Multi-dimensional stability of waves travelling through rectangular lattices in rational directions

Hoffman, A.; Hupkes, H.; Van Vleck, E.

doi:10.1090/S0002-9947-2015-06392-2

Multi-dimensional stability of waves travelling through rectangular lattices in rational directions
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by A. Hoffman, H. J. Hupkes and E. S. Van Vleck PDF

Trans. Amer. Math. Soc. 367 (2015), 8757-8808 Request permission

Abstract:

We consider general reaction diffusion systems posed on rectangular lattices in two or more spatial dimensions. We show that travelling wave solutions to such systems that propagate in rational directions are nonlinearly stable under small perturbations. We employ recently developed techniques involving pointwise Green’s functions estimates for functional differential equations of mixed type (MFDEs), allowing our results to be applied even in situations where comparison principles are not available.

References

Kate A. Abell, Christopher E. Elmer, A. R. Humphries, and Erik S. Van Vleck, Computation of mixed type functional differential boundary value problems, SIAM J. Appl. Dyn. Syst. 4 (2005), no. 3, 755–781. MR 2173468, DOI 10.1137/040603425
Peter W. Bates, Xinfu Chen, and Adam J. J. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal. 35 (2003), no. 2, 520–546. MR 2001111, DOI 10.1137/S0036141000374002
Peter W. Bates and Fengxin Chen, Spectral analysis and multidimensional stability of traveling waves for nonlocal Allen-Cahn equation, J. Math. Anal. Appl. 273 (2002), no. 1, 45–57. MR 1933014, DOI 10.1016/S0022-247X(02)00205-6
Margaret Beck, Hermen Jan Hupkes, Björn Sandstede, and Kevin Zumbrun, Nonlinear stability of semidiscrete shocks for two-sided schemes, SIAM J. Math. Anal. 42 (2010), no. 2, 857–903. MR 2644362, DOI 10.1137/090775634
M. Beck, T. Nguyen, B. Sandstede, and K. Zumbrun, Toward nonlinear stability of sources via a modified Burgers equation, Physica D 241 (2012), 382–392.
Margaret Beck and C. Eugene Wayne, Invariant manifolds and the stability of traveling waves in scalar viscous conservation laws, J. Differential Equations 244 (2008), no. 1, 87–116. MR 2373656, DOI 10.1016/j.jde.2007.10.022
Jonathan Bell and Chris Cosner, Threshold behavior and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Quart. Appl. Math. 42 (1984), no. 1, 1–14. MR 736501, DOI 10.1090/S0033-569X-1984-0736501-0
S. Benzoni-Gavage, P. Huot, and F. Rousset, Nonlinear stability of semidiscrete shock waves, SIAM J. Math. Anal. 35 (2003), no. 3, 639–707. MR 2048404, DOI 10.1137/S0036141002418054
J. W. Cahn, Theory of crystal growth and interface motion in crystalline materials, Acta Met. 8 (1960), 554–562.
John W. Cahn, John Mallet-Paret, and Erik S. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Appl. Math. 59 (1999), no. 2, 455–493. MR 1654427, DOI 10.1137/S0036139996312703
Xinfu Chen, Sheng-Chen Fu, and Jong-Shenq Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal. 38 (2006), no. 1, 233–258. MR 2217316, DOI 10.1137/050627824
I-Liang Chern and Tai-Ping Liu, Convergence to diffusion waves of solutions for viscous conservation laws, Comm. Math. Phys. 110 (1987), no. 3, 503–517. MR 891950, DOI 10.1007/BF01212425
Shui-Nee Chow, John Mallet-Paret, and Wenxian Shen, Traveling waves in lattice dynamical systems, J. Differential Equations 149 (1998), no. 2, 248–291. MR 1646240, DOI 10.1006/jdeq.1998.3478
Christopher E. Elmer and Erik S. Van Vleck, A variant of Newton’s method for the computation of traveling waves of bistable differential-difference equations, J. Dynam. Differential Equations 14 (2002), no. 3, 493–517. MR 1917648, DOI 10.1023/A:1016386414393
Christopher E. Elmer and Erik S. van Vleck, Dynamics of monotone travelling fronts for discretizations of Nagumo PDEs, Nonlinearity 18 (2005), no. 4, 1605–1628. MR 2150345, DOI 10.1088/0951-7715/18/4/010
G. Friesecke and R. L. Pego, Solitary waves on Fermi-Pasta-Ulam lattices. III. Howland-type Floquet theory, Nonlinearity 17 (2004), no. 1, 207–227. MR 2023440, DOI 10.1088/0951-7715/17/1/013
Mariana Haragus and Arnd Scheel, Corner defects in almost planar interface propagation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 23 (2006), no. 3, 283–329 (English, with English and French summaries). MR 2217654, DOI 10.1016/j.anihpc.2005.03.003
