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Stojanović Krasić M, Stojanović M, Maluckov A, Maczewsky LJ, Szameit A, Stepić M. Localized modes in a two-dimensional lattice with a pluslike geometry. Phys Rev E 2020; 102:032207. [PMID: 33075910 DOI: 10.1103/physreve.102.032207] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/28/2019] [Accepted: 08/10/2020] [Indexed: 11/07/2022]
Abstract
We investigate analytically and numerically the existence and dynamical stability of different localized modes in a two-dimensional photonic lattice comprising a square plaquette inscribed in the dodecagon lattices. The eigenvalue spectrum of the underlying linear lattice is characterized by a net formed of one flat band and four dispersive bands. By tailoring the intersite coupling coefficient ratio, opening of gaps between two pairs of neighboring dispersive bands can be induced, while the fully degenerate flat band characterized by compact eigenmodes stays nested between two inner dispersive bands. The nonlinearity destabilizes the compact modes and gives rise to unique families of localized modes in the newly opened gaps, as well as in the semi-infinite gaps. The governing mechanism of mode localization in that case is the light energy self-trapping effect. We have shown the stability of a few families of nonlinear modes in gaps. The suggested lattice model may serve for probing various artificial flat-band systems such as ultracold atoms in optical lattices, periodic electronic networks, and polariton condensates.
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Affiliation(s)
| | - Mirjana Stojanović
- Vinča Institute of Nuclear Sciences, University of Belgrade, P.O. Box 522, 11000 Belgrade, Serbia
| | - Aleksandra Maluckov
- Vinča Institute of Nuclear Sciences, University of Belgrade, P.O. Box 522, 11000 Belgrade, Serbia
| | | | | | - Milutin Stepić
- Vinča Institute of Nuclear Sciences, University of Belgrade, P.O. Box 522, 11000 Belgrade, Serbia
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Kevrekidis PG, Malomed BA, Saxena A, Bishop AR, Frantzeskakis DJ. Solitons and vortices in two-dimensional discrete nonlinear Schrödinger systems with spatially modulated nonlinearity. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:043201. [PMID: 25974604 DOI: 10.1103/physreve.91.043201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/15/2014] [Indexed: 06/04/2023]
Abstract
We consider a two-dimensional (2D) generalization of a recently proposed model [Gligorić et al., Phys. Rev. E 88, 032905 (2013)], which gives rise to bright discrete solitons supported by the defocusing nonlinearity whose local strength grows from the center to the periphery. We explore the 2D model starting from the anticontinuum (AC) limit of vanishing coupling. In this limit, we can construct a wide variety of solutions including not only single-site excitations, but also dipole and quadrupole ones. Additionally, two separate families of solutions are explored: the usual "extended" unstaggered bright solitons, in which all sites are excited in the AC limit, with the same sign across the lattice (they represent the most robust states supported by the lattice, their 1D counterparts being those considered as 1D bright solitons in the above-mentioned work), and the vortex cross, which is specific to the 2D setting. For all the existing states, we explore their stability (also analytically, when possible). Typical scenarios of instability development are exhibited through direct simulations.
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Affiliation(s)
- P G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA and Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
| | - Avadh Saxena
- Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| | - A R Bishop
- Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| | - D J Frantzeskakis
- Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece
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Gligorić G, Maluckov A, Hadžievski L, Malomed BA. Localized modes in mini-gaps opened by periodically modulated intersite coupling in two-dimensional nonlinear lattices. CHAOS (WOODBURY, N.Y.) 2014; 24:023124. [PMID: 24985438 DOI: 10.1063/1.4881678] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
Spatially periodic modulation of the intersite coupling in two-dimensional (2D) nonlinear lattices modifies the eigenvalue spectrum by opening mini-gaps in it. This work aims to build stable localized modes in the new bandgaps. Numerical analysis shows that single-peak and composite two- and four-peak discrete static solitons and breathers emerge as such modes in certain parameter areas inside the mini-gaps of the 2D superlattice induced by the periodic modulation of the intersite coupling along both directions. The single-peak solitons and four-peak discrete solitons are stable in a part of their existence domain, while unstable stationary states (in particular, two-soliton complexes) may readily transform into robust localized breathers.
