Negative Diffusion and Traveling Waves in High Dimensional Lattice Systems
Issue Date
2013Author
Hupkes, H. J.
Van Vleck, Erik S.
Publisher
Society for Industrial and Applied Mathematics
Type
Article
Article Version
Scholarly/refereed, publisher version
Metadata
Show full item recordAbstract
We consider bistable reaction diffusion systems posed on rectangular lattices in two or more spatial dimensions. The discrete diffusion term is allowed to have positive spatially periodic coefficients, and the two spatially periodic equilibria are required to be well ordered. We establish the existence of traveling wave solutions to such pure lattice systems that connect the two stable equilibria. In addition, we show that these waves can be approximated by traveling wave solutions to systems that incorporate both local and nonlocal diffusion. In certain special situations our results can also be applied to reaction diffusion systems that include (potentially large) negative coefficients. Indeed, upon splitting the lattice suitably and applying separate coordinate transformations to each sublattice, such systems can sometimes be transformed into a periodic diffusion problem that fits within our framework. In such cases, the resulting traveling structure for the original system has a separate wave profile for each sublattice and connects spatially periodic patterns that need not be well ordered. There is no direct analogue of this procedure that can be applied to reaction diffusion systems with continuous spatial variables.
Description
This is the publisher's version, also available electronically from http://epubs.siam.org/doi/abs/10.1137/120880628
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Citation
Hupkes, Van Vleck. (2013). Negative Diffusion and Traveling Waves in High Dimensional Lattice Systems. SIAM Journal on Mathematical Analysis 45:1068-1135. http://www.dx.doi.org/10.1137/120880628
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