American Institute of Mathematical Sciences

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

Continuation and bifurcations of breathers in a finite discrete NLS equation

Panayotis Panayotaros 1,

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.
Keywords: bifurcations., Discrete NLS equations, breathers.
Mathematics Subject Classification: Primary: 37J20, 35Q55; Secondary: 37J45, 37M20, 37L6.
Citation: Panayotis Panayotaros. Continuation and bifurcations of breathers in a finite discrete NLS equation. Discrete and 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), 127-150. 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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