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Affiliation(s)
- Goran Gligorić
- P* group, Vinča Institute of Nuclear Sciences, University of Belgrade, P.O.B. 522, 11001 Belgrade, Serbia
| | - Aleksandra Maluckov
- P* group, Vinča Institute of Nuclear Sciences, University of Belgrade, P.O.B. 522, 11001 Belgrade, Serbia
| | - Ljupčo Hadžievski
- P* group, Vinča Institute of Nuclear Sciences, University of Belgrade, P.O.B. 522, 11001 Belgrade, Serbia
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
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Wang C, Theocharis G, Kevrekidis PG, Whitaker N, Law KJH, Frantzeskakis DJ, Malomed BA. Two-dimensional paradigm for symmetry breaking: the nonlinear Schrödinger equation with a four-well potential. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:046611. [PMID: 19905475 DOI: 10.1103/physreve.80.046611] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/02/2009] [Revised: 08/27/2009] [Indexed: 05/28/2023]
Abstract
We study the existence and stability of localized modes in the two-dimensional (2D) nonlinear Schrödinger/Gross-Pitaevskii (NLS/GP) equation with a symmetric four-well potential. Using the corresponding four-mode approximation, we trace the parametric evolution of the trapped stationary modes, starting from the linear limit, and thus derive a complete bifurcation diagram for families of the stationary modes. This provides the picture of spontaneous symmetry breaking in the fundamental 2D setting. In a broad parameter region, the predictions based on the four-mode decomposition are found to be in good agreement with full numerical solutions of the NLS/GP equation. Stability properties of the stationary states coincide with those suggested by the corresponding discrete model in the large-amplitude limit. The dynamics of unstable modes is explored by means of direct simulations. Finally, in addition to the full analysis for the case of the self-attractive nonlinearity, the bifurcation diagram for the case of self-repulsion is briefly considered too.
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Affiliation(s)
- C Wang
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
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Zhang Y, Wu B. Composition relation between gap solitons and Bloch waves in nonlinear periodic systems. PHYSICAL REVIEW LETTERS 2009; 102:093905. [PMID: 19392522 DOI: 10.1103/physrevlett.102.093905] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/27/2008] [Indexed: 05/27/2023]
Abstract
We show with numerical computation and analysis that Bloch waves, at either the center or edge of the Brillouin zone, of a one dimensional nonlinear periodic system can be regarded as infinite chains composed of fundamental gap solitons (FGSs). This composition relation between Bloch waves and FGSs leads us to predict that there are n families of FGSs in the nth band gap of the corresponding linear periodic system, which is confirmed numerically. Furthermore, this composition relation can be extended to construct a class of solutions similar to Bloch waves but with multiple periods.
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Affiliation(s)
- Yongping Zhang
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
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Cuevas J, Malomed BA, Kevrekidis PG. Two-dimensional discrete solitons in rotating lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:046608. [PMID: 17995128 DOI: 10.1103/physreve.76.046608] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/15/2007] [Indexed: 05/25/2023]
Abstract
We introduce a two-dimensional discrete nonlinear Schrödinger (DNLS) equation with self-attractive cubic nonlinearity in a rotating reference frame. The model applies to a Bose-Einstein condensate stirred by a rotating strong optical lattice, or light propagation in a twisted bundle of nonlinear fibers. Two types of localized states are constructed: off-axis fundamental solitons (FSs), placed at distance R from the rotation pivot, and on-axis (R=0) vortex solitons (VSs), with vorticities S=1 and 2 . At a fixed value of rotation frequency Omega , a stability interval for the FSs is found in terms of the lattice coupling constant C , 0<C<C_{cr}(R) , with monotonically decreasing C_{cr}(R) . VSs with S=1 have a stability interval, C[over ]_{cr};{(S=1)}(Omega)<C<C_{cr};{(S=1)}(Omega) , which exists for Omega below a certain critical value, Omega_{cr};{(S=1)} . This implies that the VSs with S=1 are destabilized in the weak-coupling limit by the rotation. On the contrary, VSs with S=2 , that are known to be unstable in the standard DNLS equation, with Omega=0 , are stabilized by the rotation in region 0<C<C_{cr};{(S=2)} , with C_{cr};{(S=2)} growing as a function of Omega . Quadrupole and octupole on-axis solitons are considered too, their stability regions being weakly affected by Omega not equal 0 .
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Affiliation(s)
- Jesús Cuevas
- Departamento de Física Aplicada I, Escuela Universitaria Politécnica, C/ Virgen de Africa, 7, 41011 Sevilla, Spain
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Smirnov E, Rüter CE, Kip D, Kartashov YV, Torner L. Observation of higher-order solitons in defocusing waveguide arrays. OPTICS LETTERS 2007; 32:1950-2. [PMID: 17603624 DOI: 10.1364/ol.32.001950] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/16/2023]
Abstract
We observe experimentally higher-order solitons in waveguide arrays with defocusing saturable nonlinearity. Such solitons can comprise several in-phase bright spots and are stable above a critical power threshold. We elucidate the impact of the nonlinearity saturation on the domains of existence and stability of the observed complex soliton states.
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Affiliation(s)
- Eugene Smirnov
- Institute of Physics and Physical Technologies, Clausthal University of Technology, 38678 Clausthal-Zellerfeld, Germany.
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