Self-trapped spatially localized states in combined linear-nonlinear periodic potentials
Jin-Cheng Shi, Jian-Hua Zeng
Self-trapped spatially localized states in combined linear-nonlinear periodic potentials
We analyze the existence and stability of two kinds of self-trapped spatially localized gap modes, gap solitons and truncated nonlinear Bloch waves, in one- and two-dimensional optical or matter-wave media with self-focusing nonlinearity, supported by a combination of linear and nonlinear periodic lattice potentials. The former is found to be stable once placed inside a single well of the nonlinear lattice, it is unstable otherwise. Contrary to the case with constant self-focusing nonlinearity, where the latter solution is always unstable, here, we demonstrate that it nevertheless can be stabilized by the nonlinear lattice since the model under consideration combines the unique properties of both the linear and nonlinear lattices. The practical possibilities for experimental realization of the predicted solutions are also discussed.
gap solitons and gap waves / Bose–Einstein condensates / linear and nonlinear periodic potentials
[1] |
D. E. Pelinovsky, Localization in Periodic Potential: From Schrödinger Operators to the Gross–Pitaevskii Equation, Cambridge: Cambridge University Press, 2011
|
[2] |
Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic, San Diego, CA, 2003
CrossRef
ADS
Google scholar
|
[3] |
V. V. Konotop, J. Yang, and D. A. Zezyulin, Nonlinear waves in PT-symmetric systems, Rev. Mod. Phys. 88(3), 035002 (2016)
CrossRef
ADS
Google scholar
|
[4] |
C. Kittel, Introduction to Solid State Physics, 8th Ed., New York: John Wiley & Sons, 2005
|
[5] |
J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, Princeton: Princeton University Press, 2008
|
[6] |
C. Markos, J. C. Travers, A. Abdolvand, B. J. Eggleton, and O. Bang, Hybrid photonic-crystal fiber, Rev. Mod. Phys. 89(4), 045003 (2017)
CrossRef
ADS
Google scholar
|
[7] |
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(6928), 147 (2003)
CrossRef
ADS
Google scholar
|
[8] |
F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Discrete solitons in optics, Phys. Rep. 463(1–3), 1 (2008)
CrossRef
ADS
Google scholar
|
[9] |
I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Light propagation and localization in modulated photonic lattices and waveguides, Phys. Rep. 518(1–2), 1 (2012)
CrossRef
ADS
Google scholar
|
[10] |
E. A. Ostrovskaya and Y. S. Kivshar, Matter-wave gap vortices in optical lattices, Phys. Rev. Lett. 93(16), 160405 (2004)
CrossRef
ADS
Google scholar
|
[11] |
J. Shi and J. Zeng, Asymmetric localized states in periodic potentials with a domain-wall-like Kerr nonlinearity, J. Phys. Commun. 3(3), 035003 (2019)
CrossRef
ADS
Google scholar
|
[12] |
L. Zeng and J. Zeng, Gap-type dark localized modes in a Bose–Einstein condensate with optical lattices, Adv. Photon. 1(04), 046004 (2019)
CrossRef
ADS
Google scholar
|
[13] |
F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Discrete solitons in optics, Phys. Rep. 463(1–3), 1 (2008)
CrossRef
ADS
Google scholar
|
[14] |
Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Soliton shape and mobility control in optical lattices, Prog. Opt. 52, 63 (2009)
CrossRef
ADS
Google scholar
|
[15] |
P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-González (Eds.), Emergent Nonlinear Phenomena in Bose–Einstein Condensates, Berlin: Springer, 2008
CrossRef
ADS
Google scholar
|
[16] |
Z. Chen, M. Segev, and D. N. Christodoulides, Optical spatial solitons: Historical overview and recent advances, Rep. Prog. Phys. 75(8), 086401 (2012)
CrossRef
ADS
Google scholar
|
[17] |
B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Bragg grating solitons, Phys. Rev. Lett. 76(10), 1627 (1996)
CrossRef
ADS
Google scholar
|
[18] |
