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Original Articles

Travelling waves of discrete nonlinear Schrödinger equations with nonlocal interactions

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Pages 1387-1411 | Received 20 May 2009, Accepted 10 Jul 2009, Published online: 16 Dec 2009

References

  • Dodd , RK , Eilbeck , JC , Gibbon , JD and Morris , HC . 1984 . Solitons and Nonlinear Wave Equations , London : Academic Press .
  • Hasegawa , A and Kodama , Y . 1994 . Optical Fibre Solitons , Cambridge, , UK : Cambridge University Press .
  • Sulem , C and Sulem , PL . 1999 . The Nonlinear Schrödinger Equation , New York : Springer-Verlag .
  • Aceves , AB , Luther , GG , De Angelis , C , Rubenchik , AM and Turitsyn , SK . 1995 . Energy localization in nonlinear fiber arrays: Collapse-effect compressor . Phys. Rev. Lett. , 75 : 73 – 76 .
  • Eilbeck , JC , Lomsdahl , PS and Scott , AC . 1985 . The discrete self-trapping equation . Physica D , 16 : 318 – 338 .
  • Eisenberg , H , Silberberg , Y , Morandotti , R , Boyd , A and Aitchison , J . 1998 . Discrete spatial optical solitons in waveguide arrays . Phys. Rev. Lett. , 81 : 3383 – 3386 .
  • Lenz , G , Talanina , I and Martijn de Sterke , C . 1999 . Bloch oscillations in an array of curved optical waveguides . Phys. Rev. Lett. , 83 : 963 – 966 .
  • Ustinov , AV , Doderer , T , Vernik , IV , Pedersen , NF , Huebener , RP and Oboznov , VA . 1994 . Experiments with solitons in annular Josephson junctions . Physica D , 68 : 41 – 44 .
  • Wu , WP , Schrieffer , JR and Heeger , AJ . 1980 . Soliton excitations in polyacetylene . Phys. Rev. B , 22 : 2099 – 2111 .
  • Aubry , S . 1997 . Breathers in nonlinear lattices: Existence, linear stability and quantization . Physica D , 103 : 201 – 250 .
  • Aigner , AA , Champneys , AR and Rothos , VM . 2003 . A new barrier to the existence of moving kinks in Frenkel–Kontorova lattices . Physica D , 186 : 148 – 170 .
  • Davydov , AS . 1971 . Theory of Molecular Excitons , New York : Plenum .
  • Scott , AC . 1999 . Nonlinear Science: Emergence and Dynamics of Coherent Structures , UK : Oxford University Press .
  • Abdullaev , FKh , Bludov , YuV , Dmitriev , SV , Kevrekidis , PG and Konotop , VV . 2008 . Generalized neighbor-interaction models induced by nonlinear lattices . Phys. Rev. E , 77 : 016604
  • Agrotis , M , Lafortune , S and Kevrekidis , PG . 2005 . On a discrete version of the Korteweg-de Vries equation . Discrete Contin. Dyn. Sys., Suppl. , : 22 – 29 .
  • Archilla , JF , Christiansen , PL and Gaididei , YuB . 2001 . Interplay of nonlinearity and geometry in DNA-related, Klein–Gordon model with a long-range dipole–dipole interaction . Phys. Rev. E , 65 : 016609
  • Bates , PW and Chmaj , A . 1999 . A discrete convolution model for phase transitions . Arch. Rational Mech. Anal. , 150 : 281 – 305 .
  • Bates , PW , Chen , X and Chmaj , A . 2003 . Travelling waves of bistable dynamics on lattice . SIAM J. Math. Anal. , 35 : 520 – 546 .
  • Bates , PW and Zhang , C . 2006 . Travelling pulses for the Klein–Gordon equation on a lattice or continuum with long-range interaction . Disc. Cont. Dyn. Sys. , 16 : 235 – 252 .
  • Christiansen , PL , Gadidei , GI , Gadidei , YuB , Johansson , M , Mingaleev , SF and Rasmussen , KO . 1998 . Effects of noise and nonlocal interactions in nonlinear dynamics of molecular systems . Funct. Materials , 5 : 315 – 318 .
