Analytic theory of narrow lattice solitons

Nonlinearity

Volumn 21, Issue 3, 2008, Pages 509-536

  Sivan, Y ^a   Fibich, G ^a   Efremidis, N K ^b,c   Bar Ad, S ^a  

^a TEL AVIV UNIVERSITY   (Israel)

^b UNIVERSITY OF CRETE   (Greece)

^c NATIONAL TECHNICAL UNIVERSITY OF ATHENS   (Greece)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 43049136229     PISSN: 09517715     EISSN: 13616544     Source Type: Journal    
DOI: 10.1088/0951-7715/21/3/008     Document Type: Article

References (60)

1. Weinstein M I 1989 The Nonlinear Schrödinger Equation: Singularity Formation Stability and Dispersion (Providence, RI: American Mathematical Society)
2. Weinstein M I 1982/1983 Nonlinear Schrödinger equations and sharp interpolation estimates Commun. Math. Phys. 87 567-76
3. Christodoulides D N, Lederer F and Silberberg Y 2003 Discretizing light behaviour in linear and nonlinear waveguide lattices Nature 424 817-23
4. Joannopoulos J J, Meade R D and Winn J N 1995 Photonic Crystals-Molding the Flow of Light (Princeton, NJ: Princeton University Press)
5. Ostrovskaya E and Kivshar Y 2004 Photonic crystals for matter waves: Bose-Einstein condensates in optical lattices Opt. Express 12 19
6. Fleischer J W, Bartal G, Cohen O, Schwartz T, Manela O, Freedman B, Segev M, Buljan H and Efremidis N K 2005 Spatial photonics in nonlinear waveguide arrays Opt. Express 13 1780-96
7. Aceves A B, De Angelis C, Peschel T, Muschall R, Lederer F, Trillo S and Wabnitz S 1996 Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays Phys. Rev. E 53 1172-89
8. Aceves A B, Luther G G, De Angelis C, Rubenchik A M and Turitsyn S K 1995 Energy localization in nonlinear fiber arrays: collapse-effect compressor Phys. Rev. Lett. 75 73-6
9. Abdullaev F Kh, Gammal A, Kamchatnov A M and Tomio L 2005 Dynamics of bright solitons in a Bose-Einstein condensate Int. J. Mod. Phys. B 19 3415-73
10. Brazhnyi V A and Konotop V V 2004 Theory of nonlinear matter waves in optical lattices Mod. Phys. Lett. B 18 627
11. Oh Y-G 1989 Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials Commun. Math. Phys. 121 11-33
12. Rose H A and Weinstein M I 1988 On the bound states of the nonlinear Schrödinger equation with a linear potential Physica D 30 207-18
13. Christodoulides D N and Joseph R I 1988 Discrete self-focusing in nonlinear arrays of coupled waveguides Opt. Lett. 13 794-6
14. Louis P J Y, Ostrovskaya E A, Savage C M and Kivshar Y S 2003 Bose-Einstein condensates in optical lattices: band-gap structure and solitons Phys. Rev. A 67 013602
15. Efremidis N K and Christodoulides D N 2003 Lattice solitons in Bose-Einstein condensates Phys. Rev. A 67 063608
16. Pelinovsky D E, Sukhorukov A A and Kivshar Y 2004 Bifurcations and stability of gap solitons in periodic potentials Phys. Rev. E 70 036618
17. Efremidis N K, Hudock J, Christodoulides D N, Fleischer J W, Cohen O and Segev M 2003 Two-dimensional optical lattice solitons Phys. Rev. Lett. 91 213906
18. Yang J and Musslimani Z H 2003 Fundamental and vortex solitons in a two-dimensional optical lattice Opt. Lett. 21 2094-6
19. Musslimani Z H and Yang J 2004 Self trapping of light in a two dimensional photonic lattice J. Opt. Soc. Am. B 21 973-81
20. Mihalache D, Mazilu D, Lederer F, Malomed B A, Crasovan L-C, Kartashov Y V and Torner L 2005 Stable three-dimensional solitons in attractive Bose-Einstein condensates loaded in an optical lattice Phys. Rev. A 72 021601
21. Mihalache D, Mazilu D, Lederer F, Malomed B A, Crasovan L-C, Kartashov Y V and Torner L 2004 Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice Phys. Rev. E 70 055603(R)
22. Eisenberg H B, Silberberg Y, Morandotti R, Boyd A R and Aitchison J S 1998 Discrete spatial optical solitons in waveguide arrays Phys. Rev. Lett. 81 3383-6
23. Efremidis N K, Christodoulides D N, Sears S, Fleischer J W and Segev M 2002 Discrete solitons in photorefractive optically-induced photonic lattices Phys. Rev. E 66 046602
24. Fleischer J W, Carmon T, Segev M, Efremidis N K and Christodoulides D N 2003 Observation of discrete solitons in optically-induced real-time waveguide arrays Phys. Rev. Lett. 90 023902
25. Fleischer J W, Segev M, Efremidis N K and Christodoulides D N 2003 Observation of two-dimensional discrete solitons in optically-induced nonlinear photonic lattices Nature 422 147
26. Neshev D, Ostrovskaya E, Kivshar Y and Krolikowski W 2003 Spatial solitons in optically induced gratings Opt. Lett. 28 710
27. Pertsch T, Peschel U, Lederer F, Burghoff J, Will M, Nolte S and Tünnermann A 2004 Discrete diffraction in two-dimensional arrays of coupled waveguides in silica Opt. Lett. 29 468-70
