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Modulation of uniform motion in diatomic Frenkel-Kontorova model
1. | Department of Mathematics, Soochow University, Suzhou, 215006, China |
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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