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Continuation and bifurcations of breathers in a finite discrete NLS equation
1. | Depto. Matemáticas y Mecánica, I.I.M.A.S.-U.N.A.M., Apdo. Postal 20-726, 01000 México D.F., Mexico |
References:
[1] |
G. L. Alfimov, V. A. Brazhnyi and V. V. Konotop, On classification of intrinsic localized modes for the discrete nonlinear Schrödinger equation, Physica D, 194 (2004), 127-150.
doi: 10.1016/j.physd.2004.02.001. |
[2] |
Yu. V. Bludov and V. V. Konotop, Surface modes and breathers in finite arrays of nonlinear waveguides, Phys. Rev. E, 76 (2007), 046604.
doi: 10.1103/PhysRevE.76.046604. |
[3] |
D. Bambusi and D. Vella, Quasi periodic breathers in Hamiltonian lattices with symmetries, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 389-399.
doi: 10.3934/dcdsb.2002.2.389. |
[4] |
L. A. Cisneros, J. Ize and A. A. Minzoni, Modulational and numerical solutions for the steady discrete Sine-Gordon equation in two space dimensions,, Physica D, ().
|
[5] |
L. A. Cisneros and A. A. Minzoni, P. Panayotaros and N. F. Smyth, Modulation analysis of large scale discrete vortices, Phys. Rev. E, 78 (2008), 036604.
doi: 10.1103/PhysRevE.78.036604. |
[6] |
D. N. Christodoulides, F. Lederer and Y. Silberberg, Discretizing light behavior in linear and nonlinear lattices, Nature, 424 (2003), 817-823.
doi: 10.1038/nature01936. |
[7] |
J. J. Dongarra and C. B. Moler, Eispack, a package for solving eigenvalue problems, in "Sources and Development of Mathematical Software" (W. J. Cowell, ed.), Prentice-Hall, New York, 1984. |
[8] |
J. C. Eilbeck, P. S. Lomdahl and A. C. Scott, The discrete self-trapping equation, Physica D, 16 (1985), 318-338.
doi: 10.1016/0167-2789(85)90012-0. |
[9] |
H. S. Eisenberg, Y. Silberberg, R. Morandotti and J. S. Aitchison, Diffraction management, Phys. Rev. Lett., 85 (2000), 1863.
doi: 10.1103/PhysRevLett.85.1863. |
[10] |
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), 2491-2512.
doi: 10.1088/0951-7715/18/6/005. |
[11] |
T. Kapitula, P. G. Kevrekidis and Z. Chen, Three is a crowd: Solitary waves in photorefractive media with three potential wells, SIAM J. Appl. Dyn. Syst., 5 (2007), 598-633.
doi: 10.1137/05064076X. |
[12] |
V. M. Kenkre and D. K. Campbell, Self-trapping on a dimer: Time-dependent solution of a discrete nonlinear Schrödinger equation, Phys. Rev. B, 34 (1986), 4959-4961.
doi: 10.1103/PhysRevB.34.4959. |
[13] |
P. G. Kevrekidis and V. V. Konotop, Bright compact breathers, Phys. Rev. E, 65 (2002), 066614.
doi: 10.1103/PhysRevE.65.066614. |
[14] |
G. Kopidakis, S. Aubry and G. P. Tsironis, Targeted energy transfer through discrete breathers in nonlinear systems, Phys. Rev. Lett., 87 (2001), 165501.
doi: 10.1103/PhysRevLett.87.165501. |
[15] |
R. Livi, R. Franzosi and G. Oppo, Self-localization of Bose-Einstein condensates in optical lattices via boundary dissipation, Phys. Rev. Lett., 97 (2006), 060401.
doi: 10.1103/PhysRevLett.97.060401. |
[16] |
R. S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity, 7 (1994), 1623-1643.
doi: 10.1088/0951-7715/7/6/006. |
[17] |
P. Panayotaros, Linear stability of real breathers in the discrete NLS, Phys. Lett. A, 373 (2009), 957-963.
doi: 10.1016/j.physleta.2009.01.023. |
[18] |
P. Panayotaros and D. E. Pelinovsky, Periodic oscillations of discrete NLS solitons in the presence of diffraction management, Nonlinearity, 21 (2008), 1265-1279.
doi: 10.1088/0951-7715/21/6/007. |
[19] |
C. L. Pando and E. J. Doedel, Bifurcation structures and dominant modes near relative equilibria in the one dimensional discrete nonlinear Schrödinger equation, Physica D, 238 (2009), 687-698.
doi: 10.1016/j.physd.2009.01.001. |
[20] |
D. E. Pelinovsky, P. G. Kevrekidis and D. J. Frantzeskakis, Stability of discrete solitons in nolinear Schrödinger lattices, Physica D, 212 (2005), 1-19.
doi: 10.1016/j.physd.2005.07.021. |
[21] |
D. E. Pelinovsky, P. G. Kevrekidis and D. J. Frantzeskakis, Persistence and stability of discrete vortices in nonlinear Schrödinger lattices, Physica D, 212 (2005), 20-53.
doi: 10.1016/j.physd.2005.09.015. |
[22] |
M. J. D. Powell and P. Rabinowitz, ed., "A Hybrid Method for Nonlinear Equations, in Numerical Methods for Nonlinear Algebraic Equations," Gordon and Breach, New York, 1970. |
[23] |
W. Qin and X. Xiao, Homoclinic orbits and localized solutions in nonlinear Schrödinger lattices, Nonlinearity, 20 (2007), 2305-2317.
