Spectral Analysis for transition front solutions in Cahn–Hilliard systems

We consider the spectrum associated with the linear operator obtained when a Cahn–Hilliard system on R is linearized about a transition wave solution. In many cases it’s possible to show that the only non-negative eigenvalue is λ = 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.