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

Abstract

In this paper we study spectra and Fredholm properties of Ornstein-Uhlenbeck operators $$\mathcal{L}v(x)=A\triangle v(x)+\langle Sx,\nabla v(x)\rangle+Df(v_{\star}(x))v(x),\,x\in\mathbb{R}^d,\,d\geqslant 2$$ where $v_{\star}:\mathbb{R}^d\rightarrow\mathbb{R}^m$ is a rotating wave profile with $v_{\star}(x)\to v_{\infty}\in\mathbb{R}^m$ as $|x|\to\infty$, $f:\mathbb{R}^m\rightarrow\mathbb{R}^m$ is smooth, $A\in\mathbb{R}^{m,m}$ has eigenvalues with positive real parts and commutes with the limit matrix $Df(v_{\infty})$. The matrix $S\in\mathbb{R}^{d,d}$ is assumed to be skew-symmetric with eigenvalues $(λ_1,\ldots,λ_d)=(\pm iσ_1,\ldots,\pm i σ_k,0,\ldots,0)$. 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 $λ\in\mathbb{C}$ satisfying the dispersion relation $$\det\Big(λI_m + η^2 A - Df(v_{\infty}) + i\langle n,σ\rangle I_m\Big)=0\quad\text{for some $η\in\mathbb{R}$ and $n\in\mathbb{Z}^k$}$$ belongs to the essential spectrum $σ_{\mathrm{ess}}(\mathcal{L})$ in $L^p$. For values $\mathrm{Re}\,λ$ to the right of the spectral bound for $Df(v_{\infty})$ we show that the operator $λI-\mathcal{L}$ is Fredholm of index $0$, solve the identification problem for the adjoint operator $(λI-\mathcal{L})^*$, and formulate the Fredholm alternative. Moreover, we show that the set $$σ(S)\cup\{λ_i+λ_j:\;λ_i,λ_j\inσ(S),\,1\leqslant i<j\leqslant d\}$$ belongs to the point spectrum $σ_{\mathrm{pt}}(\mathcal{L})$ 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.