A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and L. Torner, Observation of two-dimensional surface solitons in asymmetric waveguid arrays, Phys. Rev. Lett. 98(17), 173903 (2007)
CrossRef
ADS
Google scholar
|
[19] |
Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Surface gap solitons, Phys. Rev. Lett. 96(7), 073901 (2006)
CrossRef
ADS
Google scholar
|
[20] |
D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Gap solitons in waveguide arrays, Phys. Rev. Lett. 92(9), 093904 (2004)
CrossRef
ADS
Google scholar
|
[21] |
O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Conical diffraction and gap solitons in honeycomb photonic lattices, Phys. Rev. Lett. 98(10), 103901 (2007)
CrossRef
ADS
Google scholar
|
[22] |
A. S. Desyatnikov, E. A. Ostrovskaya, Y. S. Kivshar, and C. Denz, Composite band-gap solitons in nonlinear optically induced lattices, Phys. Rev. Lett. 91(15), 153902 (2003)
CrossRef
ADS
Google scholar
|
[23] |
E. A. Ostrovskaya, J. Abdullaev, M. D. Fraser, A. S. Desyatnikov, and Y. S. Kivshar, Self-localization of polariton condensates in periodic potentials, Phys. Rev. Lett. 110(17), 170407 (2013)
CrossRef
ADS
Google scholar
|
[24] |
E. A. Cerda-Méndez, D. Sarkar, D. N. Krizhanovskii, S. S. Gavrilov, K. Biermann, M. S. Skolnick, and P. V. Santos, Exciton–polariton gap solitons in two-dimensional lattices, Phys. Rev. Lett. 111(14), 146401 (2013)
CrossRef
ADS
Google scholar
|
[25] |
D. Tanese, H. Flayac, D. Solnyshkov, A. Amo, A. Lemaître, E. Galopin, R. Braive, P. Senellart, I. Sagnes, G. Malpuech, and J. Bloch, Polariton condensation in solitonic gap states in a one-dimensional periodic potential, Nat. Commun. 4(1), 1749 (2013)
CrossRef
ADS
Google scholar
|
[26] |
B. Eiermann, Th. Anker, M. Albiez, M. Taglieber, P. Treutlein, K. P. Marzlin, and M. K. Oberthaler, Bright Bose–Einstein gap solitons of atoms with repulsive interaction, Phys. Rev. Lett. 92(23), 230401 (2004)
CrossRef
ADS
Google scholar
|
[27] |
Y. Sivan, G. Fibich, and M. I. Weinstein, Waves in nonlinear lattices: Ultrashort optical pulses and Bose–Einstein condensates, Phys. Rev. Lett. 97, 193902 (2006)
CrossRef
ADS
Google scholar
|
[28] |
J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities, Phys. Rev. Lett. 98(6), 064102 (2007)
CrossRef
ADS
Google scholar
|
[29] |
J. Zeng and B. A. Malomed, Stabilization of onedimensional solitons against the critical collapse by quintic nonlinear lattices, Phys. Rev. A 85(2), 023824 (2012)
CrossRef
ADS
Google scholar
|
[30] |
X. Gao and J. Zeng, Two-dimensional matter-wave solitons and vortices in competing cubic-quintic nonlinear lattices, Front. Phys. 13(1), 130501 (2018)
CrossRef
ADS
Google scholar
|
[31] |
J. Shi, J. Zeng, and B. A. Malomed, Suppression of the critical collapse for one-dimensional solitons by saturable quintic nonlinear lattices, Chaos 28(7), 075501 (2018)
CrossRef
ADS
Google scholar
|
[32] |
L. Zeng and J. Zeng, One-dimensional solitons in fractional Schrödinger equation with a spatially periodical modulated nonlinearity: Nonlinear lattice, Opt. Lett. 44(11), 2661 (2019)
CrossRef
ADS
Google scholar
|
[33] |
Y. V. Kartashov, B. A. Malomed, and L. Torner, Solitons in nonlinear lattices, Rev. Mod. Phys. 83(1), 247 (2011)
CrossRef
ADS
Google scholar
|
[34] |
R. Carretero-González, D. J. Frantzeskakis, and P. G. Kevrekidis, Nonlinear waves in Bose–Einstein condensates: Physical relevance and mathematical techniques, Nonlinearity 21(7), R139 (2008)
CrossRef
ADS
Google scholar
|
[35] |
D. L. Machacek, E. A. Foreman, Q. E. Hoq, P. G. Kevrekidis, A. Saxena, D. J. Frantzeskakis, and A. R. Bishop, Statics and dynamics of an inhomogeneously nonlinear lattice, Phys. Rev. E 74(3), 036602 (2006)
CrossRef
ADS
Google scholar
|
[36] |
R. Carretero-González, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, Three-dimensional nonlinear lattices: From oblique vortices and octupoles to discrete diamonds and vortex cubes, Phys. Rev. Lett. 94(20), 203901 (2005)
