Fredholm Properties and L-Spectra of Localized Rotating Waves in Parabolic Systems

In this paper we study spectra and Fredholm properties of Ornstein-Uhlenbeck operators Lv(x) := A4v(x) + 〈Sx,∇v(x)〉+Df(v?(x))v(x), x ∈ R, d > 2, where v? : R → R is the profile of a rotating wave satisfying v?(x) → v∞ ∈ R as |x| → ∞, the map f : R → R is smooth, the matrix A ∈ R has eigenvalues with positive real parts and commutes with the limit matrix Df(v∞). The matrix S ∈ R is assumed to be skew-symmetric with eigenvalues (λ1, . . . , λd) = (±iσ1, . . . ,±iσk, 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 λ ∈ C satisfying the dispersion relation det ( λIm + η A−Df(v∞) + i〈n, σ〉Im ) = 0 for some η ∈ R and n ∈ Z belongs to the essential spectrum σess(L) in L. For values Reλ to the right of the spectral bound for Df(v∞) we show that the operator λI −L is Fredholm of index 0, solve the identification problem for the adjoint operator (λI − L)∗, and formulate the Fredholm alternative. Moreover, we show that the set σ(S) ∪ {λi + λj : λi, λj ∈ σ(S), 1 6 i < j 6 d} belongs to the point spectrum σpt(L) in L. We determine the associated eigenfunctions and show that they decay exponentially in space. As an application we analyze 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 nonlinear stability of rotating waves in higher space dimensions and truncations to bounded domains.