Spectral analysis of localized rotating waves in parabolic systems

Abstract

In this paper, we study the spectra and Fredholm properties of Ornstein–Uhlenbeck operators where is the profile of a rotating wave satisfying as , the map is smooth and the matrix has eigenvalues with positive real parts and commutes with the limit matrix . The matrix is assumed to be skew-symmetric with eigenvalues (λ 1 ,…,λ d )=(±i σ 1 ,…,±i σ k ,0,…,0). The spectra of these linearized operators are crucial for the nonlinear stability of rotating waves in reaction–diffusion systems. We prove under appropriate conditions that every satisfying the dispersion relation belongs to the essential spectrum in L p . For values Re λ to the right of the spectral bound for , we show that the operator is Fredholm of index 0, solve the identification problem for the adjoint operator and formulate the Fredholm alternative. Moreover, we show that the set belongs to the point spectrum in L p . We determine the associated eigenfunctions and show that they decay exponentially in space. As an application, we analyse spinning soliton solutions which occur in the Ginzburg–Landau equation and compute their numerical spectra as well as associated eigenfunctions. Our results form the basis for investigating the nonlinear stability of rotating waves in higher space dimensions and truncations to bounded domains. This article is part of the themed issue ‘Stability of nonlinear waves and patterns and related topics’.