Three is a crowd: Solitary waves in photorefractive media with three potential wells

SIAM Journal on Applied Dynamical Systems

Volumn 5, Issue 4, 2006, Pages 598-633

  Kapitula, Todd ^a   Kevrekidis, P G ^b   Chen, Zhigang ^c,d  

^a UNIVERSITY OF NEW MEXICO   (United States)

^b UNIVERSITY OF MASSACHUSETTS   (United States)

^c SAN FRANCISCO STATE UNIVERSITY   (United States)

^d NANKAI UNIVERSITY   (China)

Author keywords

Hamiltonian systems; Lyapunov Schmidt reduction; Solitary waves; Stability

Indexed keywords

BIFURCATION (MATHEMATICS); CONVERGENCE OF NUMERICAL METHODS; HAMILTONIANS; NIOBIUM COMPOUNDS; NUMERICAL METHODS; PHOTOREACTIVITY; PHOTOREFRACTIVE MATERIALS;

ANALYTICAL PREDICTIONS; BIFURCATION ANALYSIS; EINSTEIN CONDENSATE; FIXED POINT METHODS; HAMILTONIAN SYSTEMS; PHOTOREFRACTIVE MEDIA; SCHMIDT; STRONTIUM BARIUM NIOBATE CRYSTALS;

SOLITONS;

EID: 34548046932     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/05064076X     Document Type: Article

References (49)

