September  2014, 34(9): 3773-3788. doi: 10.3934/dcds.2014.34.3773

Modulation of uniform motion in diatomic Frenkel-Kontorova model

1. 

Department of Mathematics, Soochow University, Suzhou, 215006, China

Received  June 2013 Revised  January 2014 Published  March 2014

We study modulated structures of uniform motion in the diatomic Frenkel-Kontorova (FK) model with alternating light and heavy particles. By applying topological method for the damped and driven case and variational approach for the conservative case, we demonstrate for the diatomic FK model the existence of two different periodic modulation functions corresponding respectively to light and heavy particles.
Citation: Wen-Xin Qin. Modulation of uniform motion in diatomic Frenkel-Kontorova model. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3773-3788. doi: 10.3934/dcds.2014.34.3773
References:
[1]

D. G. Aronson, M. Golubitsky and J. Mallet-Paret, Ponies on a merry-go-round in large arrays of Josephson junctions, Nonlinearity, 4 (1991), 903-910. doi: 10.1088/0951-7715/4/3/014.

[2]

D. G. Aronson and Y. S. Huang, Limit and uniqueness of discrete rotating waves in large arrays of Josephson junctions, Nonlinearity, 7 (1994), 777-804. doi: 10.1088/0951-7715/7/3/005.

[3]

D. G. Aronson and Y. S. Huang, Single waveform solutions for linear arrays of Josephson junctions, Physica D, 101 (1997), 157-177. doi: 10.1016/S0167-2789(96)00221-7.

[4]

O. M. Braun and Y. S. Kivshar, The Frenkel-Kontorova Model, Concepts, Methods, and Applications, Springer-Verlag, 2004.

[5]

C. Baesens and R. S. MacKay, A novel preserved partial order for cooperative networks of units with overdamped second order dynamics, and application to tilted Frenkel-Kontorova chains, Nonlinearity, 17 (2004), 567-580. doi: 10.1088/0951-7715/17/2/012.

[6]

M. Fečkan and V. Rothos, Travelling waves in Hamiltonian systems on 2D lattices with nearest neighborhood interactions, Nonlinearity, 20 (2007), 319-341. doi: 10.1088/0951-7715/20/2/005.

[7]

M. Fečkan, Bifurcation and Chaos in Discontinuous and Continuous Systems, HEP-Springer, Berlin, 2011. doi: 10.1007/978-3-642-18269-3.

[8]

A. -M. Filip and S. Venakides, Existence and modulation of traveling waves in particle chains, Comm. Pure Appl. Math., 52 (1999), 693-735. doi: 10.1002/(SICI)1097-0312(199906)52:6<693::AID-CPA2>3.0.CO;2-9.

[9]

N. Forcadel, C. Imbert and R. Monneau, Homogenization of accelerated Frenkel-Kontorova models with $n$ types of particles, Trans. Amer. Math. Soc., 364 (2012), 6187-6227. doi: 10.1090/S0002-9947-2012-05650-9.

[10]

G. Friesecke and J. Wattis, Existence theorem for solitary waves on lattices, Commun. Math. Phys., 161 (1994), 391-418. doi: 10.1007/BF02099784.

[11]

A. Georgieva, T. Kriecherbauer and S. Venakides, Wave propagation and resonance in a one-dimensional nonlinear discrete periodic medium, SIAM J. Appl. Math., 60 (2000), 272-294. doi: 10.1137/S0036139998340315.

[12]

A. Georgieva, T. Kriecherbauer and S. Venakides, 1:2 resonance mediated second harmonic generation in a 1-D nonlinear discrete periodic medium, SIAM J. Appl. Math., 61 (2001), 1802-1815. doi: 10.1137/S0036139999365341.

[13]

A. V. Gorbach and M. Johansson, Discrete gap breathers in a diatomic Klein-Gordon chain: Stability and mobility, Phys. Rev. E, 67 (2003), 066608. doi: 10.1103/PhysRevE.67.066608.

