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Travelling waves of forced discrete nonlinear Schrödinger equations
1. | Department of Mathematical Analysis and Numerical Mathematics, Comenius University, Mlynská dolina, 842 48 Bratislava |
2. | School of Mathematics, Physics and Computational Sciences, Faculty of Engineering, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece |
References:
[1] |
M. Abounouh, H. Al Moatassime, J. P. Chehab, S. Dumont and O. Goubet, Discrete Schrödinger equations and dissipative dynamical systems, Comm Pure Appl. Analysis, 7 (2008), 211-227. |
[2] |
S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization, Physica D, 103 (1997), 201-250.
doi: 10.1016/S0167-2789(96)00261-8. |
[3] |
D. Cai, A. Sánchez, A. R. Bishop, F. Falo and L. M. Floría, Possible soliton motion in ac-driven damped nonlinear lattices, Phys. Rev. B, 50 (1994), 9652-9655.
doi: 10.1103/PhysRevB.50.9652. |
[4] |
R. Carretero-González, P. G. Kevrekidis, B. A. Malomed and D. J. Frantzeskakis, Three-dimensional nonlinear lattices: from oblique vortices and octupoles to discrete diamonds and vortex cubes, Phys. Rev. Lett., 94 (2005), 203901.
doi: 10.1103/PhysRevLett.94.203901. |
[5] |
C. Chong, R. Carretero-González, B. A. Malomed and P. G. Kevrekidis, Multistable solitons in higher-dimensional cubic-quintic nonlinear Schrödinger lattices, Physica D, 238 (2009), 126-136.
doi: 10.1016/j.physd.2008.10.002. |
[6] |
D. N. Christodoulides, F. Lederer and Y. Silberberg, Discretizing light behaviour in linear and nonlinear waveguide lattices, Nature, 424 (2003), 817-823.
doi: 10.1038/nature01936. |
[7] |
M. Fečkan, Nontrivial critical points of asymptotically quadratic functions at resonance, Ann. Polonici Math., 67 (1997), 43-57. |
[8] |
M. Fečkan and V. M. Rothos, Traveling waves of discrete nonlinear Schrödinger equations with nonlocal interactions, Appl. Anal., 89 (2010), 1387-1411.
doi: 10.1080/00036810903208130. |
[9] |
M. Fečkan and V. M. Rothos, Travelling waves in Hamiltonian systems on 2d lattices with nearest neighbor interactions, Nonlinearity, 20 (2007), 319-341.
doi: 10.1088/0951-7715/20/2/005. |
[10] |
J. Garnier, F. K. Abdullaev and M. Salerno, Solitons in strongly driven discrete nonlinear Schrödinger-type models, Phys. Rev. E, 75 (2007), 016615.
doi: 10.1103/PhysRevE.75.016615. |
[11] |
J. Gómez-Gardeñes, L. M. Floría and A. R. Bishop, Discrete breathers in two-dimensional anisotropic nonlinear Schrödinger lattices, Physica D, 216 (2006), 31-43. |
[12] |
N. I. Karachalios and A. N. Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrödinger equation, J. Differential Equations, 217 (2005), 88-123.
doi: 10.1016/j.jde.2005.06.002. |
[13] |
P. G. Kevrekidis, D. E. Pelinovsky and A. Stefanov, Asymptotic stability of small bound states in the discrete nonlinear Schrödinger equation in one dimension, SIAM J. Math. Anal., 41 (2009), 2010-2030.
doi: 10.1137/080737654. |
[14] |
P. G. Kevrekidis, K.Ø. Rasmussen and A. R. Bishop, The discrete nonlinear Schrödinger equation: a survey of recent results, Int. J. Mod. Phys. B, 15 (2001), 2833-2900.
doi: 10.1142/S0217979201007105. |
[15] |
R. Khomeriki, S. Lepri and S. Ruffo, Pattern formation and localization in the forced-damped Fermi-Pasta-Ulam lattice, Phys. Rev. E, 64 (2001), 056606.
doi: 10.1103/PhysRevE.64.056606. |
[16] |
M. Kollmann, H. W. Capel and T. Bountis, Breathers and multibreathers in a periodically driven damped discrete nonlinear Schrödinger equation, Phys. Rev. E, 60 (1999), 1195-1211.
doi: 10.1103/PhysRevE.60.1195. |
[17] |
S. Li and J. Q. Liu, Morse theory and asymptotic linear Hamiltonian system, J. Differential Equations, 78 (1989), 53-73.
doi: 10.1016/0022-0396(89)90075-2. |
[18] |
S. Li and A. Szulkin, Periodic solutions for a class of nonautonomous Hamiltonian systems, J. Differential Equations, 112 (1994), 226-238.
doi: 10.1006/jdeq.1994.1102. |
[19] |
D. Mandelik, R. Morandotti, J. S. Aitchison and Y. Silberberg, Gap solitons in waveguide arrays, Phys. Rev. Lett., 92 (2004), 093904.
