Inhomogeneities in 3 dimensional oscillatory media

Gabriela Jaramillo; Gabriela Jaramillo

doi:10.3934/nhm.2015.10.387

Networks and Heterogeneous Media

2015, Volume 10, Issue 2: 387-399. doi: 10.3934/nhm.2015.10.387

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Inhomogeneities in 3 dimensional oscillatory media

Gabriela Jaramillo ^{1
,}

1.
University of Minnesota, School of Mathematics, 127 Vincent Hall, 206 Church St SE, Minneapolis, MN 55455

Received: 01 January 2014 Revised: 01 December 2014
Primary: 35B36, 35Q56; Secondary: 46E35.

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 $ \epsilon=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.
- Chemical oscillations,
- pacemakers,
- target patterns,
- Kondratiev spaces,
- complex Ginzburg-Landau
Citation: Gabriela Jaramillo. Inhomogeneities in 3 dimensional oscillatory media[J]. Networks and Heterogeneous Media, 2015, 10(2): 387-399. doi: 10.3934/nhm.2015.10.387

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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 $ \epsilon=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.

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