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The Evans function and stability criteria for degenerate viscous shock waves

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.