• Previous Article
    Mechanisms of recovery of radiation damage based on the interaction of quodons with crystal defects
  • DCDS-S Home
  • This Issue
  • Next Article
    Nonlinear lattice models for biopolymers: Dynamical coupling to a ionic cloud and application to actin filaments
October  2011, 4(5): 1129-1145. doi: 10.3934/dcdss.2011.4.1129

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

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.
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.

[2]

S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization,, Physica D, 103 (1997), 201. 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. 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). 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. 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. doi: 10.1038/nature01936.

[7]

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

[8]

M. Fečkan and V. M. Rothos, Traveling waves of discrete nonlinear Schrödinger equations with nonlocal interactions,, Appl. Anal., 89 (2010), 1387. 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. 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). 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.

[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. 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. 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. 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). 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. 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. 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. 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). doi: 10.1103/PhysRevLett.92.093904.

[20]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Springer, (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). 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. 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. 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. 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 (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. 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). 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. doi: 10.1103/PhysRevE.62.7353.

[29]

Y. Zolotaryuk and M. Salerno, Discrete soliton ratchets driven by biharmonic fields,, Phys. Rev. E, 73 (2006). 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. 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). 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.

[2]

S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization,, Physica D, 103 (1997), 201. 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. 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). 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. 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. doi: 10.1038/nature01936.

[7]

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

[8]

M. Fečkan and V. M. Rothos, Traveling waves of discrete nonlinear Schrödinger equations with nonlocal interactions,, Appl. Anal., 89 (2010), 1387. 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. 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). 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.

[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. 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. 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. 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). 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. 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. 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. 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). doi: 10.1103/PhysRevLett.92.093904.

[20]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Springer, (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). 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. 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. 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. 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 (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. 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). 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. doi: 10.1103/PhysRevE.62.7353.

[29]

Y. Zolotaryuk and M. Salerno, Discrete soliton ratchets driven by biharmonic fields,, Phys. Rev. E, 73 (2006). 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. 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). doi: 10.1103/PhysRevA.78.011804.

[1]

Olivier Goubet, Ezzeddine Zahrouni. On a time discretization of a weakly damped forced nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1429-1442. doi: 10.3934/cpaa.2008.7.1429

[2]

Mudassar Imran, Youssef Raffoul, Muhammad Usman, Chi Zhang. A study of bifurcation parameters in travelling wave solutions of a damped forced Korteweg de Vries-Kuramoto Sivashinsky type equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 691-705. doi: 10.3934/dcdss.2018043

[3]

Brenton LeMesurier. Modeling thermal effects on nonlinear wave motion in biopolymers by a stochastic discrete nonlinear Schrödinger equation with phase damping. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 317-327. doi: 10.3934/dcdss.2008.1.317

[4]

Hans Zwart, Yann Le Gorrec, Bernhard Maschke. Relating systems properties of the wave and the Schrödinger equation. Evolution Equations & Control Theory, 2015, 4 (2) : 233-240. doi: 10.3934/eect.2015.4.233

[5]

In-Jee Jeong, Benoit Pausader. Discrete Schrödinger equation and ill-posedness for the Euler equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 281-293. doi: 10.3934/dcds.2017012

[6]

Lei Jiao, Yiqian Wang. The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1585-1606. doi: 10.3934/cpaa.2009.8.1585

[7]

Tetsu Mizumachi, Dmitry Pelinovsky. On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 971-987. doi: 10.3934/dcdss.2012.5.971

[8]

Vassilis Rothos. Subharmonic bifurcations of localized solutions of a discrete NLS equation. Conference Publications, 2005, 2005 (Special) : 756-767. doi: 10.3934/proc.2005.2005.756

[9]

Panayotis Panayotaros. Continuation and bifurcations of breathers in a finite discrete NLS equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1227-1245. doi: 10.3934/dcdss.2011.4.1227

[10]

Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic & Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639

[11]

J. Cuevas, J. C. Eilbeck, N. I. Karachalios. Thresholds for breather solutions of the discrete nonlinear Schrödinger equation with saturable and power nonlinearity. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 445-475. doi: 10.3934/dcds.2008.21.445

[12]

Jibin Li. Bifurcations and exact travelling wave solutions of the generalized two-component Hunter-Saxton system. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1719-1729. doi: 10.3934/dcdsb.2014.19.1719

[13]

Hiroshi Isozaki, Hisashi Morioka. A Rellich type theorem for discrete Schrödinger operators. Inverse Problems & Imaging, 2014, 8 (2) : 475-489. doi: 10.3934/ipi.2014.8.475

[14]

Mostafa Abounouh, H. Al Moatassime, J. P. Chehab, S. Dumont, Olivier Goubet. Discrete Schrödinger equations and dissipative dynamical systems. Communications on Pure & Applied Analysis, 2008, 7 (2) : 211-227. doi: 10.3934/cpaa.2008.7.211

[15]

Davit Martirosyan. Exponential mixing for the white-forced damped nonlinear wave equation. Evolution Equations & Control Theory, 2014, 3 (4) : 645-670. doi: 10.3934/eect.2014.3.645

[16]

A. Ducrot. Travelling wave solutions for a scalar age-structured equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 251-273. doi: 10.3934/dcdsb.2007.7.251

[17]

Christopher K. R. T. Jones, Robert Marangell. The spectrum of travelling wave solutions to the Sine-Gordon equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 925-937. doi: 10.3934/dcdss.2012.5.925

[18]

Claude Bardos, François Golse, Peter Markowich, Thierry Paul. On the classical limit of the Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5689-5709. doi: 10.3934/dcds.2015.35.5689

[19]

Camille Laurent. Internal control of the Schrödinger equation. Mathematical Control & Related Fields, 2014, 4 (2) : 161-186. doi: 10.3934/mcrf.2014.4.161

[20]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]