Homotopy Method for Nonlinear Eigenvalue Pencils with Applications

Abstract

We present a new homotopy method for studying nonlinear eigenvalue pencils related to various applications in nonlinear waves, hydrodynamic stability, or electric power systems. Our technique is based on a simple geometric intuition that suggests reducing the analysis of a real part of a spectrum of a nonlinear eigenvalue pencil to the spectral analysis of an one-parameter family of linear operators without extending a dimension of an underlying space. Two distinct cases are studied. In the first case, for a class of self-adjoint operator pencils we significantly simplify a proof and generalize a theorem that guarantees existence of a sequence of eigenvalues converging to zero. We apply the theory to study damping rates of internal waves in stably stratified non-Newtonian fluids for which we prove an existence of internal modes with an arbitrary slow exponential decay. In the second case we discuss quadratic Hermitian matrix pencils and provide a new geometric interpretation of a related algebraic structure—the Krein signature. Consequently, we give a new proof of a count (inertia law) of stable and unstable eigenvalues.