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Traveling Wavefronts in Lattice Differential Equations with Time Delay and Global Interaction

  • Shiwang MaEmail author
  • Xingfu Zou
Chapter
First Online:
Part of the Fields Institute Communications book series (FIC, volume 64)

Abstract

In this paper, we study the existence of traveling wave solutions in lattice differential equations with time delay and global interaction
$$\begin{array}{rcl}{ u^{\prime}}_{n}(t)& =& D\sum\limits_{i\in {\mathit{Z}}^{q}\setminus \{0\}}J(i)[{u}_{n-i}(t) - {u}_{n}(t)] \\ & & +F\left ({u}_{n}(t),\sum\limits_{i\in {\mathit{Z}}^{q}}K(i){\int \nolimits \nolimits }_{-r}^{0}\mathrm{d}\eta (\theta )g({u}_{ n-i}(t + \theta ))\right )\end{array}$$
Following an idea in [10], we are able to relate the existence of traveling wavefront to the existence of heteroclinic connecting orbits of the corresponding ordinary delay differential equations
$$u^{\prime}(t) = F\left (u(t),{\int \nolimits \nolimits }_{-r}^{0}\mathrm{d}\eta (\theta )g(u(t + \theta ))\right ).$$
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Notes

Acknowledgments

Research partially supported by the National Natural Science Foundation of China (SM), by Natural Sciences and Engineering Research Council of Canada, and by a Premier Research Excellence Award of Ontario (XZ)

Received 2/20/2009; Accepted 6/30/2010

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of Mathematical Sciences and LPMCNankai UniversityTianjinP. R. China
  2. 2.Department of Applied MathematicsUniversity of Western OntarioLondonCanada

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