Fredholm Properties and $L^p$-Spectra of Localized Rotating Waves in Parabolic Systems (1612.07535v1)

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 $(\lambda_1,\ldots,\lambda_d)=(\pm i\sigma_1,\ldots,\pm i \sigma_k,0,\ldots,0)$. 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 $\lambda\in\mathbb{C}$ satisfying the dispersion relation $$\det\Big(\lambda I_m + \eta^2 A - Df(v_{\infty}) + i\langle n,\sigma\rangle I_m\Big)=0\quad\text{for some $\eta\in\mathbb{R}$ and $n\in\mathbb{Z}^k$}$$ belongs to the essential spectrum $\sigma_{\mathrm{ess}}(\mathcal{L})$ in $L^p$. For values $\mathrm{Re}\,\lambda$ to the right of the spectral bound for $Df(v_{\infty})$ we show that the operator $\lambda I-\mathcal{L}$ is Fredholm of index $0$, solve the identification problem for the adjoint operator $(\lambda I-\mathcal{L})^*$, and formulate the Fredholm alternative. Moreover, we show that the set $$\sigma(S)\cup{\lambda_i+\lambda_j:\;\lambda_i,\lambda_j\in\sigma(S),\,1\leqslant i<j\leqslant d}$$ belongs to the point spectrum $\sigma_{\mathrm{pt}}(\mathcal{L})$ 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.