Unbounded Asymmetric Stationary Solutions of Lattice Nagumo Equations

• Jakub Hesoun ^[1] ; Petr Stehlík ^[1] ; Jonáš Volek ^[1]
  1. [1] University of West Bohemia

     University of West Bohemia

     Chequia

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 2, 2024
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In this paper we provide a complete characterization of a class of unbounded asymmetric stationary solutions of the lattice Nagumo equations. We show that for any bistable cubic nonlinearity and arbitrary diffusion rate there exists a two-parametric set of equivalence classes of generally asymmetric stationary solutions which diverge to infinity. Our main tool is an iterative mirroring technique which could be applicable to other problems related to lattice equations. Finally, we generalize the result for a broad class of reaction functions.

• Referencias bibliográficas
  • 1. Chow, S.-N., Mallet-Paret, J., Shen, W.: Traveling waves in lattice dynamical systems. J. Differ. Equ. 149(2), 248–291 (1998). https://doi.org/10.1006/jdeq.1998.3478
  • 2. Logan, J.D.: An Introduction to Nonlinear Partial Differential Equations. Wiley, Hoboken, New Jersey (2008)
  • 3. Chow, S.-N., Shen, W.: Dynamics in a discrete Nagumo equation: spatial topological chaos. SIAM J. Appl. Math. 55(6), 1764–1781 (1995)....
  • 4. Mallet-Paret, J.: Spatial Patterns, Spatial Chaos and Traveling Waves in Lattice Differential Equations. In: Stochastic and Spatial Structures...
  • 5. Hupkes, H.J., Morelli, L., Stehlík, P., Švígler, V.: Multichromatic travelling waves for lattice Nagumo equations. Appl. Math. Comput....
  • 6. Hupkes, H.J., Morelli, L., Stehlík, P., Švígler, V.: Counting and ordering periodic stationary solutions of lattice Nagumo equations. Appl....
  • 7. Švígler, V.: Periodic stationary solutions of the Nagumo lattice differential equation: Existence regions and their number. Electron. J....
  • 8. Stehlík, P., Švígler, V., Volek, J.: Bifurcations in Nagumo Equations on Graphs and Fiedler Vectors. J. Dyn. Diff. Equat. 35(3), 2397–2412...
  • 9. Bramburger, J.J., Sandstede, B.: Spatially localized structures in lattice dynamical systems. J. Nonlinear Sci. 30(2), 603–644 (2020)....
  • 10. Bramburger, J.J.: Isolas of multi-pulse solutions to lattice dynamical systems. Proc. Royal Soc. Edinburgh: Sect. A Math. 151(3), 916–952...
  • 11. Keener, J.P.: Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. 47(3), 556–572 (1987)
  • 12. Mallet-Paret, J.: The global structure of traveling waves in spatially discrete dynamical systems. J. Dyn. Differ. Equ. 11(1), 49–127...
  • 13. Zinner, B.: Existence of traveling wavefront solutions for the discrete Nagumo equation. J. Differ. Equ. 96(1), 1–27 (1992). https://doi.org/10.1016/0022-0396(92)90142-A
  • 14. Elmer, C.E., Van Vleck, E.S.: A variant of newton’s method for the computation of traveling waves of bistable differential-difference...
  • 15. Hupkes, H.J., Lunel, S.M.V.: Analysis of newton’s method to compute travelling waves in discrete media. J. Dyn. Differ. Equ. 17(3), 523–572...
  • 16. Hupkes, H.J., Van Vleck, E.S.: Travelling waves for complete discretizations of reaction diffusion systems. J. Dyn. Differ. Equ. 28(3–4),...
  • 17. Elmer, C.E., Van Vleck, E.S.: Spatially discrete FitzHugh–Nagumo equations. SIAM J. Appl. Math. 65, 1153–1174 (2005). https://doi.org/10.1137/S003613990343687X
  • 18. Guo, J.-S., Wu, C.-H.: Wave propagation for a two-component lattice dynamical system arising in strong competition models. J. Differ....
  • 19. Hupkes, H.J., Morelli, L., Stehlík, P.: Bichromatic travelling waves for lattice Nagumo equations. SIAM J. Appl. Dyn. Syst. 18(2), 973–1014...
  • 20. Zemskov, E.P., Tsyganov, M.A., Horsthemke, W.: Multifront regime of a piecewise-linear FitzHugh– Nagumo model with cross diffusion. Phys....
  • 21. Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press,...