doi: 10.1103/PhysRevLett.92.093904. |
[20] |
J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer, New York, 1989. |
[21] |
T. R. O. Melvin, A. R. Champneys, P. G. Kevrekidis and J. Cuevas, Radiationless traveling waves in saturable nonlinear Schrödinger lattices, Phys. Rev. Lett., 97 (2006), 124101.
doi: 10.1103/PhysRevLett.97.124101. |
[22] |
T. R. O. Melvin, A. R. Champneys and D. E. Pelinovsky, Discrete traveling solitons in the Salerno model, SIAM J. Appl. Dyn. Syst., 8 (2009), 689-709.
doi: 10.1137/080715408. |
[23] |
R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg and Y. Silberberg, Dynamics of discrete solitons in optical waveguide arrays, Phys. Rev. Lett., 83 (1999), 2726-2729.
doi: 10.1103/PhysRevLett.83.2726. |
[24] |
D. E. Pelinovsky, T. R. O. Melvin and A. R. Champneys, One-parameter localized traveling waves in nonlinear Schrödinger lattices, Physica D, 236 (2007), 22-43.
doi: 10.1016/j.physd.2007.07.010. |
[25] |
P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," CBMS Reg. Conf. Ser. Math., 65, Amer. Math. Soc., Providence, RI, 1986. |
[26] |
K. Ø. Rasmussen, B. A. Malomed, A. R. Bishop and N. Grønbech-Jensen, Soliton motion in a parametrically ac-driven damped Toda lattice, Phys. Rev. E, 58 (1998), 6695-6699.
doi: 10.1103/PhysRevE.58.6695. |
[27] |
M. Syafwan, H. Susanto and S. M. Cox, Discrete solitons in electromechanical resonators, Phys. Rev. E, 81 (2010), 026207.
doi: 10.1103/PhysRevE.81.026207. |
[28] |
A. Vanossi, K. Ø. Rasmussen, A. R. Bishop, B. A. Malomed and V. Bortolani, Spontaneous pattern formation in driven nonlinear lattices, Phys. Rev. E, 62 (2000), 7353-7357.
doi: 10.1103/PhysRevE.62.7353. |
[29] |
Y. Zolotaryuk and M. Salerno, Discrete soliton ratchets driven by biharmonic fields, Phys. Rev. E, 73 (2006), 066621.
doi: 10.1103/PhysRevE.73.066621. |
[30] |
A. V. Yulin and A. R. Champneys, Discrete snaking: multiple cavity solitons in saturable media, SIAM J. Appl. Dyn. Syst., 9 (2010), 391-431.
doi: 10.1137/080734297. |
[31] |
A. V. Yulin, A. R. Champneys and D. V. Skryabin, Discrete cavity solitons due to saturable nonlinearity, Phys. Rev. A, 78 (2008), 011804.
doi: 10.1103/PhysRevA.78.011804. |
show all references
References:
[1] |
M. Abounouh, H. Al Moatassime, J. P. Chehab, S. Dumont and O. Goubet, Discrete Schrödinger equations and dissipative dynamical systems, Comm Pure Appl. Analysis, 7 (2008), 211-227. |
[2] |
S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization, Physica D, 103 (1997), 201-250.
doi: 10.1016/S0167-2789(96)00261-8. |
[3] |
D. Cai, A. Sánchez, A. R. Bishop, F. Falo and L. M. Floría, Possible soliton motion in ac-driven damped nonlinear lattices, Phys. Rev. B, 50 (1994), 9652-9655.
doi: 10.1103/PhysRevB.50.9652. |
[4] |
R. Carretero-González, P. G. Kevrekidis, B. A. Malomed and D. J. Frantzeskakis, Three-dimensional nonlinear lattices: from oblique vortices and octupoles to discrete diamonds and vortex cubes, Phys. Rev. Lett., 94 (2005), 203901.
doi: 10.1103/PhysRevLett.94.203901. |
[5] |
C. Chong, R. Carretero-González, B. A. Malomed and P. G. Kevrekidis, Multistable solitons in higher-dimensional cubic-quintic nonlinear Schrödinger lattices, Physica D, 238 (2009), 126-136.
doi: 10.1016/j.physd.2008.10.002. |
[6] |
D. N. Christodoulides, F. Lederer and Y. Silberberg, Discretizing light behaviour in linear and nonlinear waveguide lattices, Nature, 424 (2003), 817-823.
doi: 10.1038/nature01936. |
[7] |
M. Fečkan, Nontrivial critical points of asymptotically quadratic functions at resonance, Ann. Polonici Math., 67 (1997), 43-57. |
[8] |
M. Fečkan and V. M. Rothos, Traveling waves of discrete nonlinear Schrödinger equations with nonlocal interactions, Appl. Anal., 89 (2010), 1387-1411.
doi: 10.1080/00036810903208130. |
[9] |
M. Fečkan and V. M. Rothos, Travelling waves in Hamiltonian systems on 2d lattices with nearest neighbor interactions, Nonlinearity, 20 (2007), 319-341.
doi: 10.1088/0951-7715/20/2/005. |
[10] |
J. Garnier, F. K. Abdullaev and M. Salerno, Solitons in strongly driven discrete nonlinear Schrödinger-type models, Phys. Rev. E, 75 (2007), 016615.
