Two concepts, evidently very different in nature, have proved to be useful in analytical and numerical studies of spectral stability in nonlinear wave theory: (i) the Krein signature of an eigenvalue, a quantity usually defined in terms of the relative orientation of certain subspaces that is capable of detecting the structural instability of imaginary eigenvalues and hence their potential for moving into the right halfplane leading to dynamical instability under perturbation of the system, and (ii) the Evans function, an analytic function detecting the location of eigenvalues. One might expect these two concepts to be related, but unfortunately examples demonstrate that there is no way in general to deduce the Krein signature of an eigenvalue from the Evans function, for example by studying derivatives of the latter.The purpose of this paper is to recall and popularize a simple graphical interpretation of the Krein signature well-known in the spectral theory of polynomial operator pencils. Once established, this interpretation avoids altogether the need to view the Krein signature in terms of root subspaces and their relation to indefinite quadratic forms. To demonstrate the utility of this graphical interpretation of the Krein signature, we use it to define a simple generalization of the Evans function -the Evans-Krein function -that allows the calculation of Krein signatures in a way that is easy to incorporate into existing Evans function evaluation codes at virtually no additional computational cost. The graphical interpretation of the Krein signature also enables us to give elegant proofs of index theorems for linearized Hamiltonians in the finite dimensional setting: a general result implying as a corollary the generalized Vakhitov-Kolokolov criterion (or Grillakis-Shatah-Strauss criterion) and a count of real eigenvalues for linearized Hamiltonian systems in canonical form. These applications demonstrate how the simple graphical nature of the Krein signature may be easily exploited.

We investigate spectral stability of vortex solutions of the Gross-Pitaevskii equation, a mean-field approximation for Bose-Einstein condensates (BEC) in an effectively two-dimensional axisymmetric harmonic trap. We study eigenvalues of the linearization both rigorously and through computation of the Evans function, a sensitive and robust technique whose use we justify mathematically. Computational effort is significantly reduced through use of the Krein signature of purely imaginary eigenvalues. In agreement with previous studies in the physical literature we find a singly-quantized vortex spectrally stable while the stability of multi-quantized vortices depends on the diluteness of the condensate, with alternating intervals of stability and instability.

We analyse the stability of periodic, travelling-wave solutions to the Kawahara equation and some of its generalizations. We determine the parameter regime for which these solutions can exhibit resonance. By examining perturbations of small-amplitude solutions, we show that generalised resonance is a mechanism for high-frequency instabilities. We derive a quadratic equation which fully determines the stability region for these solutions. Focussing on perturbations of the small-amplitude solutions, we obtain asymptotic results for how their instabilities develop and grow. Numerical computation is used to confirm these asymptotic results and illustrate regimes where our asymptotic analysis does not apply.

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We investigate the effect of spatially localized inhomogeneities on a spatially homogeneous oscillation in a reaction-diffusion system. In dimension up to 2, we find sources and contact defects, that is, the inhomogeneity may either send out phase waves or act as a weak sink. We show that small inhomogeneities cannot act as sources in more than 2 space dimensions. We also derive asymptotics for wavenumbers and group velocities in the far field. The results are established rigorously for radially symmetric inhomogeneities in reaction-diffusion systems, and for arbitrary inhomogeneities in a modulation equation approximation.Keywords: reaction-diffusion, phase diffusion, eikonal equation, coherent structures, target patterns Running head: Inhomogeneities in oscillatory mediaCorresponding author: Arnd Scheel 1 Oscillatory reaction-diffusion systems IntroductionWe are interested in patterns that arise in dissipative, spatially extended systems far from equilibrium. The arguably simplest non-equilibrium pattern in a dynamical system is a periodic orbit. Periodic orbits are ubiquitous in dynamical systems, a fact which is partly justified by their robustness. Indeed, when studying ordinary differential equations or partial differential equations posed on bounded domains, periodic orbits are typically robust: the trivial Floquet multiplier associated with the phase of the oscillation is algebraically simple, and for any small perturbation of the system, one will find a nearby periodic orbit, with similar frequency. 1Spatially extended, large systems of oscillators have attracted attention in the physical and mathematical literature in many contexts. A classical prominent example is the BelousovZhabotinsky reaction, a reaction-diffusion system where the chemical concentrations undergo a relaxation-type oscillation which can be sustained for many cycles; see for example [25,9]. Other examples include biological systems such as cardiac tissue [24], neural systems [23], and ecological systems [6].When studying such large systems, in unbounded or in large domains, two interrelated issues complicate the concept of a robust oscillation.First, robustness turns out to be a delicate issue on a technical level. In large domains, the fixed point problem for the Poincare map is ill-conditioned due to clusters of eigenvalues of the linearization near the neutral phase mode; in unbounded domains, the neutral phase mode is even embedded into a continuum of spectrum; see Section 1.3, below. The spatial diffusive coupling, is responsible for this lack of separation between slow phase modes and the fast normal modes near the periodic orbit, since it covers a full band of possible exponential relaxation rates. As a consequence, it is often not obvious if periodic orbits are robust under changes of system parameters! Second, periodic orbits come in very different spatial flavors: spatially homogeneous oscillations, plane waves, target patterns, and spiral waves, to name but a few. A perturbation theory for spatially extended system...