Treffer: Linear stability analysis for travelling waves of second order in time PDE's

We study travelling waves ϕc of second order in time PDE's utt + Lu + N(u) = 0. The linear stability analysis for these models is reduced to the question of the stability of quadratic pencils in the form λ2Id + 2cλ∂x + ℌc, where ℌc = c2∂xx + L + N'(ϕc). If ℌc is a self-adjoint operator, with a simple negative eigenvalue and a simple eigenvalue at zero, then we completely characterize the linear stability of ϕc. More precisely, we introduce an explicitly computable index ω*(ℌc) ∈ (0, ∞], so that the wave ϕc is stable if and only if |c| ≥ ω*(ℌc). The results are applicable both in the periodic case and in the whole line case. The method of proof involves a delicate analysis of a function G, associated with ℌ, whose positive zeros are exactly the positive (unstable) eigenvalues of the pencil λ2 Id + 2cλ∂x + ℌ. We would like to emphasize that the function G is not the Evans function for the problem, but rather a new object that we define herein, which fits the situation rather well. As an application, we consider three classical models—the 'good' Boussinesq equation, the Klein―Gordon―Zakharov (KGZ) system and the fourth order beam equation. In the whole line case, for the Boussinesq case and the KGZ system (and as a direct application of the main results), we compute explicitly the set of speeds which give rise to linearly stable travelling waves (and for all powers of p in the case of Boussinesq). This result is new for the KGZ system, while it generalizes the results of Alexander et al (2012, personal communication) and Alexander and Sachs (1995 Nonlinear World 2 471-507), which apply to the case p = 2. For the beam equation, we provide an implicit formula (depending only on the function ∥ϕ'c∥L2), which works for all p and for both the periodic and the whole line cases. Our results complement (and exactly match, whenever they exist) the results of a long line of investigation regarding the related notion of orbital stability of the same waves. Informally, we have found that in all the examples that we have looked at, our theory applies, whenever the Grillakis―Shatah― Strauss (GSS) theory applies. We believe that the results in this paper (or a variation thereof) will enable the linear stability analysis as well as asymptotic stability analysis for most models in the form utt + Lu + N(u) = 0.