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  • 1.
    Belicev, P. P.
    et al.
    University of Belgrade, Serbia.
    Gligoric, G.
    University of Belgrade, Serbia.
    Maluckov, A.
    University of Belgrade, Serbia.
    Stepic, M.
    University of Belgrade, Serbia.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.
    Localized gap modes in nonlinear dimerized Lieb lattices2017In: Physical Review A: covering atomic, molecular, and optical physics and quantum information, ISSN 2469-9926, E-ISSN 2469-9934, Vol. 96, no 6, article id 063838Article in journal (Refereed)
    Abstract [en]

    Compact localized modes of ring type exist in many two-dimensional lattices with a flat linear band, such as the Lieb lattice. The uniform Lieb lattice is gapless, but gaps surrounding the flat band can be induced by various types of bond alternations (dimerizations) without destroying the compact linear eigenmodes. Here, we investigate the conditions under which such diffractionless modes can be formed and propagated also in the presence of a cubic on-site (Kerr) nonlinearity. For the simplest type of dimerization with a three-site unit cell, nonlinearity destroys the exact compactness, but strongly localized modes with frequencies inside the gap are still found to propagate stably for certain regimes of system parameters. By contrast, introducing a dimerization with a 12-site unit cell, compact (diffractionless) gap modes are found to exist as exact nonlinear solutions in continuation of flat band linear eigenmodes. These modes appear to be generally weakly unstable, but dynamical simulations show parameter regimes where localization would persist for propagation lengths much larger than the size of typical experimental waveguide array configurations. Our findings represent an attempt to realize conditions for full control of light propagation in photonic environments.

  • 2.
    Belicev, P. P.
    et al.
    University of Belgrade, Serbia.
    Gligoric, G.
    University of Belgrade, Serbia.
    Radosavljevic, A.
    University of Belgrade, Serbia.
    Maluckov, A.
    University of Belgrade, Serbia.
    Stepic, M.
    University of Belgrade, Serbia.
    Vicencio, R. A.
    University of Chile, Chile; University of Chile, Chile.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.
    Localized modes in nonlinear binary kagome ribbons2015In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 92, no 5, p. 052916-Article in journal (Refereed)
    Abstract [en]

    The localized mode propagation in binary nonlinear kagome ribbons is investigated with the premise to ensure controlled light propagation through photonic lattice media. Particularity of the linear system characterized by the dispersionless flat band in the spectrum is the opening of new minigaps due to the "binarism." Together with the presence of nonlinearity, this determines the guiding mode types and properties. Nonlinearity destabilizes the staggered rings found to be nondiffracting in the linear system, but can give rise to dynamically stable ringlike solutions of several types: unstaggered rings, low-power staggered rings, hour-glass-like solutions, and vortex rings with high power. The type of solutions, i.e., the energy and angular momentum circulation through the nonlinear lattice, can be controlled by suitable initial excitation of the ribbon. In addition, by controlling the system "binarism" various localized modes can be generated and guided through the system, owing to the opening of the minigaps in the spectrum. All these findings offer diverse technical possibilities, especially with respect to the high-speed optical communications and high-power lasers.

  • 3.
    Beličev, P.P.
    et al.
    University of Belgrade, Serbia.
    Gligorić, G.
    University of Belgrade, Serbia.
    Radosavljević, A.
    University of Belgrade, Serbia.
    Maluckov, A.
    University of Belgrade, Serbia.
    Stepić, M.
    University of Belgrade, Serbia.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.
    Light Localization in Nonlinear Binary Two-Dimensional Lieb Lattices2016In: Abstract Book of RIAO-OPTILAS 2016 / [ed] Moraga, P. and Saavedra, C, Concepción - Chile: CEFOP-UdeC , 2016, p. 80-80Conference paper (Refereed)
    Abstract [en]

    Light localization in photonic lattices (PLs) is a well-known phenomenon which has been investigated during decades. It has been shown that light localization in the linear regime can be achieved by designing PLs with specific geometries, instead of embedding defects or disorder in otherwise periodic lattices [1]. These geometries provide conditions necessary for destructive wave interference, leading to formation of a perfectly flat (dispersionless) energy band. Eigenvectors associated to the flat-band (FB) eigenfrequencies are entirely degenerate and compact states (FB modes) and any superposition of them is nondiffracting. One of the simplest FB lattice patterns is the two-dimensional (2D) Lieb lattice [2,3] in which the primitive cell contains three sites. By appropriate spatial repetition of this fundamental block, it is possible to achieve a FB in the energy spectrum. Light confinement in PLs can also be a consequence of the interplay between nonlinearity and diffraction when these effects cancel each other, leading to formation of solitons. Recently, it has been reported that nonlinearity and “binarism” in quasi-one-dimensional FB systems can increase the range of existence and stability of FB ring modes [4].

    We model a 2D binary Lieb lattice with nonlinearity of Kerr type and analyse numerically and analytically the existence, stability and dynamical properties of various localized modes found to emerge in spectrum. From the derived dispersion relation we found that binarism does not affect the FB. However, due to the presence of additional periodicity, new gaps occur in the energy spectrum above and below the FB and their widths depend on the ratio between coupling constants. Like in the uniform Lieb lattice, we found eigenmodes in the form of a staggered four-peak “ring” structure, but only under certain conditions which require a particular relation between the field amplitudes in neighbouring sites. In the nonlinear regime, ring modes survive in the uniform Lieb lattice but lose their stability moving away from the FB. On the other hand, nonlinearity destroys the existence of ring solutions in the binary Lieb lattice, leading to a new class of stable localized solutions which can be found in minigaps. As in previous kagome and ladder binary nonlinear strips [4], it is shown that the binarism increases the existence range of stable nonlinear localized solutions.

    References

    [1] R. A. Vicencio, M. Johansson, Physical Review A 87, 061803(R) (2013).

    [2] R. A. Vicencio et al., Physical Review Letters 114, 245503 (2015).

    [3] D. Leykam, O. Bahat-Treidel, A. S. Desyatnikov, Physical Review A 86, 031805(R) (2012).

