Abstract
We consider the effects of anisotropy on two types of localized states with topological charges equal to 1 in two-dimensional nonlinear lattices, using the discrete nonlinear Schrödinger equation as a paradigm model. We find that on-site-centered vortices with different propagation constants are not globally stable, and that upper and lower boundaries of the propagation constant exist. The region between these two boundaries is the domain outside of which the on-site-centered vortices are unstable. This region decreases in size as the anisotropy parameter is gradually increased. We also consider off-site-centered vortices on anisotropic lattices, which are unstable on this lattice type and either transform into stable quadrupoles or collapse. We find that the transformation of off-sitecentered vortices into quadrupoles, which occurs on anisotropic lattices, cannot occur on isotropic lattices. In the quadrupole case, a propagation-constant region also exists, outside of which the localized states cannot stably exist. The influence of anisotropy on this region is almost identical to its effects on the on-site-centered vortex case.
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References
D. N. Christodoulides, F. Lederer, and Y. Silberberg, Discretizing light behaviour in linear and nonlinear waveguide lattices, Nature 424(6950), 817 (2003)
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)
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)
Z. Chen, M. Segev, and D. N. Christodoulides, Optical spatial solitons: Historical overview and recent advances, Prog. Phys. 75(8), 086401 (2012)
C. Lou, L. Tang, D. Song, X. Wang, J. Xu, and Z. Chen, Novel spatial solitons in light-induced photonic bandgap structures, Front. Phys. 3(1), 1 (2008)
Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, Defect-mediated discrete solitons in optically induced photorefractive lattices, Phys. Rev. A 80(4), 043824 (2009)
Y. Li, B. A. Malomed, J. Wu, W. Pang, S. Wang, and J. Zhou, Quasicompactons in inverted nonlinear photonic crystals, Phys. Rev. A 84(4), 043839 (2011)
J. H. Huang, H. J. Li, X. Y. Zhang, and Y. Y. Li, Transmission, reflection, scattering, and trapping of traveling discrete solitons by c and v point defects, Front. Phys. 10(2), 104201 (2015)
Y. Zhang, K. Lu, M. Zhang, K. Li, S. Liu, and Y. Zhang, Dynamics of incoherent photovoltaic spatial solitons, Chin. Phys. Lett. 3, 132 (2009)
B. Lü and Q. Tian, Discrete breathers in a two-dimensional Morse lattice with an on-site harmonic potential, Front. Phys. 4(4), 497 (2009)
W. Pang, J. Wu, Z. Yuan, Y. Liu, and G. Chen, Lattice solitons in optical lattice controlled by electromagnetically induced transparency, J. Phys. Soc. Jpn. 80(11), 113401 (2011)
Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, Four-wave mixing dipole soliton in laser-induced atomic gratings, Phys. Rev. Lett. 106(9), 093904 (2011)
Y. Zhang, Z. Nie, Y. Zhao, C. Li, R. Wang, J. Si, and M. Xiao, Modulated vortex solitons of four-wave mixing, Opt. Express 18(11), 10963 (2010)
Y. Zhang, C. Yuan, Y. Zhang, H. Zheng, H. Chen, C. Li, Z. Wang, and M. Xiao, Surface solitons of four-wave mixing in an electromagnetically induced lattice, Laser Phys. Lett. 10(5), 055406 (2013)
R. Wang, Z. Wu, Y. Zhang, Z. Zhang, C. Yuan, H. Zheng, Y. Li, J. Zhang, and Y. Zhang, Observation of multi-component spatial vector solitons of four-wave mixing, Opt. Express 20(13), 14168 (2012)
Y. Zhang, Z. Wu, C. Yuan, X. Yao, K. Lu, M. Belíc, and Y. Zhang, Optical vortices induced in nonlinear multilevel atomic vapors, Opt. Lett. 37(21), 4507 (2012)
G. Chen, Z. Huang, and Z. Mai, Two-dimensional discrete Anderson location in waveguide matrix, J. Nonlinear Opt. Phys. 23(03), 1450033 (2014)
X. Y. Zhang, J. L. Chai, J. S. Huang, Z. Q. Chen, Y. Y. Li, and B. A. Malomed, Discrete solitons and scattering of lattice waves in guiding arrays with a nonlinear PT-symmetric defect, Opt. Express 22(11), 13927 (2014)
B. A. Malomed and P. G. Kevrekidis, Discrete vortex solitons, Phys. Rev. E 64(2), 026601 (2001)
J. Yang and Z. H. Musslimani, Fundamental and vortex solitons in a two-dimensional optical lattice, Opt. Lett. 28(21), 2094 (2003)
H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices, Phys. Rev. Lett. 92(12), 123902 (2004)
D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, Observation of discrete vortex solitons in optically induced photonic lattices, Phys. Rev. Lett. 92(12), 123903 (2004)
