American Institute of Mathematical Sciences

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January  2012, 32(1): 101-124. doi: 10.3934/dcds.2012.32.101

Traveling wave solution for a lattice dynamical system with convolution type nonlinearity

Jong-Shenq Guo 1, and  Ying-Chih Lin 2,

1. 

Department of Mathematics, Tamkang University, 151, Ying-Chuan Road, Tamsui, Taipei County 25137
2. 

Department of Mathematics, National Taiwan Normal University, 88, S-4, Ting Chou Road, Taipei 11677, Taiwan

Received  August 2010 Revised  December 2010 Published  September 2011

We study traveling wave solutions for a lattice dynamical system with convolution type nonlinearity. We consider the monostable case and discuss the asymptotic behaviors, monotonicity and uniqueness of traveling wave. First, we characterize the asymptotic behavior of wave profile at both wave tails. Next, we prove that any wave profile is strictly decreasing. Finally, we prove the uniqueness (up to translation) of wave profile for each given admissible wave speed.
Keywords: Lattice dynamical system, uniqueness., traveling wave, monotonicity.
Mathematics Subject Classification: Primary: 34K05, 34A34; Secondary: 34K60, 34E0.
Citation: Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101
References:
[1]

P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136. doi: 10.1007/s002050050037.  Google Scholar

[2]

P. W. Bates, X. 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

[3]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439. doi: 10.1090/S0002-9939-04-07432-5.  Google Scholar

[4]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Advances in Differential Equations, 2 (1997), 125-160.  Google Scholar

[5]

X. Chen, S.-C. Fu and J.-S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258. doi: 10.1137/050627824.  Google Scholar

[6]

X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.  Google Scholar

[7]

X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146. doi: 10.1007/s00208-003-0414-0.  Google Scholar

[8]

J. Coville and L. Dupaigne, Travelling fronts in integrodifferential equations, C. R. Math. Acad. Sci. Paris, 337 (2003), 25-30.  Google Scholar

[9]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60 (2005), 797-819. doi: 10.1016/j.na.2003.10.030.  Google Scholar

[10]

J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755. doi: 10.1017/S0308210504000721.  Google Scholar

[11]

S.-C. Fu, J.-S. Guo and S.-Y. Shieh, Traveling wave solutions for some discrete quasilinear parabolic equations, Nonlinear Analysis Series A, 48 (2002), 1137-1149. doi: 10.1016/S0362-546X(00)00242-X.  Google Scholar

[12]

W. Hudson and B. Zinner, Existence of traveling waves for reaction diffusion equations of Fisher type in periodic media, World Sci. Publ., 1 (1995), 187-199.  Google Scholar

[13]

G. Lv, Asymptotic behavior of traveling fronts and entire solutions for a nonlinear monostable equation, Nonlinear Analysis, 72 (2010), 3659-3668. doi: 10.1016/j.na.2009.12.047.  Google Scholar

[14]

S. Ma and X. Zou, Existence, uniqueness and stability of traveling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87.  Google Scholar

[15]

S. Ma and X. Zou, Propagation and its failure in a lattice delayed differential equation with global interaction, J. Differential Equations, 212 (2005), 129-190.  Google Scholar

[16]

S. Ma, P. Weng and X. Zou, Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation, Nonlinear Analysis, 65 (2006), 1858-1890. doi: 10.1016/j.na.2005.10.042.  Google Scholar

[17]

K. Schumacher, Travelling-front solutions for integro-differential equations, J. Reine Angew. Math., 316 (1980), 54-70. doi: 10.1515/crll.1980.316.54.  Google Scholar

[18]

P. Weng, H. Huang and J. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439. doi: 10.1093/imamat/68.4.409.  Google Scholar

[19]

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

[20]

B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Differential Equations, 96 (1992), 1-27.  Google Scholar

[21]

B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation, J. Differential Equations, 105 (1993), 46-62.  Google Scholar

show all references

References:
[1]

P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136. doi: 10.1007/s002050050037.  Google Scholar

[2]

P. W. Bates, X. 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

[3]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439. doi: 10.1090/S0002-9939-04-07432-5.  Google Scholar

[4]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Advances in Differential Equations, 2 (1997), 125-160.  Google Scholar

[5]

X. Chen, S.-C. Fu and J.-S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258. doi: 10.1137/050627824.  Google Scholar

[6]

X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.  Google Scholar

[7]

X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146. doi: 10.1007/s00208-003-0414-0.  Google Scholar

[8]

J. Coville and L. Dupaigne, Travelling fronts in integrodifferential equations, C. R. Math. Acad. Sci. Paris, 337 (2003), 25-30.  Google Scholar

[9]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60 (2005), 797-819. doi: 10.1016/j.na.2003.10.030.  Google Scholar

[10]

J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755. doi: 10.1017/S0308210504000721.  Google Scholar

[11]

S.-C. Fu, J.-S. Guo and S.-Y. Shieh, Traveling wave solutions for some discrete quasilinear parabolic equations, Nonlinear Analysis Series A, 48 (2002), 1137-1149. doi: 10.1016/S0362-546X(00)00242-X.  Google Scholar

[12]

W. Hudson and B. Zinner, Existence of traveling waves for reaction diffusion equations of Fisher type in periodic media, World Sci. Publ., 1 (1995), 187-199.  Google Scholar

[13]

G. Lv, Asymptotic behavior of traveling fronts and entire solutions for a nonlinear monostable equation, Nonlinear Analysis, 72 (2010), 3659-3668. doi: 10.1016/j.na.2009.12.047.  Google Scholar

[14]

S. Ma and X. Zou, Existence, uniqueness and stability of traveling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87.  Google Scholar

[15]

S. Ma and X. Zou, Propagation and its failure in a lattice delayed differential equation with global interaction, J. Differential Equations, 212 (2005), 129-190.  Google Scholar

[16]

S. Ma, P. Weng and X. Zou, Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation, Nonlinear Analysis, 65 (2006), 1858-1890. doi: 10.1016/j.na.2005.10.042.  Google Scholar

[17]

K. Schumacher, Travelling-front solutions for integro-differential equations, J. Reine Angew. Math., 316 (1980), 54-70. doi: 10.1515/crll.1980.316.54.  Google Scholar

[18]

P. Weng, H. Huang and J. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439. doi: 10.1093/imamat/68.4.409.  Google Scholar

[19]

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

[20]

B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Differential Equations, 96 (1992), 1-27.  Google Scholar

[21]

B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation, J. Differential Equations, 105 (1993), 46-62.  Google Scholar

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