PUB - Publikationen an der Universität Bielefeld

In this paper, we study the spectra and Fredholm properties of Ornstein-Uhlenbeck operators Lv(x): = A Delta v(x) + < Sx, del v(x)> + Df(v(x)(x))v(x), x is an element of R-d , d >= 2, where v : R-d -> R-m is the profile of a rotating wave satisfying v (x) -> v(infinity) is an element of R-m as vertical bar x vertical bar -> infinity, the map f : R-m -> R-m is smooth and the matrix A is an element of R-m,R-m eigenvalues with positive real parts and commutes with the limit matrix Df (v(infinity)). The matrix S. Rd, d is assumed to be skew- symmetric with eigenvalues (lambda(1),...,lambda(d)) = (+/- i sigma(1),...,+/- i sigma(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 lambda is an element of C satisfying the dispersion relation det(lambda I-m + eta(2)A - Df(v(infinity)) + i < n, sigma > I-m) = 0 for soe=me eta is an element of R and n is an element of Z(k), sigma = (sigma(1),...,sigma(k)) is an element of R-k belongs to the essential spectrum sigma(ess)(L) in L-p. For values Re lambda to the right of the spectral bound for Df (v(infinity)), we show that the operator lambda I - L is Fredholm of index 0, solve the identification problem for the adjoint operator (lambda I - L)* and formulate the Fredholm alternative. Moreover, we show that the set sigma(S) boolean OR {lambda(i) + lambda(i) : lambda(i), lambda(j is an element of) sigma(S), 1 <= i <= j <= d} belongs to the point spectrum sigma(pt)(L) 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 'tability of nonlinear waves and patterns and related topics'.