J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Observation of vortex-ring “discrete” solitons in 2D photonic lattices, Phys. Rev. Lett. 92(12), 123904 (2004)
P. G. Kevrekidis, B. A. Malomed, Z. Chen, and D. J. Frantzeskakis, Stable higher-order vortices and quasivortices in the discrete nonlinear Schrödinger equation, Phys. Rev. E 70(5), 056612 (2004)
T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, Asymmetric vortex solitons in nonlinear periodic lattices, Phys. Rev. Lett. 93(6), 063901 (2004)
D. E. Pelinovsky, P. G. Kevrekidis, and D. J. Frantzeskakis, Nonlinear schrödinger lattices (ii): Persistence and stability of discrete vortices, arXiv: nlin/0411016v1 (2004)
Z. Chen, J. Liu, S. Fu, Y. Li, and B. A. Malomed, Discrete solitons and vortices on two-dimensional lattices of PT-symmetric couplers, Opt. Express 22(24), 29679 (2014)
Y. Zhang, M. Belíc, Z. Wu, C. Yuan, R. Wang, K. Lu, and Y. Zhang, Multicharged optical vortices induced in a dissipative atomic vapor system, Phys. Rev. A 88(1), 013847 (2013)
H. Li, X. Zhu, Z. Shi, B. A. Malomed, T. Lai, and C. Lee, Bulk vortices and half-vortex surface modes in parity-timesymmetric media, Phys. Rev. A 89(5), 053811 (2014)
G. Chen, H. Huang, and M. Wu, Solitary vortices in twodimensional waveguide matrix, J. Nonlinear Opt. Phys. 24, 1550012 (2015)
J. Yang, I. Makasyuk, H. Martin, P. G. Kevrekidis, B. A. Malomed, D. J. Frantzeskakis, and Z. Chen, Necklace-like solitons in optically induced photonic lattices, Phys. Rev. Lett. 94(11), 113902 (2005)
P. G. Kevrekidis, B. A. Malomed, D. J. Frantzeskakis, and R. Carretero-González, Three dimensional solitary waves and vortices in a discrete nonlinear Schrödinger lattice, Phys. Rev. Lett. 93(8), 080403 (2004)
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)
T. Mayteevarunyooa, B. A. Malomed, B. B. Baizakov, and M. Salerno, Matter-wave vortices and solitons in anisotropic optical lattices, Physica D 238(15), 1439 (2009)
N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Discrete solitons in photorefractive optically induced photonic lattices, Phys. Rev. E 66(4), 046602 (2002)
A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, Spatial optical solitons in waveguide arrays, IEEE J. Quantum Electron. 39(1), 31 (2003)
J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Observation of discrete solitons in optically induced real time waveguide arrays, Phys. Rev. Lett. 90(2), 023902 (2003)
J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, Dipole solitons in optically induced two-dimensional photonic lattices, Opt. Lett. 29(14), 1662 (2004)
Z. Chen, H. Martin, E. D. Eugenieva, J. Xu, and A. Bezryadina, Anisotropic enhancement of discrete diffraction and formation of two-dimensional discrete-soliton trains, Phys. Rev. Lett. 92(14), 143902 (2004)
Z. Chen, A. Bezryadina, I. Makasyuk, and J. Yang, Observation of two-dimensional lattice vector solitons, Opt. Lett. 29(14), 1656 (2004)
Y. Li, B. A. Malomed, M. Feng, and J. Zhou, Arrayed and checkerboard optical waveguides controlled by the electromagnetically induced transparency, Phys. Rev. A 82(6), 063813 (2010)
J. Wu, M. Feng, W. Pang, S. Fu, and Y. Li, The transmission of quasi-discrete solitons in resonant waveguide arrays activated by the electromagnetically induced transparency, J. Nonlinear Opt. Phys. 20(02), 193 (2011)
P. G. Kevrekidis and D. J. Frantzeskakis, Pattern forming dynamical instabilities of Bose-Einstein condensates, Mod. Phys. Lett. B 18(05n06), 173 (2004)
V. A. Brazhnyi and V. V. Konotop, Theory of nonlinear matter waves in optical lattices, Mod. Phys. Lett. B 18(14), 627 (2004)
P. G. Kevrekidis, R. Carretero-González, D. J. Frantzeskakis, and I. G. Kevrekidis, Vortices in Bose-Einstein condensates: Some recent developments, Mod. Phys. Lett. B 18(30), 1481 (2004)
M. I. Weinstein, Excitation thresholds for nonlinear localized modes on lattices, Nonlinearity 12(3), 673 (1999)
P. G. Kevrekidis, K. O. Rasmussen, and A. R. Bishop, Twodimensional discrete breathers: Construction, stability, and bifurcations, Phys. Rev. E 61(2), 2006 (2000)
P. G. Kevrekidis, K. O. Rasmussen, and A. R. Bishop, Localized excitations and their thresholds, Phys. Rev. E 61(4), 4652 (2000)
P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations and Physical Perspectives, Springer, 2009
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Chen, GH., Wang, HC. & Chen, ZF. Discrete vortices on anisotropic lattices. Front. Phys. 10, 1–6 (2015). https://doi.org/10.1007/s11467-015-0494-9
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DOI: https://doi.org/10.1007/s11467-015-0494-9