Evans function and blow-up methods in critical eigenvalue problems
The goal of this paper is to investigate the stability properties of certain nonlinear waves that arise in dissipative, pattern-forming partial differential equations. A reaction-diffusion system \[ U_t = DU_{xx}+F(U), x\in{\mathbb R}\tag{1} \] are considered. Here \(U\in{\mathbb R}^N\) and \(D\) is a diagonal positive diffusion matrix. It is assumed that (1) supports a family of wave-train solutions \(U(x,t) = U_{wt}(kx-\omega t;k)\) that are \(2\pi\)-periodic in \(\phi = kx-\omega t\). The authors show that the Evans function can be extended across this branch point and discuss the smoothness properties of the extension. The construction utilizes blowup techniques and is quite general in nature. They also comment on known relations between roots of the Evans function and the temporal asymptotics of Green's functions, and discuss applications to algebraically decaying solitons.
scientific article; zbMATH DE number 2113143