American Institute of Mathematical Sciences

Advanced
March  2016, 15(2): 457-475. doi: 10.3934/cpaa.2016.15.457

Competing interactions and traveling wave solutions in lattice differential equations

E. S. Van Vleck 1, and  Aijun Zhang 2,

1. 

Department of Mathematics, University of Kansas, Lawrence, KS 66045, United States
2. 

Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR 72701, United States

Received  March 2015 Revised  November 2015 Published  January 2016

The existence of traveling front solutions to bistable lattice differential equations in the absence of a comparison principle is studied. The results are in the spirit of those in Bates, Chen, and Chmaj [1], but are applicable to vector equations and to more general limiting systems. An abstract result on the persistence of traveling wave solutions is obtained and is then applied to lattice differential equations with repelling first and/or second neighbor interactions and to some problems with infinite range interactions.
Keywords: Bistable, traveling waves, Fredholm operator, competing interaction, lattice differential equation..
Mathematics Subject Classification: Primary: 39A12, 34K31, 35K57, 37L6.
Citation: E. S. Van Vleck, Aijun Zhang. Competing interactions and traveling wave solutions in lattice differential equations. Communications on Pure & Applied Analysis, 2016, 15 (2) : 457-475. doi: 10.3934/cpaa.2016.15.457
References:
[1]

P. W. Bates, X. F. Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546. doi: 10.1137/S0036141000374002.  Google Scholar

[2]

M. Brucal - Hallare and E.S. Van Vleck, Traveling fronts in an antidiffusion lattice Nagumo model, SIAM J. Appl. Dyn. Sys., 10 (2011), 921-959. doi: 10.1137/100819461.  Google Scholar

[3]

J. W. Cahn, J. Mallet-Paret and E. S. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Appl. Math., 59 (1999), 455-493. doi: 10.1137/S0036139996312703.  Google Scholar

[4]

X. F. Chen, J. S. Guo and C. C. Wu, Traveling waves in discrete periodic media for bistable dynamics, Arch. Ration. Mech. Anal., 189 (2008), 189-236. doi: 10.1007/s00205-007-0103-3.  Google Scholar

[5]

S. N. Chow, J. Mallet-Paret and W. X. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291. doi: 10.1006/jdeq.1998.3478.  Google Scholar

[6]

J. Harterich, B. Sandstede and A. Scheel, Exponential dichotomies for linear non-autonomous functional differential equations of mixed type, Indiana Univ. Math. J., 51 (2002), 1081-1109. doi: 10.1512/iumj.2002.51.2188.  Google Scholar

[7]

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), 523-572. doi: 10.1007/s10884-005-5809-z.  Google Scholar

[8]

H. J. Hupkes and S. M. Verduyn-Lunel, Center manifold theory for functional differential equations of mixed type, J. Dynam. Diff. Eqns., 19 (2007), 497-560. doi: 10.1007/s10884-006-9055-9.  Google Scholar

[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), 2433-2487. doi: 10.1512/iumj.2009.58.3661.  Google Scholar

[10]

H. J. Hupkes and B. Sandstede, Traveling pulse solutions for the discrete FitzHugh-Nagumo system, SIAM J. Appl. Dyn. Syst., 9 (2010), 827-882. doi: 10.1137/090771740.  Google Scholar

[11]

H. J. Hupkes and E. S. Van Vleck, Negative diffusion and traveling waves in high dimensional lattice systems, SIAM J. Math. Anal., 45 (2013), 1068-1135. doi: 10.1137/120880628.  Google Scholar

[12]

C. Lamb and E. S. Van Vleck, Neutral mixed type functional differential equations, J. Dynam. Differential Equations, (2015), in press. Google Scholar

[13]

J. Mallet-Paret and S. M. Verduyn-Lunel, Exponential dichotomies and Wiener-Hopf factorizations for mixed-type functional differential equations,, \emph{J. of Differential Equations}, ().   Google Scholar

[14]

J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dynamics and Differential Equations, 11 (1999), 1-48. doi: 10.1023/A:1021889401235.  Google Scholar

[15]

J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, J. Dyn. Diff. Eqn., 11 (1999), 49-127. doi: 10.1023/A:1021841618074.  Google Scholar

[16]

J. Mallet-Paret, Traveling waves in spatially-discrete dynamical systems of diffusive type, Lecture Notes in Math, 1822 (2003), 231-298. doi: 10.1007/978-3-540-45204-1_4.  Google Scholar

[17]

A. Rustichini, Functional-differential equations of mixed type: the linear autonomous case, J. Dynam. Differential Equations, 1 (1989), 121-143, doi: 10.1007/BF01047828.  Google Scholar