CrossRef
ADS
Google scholar
|
[37] |
V. Achilleos, G. Theocharis, P. G. Kevrekidis, N. I. Karachalios, F. K. Diakonos, and D. J. Frantzeskakis, Stationary states of a nonlinear Schrödinger lattice with a harmonic trap, J. Math. Phys. 52(9), 092701 (2011)
CrossRef
ADS
Google scholar
|
[38] |
P. G. Kevrekidis, D. J. Frantzeskakis, R. Carretero-González, B. A. Malomed, and A. R. Bishop, Discrete solitons and vortices on anisotropic lattices, Phys. Rev. E 72(4), 046613 (2005)
CrossRef
ADS
Google scholar
|
[39] |
L. Salasnich, A. Cetoli, B. A. Malomed, F. Toigo, and L. Reatto, Bose–Einstein condensates under a spatially modulated transverse confinement, Phys. Rev. A 76(1), 013623 (2007)
CrossRef
ADS
Google scholar
|
[40] |
Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity, Opt. Lett. 33(15), 1747 (2008)
CrossRef
ADS
Google scholar
|
[41] |
Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Powerdependent shaping of vortex solitons in optical lattices with spatially modulated nonlinear refractive index, Opt. Lett. 33(19), 2173 (2008)
CrossRef
ADS
Google scholar
|
[42] |
H. Sakaguchi and B. A. Malomed, Solitons in combined linear and nonlinear lattice potentials, Phys. Rev. A 81(1), 013624 (2010)
CrossRef
ADS
Google scholar
|
[43] |
J. Zeng and B. A. Malomed, Two-dimensional solitons and vortices in media with incommensurate linear and nonlinear lattice potentials, Phys. Scr. T 149, 014035 (2012)
CrossRef
ADS
Google scholar
|
[44] |
Y. V. Bludov and V. V. Konotop, Localized modes in arrays of boson–fermion mixtures, Phys. Rev. A 74(4), 043616 (2006)
CrossRef
ADS
Google scholar
|
[45] |
Y. V. Bludov, V. A. Brazhnyi, and V. V. Konotop, Delocalizing transition in one-dimensional condensates in optical lattices due to inhomogeneous interactions, Phys. Rev. A 76(2), 023603 (2007)
CrossRef
ADS
Google scholar
|
[46] |
Z. Rapti, P. G. Kevrekidis, V. V. Konotop, and C. K. R. T. Jones, Solitary waves under the competition of linear and nonlinear periodic potentials, J. Phys. A Math. Theor. 40(47), 14151 (2007)
CrossRef
ADS
Google scholar
|
[47] |
J. Belmonte-Beitia, V. V. Konotop, V. M. Perez-García, and V. E. Vekslerchik, Localized and periodic exact solutions to the nonlinear Schrödinger equation with spatially modulated parameters: Linear and nonlinear lattices, Chaos Solitons Fractals 41(3), 1158 (2009)
CrossRef
ADS
Google scholar
|
[48] |
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(2), 020403 (2005)
CrossRef
ADS
Google scholar
|
[49] |
T. J. Alexander, E. A. Ostrovskaya, and Y. S. Kivshar, Self-trapped nonlinear matter waves in periodic potentials, Phys. Rev. Lett. 96(4), 040401 (2006)
CrossRef
ADS
Google scholar
|
[50] |
Y. Zhang and B. Wu, Composition relation between gap solitons and bloch waves in nonlinear periodic systems, Phys. Rev. Lett. 102(9), 093905 (2009)
CrossRef
ADS
Google scholar
|
[51] |
F. H. Bennet, T. J. Alexander, F. Haslinger, A. Mitchell, D. N. Neshev, and Y. S. Kivshar, Observation of nonlinear self-trapping of broad beams in defocusing waveguide arrays, Phys. Rev. Lett. 106(9), 093901 (2011)
CrossRef
ADS
Google scholar
|
[52] |
C. Bersch, G. Onishchukov, and U. Peschel, Optical gap solitons and truncated nonlinear Bloch waves in temporal lattices, Phys. Rev. Lett. 109(9), 093903 (2012)
CrossRef
ADS
Google scholar
|
[53] |
J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, Truncated-Bloch-wave solitons in optical lattices, Phys. Rev. A 79(4), 043610 (2009)
CrossRef
ADS
Google scholar
|
[54] |
Z. Shi, J. Wang, Z. Chen, and J. Yang, Linear instability of two-dimensional low-amplitude gap solitons near band edges in periodic media, Phys. Rev. A 78(6), 063812 (2008)
CrossRef
ADS
Google scholar
|
[55] |
J. Zeng and B. A. Malomed, Two-dimensional intra-band solitons in lattice potentials with local defects and selffocusing nonlinearity, J. Opt. Soc. Am. B 30(7), 1786 (2013)