  • Comech , A , Cuevas , J and Kevrekidis , PG . 2005 . Discrete peakons . Physica D , 207 : 137 – 160 .
  • Larsen , PV , Christiansen , PL , Bang , O , Archilla , JFR and Gaididei , YuB . 2004 . Energy funneling in a bent chain of Morse oscillators with a long-range coupling . Phys. Rev. E , 69 : 026603
  • Mingaleev , SF , Gaididei , YuB and Mertens , FG . 1998 . Solitons in anharmonic chains with power-law long-range interactions . Phys. Rev. E , 58 : 3833 – 3842 .
  • Carretero-González , R , Kevrekidis , PG , Malomed , BA and Frantzeskakis , DJ . 2005 . Three-dimensional nonlinear lattices: From oblique vortices and octupoles to discrete diamonds and vortex cubes . Phys. Rev. Lett. , 94 : 203901-1 – 203901-4 .
  • Chong , C , Carretero-González , R , Malomed , BA and Kevrekidis , PG . 2009 . Multistable solitons in higher-dimensional cubic-quintic nonlinear Schrödinger lattices . Physica D , 238 : 126 – 136 .
  • Gómez-Gardeñes , J , Floría , LM and Bishop , AR . 2006 . Discrete breathers in two-dimensional anisotropic nonlinear Schrödinger lattices . Physica D , 216 : 31 – 43 .
  • Lukas , M , Pelinovsky , DE and Kevrekidis , PG . 2008 . Lyapunov–Schmidt reduction algorithm for three-dimensional discrete vortices . Physica D , 237 : 339 – 350 .
  • Guo , S , Lamb , LSW and Rink , BW . 2009 . Branching patterns of wave trains in the FPU lattice . Nonlinearity , 22 : 283 – 299 .
  • Pelinovsky , DE . 2006 . Translationally invariant nonlinear Schrödinger lattices . Nonlinearity , 19 : 2695 – 2716 .
  • Chow , SN , Mallet-Paret , J and Shen , W . 1998 . Traveling waves in lattice dynamical systems . J. Differential Equations , 149 : 248 – 291 .
  • Coullet , P and Walgraef , D . 1989 . Spatial forcing of 2d wave patterns . Europhys. Lett. , 10 : 525 – 531 .
  • Fečkan , M and Rothos , VM . 2005 . Kink-like periodic travelling waves for lattice equations with on-site and intersite potentials . Dynamics of PDEs , 2 ( 4 ) : 357 – 370 .
  • Pankov , A and Pflüger , K . 2000 . Traveling waves in lattice dynamical systems . Math. Methods Appl. Sci. , 23 : 1223 – 1235 .
  • Pouget , J , Remoissenet , M and Tamga , JM . 1993 . Energy self-localization and gap local pulses in a two-dimensional nonlinear lattice . Phys. Rev. B , 47 ( 22 ) : 14866 – 14874 .
  • Smets , D and Willem , M . 1997 . Solitary waves with prescribed speed on infinite lattices . J. Funct. Anal. , 149 : 266 – 275 .
  • Tamga , JM , Remoissenet , M and Pouget , J . 1995 . Breathing solitary waves in a sine-Gordon two dimensional lattice . Phys. Rev. Letters , 75 : 357 – 361 .
  • Fečkan , M and Rothos , VM . 2007 . Traveling waves in Hamiltonian systems on 2d lattices with nearest neighbor interactions . Nonlinearity , 20 : 319 – 341 .
  • Li , S and Szulkin , A . 1994 . Periodic solutions for a class of nonautonomous Hamiltonian systems . J. Differential Equations , 112 : 226 – 238 .
  • Rabinowitz , PH . 1986 . “ Minimax Methods in Critical Point Theory with Applications to Differential Equations ” . In CBMS Regional Conference Series in Mathematics , Providence, RI : American Mathematical Society .
  • Zygmund , A . 2002 . Trigonometric Series, , 3rd , Cambridge, UK : Cambridge University Press .
  • Mawhin , J and Willem , M . 1989 . Critical Point Theory and Hamiltonian Systems , New York : Springer .

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