[14]

M. Herrmann, Unimodal wavetrains and solitons in convex Fermi-Pasta-Ulam chains, Proc. Roy. Soc. Edinburgh, Sect. A, 140 (2010), 753-785. doi: 10.1017/S0308210509000146.

[15]

G. Katriel, Existence of travelling waves in discrete sine-Gordon rings, SIAM J. Math. Anal., 36 (2005), 1434-1443. doi: 10.1137/S0036141004440174.

[16]

Y. Kivshar and N. Flytzanis, Gap solitons in diatomic lattices, Phys. Rev. A, 46 (1992), 7972-7978. doi: 10.1103/PhysRevA.46.7972.

[17]

R. Livi, M. Spicci and R. S. MacKay, Breathers on a diatomic FPU chain, Nonlinearity, 10 (1997), 1421-1434. doi: 10.1088/0951-7715/10/6/003.

[18]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989.

[19]

R. Mirollo and N. Rosen, Existence, uniqueness, and nonuniqueness of single-wave-form solutions to Josephson junction systems, SIAM J. Appl. Math., 60 (2000), 1471-1501. doi: 10.1137/S003613999834385X.

[20]

A. Pankov, Traveling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices, Imperial College Press, London, 2005. doi: 10.1142/9781860947216.

[21]

A. Pankov and V. Rothos, Traveling waves in Fermi-Pasta-Ulam lattices with saturable nonlinearities, Discrete and Continuous Dynamical Systems, 30 (2011), 835-849. doi: 10.3934/dcds.2011.30.835.

[22]

M. Peyrard, St. Pnevmatikos and N. Flytzanis, Dynamics of two-component solitary waves in hydrogen-bonded chains, Phys. Rev. A, 36 (1987), 903-914. doi: 10.1103/PhysRevA.36.903.

[23]

W. -X. Qin, Uniform sliding states in the undamped Frenkel-Kontorova model, J. Diff. Equa., 249 (2010), 1764-1776. doi: 10.1016/j.jde.2010.07.028.

[24]

W. -X. Qin, Existence and modulation of uniform sliding states in driven and overdamped particle chains, Commun. Math. Phys., 311 (2012), 513-538. doi: 10.1007/s00220-011-1385-8.

[25]

W. -X. Qin, Existence of dynamical hull functions with two variables for the ac-driven Frenkel-Kontorova model, J. Diff. Equa., 255 (2013), 3472-3490. doi: 10.1016/j.jde.2013.07.050.

[26]

P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9.

[27]

P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Vol. 65, Amer. Math. Soc., Providence, RI, 1986.

[28]

H. Schwetlick and J. Zimmer, Solitary waves for nonconvex FPU lattices, J. Nonlinear Sci., 17 (2007), 1-12. doi: 10.1007/s00332-005-0735-0.

[29]

D. Smets and M. Willem, Solitary waves with prescribed speed on infinite lattices, J. Funct. Anal., 149 (1997), 266-275. doi: 10.1006/jfan.1996.3121.

[30]

T. Strunz and F.-J. Elmer, Driven Frenkel-Kontorova model: I. Uniform sliding states and dynamical domains of different particle densities, Phy. Rev. E, 58 (1998), 1601-1611. doi: 10.1103/PhysRevE.58.1601.

[31]

A. Vainchtein and P. G. Kevrekidis, Dynamics of phase transitions in a piecewise linear diatomic chain, J. Nonlinear Sci., 22 (2012), 107-134. doi: 10.1007/s00332-011-9110-5.

[32]

J. A. D. Wattis, Solitary waves in a diatomic lattice: Analytic approximations for a wide range of speeds by quasi-continuum methods, Phys. Lett. A, 284 (2001), 16-22. doi: 10.1016/S0375-9601(01)00277-8.