    [4] P. P. Beličev et al., Physical Review E 92, 052916 (2015).

  • 4.
    Beličev, P.P.
    et al.
    University of Belgrade, Serbia..
    Gligorić, G.
    University of Belgrade, Serbia..
    Radosavljević, A.
    University of Belgrade, Serbia..
    Maluckov, A.
    University of Belgrade, Serbia..
    Stepić, M.
    University of Belgrade, Serbia..
    Vicencio, R.A.
    University of Chile, Chile..
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.
    On localized modes in nonlinear binary kagome ribbons2015Conference paper (Other academic)
    Abstract [en]

    One of the attractive two-dimensional [2D] lattice configurations is characterized by kagome geometry. The specific arrangement of its elements, i.e. waveguides, in the form of periodic hexagons renders completely flat the first energy band in linear case. As a consequence, the localized ring-like eigenmodes belonging to the lowest energy state propagate without diffraction through the system [1, 2]. Here we study kagome ribbon [3], which can be interpreted as one-dimensional counterpart of the standard 2D kagome lattice, and can be fabricated by dint of the direct femtosecond laser inscription [4, 5].

    The existence, stability and dynamical properties of various localized modes in binary kagome ribbon with defocusing Kerr type of nonlinearity have been explored, both numerically and analytically. We derived the corresponding dispersion relation and the bandgap spectrum, confirmed the opening of mini-gaps in it and found several types of stable ring-like modes to exist: staggered, unstaggered and vortex. Beside these nonlinear mode configurations occurring in a semi-infinite gap, we investigated features of "hourglass" solutions, identified in [3] as interesting structures when kagome lattice dimensionality is reduced to 1D. In nonlinear binary kagome ribbon dynamically stable propagation of unstaggered rings, vortex modes with certain topological charge and hourglass solutions are observed, while the staggered ring solutions are destabilized. In addition, we examined possibility to generate stable propagating solitary modes inside the first mini-gap and found that these mode patterns localize within sites mutually coupled by smaller coupling constant. The last feature is opposite to the nonlinear localized solutions found in the semi-infinite gap.

    REFERENCES

    [1] R. A. Vicencio, C. Mejía-Cortés, J. Opt. 16, 015706 (2014).

    [2] R. A. Vicencio, M. Johansson, Phys. Rev. A 87, R061803 (2013).

    [3] M. Molina, Phys. Lett. A 376, 3458 (2012).

    [4] K. Davies et al., Opt. Lett. 21, 1729 (1996).

    [5] K. Itoh et al., MRS Bulletin 31, 620 (2006).

  • 5. Gorbach, AV
    et al.
    Johansson, Magnus
    Linköping University, The Institute of Technology. Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics .
    Discrete gap breathers in a diatomic Klein-Gordon chain: Stability and mobility2003In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, ISSN 1063-651X, E-ISSN 1095-3787, Vol. 67, no 6Article in journal (Refereed)
    Abstract [en]

    A one-dimensional diatomic chain with harmonic intersite potential and nonlinear external potential is considered (the Klein-Gordon model). Localized solutions of the corresponding nonlinear differential equations with frequencies inside the gap of the linear wave spectrum-"gap breathers"-are studied numerically. The linear stability analysis for these solutions is performed while changing the system parameters from the anticontinuous to the continuous limit. Two different types of solutions are considered: symmetric centered at a heavy atom and antisymmetric centered at a light atom, respectively. Different mechanisms of instability, oscillatory as well as nonoscillatory, of the gap breathers are studied, and the influence of the instabilities on the breather solutions is investigated in the dynamics simulations. In particular, the presence of an "inversion of stability" regime, with simultaneous nonoscillatory instabilities of symmetric and antisymmetric solutions with respect to antisymmetric perturbations, is found, yielding practically radiationless mobility.

  • 6. Gorbach, A.V.
    et al.
    Johansson, Magnus
    Linköping University, The Institute of Technology. Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics .
    Gap and out-gap breathers in a binary modulated discrete nonlinear Schrödinger model2004In: European Physical Journal D: Atomic, Molecular and Optical Physics, ISSN 1434-6060, E-ISSN 1434-6079, Vol. 29, no 1, p. 77-93Article in journal (Refereed)
    Abstract [en]

    We consider a modulated discrete nonlinear Schrödinger (DNLS) model with alternating on-site potential, having a linear spectrum with two branches separated by a 'forbidden' gap. Nonlinear localized time-periodic solutions with frequencies in the gap and near the gap - discrete gap and out-gap breathers (DGBs and DOGBs) - are investigated. Their linear stability is studied varying the system parameters from the continuous to the anti-continuous limit, and different types of oscillatory and real instabilities are revealed. It is shown, that generally DGBs in infinite modulated DNLS chains with hard (soft) non-linearity do not possess any oscillatory instabilities for breather frequencies in the lower (upper) half of the gap. Regimes of 'exchange of stability' between symmetric and antisymmetric DGBs are observed, where an increased breather mobility is expected. The transformation from DGBs to DOGBs when the breather frequency enters the linear spectrum is studied, and the general bifurcation picture for DOGBs with tails of different wave numbers is described. Close to the anti-continuous limit, the localized linear eigenmodes and their corresponding eigenfrequencies are calculated analytically for several gap/out-gap breather configurations, yielding explicit proof of their linear stability or instability close to this limit.

  • 7.
    Jason, Peter
    et al.
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.
    Charge Flipping Vortices in DNLS trimer and hexamer2015In: PHOTONICA 2015. V International School and Conference on Photonics& COST actions: MP1204 and BM1205 & the Second international workshop "Control of light and matter waves propagation and localization in photonic lattices“, Belgrad 2015: Book of Abstracts / [ed] Suzana Petrović , Goran Gligorić and Milutin Stepić, Belgrade, Serbia: Vinča Institute of Nuclear Sciences , 2015, p. 65-65Conference paper (Other academic)
    Abstract [en]

    We examine the existence and properties of Charge Flipping Vortices (CFVs), rtices which periodically flip the topological charge, in three-site (trimer) and six-site (hexamer) discrete nonlinear Schrödinger (DNLS) lattices. We demonstrate numerically that CFVs exist as exact quasiperiodic solutions in continuous families which connect two different stationary solutions without topological charge, and that it is possible to interpret the dynamics of certain CFVs as the result of perturbations of these stationary solutions. The CFVs are calculated with high numerical accuracy and we may therefore accurately determine many of their properties, such as their energy and linear stability, and the CFVs are found to be stable over large parameter regimes.

