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Gap solitons for the repulsive Gross-Pitaevskii equation with periodic potential: Coding and method for computation
1. | Moscow Institute of Electronic Engineering, Zelenograd, Moscow, 124498, Russia |
2. | Centro de Física Teórica e Computacional and Departamento de Física, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, Ed. C8, Lisboa P-1749-016, Portugal |
The paper is devoted to nonlinear localized modes (“gap solitons”) for the spatially one-dimensional Gross-Pitaevskii equation (1D GPE) with a periodic potential and repulsive interparticle interactions. It has been recently shown (G. L. Alfimov, A. I. Avramenko, Physica D, 254, 29 (2013)) that under certain conditions all the stationary modes for the 1D GPE can be coded by bi-infinite sequences of symbols of some finite alphabet (called “codes” of the solutions). We present and justify a numerical method which allows to reconstruct the profile of a localized mode by its code. As an example, the method is applied to compute the profiles of gap solitons for 1D GPE with a cosine potential.
References:
[1] |
T. J. Alexander, E. A. Ostrovskaya and Yu. S. Kivshar, Self-Trapped Nonlinear Matter Waves in Periodic Potentials Phys. Rev. Lett. 96 (2006), 040401.
doi: 10.1103/PhysRevLett.96.040401. |
[2] |
G. L. Alfimov, V. V. Konotop and M. Salerno,
Matter solitons in Bose-Einstein Condensates with optical lattices, Europhys. Lett., 58 (2002), 7-13.
doi: 10.1209/epl/i2002-00599-0. |
[3] |
G. L. Alfimov and A. I. Avramenko,
Coding of nonlinear states for the Gross-Pitaevskii equation with periodic potential, Physica D, 254 (2013), 29-45.
doi: 10.1016/j.physd.2013.03.009. |
[4] |
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. |
[5] |
G. L. Alfimov, P. G. Kevrekidis, V. V. Konotop and M. Salerno, Wannier functions analysis of the nonlinear Schr′odinger equation with a periodic potential Phys. Rev. E 66 (2002), 046608, 6pp.
doi: 10.1103/PhysRevE.66.046608. |
[6] |
G. L. Alfimov and P. P. Kizin,
On initial data for Cauchy problem for equation $u_{xx}+Q(x)u-P(u)=0$ having no singularities on a given interval,
Ufa Mathematical Journal 2016, accepted. |
[7] |
G. L. Alfimov and M. E. Lebedev,
On regular and singular solutions for equation $u_{xx}+Q(x)u+P(x)u^3=0$, Ufa Mathematical Journal, 7 (2015), 3-16.
doi: 10.13108/2015-7-2-3. |
[8] |
V. I. Arnold,
Mathematical Methods of Classical Mechanics Springer-Verlag, 1989. |
[9] |
I. V. Barashenkov, D. E. Pelinovsky and E. V. Zemlyanaya,
Vibrations and Oscillatory Instabilities of Gap Solitons, Phys. Rev. Lett., 80 (1998), 5117-5120.
doi: 10.1103/PhysRevLett.80.5117. |
[10] |
I. V. Barashenkov and E. V. Zemlyanaya,
Oscillatory instabilities of gap solitons: A numerical study, Comp. Phys. Comm., 126 (2000), 22-27.
doi: 10.1016/S0010-4655(99)00241-6. |
[11] |
F. A. Berezin and M. A. Shubin,
The Shrödinder Equation Kluwer, Dordrech, 1991. |
[12] |
R. Fukuizumi and A. Sacchetti,
Stationary States for Nonlinear Schrödinger Equations with Periodic Potentials, J. Stat. Phys., 156 (2014), 707-738.
doi: 10.1007/s10955-014-1023-x. |
[13] |
P. G. Kevrekidis, B. A. Malomed, D. J. Frantzeskakis, A. R. Bishop, H. Nistazakis and R. Carretero-González,
Domain walls of single-component Bose-Einstein condensates in external potentials, Mathematics and Computers in Simulation, 69 (2005), 334-345.
