Spatial Decay of Rotating Waves in Reaction Diffusion Systems

TL;DR

This paper establishes exponential spatial decay properties of rotating wave solutions in reaction-diffusion systems, extending to eigenfunctions and applications like spinning solitons in the Ginzburg-Landau equation.

Contribution

It proves exponential decay of bounded solutions and eigenfunctions of nonlinear reaction-diffusion equations with rotating waves, including extensions to complex systems and higher Sobolev spaces.

Findings

Bounded solutions decay exponentially in space.

Eigenfunctions decay exponentially when spectral conditions are met.

Results apply to spinning solitons in Ginzburg-Landau equations.

Abstract

In this paper we study nonlinear problems for Ornstein-Uhlenbeck operators \begin{align*} A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle + f(v(x)) = 0,\,x\in\mathbb{R}^d,\,d\geqslant 2, \end{align*} where the matrix $A\in\mathbb{R}^{N,N}$ is diagonalizable and has eigenvalues with positive real part, the map $f:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is sufficiently smooth and the matrix $S\in\mathbb{R}^{d,d}$ in the unbounded drift term is skew-symmetric. Nonlinear problems of this form appear as stationary equations for rotating waves in time-dependent reaction diffusion systems. We prove under appropriate conditions that every bounded classical solution $v_{\star}$ of the nonlinear problem, which falls below a certain threshold at infinity, already decays exponentially in space, in the sense that $v_{\star}$ belongs to an exponentially weighted Sobolev space $ W^{1,p}_{\theta}(\mathbb{R}^d,\mathbb{R}^N)$. Several extensions of this basic result are presented: to complex-valued systems, to exponential decay in higher order Sobolev spaces and to pointwise estimates. We also prove that every bounded classical solution $v$ of the eigenvalue problem \begin{align*} A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle + Df(v_{\star}(x))v(x) = \lambda v(x),\,x\in\mathbb{R}^d,\,d\geqslant 2, \end{align*} decays exponentially in space, provided $\mathrm{Re}\,\lambda$ lies to the right of the essential spectrum. As an application we analyze spinning soliton solutions which occur in the Ginzburg-Landau equation. Our results form the basis for investigating nonlinear stability of rotating waves in higher space dimensions and truncations to bounded domains.