    We also show that, like in earlier studies for lattices with a multiple of four sites, trimer and hexamer CFVs can be obtained by perturbing stationary constant amplitude vortices with certain linear eigenmodes. However, in contrast to the former case where the perturbation could be infinitesimal, the magnitude of the perturbations for trimers and hexamers must overcome a quite large threshold value. These CFVs may be interpreted as exact quasiperiodic CFVs, with a small perturbation applied to. The concept of a charge flipping energy barrier is introduced and discussed.

    REFERENCES

    [1]P. Jason, M. Johansson, Phys. Rev. E. 91, 022910 (2015).

  • 8.
    Jason, Peter
    et al.
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
    Charge flipping vortices in the discrete nonlinear Schrodinger trimer and hexamer2015In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 91, no 2, p. 022910-Article in journal (Refereed)
    Abstract [en]

    We examine the existence and properties of charge flipping vortices (CFVs), vortices which periodically flip the topological charge, in three-site (trimer) and six-site (hexamer) discrete nonlinear Schrodinger lattices. We demonstrate numerically that CFVs exist as exact quasiperiodic solutions in continuous families which connect two different stationary solutions without topological charge, and that it is possible to interpret the dynamics of certain CFVs as the result of perturbations of these stationary solutions. The CFVs are calculated with high numerical accuracy and we may therefore accurately determine many of their properties, such as their energy and linear stability, and the CFVs are found to be stable over large parameter regimes. We also show that, like in earlier studies for lattices with a multiple of four sites, trimer and hexamer CFVs can be obtained by perturbing stationary constant amplitude vortices with certain linear eigenmodes. However, in contrast to the former case where the perturbation could be infinitesimal, the magnitude of the perturbations for trimers and hexamers must overcome a quite large threshold value. These CFVs may be interpreted as exact quasiperiodic CFVs, with a small perturbation applied. The concept of a charge flipping energy barrier is introduced and discussed.

  • 9.
    Jason, Peter
    et al.
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.
    Discrete breathers for a discrete nonlinear Schrodinger ring coupled to a central site2016In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, ISSN 1063-651X, E-ISSN 1095-3787, Vol. 93, no 1, p. 012219-Article in journal (Refereed)
    Abstract [en]

    We examine the existence and properties of certain discrete breathers for a discrete nonlinear Schrodinger model where all but one site are placed in a ring and coupled to the additional central site. The discrete breathers we focus on are stationary solutions mainly localized on one or a few of the ring sites and possibly also the central site. By numerical methods, we trace out and study the continuous families the discrete breathers belong to. Our main result is the discovery of a split bifurcation at a critical value of the coupling between neighboring ring sites. Below this critical value, families form closed loops in a certain parameter space, implying that discrete breathers with and without central-site occupation belong to the same family. Above the split bifurcation the families split up into several separate ones, which bifurcate with solutions with constant ring amplitudes. For symmetry reasons, the families have different properties below the split bifurcation for even and odd numbers of sites. It is also determined under which conditions the discrete breathers are linearly stable. The dynamics of some simpler initial conditions that approximate the discrete breathers are also studied and the parameter regimes where the dynamics remain localized close to the initially excited ring site are related to the linear stability of the exact discrete breathers.

  • 10.
    Jason, Peter
    et al.
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
    Exact localized eigenstates for an extended Bose-Hubbard model with pair-correlated hopping2012In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, Vol. 85, no 1, p. 016603(R)-Article in journal (Refereed)
    Abstract [en]

    We show that a Bose-Hubbard model extended with pair-correlated hopping has exact eigenstates, quantum lattice compactons, with complete single-site localization. These appear at parameter values where the one-particle tunneling is exactly canceled by nonlocal pair correlations, and correspond in a classical limit to compact solutions of an extended discrete nonlinear Schrödinger model. Classical compactons at other parameter values, as well as multisite compactons, generically get delocalized by quantum effects, but strong localization appears asymptotically for increasing particle number.

  • 11.
    Jason, Peter
    et al.
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
    Existence, dynamics and mobility of Quantum Compactons in an extended Bose-Hubbard model2014Conference paper (Refereed)
    Abstract [en]

    Lattice Compactons, discrete breathers with compact support, were found for a discrete nonlinear Schrödinger (DNLS) equation extended with nearest neighbour intersite nonlinearities [1], a model originally studied with waveguide arrays in mind. These compactons were shown to exhibit very good mobility if the parameters are tuned close to the compactons stability boundary. The DNLS can also be used to model the behaviour of Bose-Einstein condensates in optical lattices, and the remarkable control over the experiments in this field of research has made it possible to study the quantum mechanics of strongly correlated atoms.

    We will define the concept of a Quantum Lattice Compacton [2] and discuss the existence and dynamics, with special emphasis on mobility [3], of these in an extended Bose-Hubbard model corresponding to above-mentioned extended DNLS equation in the quantum mechanical limit. The compactons is given 'a kick' by means of a phase-gradient and it is shown that the size of this phase is crucial for the mobility of the compactons. For small phase-gradients, corresponding to a slow coherent motion in the classical model, the time-scales of the quantum tunnelings become of the same order as the time-scale of the translational motion and the classical mobility is destroyed by quantum  fluctuations. For large phase-gradients, corresponding to rapid classical motion, the classical and quantum time-scales separate so that a mobile, localized coherent quantum state can be translated many sites in the lattice already for small particle numbers of the order of 10 [3].

    Acknowledgements: This project has been financed by the Swedish Research Council.