doi: 10.1016/j.matcom.2005.01.016. |
[14] |
P. P. Kizin, D. A. Zezyulin and G. L. Alfimov,
Oscillatory instabilities of gap solitons in a repulsive Bose-Einstein condensate, Physica D, 337 (2016), 58-66.
doi: 10.1016/j.physd.2016.07.007. |
[15] |
V. V. Konotop and M. Salerno, Modulational instability in cigar-shaped Bose-Einstein condensates in optical lattices Phys. Rev. A 65 (2002), 021602.
doi: 10.1103/PhysRevA.65.021602. |
[16] |
P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage and Yu. S. Kivshar, Bose-Einstein condensates in optical lattices: Band-gap structure and solitons Phys. Rev. A 67 (2003), 013602.
doi: 10.1103/PhysRevA.67.013602. |
[17] |
B. A. Malomed and R. S. Tasgal, Vibration modes of a gap soliton in a nonlinear optical medium Phys. Rev. E 49 (1994), 5787.
doi: 10.1103/PhysRevE.49.5787. |
[18] |
T. Mayteevarunyoo and B. A. Malomed, Stability limits for gap solitons in a Bose-Einstein condensate trapped in a time-modulated optical lattice Phys. Rev. A 74} (2006), 033616.
doi: 10.1103/PhysRevA.74.033616. |
[19] |
D. E. Pelinovsky, A. A. Sukhorukov and Yu. S. Kivshar, Bifurcations and stability of gap solitons in periodic potentials Phys. Rev. E 70 (2004), 036618, 17pp.
doi: 10.1103/PhysRevE.70.036618. |
[20] |
L. Pitaevskii and S. Stringari,
Bose-Einstein Condensation, Clarendon Press, Oxford, (2003).
|
[21] |
B. Wu and Q. Niu, Landau and dynamical instabilities of the superflow of Bose-Einstein Condensates in optical lattices Phys. Rev. A 64 (2001), 061603(R).
doi: 10.1103/PhysRevA.64.061603. |
[22] |
J. Yang,
Nonlinear Waves in Integrable and Nonintegrable Systems SIAM, Philadelphia, 2010.
doi: 10.1137/1.9780898719680. |
[23] |
Yo. Zhang, Zh. Liang and B. Wu, Gap solitons and Bloch waves in nonlinear periodic systems Phys. Rev. A 80 (2009), 063815.
doi: 10.1103/PhysRevA.80.063815. |
[24] |
Yo. Zhang and B. Wu, Composition relation between gap solitons and bloch waves in nonlinear periodic systems Phys. Rev. Lett. 102 (2009), 093905.
doi: 10.1103/PhysRevLett.102.093905. |
show all references
References:
[1] |
T. J. Alexander, E. A. Ostrovskaya and Yu. S. Kivshar, Self-Trapped Nonlinear Matter Waves in Periodic Potentials Phys. Rev. Lett. 96 (2006), 040401.
doi: 10.1103/PhysRevLett.96.040401. |
[2] |
G. L. Alfimov, V. V. Konotop and M. Salerno,
Matter solitons in Bose-Einstein Condensates with optical lattices, Europhys. Lett., 58 (2002), 7-13.
doi: 10.1209/epl/i2002-00599-0. |
[3] |
G. L. Alfimov and A. I. Avramenko,
Coding of nonlinear states for the Gross-Pitaevskii equation with periodic potential, Physica D, 254 (2013), 29-45.
doi: 10.1016/j.physd.2013.03.009. |
[4] |
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. |
[5] |
G. L. Alfimov, P. G. Kevrekidis, V. V. Konotop and M. Salerno, Wannier functions analysis of the nonlinear Schr′odinger equation with a periodic potential Phys. Rev. E 66 (2002), 046608, 6pp.
doi: 10.1103/PhysRevE.66.046608. |
[6] |
G. L. Alfimov and P. P. Kizin,
On initial data for Cauchy problem for equation $u_{xx}+Q(x)u-P(u)=0$ having no singularities on a given interval,
Ufa Mathematical Journal 2016, accepted. |
[7] |
G. L. Alfimov and M. E. Lebedev,
On regular and singular solutions for equation $u_{xx}+Q(x)u+P(x)u^3=0$, Ufa Mathematical Journal, 7 (2015), 3-16.
