2012/10/15 by Wolf-Jüergen Beyn, Beyn, Wolf-Juergen, Yuri Latushkin +3 · 1 citation
Mathematics · Engineering · Physics and Astronomy · #Numerical methods for differential equations #Electromagnetic Simulation and Numerical Methods #Electromagnetic Scattering and Analysis
paper · pdf · doi:10.48550/arxiv.1210.3952
Investigating the stability of nonlinear waves often leads to linear or\nnonlinear eigenvalue problems for differential operators on unbounded domains.\nIn this paper we propose to detect and approximate the point spectra of such\noperators (and the associated eigenfunctions) via contour integrals of\nsolutions to resolvent equations. The approach is based on Keldysh' theorem and\nextends a recent method for matrices depending analytically on the eigenvalue\nparameter. We show that errors are well-controlled under very general\nassumptions when the resolvent equations are solved via boundary value problems\non finite domains. Two applications are presented: an analytical study of\nSchr "odinger operators on the real line as well as on bounded intervals and a\nnumerical study of the FitzHugh-Nagumo system. We also relate the contour\nmethod to the well-known Evans function and show that our approach provides an\nalternative to evaluating and computing its zeroes.\n