[33]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

[34]

A. Xu, G. Wang, S. Chen and B. Hu, Generalized Frenkel-Kontorova model: A diatomic chain in a sinusoidal potential, Phys. Rev. B, 58 (1998), 721-733. doi: 10.1103/PhysRevB.58.721.

show all references

References:
[1]

D. G. Aronson, M. Golubitsky and J. Mallet-Paret, Ponies on a merry-go-round in large arrays of Josephson junctions, Nonlinearity, 4 (1991), 903-910. doi: 10.1088/0951-7715/4/3/014.

[2]

D. G. Aronson and Y. S. Huang, Limit and uniqueness of discrete rotating waves in large arrays of Josephson junctions, Nonlinearity, 7 (1994), 777-804. doi: 10.1088/0951-7715/7/3/005.

[3]

D. G. Aronson and Y. S. Huang, Single waveform solutions for linear arrays of Josephson junctions, Physica D, 101 (1997), 157-177. doi: 10.1016/S0167-2789(96)00221-7.

[4]

O. M. Braun and Y. S. Kivshar, The Frenkel-Kontorova Model, Concepts, Methods, and Applications, Springer-Verlag, 2004.

[5]

C. Baesens and R. S. MacKay, A novel preserved partial order for cooperative networks of units with overdamped second order dynamics, and application to tilted Frenkel-Kontorova chains, Nonlinearity, 17 (2004), 567-580. doi: 10.1088/0951-7715/17/2/012.

[6]

M. Fečkan and V. Rothos, Travelling waves in Hamiltonian systems on 2D lattices with nearest neighborhood interactions, Nonlinearity, 20 (2007), 319-341. doi: 10.1088/0951-7715/20/2/005.

[7]

M. Fečkan, Bifurcation and Chaos in Discontinuous and Continuous Systems, HEP-Springer, Berlin, 2011. doi: 10.1007/978-3-642-18269-3.

[8]

A. -M. Filip and S. Venakides, Existence and modulation of traveling waves in particle chains, Comm. Pure Appl. Math., 52 (1999), 693-735. doi: 10.1002/(SICI)1097-0312(199906)52:6<693::AID-CPA2>3.0.CO;2-9.

[9]

N. Forcadel, C. Imbert and R. Monneau, Homogenization of accelerated Frenkel-Kontorova models with $n$ types of particles, Trans. Amer. Math. Soc., 364 (2012), 6187-6227. doi: 10.1090/S0002-9947-2012-05650-9.

[10]

G. Friesecke and J. Wattis, Existence theorem for solitary waves on lattices, Commun. Math. Phys., 161 (1994), 391-418. doi: 10.1007/BF02099784.

[11]

A. Georgieva, T. Kriecherbauer and S. Venakides, Wave propagation and resonance in a one-dimensional nonlinear discrete periodic medium, SIAM J. Appl. Math., 60 (2000), 272-294. doi: 10.1137/S0036139998340315.

[12]

A. Georgieva, T. Kriecherbauer and S. Venakides, 1:2 resonance mediated second harmonic generation in a 1-D nonlinear discrete periodic medium, SIAM J. Appl. Math., 61 (2001), 1802-1815. doi: 10.1137/S0036139999365341.

[13]

A. V. Gorbach and M. Johansson, Discrete gap breathers in a diatomic Klein-Gordon chain: Stability and mobility, Phys. Rev. E, 67 (2003), 066608. doi: 10.1103/PhysRevE.67.066608.

[14]

M. Herrmann, Unimodal wavetrains and solitons in convex Fermi-Pasta-Ulam chains, Proc. Roy. Soc. Edinburgh, Sect. A, 140 (2010), 753-785. doi: 10.1017/S0308210509000146.

[15]

G. Katriel, Existence of travelling waves in discrete sine-Gordon rings, SIAM J. Math. Anal., 36 (2005), 1434-1443. doi: 10.1137/S0036141004440174.

[16]

Y. Kivshar and N. Flytzanis, Gap solitons in diatomic lattices, Phys. Rev. A, 46 (1992), 7972-7978. doi: 10.1103/PhysRevA.46.7972.