doi: 10.13108/2015-7-2-3. |
[8] |
V. I. Arnold,
Mathematical Methods of Classical Mechanics Springer-Verlag, 1989. |
[9] |
I. V. Barashenkov, D. E. Pelinovsky and E. V. Zemlyanaya,
Vibrations and Oscillatory Instabilities of Gap Solitons, Phys. Rev. Lett., 80 (1998), 5117-5120.
doi: 10.1103/PhysRevLett.80.5117. |
[10] |
I. V. Barashenkov and E. V. Zemlyanaya,
Oscillatory instabilities of gap solitons: A numerical study, Comp. Phys. Comm., 126 (2000), 22-27.
doi: 10.1016/S0010-4655(99)00241-6. |
[11] |
F. A. Berezin and M. A. Shubin,
The Shrödinder Equation Kluwer, Dordrech, 1991. |
[12] |
R. Fukuizumi and A. Sacchetti,
Stationary States for Nonlinear Schrödinger Equations with Periodic Potentials, J. Stat. Phys., 156 (2014), 707-738.
doi: 10.1007/s10955-014-1023-x. |
[13] |
P. G. Kevrekidis, B. A. Malomed, D. J. Frantzeskakis, A. R. Bishop, H. Nistazakis and R. Carretero-González,
Domain walls of single-component Bose-Einstein condensates in external potentials, Mathematics and Computers in Simulation, 69 (2005), 334-345.
doi: 10.1016/j.matcom.2005.01.016. |
[14] |
P. P. Kizin, D. A. Zezyulin and G. L. Alfimov,
Oscillatory instabilities of gap solitons in a repulsive Bose-Einstein condensate, Physica D, 337 (2016), 58-66.
doi: 10.1016/j.physd.2016.07.007. |
[15] |
V. V. Konotop and M. Salerno, Modulational instability in cigar-shaped Bose-Einstein condensates in optical lattices Phys. Rev. A 65 (2002), 021602.
doi: 10.1103/PhysRevA.65.021602. |
[16] |
P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage and Yu. S. Kivshar, Bose-Einstein condensates in optical lattices: Band-gap structure and solitons Phys. Rev. A 67 (2003), 013602.
doi: 10.1103/PhysRevA.67.013602. |
[17] |
B. A. Malomed and R. S. Tasgal, Vibration modes of a gap soliton in a nonlinear optical medium Phys. Rev. E 49 (1994), 5787.
doi: 10.1103/PhysRevE.49.5787. |
[18] |
T. Mayteevarunyoo and B. A. Malomed, Stability limits for gap solitons in a Bose-Einstein condensate trapped in a time-modulated optical lattice Phys. Rev. A 74} (2006), 033616.
doi: 10.1103/PhysRevA.74.033616. |
[19] |
D. E. Pelinovsky, A. A. Sukhorukov and Yu. S. Kivshar, Bifurcations and stability of gap solitons in periodic potentials Phys. Rev. E 70 (2004), 036618, 17pp.
doi: 10.1103/PhysRevE.70.036618. |
[20] |
L. Pitaevskii and S. Stringari,
Bose-Einstein Condensation, Clarendon Press, Oxford, (2003).
|
[21] |
B. Wu and Q. Niu, Landau and dynamical instabilities of the superflow of Bose-Einstein Condensates in optical lattices Phys. Rev. A 64 (2001), 061603(R).
doi: 10.1103/PhysRevA.64.061603. |
[22] |
J. Yang,
Nonlinear Waves in Integrable and Nonintegrable Systems SIAM, Philadelphia, 2010.
doi: 10.1137/1.9780898719680. |
[23] |
Yo. Zhang, Zh. Liang and B. Wu, Gap solitons and Bloch waves in nonlinear periodic systems Phys. Rev. A 80 (2009), 063815.
doi: 10.1103/PhysRevA.80.063815. |
[24] |
Yo. Zhang and B. Wu, Composition relation between gap solitons and bloch waves in nonlinear periodic systems Phys. Rev. Lett. 102 (2009), 093905.
doi: 10.1103/PhysRevLett.102.093905. |










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