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

TL;DR

This paper analyzes the spectral and Fredholm properties of linearized operators around rotating wave solutions in parabolic systems, providing conditions for essential and point spectra, and applying results to spinning solitons in Ginzburg-Landau equations.

Contribution

It offers a detailed spectral analysis of Ornstein-Uhlenbeck operators with rotating wave profiles, including criteria for essential and point spectra, and applies findings to spinning solitons in reaction-diffusion systems.

Findings

Characterization of essential spectrum via dispersion relation.

Fredholm properties of the linearized operator in $L^p$.

Identification of eigenfunctions with exponential decay.

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.