Abstract and Applied Analysis

Monostable-Type Travelling Wave Solutions of the Diffusive FitzHugh-Nagumo-Type System in ${\mathbf{R}}^{N}$

Chih-Chiang Huang

Abstract

This paper is concerned with monostable-type travelling wave solutions of the diffusive FitzHugh-Nagumo-type system (FHN) in ${\mathbf{R}}^{N}$ for the two components $u$ and $v$. By solving $v$ in terms of $u$, this system can be reduced to a nonlocal single equation for $u$. When the diffusion coefficients in the system are equal, we construct travelling wave solutions for the non-local equation by the method of super- and subsolutions developed by Morita and Ninomiya (2008) Moreover, we propose a condition for $\gamma$, which is similar to the condition Reinecke and Sweers (1999) used to transform (FHN) into a quasimonotone system.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 735675, 9 pages.

Dates
First available in Project Euclid: 14 December 2012

https://projecteuclid.org/euclid.aaa/1355495773

Digital Object Identifier
doi:10.1155/2012/735675

Mathematical Reviews number (MathSciNet)
MR2947740

Zentralblatt MATH identifier
1246.35160

Citation

Huang, Chih-Chiang. Monostable-Type Travelling Wave Solutions of the Diffusive FitzHugh-Nagumo-Type System in ${\mathbf{R}}^{N}$. Abstr. Appl. Anal. 2012 (2012), Article ID 735675, 9 pages. doi:10.1155/2012/735675. https://projecteuclid.org/euclid.aaa/1355495773

References

• J. D. Dockery, “Existence of standing pulse solutions for an excitable activator-inhibitory system,” Journal of Dynamics and Differential Equations, vol. 4, no. 2, pp. 231–257, 1992.
• G. B. Ermentrout, S. P. Hastings, and W. C. Troy, “Large amplitude stationary waves in an excitable lateral-inhibitory medium,” SIAM Journal on Applied Mathematics, vol. 44, no. 6, pp. 1133–1149, 1984.
• G. A. Klaasen and W. C. Troy, “Stationary wave solutions of a system of reaction-diffusion equations derived from the FitzHugh-Nagumo equations,” SIAM Journal on Applied Mathematics, vol. 44, no. 1, pp. 96–110, 1984.
• H. Ikeda, M. Mimura, and Y. Nishiura, “Global bifurcation phenomena of travelling wave solutions for some bistable reaction-diffusion systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 13, no. 5, pp. 507–526, 1989.
• H. Kokubu, Y. Nishiura, and H. Oka, “Heteroclinic and homoclinic bifurcations in bistable reaction diffusion systems,” Journal of Differential Equations, vol. 86, no. 2, pp. 260–341, 1990.
• C. Reinecke and G. Sweers, “A positive solution on ${R}^{N}$ to a system of elliptic equations of FitzHugh-Nagumo type,” Journal of Differential Equations, vol. 153, no. 2, pp. 292–312, 1999.
• J. Wei and M. Winter, “Standing waves in the FitzHugh-Nagumo system and a problem in combinatorial geometry,” Mathematische Zeitschrift, vol. 254, no. 2, pp. 359–383, 2006.
• F. Hamel, R. Monneau, and J.-M. Roquejoffre, “Existence and qualitative properties of multidimensional conical bistable fronts,” Discrete and Continuous Dynamical Systems A, vol. 13, no. 4, pp. 1069–1096, 2005.
• H. Ninomiya and M. Taniguchi, “Existence and global stability of traveling curved fronts in the Allen-Cahn equations,” Journal of Differential Equations, vol. 213, no. 1, pp. 204–233, 2005.
• Y. Kurokawa and M. Taniguchi, “Multi-dimensional pyramidal travelling fronts in the Allen-Cahn equations,” Proceedings of the Royal Society of Edinburgh, vol. 141, no. 5, pp. 1031–1054, 2011.
• M. Taniguchi, “Traveling fronts of pyramidal shapes in the Allen-Cahn equations,” SIAM Journal on Mathematical Analysis, vol. 39, no. 1, pp. 319–344, 2007.
• F. Hamel and J.-M. Roquejoffre, “Heteroclinic connections for multidimensional bistable reaction-diffusion equations,” Discrete and Continuous Dynamical Systems S, vol. 4, no. 1, pp. 101–123, 2011.
• Y. Morita and H. Ninomiya, “Monostable-type traveling waves of bistable reaction-diffusion equations in the multi-dimensional space,” Bulletin of the Institute of Mathematics, vol. 3, no. 4, pp. 567–584, 2008.