Existence and homogenisation of travelling waves bifurcating from resonances of reaction-diffusion equations in periodic media

Adam Boden, Karsten Matthies

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Abstract

The existence of travelling wave type solutions is studied for a scalar reaction diffusion equation in R2 with a nonlinearity which depends periodically on the spatial variable. We treat the coefficient of the linear term as a parameter and we formulate the problem as an infinite spatial dynamical system. Using a centre manifold reduction we obtain a finite dimensional dynamical system on the centre manifold with fully degenerate linear part. By phase space analysis and Conley index methods we find conditions on the parameter and nonlinearity for the existence of travelling wave type solutions with particular wave speeds. The analysis provides an approach to the homogenisation problem as the period of the periodic dependence in the nonlinearity tends to zero.
LanguageEnglish
Pages405-459
JournalJournal of Dynamics and Differential Equations
Volume26
Issue number3
DOIs
StatusPublished - Sep 2014

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Periodic Media
Reaction-diffusion Equations
Homogenization
Traveling Wave
Nonlinearity
Dynamical system
Conley Index
Center Manifold Reduction
Center Manifold
Wave Speed
Phase Space
Scalar
Tend
Zero
Coefficient
Term

Cite this

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abstract = "The existence of travelling wave type solutions is studied for a scalar reaction diffusion equation in R2 with a nonlinearity which depends periodically on the spatial variable. We treat the coefficient of the linear term as a parameter and we formulate the problem as an infinite spatial dynamical system. Using a centre manifold reduction we obtain a finite dimensional dynamical system on the centre manifold with fully degenerate linear part. By phase space analysis and Conley index methods we find conditions on the parameter and nonlinearity for the existence of travelling wave type solutions with particular wave speeds. The analysis provides an approach to the homogenisation problem as the period of the periodic dependence in the nonlinearity tends to zero.",
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AB - The existence of travelling wave type solutions is studied for a scalar reaction diffusion equation in R2 with a nonlinearity which depends periodically on the spatial variable. We treat the coefficient of the linear term as a parameter and we formulate the problem as an infinite spatial dynamical system. Using a centre manifold reduction we obtain a finite dimensional dynamical system on the centre manifold with fully degenerate linear part. By phase space analysis and Conley index methods we find conditions on the parameter and nonlinearity for the existence of travelling wave type solutions with particular wave speeds. The analysis provides an approach to the homogenisation problem as the period of the periodic dependence in the nonlinearity tends to zero.

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