Preprint of the project: miscellaneous activities
01/18 (revised version of sfb701b3 16-043) Wolf-Jürgen Beyn, Denny Otten.
Spectral Analysis of Localized Rotating Waves in Parabolic Systems
In this paper we study spectra and Fredholm properties of Ornstein-Uhlenbeck operators
\[
\mathcal{L} v(x) = A\triangle v(x) + \langle Sx,\nabla v(x)\rangle + Df(v_{\star}(x))v(x), \,x\in\mathbb{R}^d,\,d\geq 2,
\]
where \(v_{\star}:\mathbb{R}^d\rightarrow\mathbb{R}^m\) is the profile of a rotating wave satisfying \(v_{\star}(x)\to v_{\infty}\in\mathbb{R}^m\) as \(|x|\to\infty\), the map \(f:\mathbb{R}^m\rightarrow\mathbb{R}^m\) is smooth, the matrix \(A\in\mathbb{R}^{m,m}\) 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 appropriate 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
\[
\sigma(S)\cup\{\lambda_i+\lambda_j:\;\lambda_i,\lambda_j\in\sigma(S),\,1\leq i \lt j\leq d\}
\]
belongs to the point spectrum \(\sigma_{\mathrm{pt}}(\mathcal{L})\) in \(L^p\). 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.