1. M. Albiez, R. Gati, J. Fölling, S. Hunsmann, T. Cristiani, and M. K. Oberthaler, Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction, Phys. Rev. Lett., 95 (2005), paper 010402.
2. G. L. Alfimov, V. A. Brazhnyi, and V. V. Konotop, On classification of intrinsic localized modes for the discrete nonlinear Schrödinger equation, Phys. D, 194 (2004), pp. 127–150.
3. Th. Anker, M. Albiez, R. Gati, S. Hunsmann, B. Eiermann, A. Trombettoni, and M. K. Oberthaler, Nonlinear self-trapping of matter waves in periodic potentials, Phys. Rev. Lett., 94 (2005), paper 020403.
4. V. A. Brazhnyi and V. V. Konotop, Theory of nonlinear matter waves in optical lattices, Mod. Phys. Lett. B, 18 (2004), pp. 627–651.
5. R. Brout, A Brief Course in Spontaneous Symmetry Breaking I: The Palelolitic Age, hep-th/0203096, Los Alamos National Laboratory archives, Los Alamos, NM, 2002.
6. C. Cambournac, T. Sylvestre, H. Maillotte, B. Vanderlinden, P. Kockaert, Ph. Emplit, and M. Haelterman, Symmetry-breaking instability of multimode vector solitons, Phys. Rev. Lett., 89 (2002), paper 083901.
7. Z. Chen and K. McCarthy, Spatial soliton pixels from partially coherent light, Opt. Lett., 27 (2002), pp. 2019–2021.
8. Z. Chen, A. Bezryadina, I. Makasyuk, and J. Yang, Observation of two-dimensional lattice vector solitons, Opt. Lett., 29 (2004), pp. 1656–1658.
9. Z. Chen, H. Martin, E. D. Eugenieva, J. Xu, and A. Bezryadina, Anisotropic enhancement of discrete diffraction and formation of two-dimensional discrete-soliton trains, Phys. Rev. Lett., 92 (2004), paper 143902.
10. D. N. Christodoulides, F. Lederer, and Y. Silberberg, Discretizing light behavior in linear and nonlinear waveguide lattices, Nature, 424 (2003), pp. 817–823.
11. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Discrete solitons in photorefractive optically induced photonic lattices, Phys. Rev. E, 66 (2002), paper 046602.
12. J. C. Eilbeck, P. S. Lomdahl, and A. C. Scott, The discrete self-trapping equation, Phys. D, 16 (1985), pp. 318–338.
13. F. Englert, A Brief Course in Spontaneous Symmetry Breaking II: Modern Times: The BEH Mechanism, hep-th/0203097, Los Alamos National Laboratory archives, Los Alamos, NM, 2002.
14. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Observation of discrete solitons in optically induced real time waveguide arrays, Phys. Rev. Lett., 90 (2003), paper 023902.
15. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices, Nature, 422 (2003), pp. 147–150.
16. J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Observation of vortex-ring “discrete” solitons in 2D photonic lattices, Phys. Rev. Lett., 92 (2004), paper 123904.
17. R. Franzosi and V. Penna, Chaotic behavior, collective modes and self-trapping in the dynamics of three coupled Bose-Einstein condensates, Phys. Rev. E, 67 (2003), paper 046227.
18. M. Golubitsky and D. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. 1, Springer-Verlag, New York, Berlin, 1985.
19. A. Hasegawa and Y. Kodama, Solitons in Optical Communications, Clarendon Press, Oxford, UK, 1995.
20. R. K. Jackson and M. I. Weinstein, Geometric analysis of bifurcation and symmetry breaking in a Gross-Pitaevskii equation, J. Statist. Phys., 116 (2004), pp. 881–905.
21. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, eds., Photonic Crystals: Modeling the Flow of Light, Princeton University Press, Princeton, NJ, 1995.
22. M. Johansson, Hamiltonian Hopf bifurcations in the discrete nonlinear Schrödinger trimer: Oscillatory instabilities, quasi-periodic solutions and a “new” type of self-trapping transition, J. Phys. A, 37 (2004), pp. 2201–2222.
23. T. Kapitula and P. G. Kevrekidis, Bose-Einstein condensates in the presence of a magnetic trap and an optical lattice, Chaos, 15 (2005), paper 037114.
24. T. Kapitula and P. G. Kevrekidis, Bose-Einstein condensates in the presence of a magnetic trap and optical lattice: Two-mode approximation, Nonlinearity, 18 (2005), pp. 2491–2512.
25. T. Kapitula, P. G. Kevrekidis, and B. A. Malomed, Stability of multiple pulses in discrete systems, Phys. Rev. E, 63 (2001), paper 036602.
26. T. Kapitula, P. Kevrekidis, and B. Sandstede, Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems, Phys. D, 195 (2004), pp. 263–282.
27. T. Kapitula, P. Kevrekidis, and B. Sandstede, Addendum: Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems, Phys. D, 201 (2005), pp. 199–201.
28. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1980.
29. P. G. Kevrekidis and D. J. Frantzeskakis, Pattern forming dynamical instabilities of Bose-Einstein condensates, Mod. Phys. Lett. B, 18 (2004), pp. 173–202.
30. P. G. Kevrekidis, A. R. Bishop, and K. Ø. Rasmussen, Twisted localized modes, Phys. Rev. E, 18 (2001), paper 036603.
31. P. G. Kevrekidis, Z. Chen, B. A. Malomed, D. J. Frantzeskakis, and M. I. Weinstein, Symmetry-breaking instability of multimode vector solitons, Phys. Lett. A, 340 (2005), pp. 275–280.
32. Yu. S. Kivshar and G. P. Agrawal, eds., Optical Solitons: From Fibers to Photonic Crystals, Academic Press, New York, 2003.
33. Y. Li and K. Promislow, Structural stability of non-ground state traveling waves of coupled nonlinear Schrödinger equations, Phys. D, 124 (1998), pp. 137–165.
34. K. W. Mahmud, J. N. Kutz, and W. P. Reinhardt, Bose-Einstein condensates in a one-dimensional double square well: Analytical solutions of the nonlinear Schrödinger equation, Phys. Rev. A, 66 (2002), paper 063607.
35. H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices, Phys. Rev. Lett., 92 (2004), paper 123902.
36. A. Morgante, M. Johansson, G. Kopidakis, and S. Aubry, Standing wave instabilities in a chain of nonlinear coupled oscillators, Phys. D, 162 (2002), pp. 53–94.
37. O. Morsch and M. Oberthaler, Bose-Einstein condensates in optical lattices, Rev. Mod. Phys., 78 (2006), pp. 179–215.
38. D. Neshev, E. Ostrovskaya, Yu. Kivshar, and W. Krolikowski, Spatial solitons in optically induced gratings, Opt. Lett., 28 (2003), pp. 710–712.
39. D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Yu. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, Observation of discrete vortex solitons in optically induced photonic lattices, Phys. Rev. Lett., 92 (2004), paper 123903.
40. D. Pelinovsky, Inertia law for spectral stability of solitary waves in coupled nonlinear Schrödinger equations, Proc. Royal Soc. London A, 461 (2005), pp. 783–812.
41. D. Pelinovsky and J. Yang, Instabilities of multihump vector solitons in coupled nonlinear Schrödinger equations, Stud. Appl. Math., 115 (2005), pp. 109–137.
42. D. Pelinovsky, P. Kevrekidis, and D. Frantzeskakis, Stability of discrete solitons in nonlinear Schrödinger lattices, Phys. D, 212 (2005), pp. 1–19.
43. D. Pelinovsky, P. Kevrekidis, and D. Frantzeskakis, Persistence and stability of discrete vortices in nonlinear Schrödinger lattices, Phys. D, 212 (2005), pp. 20–53.
44. A. Sacchetti, Nonlinear time-dependent Schrödinger equations: The Gross-Pitaevskii equation with double-well potential, J. Evol. Equ., 4 (2004), pp. 345–369.
45. A. Sacchetti, Nonlinear time-dependent one-dimensional Schrödinger equation with double-well potential, SIAM J. Math. Anal., 35 (2004), pp. 1160–1176.
46. B. Sandstede, Stability of multiple-pulse solutions, Trans. Amer. Math. Soc., 350 (1998), pp. 429–472.
47. C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Springer-Verlag, New York, Berlin, 1999.
48. M. Weinstein, Excitation thresholds for nonlinear localized modes on lattices, Nonlinearity, 12 (1999), pp. 673–691.
49. J. Yang, Stability of vortex solitons in a photorefractive optical lattice, New J. Phys., 6 (2004), p. 47.