[17]

R. Livi, M. Spicci and R. S. MacKay, Breathers on a diatomic FPU chain, Nonlinearity, 10 (1997), 1421-1434. doi: 10.1088/0951-7715/10/6/003.

[18]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989.

[19]

R. Mirollo and N. Rosen, Existence, uniqueness, and nonuniqueness of single-wave-form solutions to Josephson junction systems, SIAM J. Appl. Math., 60 (2000), 1471-1501. doi: 10.1137/S003613999834385X.

[20]

A. Pankov, Traveling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices, Imperial College Press, London, 2005. doi: 10.1142/9781860947216.

[21]

A. Pankov and V. Rothos, Traveling waves in Fermi-Pasta-Ulam lattices with saturable nonlinearities, Discrete and Continuous Dynamical Systems, 30 (2011), 835-849. doi: 10.3934/dcds.2011.30.835.

[22]

M. Peyrard, St. Pnevmatikos and N. Flytzanis, Dynamics of two-component solitary waves in hydrogen-bonded chains, Phys. Rev. A, 36 (1987), 903-914. doi: 10.1103/PhysRevA.36.903.

[23]

W. -X. Qin, Uniform sliding states in the undamped Frenkel-Kontorova model, J. Diff. Equa., 249 (2010), 1764-1776. doi: 10.1016/j.jde.2010.07.028.

[24]

W. -X. Qin, Existence and modulation of uniform sliding states in driven and overdamped particle chains, Commun. Math. Phys., 311 (2012), 513-538. doi: 10.1007/s00220-011-1385-8.

[25]

W. -X. Qin, Existence of dynamical hull functions with two variables for the ac-driven Frenkel-Kontorova model, J. Diff. Equa., 255 (2013), 3472-3490. doi: 10.1016/j.jde.2013.07.050.

[26]

P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9.

[27]

P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Vol. 65, Amer. Math. Soc., Providence, RI, 1986.

[28]

H. Schwetlick and J. Zimmer, Solitary waves for nonconvex FPU lattices, J. Nonlinear Sci., 17 (2007), 1-12. doi: 10.1007/s00332-005-0735-0.

[29]

D. Smets and M. Willem, Solitary waves with prescribed speed on infinite lattices, J. Funct. Anal., 149 (1997), 266-275. doi: 10.1006/jfan.1996.3121.

[30]

T. Strunz and F.-J. Elmer, Driven Frenkel-Kontorova model: I. Uniform sliding states and dynamical domains of different particle densities, Phy. Rev. E, 58 (1998), 1601-1611. doi: 10.1103/PhysRevE.58.1601.

[31]

A. Vainchtein and P. G. Kevrekidis, Dynamics of phase transitions in a piecewise linear diatomic chain, J. Nonlinear Sci., 22 (2012), 107-134. doi: 10.1007/s00332-011-9110-5.

[32]

J. A. D. Wattis, Solitary waves in a diatomic lattice: Analytic approximations for a wide range of speeds by quasi-continuum methods, Phys. Lett. A, 284 (2001), 16-22. doi: 10.1016/S0375-9601(01)00277-8.

[33]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

[34]

A. Xu, G. Wang, S. Chen and B. Hu, Generalized Frenkel-Kontorova model: A diatomic chain in a sinusoidal potential, Phys. Rev. B, 58 (1998), 721-733. doi: 10.1103/PhysRevB.58.721.

