Inhomogeneities in 3 dimensional oscillatory media

Abstract

We consider localized perturbations to spatially homogeneous oscillations in dimension 3 using the complex Ginzburg-Landau equation as a prototype. In particular, we will focus on inhomogeneities that locally change the phase of the oscillations. In the usual translation invariant spaces and at ε = 0 the linearization about these spatially homogeneous solutions result in an operator with zero eigenvalue embedded in the essential spectrum. In contrast, we show that when considered as an operator between Kondratiev spaces, the linearization is a Fredholm operator. These spaces consist of functions with algebraical localization that increases with each derivative. We use this result to construct solutions close to the equilibrium via the Implicit Function Theorem and derive asymptotics for wavenumbers in the far field.1991 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35.

The effect of impurities on striped phases

“…From a phenomenological point of view, the one-dimensional case is the most difficult, since effective diffusion of the neutral mode is weakest in one spatial dimension, such that the effect of inhomogeneity on the far-field is most significant. This phenomenon is well understood in the case of diffusive stability, where decay of localized data is faster in n spatial dimensions (t −n/2 ), or in the case of impurities in oscillatory media, where small impurities can generate wave sources only in dimensions n 2 [9,11,14]. On the other hand, From a technical point of view, the one-dimensional case is easiest since the problem of finding stationary solutions can be cast as an ordinary differential equation (see, for example, [18,24] for this point of view).…”

“…Technically, our work is following up on recent studies of inhomogeneities in a variety of contexts [11,9,10], where Kondratiev spaces were used to study perturbations of spatio-temporally periodic patterns by inhomogeneities. The present work goes however significantly past those techniques by treating non-normal form, actual periodic patterns, where in [11,9,10] the periodic patterns were, after appropriate transformations, constant in space.…”

Pacemakers in large arrays of oscillators with nonlocal coupling

“…We will denote the dual space to M k,p γ by M −k,q −γ , where 1/p + 1/q = 1. Such spaces have been used extensively in regularity theory [8], fluids [14,19], and in our prior work on inhomogeneities [6,5]. Fredholm properties for the Laplacian and various generalizations have been established in [15,10,11,12].…”

“…Previous results have studied phase-dynamics, formally derived from the complex Ginzburg-Landau equation for amplitude-phase oscillators [20], and general stable periodic orbits with diffusive coupling, but in a one-dimensional context [16] or with radial symmetry [7]. Radial symmetry was removed as an assumption in [5] in the complex Ginzburg-Landau equation in 3 dimensions.…”

“…Linear problems similar to (1.7) have been studied in [3] using spaces of exponentially localized functions. As demonstrated in [5,6], this approach is not easily viable in higher space dimensions. A more robust approach relies on algebraic weights and will be pursued here.…”