    References

    [1] M. Öster, M. Johansson, and A. Eriksson 2003 Phys. Rev. E 67 056606

    [2] P. Jason and M. Johansson 2012 Phys. Rev. A 85 011603(R)

    [3] P. Jason and M. Johansson 2013 Phys. Rev. A 88 033605

  • 12.
    Jason, Peter
    et al.
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
    Existence, dynamics and mobility of Quantum Compactons in an extended Bose-Hubbard model2013Conference paper (Other academic)
    Abstract [en]

    Lattice Compactons, discrete breathers with compact support, were found for a discrete nonlinear Schrödinger (DNLS) equation extended with nearest neighbour intersite nonlinearities, a model originally studied with waveguide arrays in mind. These compactons were shown to exhibit very good mobility if the parameters are tuned close to the compactons stability boundary. The DNLS can also be used to model the behaviour of Bose-Einstein condensates in optical lattices, and the remarkable control over the experiments in this field of research has made it possible to study the quantum mechanics of strongly correlated atoms.

    We will define the concept of a Quantum Lattice Compacton and discuss the existence and dynamics, with special emphasis on mobility, of these in an extended Bose-Hubbard model corresponding to above-mentioned extended DNLS equation in the quantum mechanical limit.

    The compactons is given 'a kick' by means of a phase-gradient and it is shown that the size of this phase is crucial for the mobility of the compactons. For small phase-gradients, corresponding to a slow coherent motion in the classical model, the time-scales of the quantum tunnelings become of the same order as the time-scale of the translational motion and the classical mobility is destroyed by quantum fluctuations. For large phase-gradients, corresponding to rapid classical motion, the classical and quantum time-scales separate so that a mobile, localized coherent quantum state can be translated many sites in the lattice already for small particle numbers of the order of 10.

  • 13.
    Jason, Peter
    et al.
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
    Quantum dynamics of lattice states with compact support in an extended Bose-Hubbard model2013In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, Vol. 88, no 3, p. 033605-Article in journal (Refereed)
    Abstract [en]

    We study the dynamical properties, with special emphasis on mobility, of quantum lattice compactons (QLCs) in a one-dimensional Bose-Hubbard model extended with pair-correlated hopping. These are quantum counterparts of classical lattice compactons (localized solutions with exact zero amplitude outside a given region) of an extended discrete nonlinear Schrödinger equation, which can be derived in the classical limit from the extended Bose-Hubbard model. While an exact one-site QLC eigenstate corresponds to a classical one-site compacton, the compact support of classical several-site compactons is destroyed by quantum fluctuations. We show that it is possible to reproduce the stability exchange regions of the one-site and two-site localized solutions in the classical model with properly chosen quantum states. Quantum dynamical simulations are performed for two different types of initial conditions: “localized ground states” which are localized wave packets built from superpositions of compactonlike eigenstates, and SU(4) coherent states corresponding to classical two-site compactons. Clear signatures of the mobility of classical lattice compactons are seen, but this crucially depends on the magnitude of the applied phase gradient. For small phase gradients, which classically correspond to a slow coherent motion, the quantum time scale is of the same order as the time scale of the translational motion, and the classical mobility is therefore destroyed by quantum fluctuations. For a large phase instead, corresponding to fast classical motion, the time scales separate so that a mobile, localized, coherent quantum state can be translated many sites for particle numbers already of the order of 10.

  • 14.
    Jason, Peter
    et al.
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.
    Quantum signatures of charge flipping vortices in the Bose-Hubbard trimer2016In: PHYSICAL REVIEW E, ISSN 2470-0045, Vol. 94, no 5, article id 052215Article in journal (Refereed)
    Abstract [en]

    In this work we study quantum signatures of charge flipping vortices, found in the classical discrete nonlinear Schrodinger trimer, by use of the Bose-Hubbard model. We are able to identify such signatures in the quantum energy eigenstates, for instance when comparing the site amplitudes of the classical charge flipping vortices with the probability distribution over different particle configurations. It is also discussed how to construct quantum states that correspond to the classical charge flipping vortices and which effects can lead to deviations between the classical and quantum dynamics. We also examine properties of certain coherent states: classical-like quantum states that can be used to derive the classical model. Several quantum signatures are identified when studying the dynamics of these coherent states, for example, when comparing the average number of particles on a site with the classical site amplitude, when comparing the quantum and classical currents and topological charge, and when studying the evolution of the quantum probability amplitudes. The flipping of the quantum currents are found to be an especially robust feature of these states.

  • 15.
    Jason, Peter
    et al.
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.
    Quantum Signatures Of Charge-Flipping Vortices In The Bose-HubbardTrimer2016In: Abstract Book of RIAO-OPTILAS 2016 / [ed] Moraga, P. and Saavedra, C, Concepción - Chile: CEFOP-UdeC , 2016, p. 99-99Conference paper (Refereed)
    Abstract [en]

    In this work we study quantum signatures of charge flipping vortices[1], found in the classical discrete nonlinear Schrödinger trimer[2], by use of the Bose-Hubbard model. We are able to identify such signatures in the quantum energy eigenstates, for instance when comparing the site amplitudes of the classical charge flipping vortices with the probability distribution over different particle configurations. It is also discussed how to construct quantum states that correspond to the classical charge flipping vortices, and which effects that can lead to deviations between the classical and quantum dynamics.

    We also examine properties of certain coherent states: classical-like quantum states that can be used to derive the classical model. Several quantum signatures are identified when studying the dynamics of these coherent states, for example when comparing the average number of particles on a site with the classical site amplitude, when comparing the quantum and classical currents and topological charge, and when studying the evolution of the quantum probability amplitudes. The flipping of the quantum currents are found to be an especially robust feature of these states.

    References

    [1] A.S. Desyatnikov, M.R. Dennis, A. Ferrando, Physical Review A 83, 063822 (2011)

    [2] P. Jason, M. Johansson, Physical Review E 91, 022910 (2015).