doi: 10.1088/0951-7715/20/10/002. |
[24] |
M. I. Weinstein, Excitation thresholds for nonlinear localized modes on lattices, Nonlinearity, 12 (1999), 673-691.
doi: 10.1088/0951-7715/12/3/314. |
show all references
References:
[1] |
G. L. Alfimov, V. A. Brazhnyi and V. V. Konotop, On classification of intrinsic localized modes for the discrete nonlinear Schrödinger equation, Physica D, 194 (2004), 127-150.
doi: 10.1016/j.physd.2004.02.001. |
[2] |
Yu. V. Bludov and V. V. Konotop, Surface modes and breathers in finite arrays of nonlinear waveguides, Phys. Rev. E, 76 (2007), 046604.
doi: 10.1103/PhysRevE.76.046604. |
[3] |
D. Bambusi and D. Vella, Quasi periodic breathers in Hamiltonian lattices with symmetries, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 389-399.
doi: 10.3934/dcdsb.2002.2.389. |
[4] |
L. A. Cisneros, J. Ize and A. A. Minzoni, Modulational and numerical solutions for the steady discrete Sine-Gordon equation in two space dimensions,, Physica D, ().
|
[5] |
L. A. Cisneros and A. A. Minzoni, P. Panayotaros and N. F. Smyth, Modulation analysis of large scale discrete vortices, Phys. Rev. E, 78 (2008), 036604.
doi: 10.1103/PhysRevE.78.036604. |
[6] |
D. N. Christodoulides, F. Lederer and Y. Silberberg, Discretizing light behavior in linear and nonlinear lattices, Nature, 424 (2003), 817-823.
doi: 10.1038/nature01936. |
[7] |
J. J. Dongarra and C. B. Moler, Eispack, a package for solving eigenvalue problems, in "Sources and Development of Mathematical Software" (W. J. Cowell, ed.), Prentice-Hall, New York, 1984. |
[8] |
J. C. Eilbeck, P. S. Lomdahl and A. C. Scott, The discrete self-trapping equation, Physica D, 16 (1985), 318-338.
doi: 10.1016/0167-2789(85)90012-0. |
[9] |
H. S. Eisenberg, Y. Silberberg, R. Morandotti and J. S. Aitchison, Diffraction management, Phys. Rev. Lett., 85 (2000), 1863.
doi: 10.1103/PhysRevLett.85.1863. |
[10] |
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), 2491-2512.
doi: 10.1088/0951-7715/18/6/005. |
[11] |
T. Kapitula, P. G. Kevrekidis and Z. Chen, Three is a crowd: Solitary waves in photorefractive media with three potential wells, SIAM J. Appl. Dyn. Syst., 5 (2007), 598-633.
doi: 10.1137/05064076X. |
[12] |
V. M. Kenkre and D. K. Campbell, Self-trapping on a dimer: Time-dependent solution of a discrete nonlinear Schrödinger equation, Phys. Rev. B, 34 (1986), 4959-4961.
doi: 10.1103/PhysRevB.34.4959. |
[13] |
P. G. Kevrekidis and V. V. Konotop, Bright compact breathers, Phys. Rev. E, 65 (2002), 066614.
doi: 10.1103/PhysRevE.65.066614. |
[14] |
G. Kopidakis, S. Aubry and G. P. Tsironis, Targeted energy transfer through discrete breathers in nonlinear systems, Phys. Rev. Lett., 87 (2001), 165501.
doi: 10.1103/PhysRevLett.87.165501. |
[15] |
R. Livi, R. Franzosi and G. Oppo, Self-localization of Bose-Einstein condensates in optical lattices via boundary dissipation, Phys. Rev. Lett., 97 (2006), 060401.
doi: 10.1103/PhysRevLett.97.060401. |
[16] |
R. S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity, 7 (1994), 1623-1643.
doi: 10.1088/0951-7715/7/6/006. |
[17] |
P. Panayotaros, Linear stability of real breathers in the discrete NLS, Phys. Lett. A, 373 (2009), 957-963.
doi: 10.1016/j.physleta.2009.01.023. |
[18] |
P. Panayotaros and D. E. Pelinovsky, Periodic oscillations of discrete NLS solitons in the presence of diffraction management, Nonlinearity, 21 (2008), 1265-1279.
doi: 10.1088/0951-7715/21/6/007. |
[19] |
C. L. Pando and E. J. Doedel, Bifurcation structures and dominant modes near relative equilibria in the one dimensional discrete nonlinear Schrödinger equation, Physica D, 238 (2009), 687-698.
doi: 10.1016/j.physd.2009.01.001. |
[20] |
D. E. Pelinovsky, P. G. Kevrekidis and D. J. Frantzeskakis, Stability of discrete solitons in nolinear Schrödinger lattices, Physica D, 212 (2005), 1-19.
doi: 10.1016/j.physd.2005.07.021. |
[21] |
D. E. Pelinovsky, P. G. Kevrekidis and D. J. Frantzeskakis, Persistence and stability of discrete vortices in nonlinear Schrödinger lattices, Physica D, 212 (2005), 20-53.
doi: 10.1016/j.physd.2005.09.015. |
[22] |
M. J. D. Powell and P. Rabinowitz, ed., "A Hybrid Method for Nonlinear Equations, in Numerical Methods for Nonlinear Algebraic Equations," Gordon and Breach, New York, 1970. |
[23] |
W. Qin and X. Xiao, Homoclinic orbits and localized solutions in nonlinear Schrödinger lattices, Nonlinearity, 20 (2007), 2305-2317.
doi: 10.1088/0951-7715/20/10/002. |
[24] |
M. I. Weinstein, Excitation thresholds for nonlinear localized modes on lattices, Nonlinearity, 12 (1999), 673-691.
doi: 10.1088/0951-7715/12/3/314. |
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