American Institute of Mathematical Sciences

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

Modulation of uniform motion in diatomic Frenkel-Kontorova model

Wen-Xin Qin 1,

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.
Keywords: uniform sliding states., Diatomic FK model, modulation functions.
Mathematics Subject Classification: 34C15, 34C60, 34K13, 37K60, 49J35, 70F4.
Citation: Wen-Xin Qin. Modulation of uniform motion in diatomic Frenkel-Kontorova model. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

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