  • 16.
    Jason, Peter
    et al.
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
    Kirr, Katarina
    Linköping University, Department of Physics, Chemistry and Biology. Linköping University, The Institute of Technology.
    Quantum signatures of an oscillatory instability in the Bose-Hubbard trimer2012In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 86, no 1, p. 016214-Article in journal (Refereed)
    Abstract [en]

    We study the Bose-Hubbard model for three sites in a symmetric, triangular configuration and search for quantum signatures of the classical regime of oscillatory instabilities, known to appear through Hamiltonian Hopf bifurcations for the "single-depleted-well" family of stationary states in the discrete nonlinear Schrodinger equation. In the regimes of classical stability, single quantum eigenstates with properties analogous to those of the classical stationary states can be identified already for rather small particle numbers. On the other hand, in the instability regime the interaction with other eigenstates through avoided crossings leads to strong mixing, and no single eigenstate with classical-like properties can be seen. We compare the quantum dynamics resulting from initial conditions taken as perturbed quantum eigenstates and SU(3) coherent states, respectively, in a quantum-semiclassical transitional regime of 10-100 particles. While the perturbed quantum eigenstates do not show a classical-like behavior in the instability regime, a coherent state behaves analogously to a perturbed classical stationary state, and exhibits initially resonant oscillations with oscillation frequencies well described by classical internal-mode oscillations.

  • 17.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
    Breather mobility and the PN potential - Brief review and recent progress2014Conference paper (Other academic)
    Abstract [en]

    The question whether a nonlinear localized mode (discrete soliton/breather) can be mobile in a lattice has a standard interpretation in terms of the Peierls-Nabarro (PN) potential barrier. For the most commonly studied cases, the PN barrier for strongly localized solutions becomes large, rendering these essentially immobile. Several ways to improve the mobility by reducing the PN-barrier have been proposed during the last decade, and the first part gives a brief review of such scenarios in 1D and 2D. We then proceed to discuss two recently discovered novel mobility scenarios. The first example is the 2D Kagome lattice, where the existence of a highly degenerate, flat linear band allows for a very small PN-barrier and mobility of highly localized modes in a small-power regime. The second example is a 1D waveguide array in an active medium with intrinsic (saturable) gain and damping, where exponentially localized, travelling discrete dissipative solitons may exist as stable attractors.

  • 18.
    Johansson, Magnus
    Linköping University, The Institute of Technology. Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics .
    Discrete nonlinear Schrödinger approximation of a mixed Klein-Gordon/Fermi-Pasta-Ulam chain: Modulational instability and a statistical condition for creation of thermodynamic breathers2006In: Physica D: Non-linear phenomena, ISSN 0167-2789, E-ISSN 1872-8022, Vol. 216, no 1 SPEC. ISS., p. 62-70Article in journal (Refereed)
    Abstract [en]

    We analyze certain aspects of the classical dynamics of a one-dimensional discrete nonlinear Schrödinger model with inter-site as well as on-site nonlinearities. The equation is derived from a mixed Klein-Gordon/Fermi-Pasta-Ulam chain of anharmonic oscillators coupled with anharmonic inter-site potentials, and approximates the slow dynamics of the fundamental harmonic of discrete small-amplitude modulational waves. We give explicit analytical conditions for modulational instability of travelling plane waves, and find in particular that sufficiently strong inter-site nonlinearities may change the nature of the instabilities from long-wavelength to short-wavelength perturbations. Further, we describe thermodynamic properties of the model using the grand-canonical ensemble to account for two conserved quantities: norm and Hamiltonian. The available phase space is divided into two separated parts with qualitatively different properties in thermal equilibrium: one part corresponding to a normal thermalized state with exponentially small probabilities for large-amplitude excitations, and another part typically associated with the formation of high-amplitude localized excitations, interacting with an infinite-temperature phonon bath. A modulationally unstable travelling wave may exhibit a transition from one region to the other as its amplitude is varied, and thus modulational instability is not a sufficient criterion for the creation of persistent localized modes in thermal equilibrium. For pure on-site nonlinearities the created localized excitations are typically pinned to particular lattice sites, while for significant inter-site nonlinearities they become mobile, in agreement with well-known properties of localized excitations in Fermi-Pasta-Ulam-type chains. © 2006 Elsevier Ltd. All rights reserved.

  • 19.
    Johansson, Magnus
    Linköping University, The Institute of Technology. Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics.
    Discrete nonlinear Schrödinger equations2005In: Encyclopedia of Nonlinear Science / [ed] Alwyn Scott, New York and London: Routledge , 2005, p. 213-215Chapter in book (Other academic)
    Abstract [en]

    In 438 alphabetically-arranged essays, this work provides a useful overview of the core mathematical background for nonlinear science, as well as its applications to key problems in ecology and biological systems, chemical reaction-diffusion problems, geophysics, economics, electrical and mechanical oscillations in engineering systems, lasers and nonlinear optics, fluid mechanics and turbulence, and condensed matter physics, among others.

  • 20.
    Johansson, Magnus
    Linköping University, The Institute of Technology. Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics .
    Discrete reduced-symmetry solitons in two dimensional nonlinear waveguide arrays2007In: Nonlinear Physics in Periodic Structures and Metamaterials,2007, 2007Conference paper (Refereed)
    Abstract [en]

       

  • 21.
    Johansson, Magnus
    Linköping University, The Institute of Technology. Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics .
    Discrete solitons, breathers and vortices in nonlinear Schroedinger-type lattices: stability, mobility and thermodynamics.2008In: WEH Seminar Discrete Optics and Beyond,2008, 2008Conference paper (Other academic)
    Abstract [en]

      

  • 22.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
    Dynamical properties of the DNLS2014Conference paper (Other academic)
  • 23.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.
    Editorial Material: Comment on "Localization-delocalization transition in self-dual quasi-periodic lattices" by Sun M. L. et al. in EPL, vol 112, issue 1, pp2015In: Europhysics letters, ISSN 0295-5075, E-ISSN 1286-4854, Vol. 112, no 1, p. 17002-Article in journal (Other academic)
    Abstract [en]

    n/a

  • 24.
    Johansson, Magnus
    Linköping University, The Institute of Technology. Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics .
    Effects of nonlinear coupling in nonlinear Schrödinger lattices2008In: BICS Workshop: Lattice Models,2008, Bath: University of Bath , 2008Conference paper (Other academic)
    Abstract [en]

      

