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DCDS

It is well known that the stability of certain distinguished waves arising
in evolutionary PDE can be determined by the spectrum of the linear operator
found by linearizing the PDE about the wave. Indeed, work over the last
fifteen years has shown that

*spectral stability*implies nonlinear stability in a broad range of cases, including asymptotically constant traveling waves in both reaction--diffusion equations and viscous conservation laws. A critical step toward analyzing the spectrum of such operators was taken in the late eighties by Alexander, Gardner, and Jones, whose*Evans function*(generalizing earlier work of John W. Evans) serves as a characteristic function for the above-mentioned operators. Thus far, results obtained through working with the Evans function have made critical use of the function's analyticity at the origin (or its analyticity over an appropriate Riemann surface). In the case of degenerate (or sonic) viscous shock waves, however, the Evans function is certainly not analytic in a neighborhood of the origin, and does not appear to admit analytic extension to a Riemann manifold. We surmount this obstacle by dividing the Evans function (plus related objects) into two pieces: one analytic in a neighborhood of the origin, and one sufficiently small.
DCDS

We consider the spectrum associated with the linear operator
obtained when a Cahn--Hilliard system on $\mathbb{R}$ is linearized about
a transition wave solution. In many cases it's possible
to show that the only non-negative eigenvalue is $\lambda = 0$,
and so stability depends entirely on the nature of this
neutral eigenvalue. In such cases, we identify a stability condition
based on an appropriate Evans function, and we verify this
condition under strong structural conditions on our equations.
More generally, we discuss and implement a straightforward
numerical check of our condition, valid under mild structural
conditions.

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