[1]

Cecilia Cavaterra, Denis Enăchescu, Gabriela Marinoschi. Sliding mode control of the Hodgkin–Huxley mathematical model. Evolution Equations and Control Theory, 2019, 8 (4) : 883-902. doi: 10.3934/eect.2019043

[2]

Lei Jing, Jiawei Sun. Global existence and long time behavior of the Ellipsoidal-Statistical-Fokker-Planck model for diatomic gases. Kinetic and Related Models, 2020, 13 (2) : 373-400. doi: 10.3934/krm.2020013

[3]

Shu Zhang, Yuan Yuan. The Filippov equilibrium and sliding motion in an internet congestion control model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 1189-1206. doi: 10.3934/dcdsb.2017058

[4]

Jeremy Levesley, Xinping Sun, Fahd Jarad, Alexander Kushpel. Interpolation of exponential-type functions on a uniform grid by shifts of a basis function. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2399-2416. doi: 10.3934/dcdss.2020403

[5]

Yajuan Zang, Guangzhou Chen, Kejun Chen, Zihong Tian. Further results on 2-uniform states arising from irredundant orthogonal arrays. Advances in Mathematics of Communications, 2022, 16 (2) : 231-247. doi: 10.3934/amc.2020109

[6]

Anne Nouri, Christian Schmeiser. Aggregated steady states of a kinetic model for chemotaxis. Kinetic and Related Models, 2017, 10 (1) : 313-327. doi: 10.3934/krm.2017013

[7]

Zhi-An Wang. A kinetic chemotaxis model with internal states and temporal sensing. Kinetic and Related Models, 2022, 15 (1) : 27-48. doi: 10.3934/krm.2021043

[8]

Johannes Giannoulis. Transport and generation of macroscopically modulated waves in diatomic chains. Conference Publications, 2011, 2011 (Special) : 485-494. doi: 10.3934/proc.2011.2011.485

[9]

Claude Carlet, Yousuf Alsalami. A new construction of differentially 4-uniform $(n,n-1)$-functions. Advances in Mathematics of Communications, 2015, 9 (4) : 541-565. doi: 10.3934/amc.2015.9.541

[10]

Linda J. S. Allen, B. M. Bolker, Yuan Lou, A. L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1

[11]

Seung-Yeal Ha, Jinyeong Park, Sang Woo Ryoo. Emergence of phase-locked states for the Winfree model in a large coupling regime. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3417-3436. doi: 10.3934/dcds.2015.35.3417

[12]

Shanshan Chen. Nonexistence of nonconstant positive steady states of a diffusive predator-prey model. Communications on Pure and Applied Analysis, 2018, 17 (2) : 477-485. doi: 10.3934/cpaa.2018026

[13]

Xinfu Chen, Yuanwei Qi, Mingxin Wang. Steady states of a strongly coupled prey-predator model. Conference Publications, 2005, 2005 (Special) : 173-180. doi: 10.3934/proc.2005.2005.173

[14]

Wenshu Zhou, Hongxing Zhao, Xiaodan Wei, Guokai Xu. Existence of positive steady states for a predator-prey model with diffusion. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2189-2201. doi: 10.3934/cpaa.2013.12.2189

[15]

Yunfeng Jia, Yi Li, Jianhua Wu. Qualitative analysis on positive steady-states for an autocatalytic reaction model in thermodynamics. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4785-4813. doi: 10.3934/dcds.2017206

[16]

Kousuke Kuto, Yoshio Yamada. Coexistence states for a prey-predator model with cross-diffusion. Conference Publications, 2005, 2005 (Special) : 536-545. doi: 10.3934/proc.2005.2005.536

[17]

Alan D. Rendall. Multiple steady states in a mathematical model for interactions between T cells and macrophages. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 769-782. doi: 10.3934/dcdsb.2013.18.769

[18]

Yongli Cai, Yun Kang, Weiming Wang. Global stability of the steady states of an epidemic model incorporating intervention strategies. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1071-1089. doi: 10.3934/mbe.2017056

[19]

Yan'e Wang, Jianhua Wu. Stability of positive constant steady states and their bifurcation in a biological depletion model. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 849-865. doi: 10.3934/dcdsb.2011.15.849

[20]

Junhyeok Byeon, Seung-Yeal Ha, Hansol Park. Asymptotic interplay of states and adaptive coupling gains in the Lohe Hermitian sphere model. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022007

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (77)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]