  • 25.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics . Linköping University, The Institute of Technology.
    Hamiltonian Hopf bifurcations in the discrete nonlinear Schrodinger trimer: oscillatory instabilities, quasi-periodic solutions and a new type of self-trapping transition2004In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 37, no 6, p. 2201-2222Article in journal (Refereed)
    Abstract [en]

    Oscillatory instabilities in Hamiltonian anharmonic lattices are known to appear through Hamiltonian Hopf bifurcations of certain time-periodic solutions of multibreather type. Here, we analyse the basic mechanisms for this scenario by considering the simplest possible model system of this kind where they appear: the three-site discrete nonlinear Schrodinger model with periodic boundary conditions. The stationary solution having equal amplitude and opposite phases on two sites and zero amplitude on the third is known to be unstable for an interval of intermediate amplitudes. We numerically analyse the nature of the two bifurcations leading to this instability and find them to be of two different types. Close to the lower-amplitude threshold stable two-frequency quasi-periodic solutions exist surrounding the unstable stationary solution, and the dynamics remains trapped around the latter so that in particular the amplitude of the originally unexcited site remains small. By contrast, close to the higher-amplitude threshold all two-frequency quasi-periodic solutions are detached from the unstable stationary solution, and the resulting dynamics is of population-inversion type involving also the originally unexcited site.

  • 26.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.
    Nonlinear localized modes in flatband lattices2017Conference paper (Other academic)
  • 27.
    Johansson, Magnus
    Linköping University, The Institute of Technology. Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics .
    Nonlinearity, discreteness and disorder: localization, delocalization and transmission threshold in time-periodically driven systems2008In: Aspectos en fisica no lineal, Mini-workshop internacional,2008, 2008Conference paper (Other academic)
  • 28.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics . Linköping University, The Institute of Technology.
    Nonlinearity, discreteness and disorder: localization, delocalization and transmission threshold in time-periodically driven systems2009Conference paper (Other academic)
    Abstract [en]

    After a brief introduction to the physics of nonlinear localization and Anderson localization, the talk will focus on discussing how these two effects may compete with each other. Examples will be given showing that in different circumstances, nonlinearity may either enhance Anderson localization or lead to delocalization in a disordered system. In the end, we will discuss recent results showing that a locally acting time-periodic driving force may destroy Anderson localization, and due to nonlinear effects yield transmission of energy through a disordered system if the force strength is above some threshold value. The application of these results to experiments on light waves in optical wave guide arrays, as well as to quantum mechanical matter waves of Bose-Einstein condensates trapped in optical lattice potentials, will also be briefly discussed.

  • 29.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.
    Quantum signatures of charge-flipping vortices in the Bose-Hubbard trimer2016Conference paper (Other academic)
  • 30.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.
    Rotational energy barriers: Charge-flipping of discrete vortices and rotation of dipole discrete gap solitons2015Conference paper (Other academic)
  • 31.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
    Strongly localized moving discrete solitons (breathers): new ways to beat the Peierls-Nabarro barrier2013Conference paper (Other academic)
    Abstract [en]

    The question whether a nonlinear localized mode (discrete soliton/breather) can be mobile in a lattice has a standard interpretation in terms of the Peierls-Nabarro (PN) potential barrier. In particular, for any system modelled by a Discrete Nonlinear Schrödinger (DNLS) type equation, this concept can be defined as the maximum difference in energy (Hamiltonian) between solutions at fixed power (norm), centered at different lattice positions. For the most commonly studied case with on-site, cubic (Kerr) nonlinearity, the PN barrier for strongly localized solutions becomes large, rendering these essentially immobile.

    Several ways to improve the mobility by reducing the PN-barrier for strongly localized modes have been proposed during the last decade, and the first part of this talk will give a brief review of two such scenarios. In 1D, one option is to utilize a competition between on-site and inter-site nonlinearities. In 2D, the mobility is normally much worse than in 1D, due to the fact that also broad solitons are prone to excitation thresholds and quasicollapse instabilities. Utilizing a saturable nonlinearity was found to considerably improve the 2D mobility by reducing the PN barrier in certain parameter regimes for large power.

    We then proceed to discuss two (if time allows) recently discovered novel mobility scenarios. The first example discussed is the 2D Kagome lattice, where the existence of a highly degenerate, flat linear band allows small-power, strongly localized nonlinear modes to appear without excitation threshold. The nonlinearity lifts the degeneracy of linear modes and causes a small energy shift between modes centered at different lattice positions, yielding a very small PN-barrier and mobility of highly localized modes in a small-power regime.

    The second example discusses a 1D waveguide array in an active medium with intrinsic (saturable) gain and damping. It is shown that exponentially localized, travelling discrete dissipative solitons may exist as stable attractors, supported only by intrinsic properties of the medium (i.e., in absence of any external field or symmetry-breaking perturbations). With a standard, on-site Kerr-nonlinearity the solitons are pinned by the PN-barrier, but decreasing the barrier with inter-site nonlinearities allows for the existence of breathing (i.e., with oscillating size) solitons as stable attractors at certain velocities, related to lattice commensurability effects. The stable moving breathers also survive in presence of weak disorder.

  • 32.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics . Linköping University, The Institute of Technology.
    Transmission thresholds in time-periodically driven nonlinear disordered systems2009Conference paper (Other academic)
    Abstract [en]

    (Joint work with G. Kopidakis, S. Lepri, and S. Aubry) We study energy propagation in locally time-periodically driven disordered nonlinear chains. For frequencies inside the band of linear Anderson modes, three different regimes are observed with increasing driver amplitude: 1) Below threshold, localized quasiperiodic oscillations and no spreading; 2) Three different regimes in time close to threshold, with almost regular oscillations initially, weak chaos and slow spreading for intermediate times, and finally strong diffusion; 3) Immediate spreading for strong driving. The thresholds are due to simple bifurcations, obtained analytically for a single oscillator, and numerically as turning-points of the nonlinear response manifold for a full chain. Generically, the threshold is nonzero also for infinite chains.