CrossRef
ADS
Google scholar
|
[56] |
N. Dror and B. A. Malomed, Stability of two-dimensional gap solitons in periodic potentials: Beyond the fundamental modes, Phys. Rev. E 87(6), 063203 (2013)
CrossRef
ADS
Google scholar
|
[57] |
H. Sakaguchi and B. A. Malomed, Solitons in combined linear and nonlinear lattice potentials, Phys. Rev. A 81(1), 013624 (2010)
CrossRef
ADS
Google scholar
|
[58] |
P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage, and Y. S. Kivshar, Bose–Einstein condensates in optical lattices: Bandgap structure and solitons, Phys. Rev. A 67(1), 013602 (2003)
CrossRef
ADS
Google scholar
|
[59] |
N. K. Efremidis and D. N. Christodoulides, Lattice solitons in Bose–Einstein condensates, Phys. Rev. A 67(6), 063608 (2003)
CrossRef
ADS
Google scholar
|
[60] |
Z. Shi and J. Yang, Solitary waves bifurcated from Blochband edges in two-dimensional periodic media, Phys. Rev. E 75(5), 056602 (2007)
CrossRef
ADS
Google scholar
|
[61] |
E. A. Ostrovskaya and Y. S. Kivshar, Matter-wave gap solitons in atomic band-gap structures, Phys. Rev. Lett. 90(16), 160407 (2003)
CrossRef
ADS
Google scholar
|
[62] |
T. Mayteevarunyoo, B. A. Malomed, B. B. Baizakov, and M. Salerno, Matter-wave vortices and solitons in anisotropic optical lattices, Physica D 238(15), 1439 (2009)
CrossRef
ADS
Google scholar
|
[63] |
M. Vakhitov, and A. Kolokolov, Stationary solutions of the wave equation in a medium with nonlinearity saturation, Radiophys. Quantum Electron. 16(7), 783 (1973)
CrossRef
ADS
Google scholar
|
[64] |
V. A. Brazhnyi and V. V. Konotop, Theory of nonlinear matter waves in optical lattice, Mod. Phys. Lett. B 18(14), 627 (2004)
CrossRef
ADS
Google scholar
|
[65] |
O. Morsch and M. Oberthaler, Dynamics of Bose– Einstein condensates in optical lattices, Rev. Mod. Phys. 78(1), 179 (2006)
CrossRef
ADS
Google scholar
|
[66] |
R. Yamazaki, S. Taie, S. Sugawa, and Y. Takahashi, Submicron spatial modulation of an interatomic interaction in a Bose–Einstein condensate, Phys. Rev. Lett. 105(5), 050405 (2010)
CrossRef
ADS
Google scholar
|
[67] |
M. Yan, B. J. DeSalvo, B. Ramachandhran, H. Pu, and T. C. Killian, Controlling condensate collapse and expansion with an optical Feshbach resonance, Phys. Rev. Lett. 110(12), 123201 (2013)
CrossRef
ADS
Google scholar
|
[68] |
L. W. Clark, L. C. Ha, C. Y. Xu, and C. Chin, Quantum dynamics with spatiotemporal control of interactions in a stable Bose–Einstein condensate, Phys. Rev. Lett. 115(15), 155301 (2015)
CrossRef
ADS
Google scholar
|
[69] |
S. Ghanbari, T. D. Kieu, A. Sidorov, and P. Hannaford, Permanent magnetic lattices for ultracold atoms and quantum degenerate gases, J. Phys. At. Mol. Opt. Phys. 39(4), 847 (2006)
CrossRef
ADS
Google scholar
|
[70] |
O. Romero-Isart, C. Navau, A. Sanchez, P. Zoller, and J. I. Cirac, Superconducting vortex lattices for ultracold atoms, Phys. Rev. Lett. 111(14), 145304 (2013)
CrossRef
ADS
Google scholar
|
[71] |
C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Rev. Mod. Phys. 82(2), 1225 (2010)
CrossRef
ADS
Google scholar
|
[72] |
P. G. Kevrekidis, G. Theocharis, D. J. Frantzeskakis, and B. A. Malomed, Feshbach resonance management for Bose–Einstein condensates, Phys. Rev. Lett. 90(23), 230401 (2003)
CrossRef
ADS
Google scholar
|
[73] |
I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80(3), 885 (2008)
CrossRef
ADS
Google scholar
|
[74] |
Y. Li, Z. Fan, Z. Luo, Y. Liu, H. He, J. Lü, J. Xie, C. Huang, and H. Tan, Cross-symmetry breaking of twocomponent discrete dipolar matter-wave solitons, Front. Phys. 12(5), 124206 (2017)
CrossRef
ADS
Google scholar
|
[75] |
R. Zhong, Z. Chen, C. Huang, Z. Luo, H. Tan, B. A. Malomed, and Y. Li, Self-trapping under two-dimensional spin–orbit coupling and spatially growing repulsive nonlinearity, Front. Phys. 13(4), 130311 (2018)
CrossRef
ADS
Google scholar
|
/
〈 | 〉 |