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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

M. Fečkan, Nontrivial critical points of asymptotically quadratic functions at resonance, Ann. Polonici Math., 67 (1997), 43-57.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

M. Fečkan, Nontrivial critical points of asymptotically quadratic functions at resonance, Ann. Polonici Math., 67 (1997), 43-57.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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