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Stability of Fronts in Inhomogeneous Wave Equations

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Abstract

This paper presents an introduction to the existence and stability of stationary fronts in wave equations with finite length spatial inhomogeneities. The main focus will be on wave equations with one or two inhomogeneities. It will be shown that the fronts come in families. The front solutions provide a parameterisation of the length of the inhomogeneities in terms of the local energy of the potential in the inhomogeneity. The stability of the fronts is determined by analysing (constrained) critical points of those length functions. Amongst others, it will shown that inhomogeneities can stabilise non-monotonic fronts. Furthermore it is demonstrated that bi-stability can occur in such systems.

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Acknowledgements

I would like to thank Arjen Doelman, Stephan van Gils, Chris Knight and Hadi Susanto for sharing their enthusiasm and ideas in our investigations of wave equations with inhomogeneities.

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  1. Department of Mathematics, University of Surrey, Guildford, Surrey, GU2 7XH, UK

    Gianne Derks

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  1. Gianne Derks
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Derks, G. Stability of Fronts in Inhomogeneous Wave Equations. Acta Appl Math 137, 61–78 (2015). https://doi.org/10.1007/s10440-014-9991-z

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  • DOI: https://doi.org/10.1007/s10440-014-9991-z

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