  • 33.
    Johansson, Magnus
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics .
    Gorbach, A.V.
    Max-Planck-Inst. Phys. Komplexer S., Nöthnitzer Strasse 38, 01187 Dresden, Germany.
    Quasiperiodic localized oscillating solutions in the discrete nonlinear Schrödinger equation with alternating on-site potential2004In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, Vol. 70, no 5 2Article in journal (Refereed)
    Abstract [en]

    An explicit examples of exact stable quasiperiodic localized stable solution with spatially symmetric large amplitude oscillations in a nonintegrable Hamiltonian lattice model were presented. The proposed model as observed, is a one-dimensional discrete nonlinear Schro°dinger equation with alternating on-site energies. It was observed that the pulson solutions exists for other types of multicomponent lattices with two conserved quantities. It was also observed that the pulson character of the two-frequency solution appeared when the absolute value of the minimum value of ? n0±1 exceeds the minimum value of ?n0.

  • 34.
    Johansson, Magnus
    et al.
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.
    Jason, Peter
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.
    Breather mobility and the Peierls-Nabarro potential: brief review and recent progress2015In: Quodons in Mica: nonlinear localized travelling excitations in crystals / [ed] Juan F. R. Archilla, Noé Jiménez, Victor J. Sánchez-Morcillo, Luis M. García-Raffi, Cham: Springer, 2015, p. 147-178Chapter in book (Refereed)
    Abstract [en]

    The question whether a nonlinear localized mode (discrete soliton/breather) can be mobile in a lattice has a standard interpretation in terms of the Peierls-Nabarro (PN) potential barrier. For the most commonly studied cases, the PN barrier for strongly localized solutions becomes large, rendering these essentially immobile. Several ways to improve the mobility by reducing the PN-barrier have been proposed during the last decade, and the first part gives a brief review of such scenarios in 1D and 2D. We then proceed to discuss two recently discovered novel mobility scenarios. The first example is the 2D Kagome lattice, where the existence of a highly degenerate, flat linear band allows for a very small PN-barrier and mobility of highly localized modes in a small-power regime. The second example is a 1D waveguide array in an active medium with intrinsic (saturable) gain and damping, where exponentially localized, travelling discrete dissipative solitons may exist as stable attractors. Finally, using the framework of an extended Bose-Hubbard model, we show that while quantum fluctuations destroy the mobility of slowly moving, strongly localized classical modes, coherent mobility of rapidly moving states survives even in a strongly quantum regime

  • 35.
    Johansson, Magnus
    et al.
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
    Jason, Peter
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
    Kirr, Katarina
    Institute of Electrophysics and Radiation Technologies, Kharkiv, Ukraine.
    Quantum signatures of an oscillatory instability in the Bose-Hubbard trimer2012Conference paper (Other academic)
    Abstract [en]

    We study the Bose-Hubbard model for three sites in a symmetric, triangular configuration, and search for quantumsignatures of the classical regime of oscillatory instabilities, known to appear through Hamiltonian Hopfbifurcations for the single-depleted well family of stationary states in the Discrete Nonlinear Schrödinger equation.In the regimes of classical stability, single quantum eigenstates with properties analogous to the classicalstationary states can be identified already for rather small particle numbers. On the other hand, in the instabilityregime the interaction with other eigenstates through avoided crossings leads to strong mixing, andno single eigenstate with classical-like properties can be seen. We compare the quantum dynamics resultingfrom initial conditions taken as perturbed quantum eigenstates, and SU(3) coherent states, respectively, in aquantum-semiclassical transitional regime of 10-100 particles. While the perturbed quantum eigenstates donot show a classical-like behaviour in the instability regime, a coherent state behaves analogously to a perturbedclassical stationary state, and exhibits initially resonant oscillations with oscillation frequencies welldescribed by classical internal-mode oscillations.

  • 36.
    Johansson, Magnus
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics.
    Kevrekidis, P.G.
    Theoretical Division, Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545.
    Malomed, B.A.
    Theoretical Division, Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545.
    Bishop, A.R.
    Theoretical Division, Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545.
    Frantzeskakis, D.J.
    Department of Physics, Univ. of Athens Panepistimiopolis, Zografos, Athens 15784, Greece.
    Comment on "localized vortices with a semi-integer charge in nonlinear dynamical lattices"2002In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 66, no 4, p. 048601-Article in journal (Other academic)
    Abstract [en]

    In a recent paper by Kevrekidis, Malomed, Bishop, and Frantzeskakis [Phys. Rev. E 65, 016605 (2001)] the existence of localized vortices with semi-integer topological charge as exact stationary solutions in a two-dimensional discrete nonlinear Schrödinger model is claimed, as well as the existence of an analog solution in the one-dimensional model. We point out that the existence of such exact stationary solutions would violate fundamental conservation laws, and therefore these claims are erroneous and appear as a consequence of inaccurate numerics. We illustrate the origin of these errors by performing similar numerical calculations using more accurate numerics.

  • 37.
    Johansson, Magnus
    et al.
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
    Kirr, K
    Linköping University, The Institute of Technology. Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics.
    Kovalev, A S
    National Academy of Science Ukraine.
    Kroon, Lars
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
    Gap and out-gap solitons in modulated systems of finite length: exact solutions in the slowly varying envelope limit2011In: PHYSICA SCRIPTA, ISSN 0031-8949, Vol. 83, no 6Article in journal (Refereed)
    Abstract [en]

    We discuss nonlinear excitations in finite-size one-dimensional modulated systems. Considering a binary modulated discrete nonlinear Schrodinger chain of large but finite length with periodic boundary conditions, we obtain exact elliptic-function solutions corresponding to stationary excitations in the slowly varying envelope limit. From these solutions, we analyze how the transformation between (localized) gap and (delocalized) out-gap solitons manifests itself in a system of finite length. The analogue of a localized gap soliton appears through a bifurcation at a critical point, so that gap soliton analogues exist only for chains longer than a critical value, which scales inversely proportional to the modulation depth. The total norm of these gap-out-gap states is found to be a monotonic function of the frequency, always inside a nonlinear gap with edges defined by the main nonlinear modes which approach the linear spectrum gap boundaries in the small-amplitude limit. The transformation from a gap to an out-gap state is associated with a particular frequency, close to the lower boundary of the linear gap; at this point the elliptic functions become trigonometric, corresponding to a finite-size analogue of an algebraic soliton. We compare the scenario with earlier results obtained numerically for purely discrete chains with few degrees of freedom.

