Abstract
We study traveling waves for reaction diffusion equations on the spatially discrete domain \({\mathbb Z^2}\). The phenomenon of crystallographic pinning occurs when traveling waves become pinned in certain directions despite moving with non-zero wave speed in nearby directions. In [19] it was shown that crystallographic pinning occurs for all rational directions, so long as the nonlinearity is close to the sawtooth, which itself was considered in [6]. In this paper we show that crystallographic pinning holds in the horizontal and vertical directions for bistable nonlinearities which satisfy a specific computable generic condition. The proof is based on dynamical systems. In particular, it relies on an examination of the heteroclinic chains which occur as singular limits of wave profiles on the boundary of the pinning region.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abraham, R., Robbin, J.: Transversal mappings and flows. An appendix by Al Kelley. W.A. Benjamin Inc., New York, Amsterdam (1967)
Bangert V.: On minimal laminations of the torus. Ann. Inst. H. Poincaré Anal. Non Linéaire 6(2), 95–138 (1989)
Bates P.W., Chmaj A.: A discrete convolution model for phase transitions. Arch. Ration. Mech. Anal. 150(4), 281–305 (1999)
Bell J.: Some threshold results for models of myelinated nerves. Math. Biosci. 54(3-4), 181–190 (1981)
Bell J., Cosner C.: Threshold behavior and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons. Quart. Appl. Math. 42(1), 1–14 (1984)
Cahn J.W., Mallet-Paret J., Van Vleck E.S.: Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice. SIAM J. Appl. Math. 59(2), 455–493 (1999)
Carpio A., Bonilla L.L.: Pulse propagation in discrete systems of coupled excitable cells. SIAM J. Appl. Math. 63(2), 619–635 (2002)
Chow S.-N., Hale J.K.: Methods of bifurcation theory. Fundamental principles of mathematical science, vol. 251. Springer-Verlag, New York (1982)
Chow S.-N., Mallet-Paret J., Van Vleck E.S.: Pattern formation and spatial chaos in spatially discrete evolution equations. Random Comput. Dynam. 4(2–3), 109–178 (1996)
Coddington E.A., Levinson N.: Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New York, Toronto, London (1955)
Elmer C.E., Van Vleck E.S.: Analysis and computation of traveling wave solutions of bistable differential-difference equations. Nonlinearity 12(4), 771–798 (1999)
Erneux T., Nicolis G.: Propagating waves in discrete bistable reaction-diffusion systems. Phys. D 67(1–3), 237–244 (1993)
Fáth G.: Propagation failure of traveling waves in a discrete bistable medium. Phys. D 116(1-2), 176–190 (1998)
Firth W.J.: Optical memory and spatial chaos. Phys. Rev. Lett. 61(3), 329–332 (1988)
Hirsch, M.W.: Differential topology. In: Graduate texts in mathematics, vol. 33. Springer-Verlag, New York (1994), (Corrected reprint of the 1976 original)
Hupkes H.J., Lunel S.M.V.: Analysis of Newton’s method to compute travelling waves in discrete media. J. Dynam. Differ. Equ. 17(3), 523–572 (2005)
James G.: Centre manifold reduction for quasilinear discrete systems. J. Nonlinear Sci. 13(1), 27–63 (2003)
Keener J.P.: Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. 47(3), 556–572 (1987)
Mallet-Paret, J.: Crystallographic pinning: direction dependent pinning in lattice differential equations. J. Differ. Equ. (to appear)
Mallet-Paret J.: The global structure of traveling waves in spatially discrete dynamical systems. J. Dynam. Differ. Equ. 11(1), 49–127 (1999)
Mallet-Paret, J.: Traveling waves in spatially discrete dynamical systems of diffusive type. In: Dynamical systems. Lecture Notes in Math., vol. 1822, pp. 231–298. Springer, Berlin (2003)
Matthies K., Wayne C.E.: Wave pinning in strips. Proc. R. Soc. Edinb. A 136(5), 971–995 (2006)
Moser J.: Minimal solutions of variational problems on a torus. Ann. Inst. H. Poincaré Anal. Non Linéaire 3(3), 229–272 (1986)
Nussbaum R.D.: Positive operators and elliptic eigenvalue problems. Math. Z. 186(2), 247–264 (1984)
Palmer K.J.: Exponential dichotomies and transversal homoclinic points. J. Differ. Equ. 55(2), 225–256 (1984)
Pérez-Muñuzuri V., Pérez-Villar V., Chua L.O.: Propagation failure in linear arrays of Chua’s circuits. Internat. J. Bifur. Chaos Appl. Sci. Eng. 2(2), 403–406 (1992)
Rabinowitz P.H., Stredulinsky E.: On a class of infinite transition solutions for an Allen-Cahn model equation. Discret. Cont. Dyn. Syst. 21(1), 319–332 (2008)
Smale S.: An infinite dimensional version of Sard’s theorem. Am. J. Math. 87, 861–866 (1965)
Thieme H.R.: Remarks on resolvent positive operators and their perturbation. Discret. Cont. Dyn. Syst. 4(1), 73–90 (1998)
Vanderbauwhede, A.: Centre manifolds, normal forms and elementary bifurcations. In: Dynamics reported. Dynam. Report. Ser. Dynam. Systems Appl., vol. 2, pp. 89–169. Wiley, Chichester (1989)
Vanderbauwhede, A., Iooss, G.: Center manifold theory in infinite dimensions. In: Dynamics reported: expositions in dynamical systems. Dynam. Report. Expositions Dynam. Systems (N.S.) vol. 1, pp. 125–163. Springer, Berlin (1992)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Jack K. Hale on the occasion of his 80th birthday.
Rights and permissions
About this article
Cite this article
Hoffman, A., Mallet-Paret, J. Universality of Crystallographic Pinning. J Dyn Diff Equat 22, 79–119 (2010). https://doi.org/10.1007/s10884-010-9157-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-010-9157-2