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October  2011, 4(5): 1129-1145. doi: 10.3934/dcdss.2011.4.1129

Travelling waves of forced discrete nonlinear Schrödinger equations

Michal Fečkan 1, and  Vassilis M. Rothos 2,

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

Received  September 2009 Revised  January 2010 Published  December 2010

We address the existence and bifurcation of periodic travelling wave solutions in forced spatially discrete nonlinear Schrödinger equations with local interactions. We consider polynomial type and bounded nonlinearities. The mathematical methods are based in using Palais-Smale conditions and variational methods. Some generalizations are also discussed.
Keywords: Travelling wave, Bifurcations., Forced discrete Schrödinger equation.
Mathematics Subject Classification: Primary: 34K14, 37K60; Secondary: 37L6.
Citation: Michal Fečkan, Vassilis M. Rothos. Travelling waves of forced discrete nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1129-1145. doi: 10.3934/dcdss.2011.4.1129
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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