Evans Function Analysis and Stability Calculations in the Continous Spectrum
Tuesday, November 16, 1999 - 11:00am - 11:50am
Rob Gardner (University of Massachusetts)
The Evans function is a useful tool for locating the eigenvalues of nonlinear waves. It is an analytic function of the eigenvalue parameter, and in the most general setting, the domain of analyticity of the Evans function is the region of the spectral plane outside the continuous spectrum. In many interesting physical problems, the crucial portion of the spectrum that determines whether a wave will be stable is either inside or close to the continuous spectrum of the wave. In such situations, it had been difficult or impossible to use the Evans function in stability calculations. However, recently, it has been shown that the Evans function can be analytically continued some finite distance into the continuous spectrum. We describe how this continuation theorem, (the Gap Lemma), can be used in stability analyses of viscous shock waves and also, of travelling wave solutions of certain reaction-diffusion systems. Although we do not specifically address the question of the stability of combustion fronts, similar issues arise in the stability analysis of both fast and slow combustion waves.