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Dynamics in \(\mathcal{PT}\)-symmetric honeycomb lattices with nonlinearity. (English) Zbl 1333.35248

The authors consider nontrivial solutions of the nonlinear Schrödinger equation represented in the form of the Gross-Pitaevskii equation defined on a honeycomb lattice (“optical graphene”). The lattice structure allows the consideration within a Floquet-Bloch 2D spectral decomposition supplied with the consideration of effects of a combined spatial inversion (\(P\)) and a tine reversal (\(T\)). This construct has resulted in the revealing of special dispersive nonlinear wave solutions, e.g., semilocalized gap solutions, a localizing effect due to \(PT\)-perturbations, etc.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q82 PDEs in connection with statistical mechanics
81V80 Quantum optics
82D80 Statistical mechanics of nanostructures and nanoparticles
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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[1] Peleg, Conical diffraction and gap solitons in honeycomb photonic lattices, Phys. Rev. Lett. 98 pp 103901– (2007) · doi:10.1103/PhysRevLett.98.103901
[2] Rechtsman, Strain-induced pseudomagnetic field and Landau levels in photonic structures, Nat. Photon 7 pp 153– (2013) · doi:10.1038/nphoton.2012.302
[3] Rechtsman, Topological creation and destruction of edge states in photonic graphene, Phys. Rev. Lett. 111 pp 103901– (2013) · doi:10.1103/PhysRevLett.111.103901
[4] Rechtsman, Photonic Floquet topological insulators, Nature 496 pp 196– (2013) · doi:10.1038/nature12066
[5] Plotnik, Observation of unconventional edge states in ’photonic graphene, Nat. Mat. 13 pp 57– (2013) · doi:10.1038/nmat3783
[6] Zanderbergen, Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene, Phys. Rev. Lett. 104 pp 043903– (2010) · doi:10.1103/PhysRevLett.104.043903
[7] Bittner, Observation of a dirac point in microwave experiments with a photonic cyrstal modeling graphene, Phys. Rev. B 82 pp 014301– (2010) · doi:10.1103/PhysRevB.82.014301
[8] Kuhl, Dirac point and edge states in a microwave realization of tight-binding graphene-like structures, Phys. Rev. B 82 pp 094308– (2010) · doi:10.1103/PhysRevB.82.094308
[9] Bellec, Tight-binding couplings in microwave artificial graphene, Phys. Rev. B 88 pp 115437– (2013) · doi:10.1103/PhysRevB.88.115437
[10] Bellec, Topological transition of Dirac points in a microwave experiment, Phys. Rev. Lett. 110 pp 033902– (2013) · doi:10.1103/PhysRevLett.110.033902
[11] Ablowitz, Evolution of Bloch-mode envelopes in two-dimensional generalized honeycomb lattices, Phys. Rev. A 82 pp 013840– (2010) · doi:10.1103/PhysRevA.82.013840
[12] Ablowitz, On tight-binding approximations in optical lattices, Stud. Appl. Math. 129 pp 362– (2012) · Zbl 1297.35212 · doi:10.1111/j.1467-9590.2012.00558.x
[13] Ablowitz, Conical diffraction in honeycomb lattices, Phys. Rev. A 79 pp 053830– (2009) · doi:10.1103/PhysRevA.79.053830
[14] Haddad, The nonlinear Dirac equation in Bose-Einstein condensates: Foundation and symmetries, Physica D 238 pp 1413– (2009) · Zbl 1167.82306 · doi:10.1016/j.physd.2009.02.001
[15] Fefferman, Honeycomb lattice potentials and Dirac points, J. Am. Math. Soc. 25 pp 1169– (2012) · Zbl 1316.35214 · doi:10.1090/S0894-0347-2012-00745-0
[16] Fefferman, Wave packets in honeycomb structures and two-dimensional Dirac equations, Comm. Math. Phys. 326 pp 251– (2014) · Zbl 1292.35195 · doi:10.1007/s00220-013-1847-2
[17] Fefferman, Topologically protected states in one-dimensional continuous systems and Dirac points, PNAS 111 pp 8759– (2014) · Zbl 1355.34124 · doi:10.1073/pnas.1407391111
[18] Haldane, Possible realization of directional optical wavesguides in photonic crystals with broken time-reversal symmetry, Phys. Rev. Lett. 100 pp 013904– (2008) · doi:10.1103/PhysRevLett.100.013904
[19] Raghu, Analogs of quantum-Hall-effect edges states in photonic crystals, Phys. Rev. A 78 pp 033834– (2008) · doi:10.1103/PhysRevA.78.033834
[20] Bender, Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 pp 5243– (1998) · Zbl 0947.81018 · doi:10.1103/PhysRevLett.80.5243
[21] Barashenkov, PT-symmetry breaking in a necklace of coupled optical waveguides, Phys. Rev. A 87 pp 033819– (2013) · doi:10.1103/PhysRevA.87.033819
[22] Ruschhaupt, Physical realization of PT-symmetric potential scattering in a planar slab waveguide, J. Phys. A 38 pp L171– (2005) · Zbl 1073.81073 · doi:10.1088/0305-4470/38/9/L03
[23] Ganainy, Theory of coupled optical PT symmetric structures, Opt. Lett. 32 pp 3185– (2007)