doi: 10.1103/PhysRevE.75.016615. |
[11] |
J. Gómez-Gardeñes, L. M. Floría and A. R. Bishop, Discrete breathers in two-dimensional anisotropic nonlinear Schrödinger lattices, Physica D, 216 (2006), 31-43. |
[12] |
N. I. Karachalios and A. N. Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrödinger equation, J. Differential Equations, 217 (2005), 88-123.
doi: 10.1016/j.jde.2005.06.002. |
[13] |
P. G. Kevrekidis, D. E. Pelinovsky and A. Stefanov, Asymptotic stability of small bound states in the discrete nonlinear Schrödinger equation in one dimension, SIAM J. Math. Anal., 41 (2009), 2010-2030.
doi: 10.1137/080737654. |
[14] |
P. G. Kevrekidis, K.Ø. Rasmussen and A. R. Bishop, The discrete nonlinear Schrödinger equation: a survey of recent results, Int. J. Mod. Phys. B, 15 (2001), 2833-2900.
doi: 10.1142/S0217979201007105. |
[15] |
R. Khomeriki, S. Lepri and S. Ruffo, Pattern formation and localization in the forced-damped Fermi-Pasta-Ulam lattice, Phys. Rev. E, 64 (2001), 056606.
doi: 10.1103/PhysRevE.64.056606. |
[16] |
M. Kollmann, H. W. Capel and T. Bountis, Breathers and multibreathers in a periodically driven damped discrete nonlinear Schrödinger equation, Phys. Rev. E, 60 (1999), 1195-1211.
doi: 10.1103/PhysRevE.60.1195. |
[17] |
S. Li and J. Q. Liu, Morse theory and asymptotic linear Hamiltonian system, J. Differential Equations, 78 (1989), 53-73.
doi: 10.1016/0022-0396(89)90075-2. |
[18] |
S. Li and A. Szulkin, Periodic solutions for a class of nonautonomous Hamiltonian systems, J. Differential Equations, 112 (1994), 226-238.
doi: 10.1006/jdeq.1994.1102. |
[19] |
D. Mandelik, R. Morandotti, J. S. Aitchison and Y. Silberberg, Gap solitons in waveguide arrays, Phys. Rev. Lett., 92 (2004), 093904.
doi: 10.1103/PhysRevLett.92.093904. |
[20] |
J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer, New York, 1989. |
[21] |
T. R. O. Melvin, A. R. Champneys, P. G. Kevrekidis and J. Cuevas, Radiationless traveling waves in saturable nonlinear Schrödinger lattices, Phys. Rev. Lett., 97 (2006), 124101.
doi: 10.1103/PhysRevLett.97.124101. |
[22] |
T. R. O. Melvin, A. R. Champneys and D. E. Pelinovsky, Discrete traveling solitons in the Salerno model, SIAM J. Appl. Dyn. Syst., 8 (2009), 689-709.
doi: 10.1137/080715408. |
[23] |
R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg and Y. Silberberg, Dynamics of discrete solitons in optical waveguide arrays, Phys. Rev. Lett., 83 (1999), 2726-2729.
doi: 10.1103/PhysRevLett.83.2726. |
[24] |
D. E. Pelinovsky, T. R. O. Melvin and A. R. Champneys, One-parameter localized traveling waves in nonlinear Schrödinger lattices, Physica D, 236 (2007), 22-43.
doi: 10.1016/j.physd.2007.07.010. |
[25] |
P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," CBMS Reg. Conf. Ser. Math., 65, Amer. Math. Soc., Providence, RI, 1986. |
[26] |
K. Ø. Rasmussen, B. A. Malomed, A. R. Bishop and N. Grønbech-Jensen, Soliton motion in a parametrically ac-driven damped Toda lattice, Phys. Rev. E, 58 (1998), 6695-6699.
doi: 10.1103/PhysRevE.58.6695. |
[27] |
M. Syafwan, H. Susanto and S. M. Cox, Discrete solitons in electromechanical resonators, Phys. Rev. E, 81 (2010), 026207.
doi: 10.1103/PhysRevE.81.026207. |
[28] |
A. Vanossi, K. Ø. Rasmussen, A. R. Bishop, B. A. Malomed and V. Bortolani, Spontaneous pattern formation in driven nonlinear lattices, Phys. Rev. E, 62 (2000), 7353-7357.
doi: 10.1103/PhysRevE.62.7353. |
[29] |
Y. Zolotaryuk and M. Salerno, Discrete soliton ratchets driven by biharmonic fields, Phys. Rev. E, 73 (2006), 066621.
doi: 10.1103/PhysRevE.73.066621. |
[30] |
A. V. Yulin and A. R. Champneys, Discrete snaking: multiple cavity solitons in saturable media, SIAM J. Appl. Dyn. Syst., 9 (2010), 391-431.
doi: 10.1137/080734297. |
[31] |
A. V. Yulin, A. R. Champneys and D. V. Skryabin, Discrete cavity solitons due to saturable nonlinearity, Phys. Rev. A, 78 (2008), 011804.
doi: 10.1103/PhysRevA.78.011804. |
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