derivatives. The essence of the method proposed is the use of Newton iterates for the construction of an asymptotic sequence. 

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