Asymptotic Analysis of a Nonlinear Eigenvalue Problem Arising in the Waveguide Theory

Abstract

We consider a nonlinear eigenvalue problem for a system of ordinary differential equations arising in the waveguide theory. The nonlinearity is characterized by two nonnegative parameters α and β. For α = β = 0, we arrive at a linear problem that has finitely many (positive) eigenvalues. It is proved that for α > 0 and β ≥ 0 there exist infinitely many positive eigenvalues; their asymptotics is indicated. It is also proved that for α = 0 and β > 0 there exist finitely many eigenvalues. A comparison theorem for the eigenvalues is obtained for α, βs > 0. It is shown that perturbation theory methods cannot be used to study the nonlinear problem completely.