[18]

A. Rustichini, Hopf bifurcation for functional-differential equations of mixed type, J. Dynam. Differential Equations, 1 (1989), 145-177. doi: 10.1007/BF01047829.  Google Scholar

[19]

W. Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities: I. Stability and uniqueness, J. Differential Equations, 159 (1999), 1-54. doi: 10.1006/jdeq.1999.3651.  Google Scholar

[20]

W. Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities: II. Existence, J. Differential Equations, 159 (1999), 55-101. doi: 10.1006/jdeq.1999.3652.  Google Scholar

[21]

A. Vainchtein and E. S. Van Vleck, Nucleation and propagation of phase mixtures in a bistable chain, Phys. Rev. B., 79 (2009), 144123. Google Scholar

[22]

A. Vainchtein, E. S. Van Vleck and A. Zhang, Propagation of periodic patterns in a discrete system with competing interactions, SIAM J. Appl. Dyn. Sys., 14 (2015), 523-555. Google Scholar

[23]

B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equation, SIAM J. Math. Anal., 22 (1991), 1016-1020. doi: 10.1137/0522066.  Google Scholar

[24]

B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Diff. Eqn, 96 (1992), 1-27. doi: 10.1016/0022-0396(92)90142-A.  Google Scholar

show all references

References:
[1]

P. W. Bates, X. F. Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546. doi: 10.1137/S0036141000374002.  Google Scholar

[2]

M. Brucal - Hallare and E.S. Van Vleck, Traveling fronts in an antidiffusion lattice Nagumo model, SIAM J. Appl. Dyn. Sys., 10 (2011), 921-959. doi: 10.1137/100819461.  Google Scholar

[3]

J. W. Cahn, J. Mallet-Paret and E. S. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Appl. Math., 59 (1999), 455-493. doi: 10.1137/S0036139996312703.  Google Scholar

[4]

X. F. Chen, J. S. Guo and C. C. Wu, Traveling waves in discrete periodic media for bistable dynamics, Arch. Ration. Mech. Anal., 189 (2008), 189-236. doi: 10.1007/s00205-007-0103-3.  Google Scholar

[5]

S. N. Chow, J. Mallet-Paret and W. X. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291. doi: 10.1006/jdeq.1998.3478.  Google Scholar

[6]

J. Harterich, B. Sandstede and A. Scheel, Exponential dichotomies for linear non-autonomous functional differential equations of mixed type, Indiana Univ. Math. J., 51 (2002), 1081-1109. doi: 10.1512/iumj.2002.51.2188.  Google Scholar

[7]

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), 523-572. doi: 10.1007/s10884-005-5809-z.  Google Scholar

[8]

H. J. Hupkes and S. M. Verduyn-Lunel, Center manifold theory for functional differential equations of mixed type, J. Dynam. Diff. Eqns., 19 (2007), 497-560. doi: 10.1007/s10884-006-9055-9.  Google Scholar

[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), 2433-2487. doi: 10.1512/iumj.2009.58.3661.  Google Scholar

[10]

H. J. Hupkes and B. Sandstede, Traveling pulse solutions for the discrete FitzHugh-Nagumo system, SIAM J. Appl. Dyn. Syst., 9 (2010), 827-882. doi: 10.1137/090771740.  Google Scholar

[11]

H. J. Hupkes and E. S. Van Vleck, Negative diffusion and traveling waves in high dimensional lattice systems, SIAM J. Math. Anal., 45 (2013), 1068-1135. doi: 10.1137/120880628.  Google Scholar

[12]

C. Lamb and E. S. Van Vleck, Neutral mixed type functional differential equations, J. Dynam. Differential Equations, (2015), in press. Google Scholar

[13]

J. Mallet-Paret and S. M. Verduyn-Lunel, Exponential dichotomies and Wiener-Hopf factorizations for mixed-type functional differential equations,, \emph{J. of Differential Equations}, ().   Google Scholar

[14]

J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dynamics and Differential Equations, 11 (1999), 1-48. doi: 10.1023/A:1021889401235.  Google Scholar

[15]

J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, J. Dyn. Diff. Eqn., 11 (1999), 49-127. doi: 10.1023/A:1021841618074.  Google Scholar

[16]

J. Mallet-Paret, Traveling waves in spatially-discrete dynamical systems of diffusive type, Lecture Notes in Math, 1822 (2003), 231-298. doi: 10.1007/978-3-540-45204-1_4.  Google Scholar

[17]

A. Rustichini, Functional-differential equations of mixed type: the linear autonomous case, J. Dynam. Differential Equations, 1 (1989), 121-143, doi: 10.1007/BF01047828.  Google Scholar

