Spectral analysis of localized rotating waves in parabolic systems

In this paper, we study the spectra and Fredholm properties of Ornstein–Uhlenbeck operatorswhereis the profile of a rotating wave satisfyingas, the mapis smooth and the matrixhas eigenvalues with positive real parts and commutes with the limit matrix. The matrixis assumed to be skew-symmetric with eigenvalues (λ1 ,…,λd )=(±iσ 1 ,…,±iσ 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 everysatisfying the dispersion relationbelongs to the essential spectruminL p . For values Re λ to the right of the spectral bound for, we show that the operatoris Fredholm of index 0, solve the identification problem for the adjoint operatorand formulate the Fredholm alternative. Moreover, we show that the setbelongs to the point spectruminL 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’.