Jörg Härterich, Björn Sandstede, and Arnd Scheel, Exponential dichotomies for linear non-autonomous functional differential equations of mixed type, Indiana Univ. Math. J. 51 (2002), no. 5, 1081–1109. MR 1947869, DOI 10.1512/iumj.2002.51.2188
A. Hoffman, H. J. Hupkes, and E. S. Van Vleck, Entire solutions for bistable lattice differential equations with obstacles, Memoirs of the AMS, to appear.
Aaron Hoffman and John Mallet-Paret, Universality of crystallographic pinning, J. Dynam. Differential Equations 22 (2010), no. 2, 79–119. MR 2665430, DOI 10.1007/s10884-010-9157-2
H. J. Hupkes, D. Pelinovsky, and B. Sandstede, Propagation failure in the discrete Nagumo equation, Proc. Amer. Math. Soc. 139 (2011), no. 10, 3537–3551. MR 2813385, DOI 10.1090/S0002-9939-2011-10757-3
H. J. Hupkes and B. Sandstede, Stability of pulse solutions for the discrete FitzHugh-Nagumo system, Trans. Amer. Math. Soc. 365 (2013), no. 1, 251–301. MR 2984059, DOI 10.1090/S0002-9947-2012-05567-X
H. J. Hupkes and S. M. Verduyn-Lunel, Analysis of Newton’s method to compute travelling wave solutions to lattice differential equations, Tech. Report 2003–09, Mathematical Institute Leiden, 2003.
H. J. Hupkes and S. M. Verduyn Lunel, Analysis of Newton’s method to compute travelling waves in discrete media, J. Dynam. Differential Equations 17 (2005), no. 3, 523–572. MR 2165558, DOI 10.1007/s10884-005-5809-z
H. J. Hupkes and S. M. Verduyn Lunel, Center manifold theory for functional differential equations of mixed type, J. Dynam. Differential Equations 19 (2007), no. 2, 497–560. MR 2333418, DOI 10.1007/s10884-006-9055-9
H. J. Hupkes and S. M. Verduyn Lunel, Lin’s method and homoclinic bifurcations for functional differential equations of mixed type, Indiana Univ. Math. J. 58 (2009), no. 6, 2433–2487. MR 2603755, DOI 10.1512/iumj.2009.58.3661
Hermen Jan Hupkes and Björn Sandstede, Traveling pulse solutions for the discrete FitzHugh-Nagumo system, SIAM J. Appl. Dyn. Syst. 9 (2010), no. 3, 827–882. MR 2665452, DOI 10.1137/090771740
Todd Kapitula, Multidimensional stability of planar travelling waves, Trans. Amer. Math. Soc. 349 (1997), no. 1, 257–269. MR 1360225, DOI 10.1090/S0002-9947-97-01668-1
C. D. Levermore and J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation. II, Comm. Partial Differential Equations 17 (1992), no. 11-12, 1901–1924. MR 1194744, DOI 10.1080/03605309208820908
John Mallet-Paret, The Fredholm alternative for functional-differential equations of mixed type, J. Dynam. Differential Equations 11 (1999), no. 1, 1–47. MR 1680463, DOI 10.1023/A:1021889401235
John Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, J. Dynam. Differential Equations 11 (1999), no. 1, 49–127. MR 1680459, DOI 10.1023/A:1021841618074
J. Mallet-Paret, Crystallographic pinning: Direction dependent pinning in lattice differential equations, preprint (2001).
J. Mallet-Paret and S. M. Verduyn-Lunel, Exponential dichotomies and Wiener-Hopf factorizations for mixed-type functional differential equations, J. Diff. Eq., to appear.
Hiroshi Matano, Mitsunori Nara, and Masaharu Taniguchi, Stability of planar waves in the Allen-Cahn equation, Comm. Partial Differential Equations 34 (2009), no. 7-9, 976–1002. MR 2560308, DOI 10.1080/03605300902963500
Wenxian Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities. I. Stability and uniqueness, J. Differential Equations 159 (1999), no. 1, 1–54. MR 1726918, DOI 10.1006/jdeq.1999.3651
Masaharu Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Equations 246 (2009), no. 5, 2103–2130. MR 2494701, DOI 10.1016/j.jde.2008.06.037
J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation. I, Comm. Partial Differential Equations 17 (1992), no. 11-12, 1889–1899. MR 1194743, DOI 10.1080/03605309208820907
Kevin Zumbrun, Instantaneous shock location and one-dimensional nonlinear stability of viscous shock waves, Quart. Appl. Math. 69 (2011), no. 1, 177–202. MR 2807984, DOI 10.1090/S0033-569X-2011-01221-6
Kevin Zumbrun and Peter Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J. 47 (1998), no. 3, 741–871. MR 1665788, DOI 10.1512/iumj.1998.47.1604