Abstract. A spectral mapping theorem is proved that resolves a key problem in applying invariant manifold theorems to nonlinear Schrödinger type equations. The theorem is applied to the operator that arises as the linearization of the equation around a standing wave solution. We cast the problem in the context of space-dependent nonlinearities that arise in optical waveguide problems. The result is, however, more generally applicable including to equations in higher dimensions and even systems. The consequence is that stable, unstable, and center manifolds exist in the neighborhood of a (stable or unstable) standing wave, such as a waveguide mode, under simple and commonly verifiable spectral conditions. Main ResultsThe local behavior near some distinguished solution, such as a steady state, of an evolution equation, can be determined through a decomposition into invariant manifolds, that is, stable, unstable and center manifolds. These (locally invariant) manifolds are characterized by decay estimates. While the flows on the stable and unstable manifolds are determined by exponential decay in forward and backward time respectively, that on the center manifold is ambiguous. Nevertheless, a determination of the flow on the center manifold can lead to a complete characterization of the local flow and thus this decomposition, when possible, leads to a reduction of this problem to one of identifying the flow on the center manifold.This strategy has a long history for studying the local behavior near a critical point of an ordinary differential equation, or a fixed point of a map, and it has gained momentum in the last few decades in the context of nonlinear wave solutions of evolutionary partial differential equations. Extending the ideas to partial differential equations has, however, introduced a number of new issues. In infinite dimensions, the relation between the linearization and the full nonlinear equations is more delicate. This issue, however, turns out to be not so difficult for the invariant manifold decomposition and has largely been resolved, see, for instance, [2], [3]. A more subtle issue arises at the linear level. All of the known proofs for the existence of invariant manifolds are based upon the use of the group (or semigroup) generated by the linearization. The hypotheses of the relevant theorems are then formulated in terms of estimates on the appropriate projections of these groups onto stable, unstable and center subspaces. These amount to spectral estimates that come directly from a determination of the spectrum of the group. However, in any actual problem, the information available will, at best, be of the spectrum of the infinitesimal generator, that is, the linearized equation
Abstract. We study the spectral properties of the linearized Euler operator obtained by linearizing the equations of incompressible two dimensional fluid at a steady state with the vorticity that contains only two nonzero complex conjugate Fourier modes. We prove that the essential spectrum coincides with the imaginary axis, and give an estimate from above for the number of isolated nonimaginary eigenvalues. In addition, we prove that the spectral mapping theorem holds for the group generated by the linearized 2D Euler operator.
We study traveling 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 for stability of quadratic pencils in the form λ 2 Id + 2cλ∂x + Hc, where Hc = c 2 ∂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 λ 2 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 traveling 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 [2] and [1], which apply to the case p = 2. For the beam equation, we provide an explicit formula (depending of the function ϕ ′ c L 2 ), 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 goes through, whenever the Grillakis-Shatah-Strauss 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.
Abstract. We consider the Ostrovsky and short pulse models in a symmetric spatial interval, subject to periodic boundary conditions. For the Ostrovsky case, we revisit the classical periodic traveling waves and for the short pulse model, we explicitly construct traveling waves in terms of Jacobi elliptic functions. For both examples, we show spectral stability, for all values of the parameters. This is achieved by studying the non-standard eigenvalue problems in the form Lu = λu , where L is a Hill operator.
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