Evans function and blow-up methods in critical eigenvalue problems
Björn Sandstede, Arnd Scheel · Discrete and Continuous Dynamical Systems · 2004
Contact defects are one of several types of defects that arisegenerically in oscillatory media modelled by reaction-diffusionsystems. An interesting property of these defects is that theasymptotic spatial wavenumber is approached only with algebraic orderO$(1/x)$ (the associated phase diverges logarithmically). Theessential spectrum of the PDE linearization about a contact defectalways has a branch point at the origin. We show that the Evansfunction can be extended across this branch point and discuss thesmoothness properties of the extension. The construction utilizesblow-up techniques and is quite general in nature. We also comment onknown relations between roots of the Evans function and the temporalasymptotics of Green's functions, and discuss applications toalgebraically decaying solitons.