[18]

A. Rustichini, Hopf bifurcation for functional-differential equations of mixed type, J. Dynam. Differential Equations, 1 (1989), 145-177. doi: 10.1007/BF01047829.  Google Scholar

[19]

W. Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities: I. Stability and uniqueness, J. Differential Equations, 159 (1999), 1-54. doi: 10.1006/jdeq.1999.3651.  Google Scholar

[20]

W. Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities: II. Existence, J. Differential Equations, 159 (1999), 55-101. doi: 10.1006/jdeq.1999.3652.  Google Scholar

[21]

A. Vainchtein and E. S. Van Vleck, Nucleation and propagation of phase mixtures in a bistable chain, Phys. Rev. B., 79 (2009), 144123. Google Scholar

[22]

A. Vainchtein, E. S. Van Vleck and A. Zhang, Propagation of periodic patterns in a discrete system with competing interactions, SIAM J. Appl. Dyn. Sys., 14 (2015), 523-555. Google Scholar

[23]

B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equation, SIAM J. Math. Anal., 22 (1991), 1016-1020. doi: 10.1137/0522066.  Google Scholar

[24]

B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Diff. Eqn, 96 (1992), 1-27. doi: 10.1016/0022-0396(92)90142-A.  Google Scholar

[1]

Aaron Hoffman, Benjamin Kennedy. Existence and uniqueness of traveling waves in a class of unidirectional lattice differential equations. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 137-167. doi: 10.3934/dcds.2011.30.137
[2]

Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013
[3]

Peter Bates, Chunlei Zhang. Traveling pulses for the Klein-Gordon equation on a lattice or continuum with long-range interaction. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 235-252. doi: 10.3934/dcds.2006.16.235
[4]

Carl-Friedrich Kreiner, Johannes Zimmer. Heteroclinic travelling waves for the lattice sine-Gordon equation with linear pair interaction. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 915-931. doi: 10.3934/dcds.2009.25.915
[5]

Alejandro B. Aceves, Luis A. Cisneros-Ake, Antonmaria A. Minzoni. Asymptotics for supersonic traveling waves in the Morse lattice. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 975-994. doi: 10.3934/dcdss.2011.4.975
[6]

Masaharu Taniguchi. Instability of planar traveling waves in bistable reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 21-44. doi: 10.3934/dcdsb.2003.3.21
[7]

Fengxin Chen. Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 659-673. doi: 10.3934/dcds.2009.24.659
[8]

Je-Chiang Tsai. Global exponential stability of traveling waves in monotone bistable systems. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 601-623. doi: 10.3934/dcds.2008.21.601
[9]

Alexander Pankov. Traveling waves in Fermi-Pasta-Ulam chains with nonlocal interaction. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2097-2113. doi: 10.3934/dcdss.2019135
[10]

Cui-Ping Cheng, Ruo-Fan An. Global stability of traveling wave fronts in a two-dimensional lattice dynamical system with global interaction. Electronic Research Archive, , () : -. doi: 10.3934/era.2021051
[11]

Melvin Faierman. Fredholm theory for an elliptic differential operator defined on $ \mathbb{R}^n $ and acting on generalized Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1463-1483. doi: 10.3934/cpaa.2020074
[12]

Joseph Thirouin. Classification of traveling waves for a quadratic Szegő equation. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3099-3122. doi: 10.3934/dcds.2019128
[13]

Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382
[14]

Cheng-Hsiung Hsu, Ting-Hui Yang. Traveling plane wave solutions of delayed lattice differential systems in competitive Lotka-Volterra type. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 111-128. doi: 10.3934/dcdsb.2010.14.111
[15]

Xiao-Biao Lin, Stephen Schecter. Traveling waves and shock waves. Discrete & Continuous Dynamical Systems, 2004, 10 (4) : i-ii. doi: 10.3934/dcds.2004.10.4i
[16]

Emile Franc Doungmo Goufo, Abdon Atangana. Dynamics of traveling waves of variable order hyperbolic Liouville equation: Regulation and control. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 645-662. doi: 10.3934/dcdss.2020035
[17]

Yuqian Zhou, Qian Liu. Reduction and bifurcation of traveling waves of the KdV-Burgers-Kuramoto equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 2057-2071. doi: 10.3934/dcdsb.2016036
[18]

Rui Huang, Ming Mei, Yong Wang. Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3621-3649. doi: 10.3934/dcds.2012.32.3621
[19]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973
[20]

Zhi-Cheng Wang. Traveling curved fronts in monotone bistable systems. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2339-2374. doi: 10.3934/dcds.2012.32.2339

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences