October  2011, 4(5): 1227-1245. doi: 10.3934/dcdss.2011.4.1227

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

Received  April 2009 Revised  October 2009 Published  December 2010

We present results on the continuation of breathers in the discrete cubic nonlinear Schrödinger equation in a finite one-dimensional lattice with Dirichlet boundary conditions. In the limit of small inter-site coupling the equation has a finite number of breather solutions and as we increase the coupling we see numerically that all breather branches undergo either fold or pitchfork bifurcations. We also see branches that persist for arbitrarily large coupling and converge to the linear normal modes of the system. The stability of the breathers that persist generally changes as the coupling is varied, although there are at least two branches that preserve their linear and nonlinear stability properties throughout the continuation.
Citation: 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
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. doi: 10.1016/j.physd.2004.02.001. Google Scholar

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Yu. V. Bludov and V. V. Konotop, Surface modes and breathers in finite arrays of nonlinear waveguides,, Phys. Rev. E, 76 (2007). doi: 10.1103/PhysRevE.76.046604. Google Scholar

[3]

D. Bambusi and D. Vella, Quasi periodic breathers in Hamiltonian lattices with symmetries,, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 389. doi: 10.3934/dcdsb.2002.2.389. Google Scholar

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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, (). Google Scholar

[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). doi: 10.1103/PhysRevE.78.036604. Google Scholar

[6]

D. N. Christodoulides, F. Lederer and Y. Silberberg, Discretizing light behavior in linear and nonlinear lattices,, Nature, 424 (2003), 817. doi: 10.1038/nature01936. Google Scholar

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J. J. Dongarra and C. B. Moler, Eispack, a package for solving eigenvalue problems,, in, (1984). Google Scholar

[8]

J. C. Eilbeck, P. S. Lomdahl and A. C. Scott, The discrete self-trapping equation,, Physica D, 16 (1985), 318. doi: 10.1016/0167-2789(85)90012-0. Google Scholar

[9]

H. S. Eisenberg, Y. Silberberg, R. Morandotti and J. S. Aitchison, Diffraction management,, Phys. Rev. Lett., 85 (2000). doi: 10.1103/PhysRevLett.85.1863. Google Scholar

[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. doi: 10.1088/0951-7715/18/6/005. Google Scholar

[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. doi: 10.1137/05064076X. Google Scholar

[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. doi: 10.1103/PhysRevB.34.4959. Google Scholar

[13]

P. G. Kevrekidis and V. V. Konotop, Bright compact breathers,, Phys. Rev. E, 65 (2002). doi: 10.1103/PhysRevE.65.066614. Google Scholar

[14]

G. Kopidakis, S. Aubry and G. P. Tsironis, Targeted energy transfer through discrete breathers in nonlinear systems,, Phys. Rev. Lett., 87 (2001). doi: 10.1103/PhysRevLett.87.165501. Google Scholar

[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). doi: 10.1103/PhysRevLett.97.060401. Google Scholar

[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. doi: 10.1088/0951-7715/7/6/006. Google Scholar

[17]

P. Panayotaros, Linear stability of real breathers in the discrete NLS,, Phys. Lett. A, 373 (2009), 957. doi: 10.1016/j.physleta.2009.01.023. Google Scholar

[18]

P. Panayotaros and D. E. Pelinovsky, Periodic oscillations of discrete NLS solitons in the presence of diffraction management,, Nonlinearity, 21 (2008), 1265. doi: 10.1088/0951-7715/21/6/007. Google Scholar

[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. doi: 10.1016/j.physd.2009.01.001. Google Scholar

[20]

D. E. Pelinovsky, P. G. Kevrekidis and D. J. Frantzeskakis, Stability of discrete solitons in nolinear Schrödinger lattices,, Physica D, 212 (2005), 1. doi: 10.1016/j.physd.2005.07.021. Google Scholar

[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. doi: 10.1016/j.physd.2005.09.015. Google Scholar

[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, (1970). Google Scholar

[23]

W. Qin and X. Xiao, Homoclinic orbits and localized solutions in nonlinear Schrödinger lattices,, Nonlinearity, 20 (2007), 2305. doi: 10.1088/0951-7715/20/10/002. Google Scholar

[24]

M. I. Weinstein, Excitation thresholds for nonlinear localized modes on lattices,, Nonlinearity, 12 (1999), 673. doi: 10.1088/0951-7715/12/3/314. Google Scholar

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. doi: 10.1016/j.physd.2004.02.001. Google Scholar

[2]

Yu. V. Bludov and V. V. Konotop, Surface modes and breathers in finite arrays of nonlinear waveguides,, Phys. Rev. E, 76 (2007). doi: 10.1103/PhysRevE.76.046604. Google Scholar

[3]

D. Bambusi and D. Vella, Quasi periodic breathers in Hamiltonian lattices with symmetries,, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 389. doi: 10.3934/dcdsb.2002.2.389. Google Scholar

[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, (). Google Scholar

[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). doi: 10.1103/PhysRevE.78.036604. Google Scholar

[6]

D. N. Christodoulides, F. Lederer and Y. Silberberg, Discretizing light behavior in linear and nonlinear lattices,, Nature, 424 (2003), 817. doi: 10.1038/nature01936. Google Scholar

[7]

J. J. Dongarra and C. B. Moler, Eispack, a package for solving eigenvalue problems,, in, (1984). Google Scholar

[8]

J. C. Eilbeck, P. S. Lomdahl and A. C. Scott, The discrete self-trapping equation,, Physica D, 16 (1985), 318. doi: 10.1016/0167-2789(85)90012-0. Google Scholar

[9]

H. S. Eisenberg, Y. Silberberg, R. Morandotti and J. S. Aitchison, Diffraction management,, Phys. Rev. Lett., 85 (2000). doi: 10.1103/PhysRevLett.85.1863. Google Scholar

[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. doi: 10.1088/0951-7715/18/6/005. Google Scholar

[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. doi: 10.1137/05064076X. Google Scholar

[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. doi: 10.1103/PhysRevB.34.4959. Google Scholar

[13]

P. G. Kevrekidis and V. V. Konotop, Bright compact breathers,, Phys. Rev. E, 65 (2002). doi: 10.1103/PhysRevE.65.066614. Google Scholar

[14]

G. Kopidakis, S. Aubry and G. P. Tsironis, Targeted energy transfer through discrete breathers in nonlinear systems,, Phys. Rev. Lett., 87 (2001). doi: 10.1103/PhysRevLett.87.165501. Google Scholar

[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). doi: 10.1103/PhysRevLett.97.060401. Google Scholar

[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. doi: 10.1088/0951-7715/7/6/006. Google Scholar

[17]

P. Panayotaros, Linear stability of real breathers in the discrete NLS,, Phys. Lett. A, 373 (2009), 957. doi: 10.1016/j.physleta.2009.01.023. Google Scholar

[18]

P. Panayotaros and D. E. Pelinovsky, Periodic oscillations of discrete NLS solitons in the presence of diffraction management,, Nonlinearity, 21 (2008), 1265. doi: 10.1088/0951-7715/21/6/007. Google Scholar

[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. doi: 10.1016/j.physd.2009.01.001. Google Scholar

[20]

D. E. Pelinovsky, P. G. Kevrekidis and D. J. Frantzeskakis, Stability of discrete solitons in nolinear Schrödinger lattices,, Physica D, 212 (2005), 1. doi: 10.1016/j.physd.2005.07.021. Google Scholar

[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. doi: 10.1016/j.physd.2005.09.015. Google Scholar

[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, (1970). Google Scholar

[23]

W. Qin and X. Xiao, Homoclinic orbits and localized solutions in nonlinear Schrödinger lattices,, Nonlinearity, 20 (2007), 2305. doi: 10.1088/0951-7715/20/10/002. Google Scholar

[24]

M. I. Weinstein, Excitation thresholds for nonlinear localized modes on lattices,, Nonlinearity, 12 (1999), 673. doi: 10.1088/0951-7715/12/3/314. Google Scholar

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