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

In this paper we study spectra and Fredholm properties of Ornstein-Uhlenbeck operators $$\mathcal{L}v(x)=A\mathrm{△}v(x)+⟨Sx,\mathrm{\nabla }v(x)⟩+Df(v_{\star }(x))v(x),x\in {\mathbb{R}}^d,d⩾2$$ where $v_{\star }:{\mathbb{R}}^d\to {\mathbb{R}}^m$ is a rotating wave profile with $v_{\star }(x)\to v_{\mathrm{\infty }}\in {\mathbb{R}}^m$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$, $f:{\mathbb{R}}^m\to {\mathbb{R}}^m$ is smooth, $A\in {\mathbb{R}}^{m,m}$ has eigenvalues with positive real parts and commutes with the limit matrix $Df(v_{\mathrm{\infty }})$. The matrix $S\in {\mathbb{R}}^{d,d}$ is assumed to be skew-symmetric with eigenvalues $({\lambda }_1,\dots ,{\lambda }_d)=(\pm i{\sigma }_1,\dots ,\pm i{\sigma }_k,0,\dots ,0)$. The spectra of these linearized operators are crucial for the nonlinear stability of rotating waves in reaction diffusion systems. We prove under suitable conditions that every $\lambda \in \mathbb{C}$ satisfying the dispersion relation $$\det (\lambda I_m+{\eta }^{2}A-Df(v_{\mathrm{\infty }})+i⟨n,\sigma ⟩I_m)=0\text{for some }\eta \in \mathbb{R}\text{ and }n\in {\mathbb{Z}}^k$$ belongs to the essential spectrum ${\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s}}(\mathcal{L})$ in $L^p$. For values $\mathrm{R}\mathrm{e}\lambda$ to the right of the spectral bound for $Df(v_{\mathrm{\infty }})$ we show that the operator $\lambda I-\mathcal{L}$ is Fredholm of index $0$, solve the identification problem for the adjoint operator $(\lambda I-\mathcal{L}{)}^{\ast }$, and formulate the Fredholm alternative. Moreover, we show that the set $$\sigma (S)\cup \{{\lambda }_i+{\lambda }_j:{\lambda }_i,{\lambda }_j\in \sigma (S),1⩽i<j⩽d\}$$ belongs to the point spectrum ${\sigma }_{\mathrm{p}\mathrm{t}}(\mathcal{L})$ in $L^p$. We determine their 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.