Denny Otten

Wolf-Jürgen Beyn, Denny Otten, Jens Rottmann-Matthes

In this paper we investigate the implementation of the so-called freezing method for second order wave equations in one and several space dimensions. The method converts the given PDE into a partial differential algebraic equation which is then solved numerically. The reformulation aims at separating the motion of a solution into a co-moving frame and a profile which varies as little as possible. Numerical examples demonstrate the feasability of this approach for semilinear wave equations with sufficient damping. We treat the case of a traveling wave in one space dimension and of a rotating wave in two space dimensions. In addition, we investigate in arbitrary space dimensions the point spectrum and the essential spectrum of operators obtained by linearizing about the profile, and we indicate the consequences for the nonlinear stability of the wave.

Wolf-Jürgen Beyn, Denny Otten, Jens Rottmann-Matthes

The topic of this paper are nonlinear traveling waves occuring in a system of damped waves equations in one space variable. We extend the freezing method from first to second order equations in time. When applied to a Cauchy problem, this method generates a comoving frame in which the solution becomes stationary. In addition it generates an algebraic variable which converges to the speed of the wave, provided the original wave satisfies certain spectral conditions and initial perturbations are sufficiently small. We develop a rigorous theory for this effect by recourse to some recent nonlinear stability results for waves in first order hyperbolic systems. Numerical computations illustrate the theory for examples of Nagumo and FitzHugh-Nagumo type.

Wolf-Jürgen Beyn, Denny Otten

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.

Wolf-Jürgen Beyn, Denny Otten

Travelling and rotating waves are ubiquitous phenomena observed in time dependent PDEs modelling the combined effect of dissipation and non-linear interaction. From an abstract viewpoint they appear as relative equilibria of an equivariant evolution equa- tion. In numerical computations the freezing method takes advantage of this structure by splitting the evolution of the PDE into the dynamics on the underlying Lie group and on some reduced phase space. The approach raises a series of questions which were answered to a certain degree by the project: linear stability implies non-linear (asymp- totic) stability, persistence of stability under discretisation, analysis and computation of spectral structures, first versus second order evolution systems, well-posedness of partial differential algebraic equations, spatial decay of wave profiles and truncation to bounded domains, analytical and numerical treatment of wave interactions, relation to connecting orbits in dynamical systems. A further numerical problem related to this topic will be discussed, namely the solution of non-linear eigenvalue problems via a contour method.