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 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.

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