28. Baizakov B B, Malomed B A and Salerno M 2004 Multidimensional solitons in a low-dimensional periodic potential Phys. Rev. A 70 053613
29. Alfimov G L, Konotop V V and Pacciani P 2007 Stationary localized modes for the quintic nonlinear Schrödinger equation with a periodic potential Phys. Rev. A 75 023624
30. Fibich G, Sivan Y and Weinstein M I 2006 Bound states of NLS equations with a periodic nonlinear microstructure Physica D 217 31-57
31. Sivan Y, Fibich G and Weinstein M I 2006 Waves in nonlinear lattices-ultrashort optical pulses and Bose-Einstein condensates Phys. Rev. Lett. 97 193902
32. Cheskis D, Bar-Ad S, Morandotti R, Aitchison J S, Heisenberg H S, Silberberg Y and Ross D 2003 Strong spatiotemporal localization in a silica nonlinear waveguide array Phys. Rev. Lett. 91 223901
33. Pertsch T, Peschel U, Kobelke J, Schuster K, Bartelt H, Nolte S, Tünnermann A and Lederer F 2004 Nonlinearity and disorder in fiber arrays Phys. Rev. Lett. 93 053901
34. Lin T C and Wei J Orbital stability of bound states of semi-classical nonlinear Schrödinger equation with critical nonlinearity SIAM J. Math. Anal. at press
35. Fukuizumi R 2005 Stability of standing waves for nonlinear Schrödinger equations with critical power nonlinearities and potentials Adv. Diff. Eqns 10 259
36. Vakhitov N G and Kolokolov A A 1973 Stationary solutions of the wave equation in a medium with nonlinearity saturation Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16 1020
37. Weinstein M I 1986 Lyapunov stability of ground states of nonlinear dispersive evolution equations Commun. Pure Appl. Math. 39 51-68
38. Fibich G and Wang X P 2003 Stability of solitary waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities Physica D 175 96-108
39. Grillakis M G 1988 Linearized instability for nonlinear Schrödinger and Klein-Gordon equations Commun. Pure Appl. Math. 41 747-74
40. Le-Coz S, Fukuizumi R, Fibich G, Ksherim B and Sivan Y Instability of bound states for a nonlinear Schrödinger equation with a Dirac potential Physica D at press
41. Trombettoni A and Smerzi A 2001 Discrete solitons and breathers with dilute Bose-Einstein condensates Phys. Rev. Lett. 86 2353-6
42. Carr L D, Mahmud K W and Reinhardt W P 2001 Tunable tunneling: an application of stationary states of Bose-Einstein condensates in traps of finite depth Phys. Rev. A 64 033603
43. Fibich G and Merle F 2001 Self-focusing on bounded domains Physica D 155 132-58
44. Malomed B A, Wang Z H, Chu P L and Peng G D 1999 Multichannel switchable system of spatial solitons J. Opt. Soc. Am. B 16 1197
45. Lieb E H, Seiringer R and Yngvason J 2003 One-dimensional bosons in three-dimensional traps Phys. Rev. Lett. 91 150401
46. Bao W, Ge Y, Jaksch D, Markowich P A and Weishäupl R M 2007 Convergence rate of dimension reduction in Bose-Einstein condensates Comput. Phys. Commun. 177 832
47. Grossi M 2002 On the number of single peaked solitos of the nonlinear Schrödinger equations Ann. Inst. H Poincaré-Anal. Non Linéaire 19 261
48. Smerzi A and Trombettoni A 2003 Discrete nonlinear dynamics of weakly coupled Bose-Einstein condensates Chaos 13 766-76
49. Weinstein M I 1985 Modulational stability of ground states of nonlinear Schrödinger equations SIAM J. Math. Anal. 16 472-91
50. Ablowitz M J, Ilan B, Schonbrun E and Piestun R 2006 Solitons in two-dimensional lattices possessing defects, dislocations, and quasicrystal structures Phys. Rev. E 74 035601(R)
51. Fibich G and Gaeta A L 2000 Critical power for self-focusing in bulk media and in hollow waveguides Opt. Lett. 25 335
52. Grillakis M G, Shatah J and Strauss W A 1987 Stability theory of solitary waves in the presence of symmetry: I J. Funct. Anal. 74 160-97
53. Grillakis M G, Shatah J and Strauss W A 1990 Stability theory of solitary waves in the presence of symmetry: II J. Funct. Anal. 94 308-48
54. Bergé L, Alexander T J and Kivshar Y S 2000 Stability criterion for attractive Bose-Einstein condensates Phys. Rev. A 62 023607
55. Fukuizumi R 2001 Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential Discrete Contin. Dyn. Syst. 7 525-44
56. Fukuizumi R and Ohta M 2003 Instability of standing waves for nonlinear Schrödinger equations with potentials Diff. Integr. Eqns 16 691-706
57. Sivan Y, Fibich G and Ilan B Phys. Rev. Lett. submitted
58. Trefethen L N and Bau D 1997 Numerical Linear Algebra (Philadelphia, PA: SIAM)
59. Sulem C and Sulem P L 1999 The Nonlinear Schrödinger Equation (New York, NY: Springer)
60. Lehoucq R B, Sorensen D C and Yang C 1998 ARPACK User's Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (Philadelphia, PA: SIAM)