[24] Makris, Beam dynamics in PT symmetric optical lattices, Phys. Rev. Lett 100 pp 103904– (2008) · doi:10.1103/PhysRevLett.100.103904
[25] Rüter, Observation of parity-time symmetry in optics, Nat. Phys 6 pp 192– (2010) · doi:10.1038/nphys1515
[26] Guo, Observation of PT-symmetry breaking in complex optical potentials, Phys. Rev. Lett 103 pp 093902– (2009) · doi:10.1103/PhysRevLett.103.093902
[27] Lin, Unidirectional invisibility induced by PT-symmetric periodic structures, Phys. Rev. Lett 106 pp 213901– (2011) · doi:10.1103/PhysRevLett.106.213901
[28] Makris, PT-symmetric periodic optical potentials, Int. J. Theor. Phys 50 pp 1019– (2011) · Zbl 1215.81036 · doi:10.1007/s10773-010-0625-6
[29] Nixon, Stability analysis for solitons in PT-symmetric optical lattices, Phys. Rev. A 85 pp 023822– (2012) · doi:10.1103/PhysRevA.85.023822
[30] Pelinovsky, PT-symmetric lattices with spatially extended gain/loss are generically unstable, Europhys. Lett 101 pp 11002– (2013) · doi:10.1209/0295-5075/101/11002
[31] Caliceti, Reality and non-reality of the spectrum of PT-symmetric operators: Operator-theoretic criteria, Pramana 73 pp 241– (2009) · doi:10.1007/s12043-009-0115-7
[32] Caliceti, Spectra of PT-symmetric operators and perturbation theory, J. Phys. A 38 pp 185– (2005) · Zbl 1065.81054 · doi:10.1088/0305-4470/38/1/013
[33] Caliceti, PT-symmetric non-self adjoint operators, diagonalizable and non-diagonalizable, with a real discrete spectrum, J. Phys. A 40 pp 10155– (2007) · Zbl 1124.81018 · doi:10.1088/1751-8113/40/33/014
[34] Ramezani, Exceptional-point dynamics in photonic honeycomb lattices with PT symmetry, Phys. Rev. A 85 pp 013818– (2012) · doi:10.1103/PhysRevA.85.013818
[35] Szameit, PT-symmetry in honeycomb photonic lattices, Phys. Rev. A 84 pp 021806– (2011) · doi:10.1103/PhysRevA.84.021806
[36] Schomerus, Parity anomaly and Landau-level laising in strained photonic honeycomb lattices, Phys. Rev. Lett 110 pp 013903– (2013) · doi:10.1103/PhysRevLett.110.013903
[37] Curtis, On the existence of real spectra in PT-symmetric honeycomb optical lattices, J. Phys. A 47 pp 225205– (2014) · Zbl 1291.81151 · doi:10.1088/1751-8113/47/22/225205
[38] Ablowitz, Nonlinear wave packets in deformed honeycomb lattices, SIAM J. Appl. Math 73 pp 1959– (2013) · Zbl 1295.41030 · doi:10.1137/120887618
[39] Agrawal, Optical Solitons: From Fibers to Photonic Crystals (2003)
[40] Creutz, Emergent spin, Ann. Phys 342 pp 21– (2014) · Zbl 1342.81371 · doi:10.1016/j.aop.2013.12.002
[41] Mecklenburg, Spin and the honeycomb lattice: Lessons from graphene, Phys. Rev. Lett 106 pp 116803– (2011) · doi:10.1103/PhysRevLett.106.116803
[42] Song, Direct observation of pseudospin-mediated vortex generation in photonic graphene, CLEO:QELS Fundamental Science (2014)
[43] Bahat-Treidel, Breakdown of Dirac dynamics in honeycomb lattices due to nonlinear interactions, Phys. Rev. A 82 pp 013830– (2010) · doi:10.1103/PhysRevA.82.013830
[44] Ablowitz, Nonlinear diffraction in photonic graphene, Opt. Lett 36 pp 762– (2011) · doi:10.1364/OL.36.003762
[45] Reed, Methods of Mathematical Physics IV: Analysis of Operators (1978) · Zbl 0401.47001
[46] Hislop, Introduction to Spectral Theory with Applications to Schrödinger Operators (1996) · Zbl 0855.47002
[47] Dimassi, Spectral Asymptotics in the Semi-Classical Limit (1999) · Zbl 0926.35002 · doi:10.1017/CBO9780511662195
[48] Calvo, On the formation of bound states by interacting solitary waves, Physica D 101 pp 270– (1997) · Zbl 0899.76090 · doi:10.1016/S0167-2789(96)00229-1
[49] Yang, Continuous families of embedded solitons in the third-order nonlinear Schrödinger equation, Stud. Appl. Math 111 pp 359– (2003) · Zbl 1141.35460 · doi:10.1111/1467-9590.t01-1-00238
[50] Cox, Exponential time differencing for stiff systems, J. Comput. Phys 176 pp 430– (2002) · Zbl 1005.65069 · doi:10.1006/jcph.2002.6995
[51] Kassam, Fourth-order time-stepping for stiff pdes, SIAM J. Sci. Comput 26 pp 1214– (2006) · Zbl 1077.65105 · doi:10.1137/S1064827502410633
[52] Ablowitz, Conservation laws and web-solutions for the Benney-Luke equation, Proc. Roy. Soc. A 2151 pp 20120690– (2013) · Zbl 1320.76018 · doi:10.1098/rspa.2012.0690
[53] Sakurai, Modern Quantum Mechanics (1994)
[54] Simon, Semiclassical analysis of low lying eigenvalues I: Non-degenerate minima: Asymptotic expansions, Ann. Inst. Henri Poincaré 38 pp 295– (1983)
[55] Helffer, Introduction to the Semi-classical Analysis for the Schrödinger Operator and Applications (1986)
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