  • 38.
    Johansson, Magnus
    et al.
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics . Linköping University, The Institute of Technology.
    Kopidakis, G
    University of Crete.
    Aubry, S
    CEA Saclay.
    KAM tori in 1D random discrete nonlinear Schrodinger model?2010In: EPL, ISSN 0295-5075, Vol. 91, no 5Article in journal (Refereed)
    Abstract [en]

    We suggest that KAM theory could be extended for certain infinite-dimensional systems with purely discrete linear spectrum. We provide empirical arguments for the existence of square summable infinite-dimensional invariant tori in the random discrete nonlinear Schrodinger equation, appearing with a finite probability for a given initial condition with sufficiently small norm. Numerical support for the existence of a fat Cantor set of initial conditions generating almost periodic oscillations is obtained by analyzing i) sets of recurrent trajectories over successively larger time scales, and ii) finite-time Lyapunov exponents. The norm region where such KAM-like tori may exist shrinks to zero when the disorder strength goes to zero and the localization length diverges.

  • 39.
    Johansson, Magnus
    et al.
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics . Linköping University, The Institute of Technology.
    Kopidakis, G
    University of Crete.
    Lepri, S
    CNR.
    Aubry , S
    CEA Saclay.
    Transmission thresholds in time-periodically driven nonlinear disordered systems2009In: EPL, ISSN 0295-5075 , Vol. 86, no 1, p. 10009-Article in journal (Refereed)
    Abstract [en]

    We study energy propagation in locally time-periodically driven disordered nonlinear chains. For frequencies inside the band of linear Anderson modes, three different regimes are observed with increasing driver amplitude: 1) below threshold, localized quasiperiodic oscillations and no spreading; 2) three different regimes in time close to threshold, with almost regular oscillations initially, weak chaos and slow spreading for intermediate times and finally strong diffusion; 3) immediate spreading for strong driving. The thresholds are due to simple bifurcations, obtained analytically for a single oscillator, and numerically as turning points of the nonlinear response manifold for a full chain. Generically, the threshold is nonzero also for infinite chains.

  • 40.
    Johansson, Magnus
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics .
    Kopidakis, Georgios
    Lepri, Stefano
    Aubry, Serge
    Transmission thresholds in time-periodically driven nonlinear disordered systems2008Report (Other academic)
    Abstract [en]

      

  • 41.
    Johansson, Magnus
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics .
    Morgante, A.M.
    Laboratoire Léon Brillouin, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France.
    Aubry, S.
    Laboratoire Léon Brillouin, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France.
    Kopidakis, G.
    Department of Physics, University of Crete, PO Box 2208, 71003, Heraklion, Crete, Greece.
    Standing wave instabilities, breather formation and thermalization in a Hamiltonian anharmonic lattice2002In: European Physical Journal B: Condensed Matter Physics, ISSN 1434-6028, E-ISSN 1434-6036, Vol. 29, no 2, p. 279-283Article in journal (Refereed)
    Abstract [en]

    Modulational instability of travelling plane waves is often considered as the first step in the formation of intrinsically localized modes (discrete breathers) in anharmonic lattices. Here, we consider an alternative mechanism for breather formation, originating in oscillatory instabilities of spatially periodic or quasiperiodic nonlinear standing waves (SWs). These SWs are constructed for Klein-Gordon or Discrete Nonlinear Schrödinger lattices as exact time periodic and time reversible multibreather solutions from the limit of uncoupled oscillators, and merge into harmonic SWs in the small-amplitude limit. Approaching the linear limit, all SWs with nontrivial wave vectors (0 < Q < p) become unstable through oscillatory instabilities, persisting for arbitrarily small amplitudes in infinite lattices. The dynamics resulting from these instabilities is found to be qualitatively different for wave vectors smaller than or larger than p/2, respectively. In one regime persisting breathers are found, while in the other regime the system thermalizes.

  • 42.
    Johansson, Magnus
    et al.
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.
    Naether, Uta
    University of Zaragoza, Spain; University of Zaragoza, Spain.
    Vicencio, Rodrigo A.
    University of Chile, Chile; University of Chile, Chile.
    Compactification tuning for nonlinear localized modes in sawtooth lattices2015In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 92, no 3-1, p. 032912-Article in journal (Refereed)
    Abstract [en]

    We discuss the properties of nonlinear localized modes in sawtooth lattices, in the framework of a discrete nonlinear Schrodinger modelwith general on-site nonlinearity. Analytic conditions for existence of exact compact three-site solutions are obtained, and explicitly illustrated for the cases of power-law (cubic) and saturable nonlinearities. These nonlinear compact modes appear as continuations of linear compact modes belonging to a flat dispersion band. While for the linear system a compact mode exists only for one specific ratio of the two different coupling constants, nonlinearity may lead to compactification of otherwise noncompact localized modes for a range of coupling ratios, at some specific power. For saturable lattices, the compactification power can be tuned by also varying the nonlinear parameter. Introducing different on-site energies and anisotropic couplings yields further possibilities for compactness tuning. The properties of strongly localized modes are investigated numerically for cubic and saturable nonlinearities, and in particular their stability over large parameter regimes is shown. Since the linear flat band is isolated, its compact modes may be continued into compact nonlinear modes both for focusing and defocusing nonlinearities. Results are discussed in relation to recent realizations of sawtooth photonic lattices.

  • 43.
    Johansson, Magnus
    et al.
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
    Prilepsky, Jaroslaw E.
    Aston University, Birmingham, UK.
    Derevyanko, Stanislav A.
    Weizmann Institute Science, Rehovot, Israel .
    Strongly localized moving discrete dissipative breather-solitons in Kerr nonlinear media supported by intrinsic gain2014In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 89, no 4, p. 042912-1-042912-9Article in journal (Refereed)
    Abstract [en]

    We investigate the mobility of nonlinear localized modes in a generalized discrete Ginzburg-Landau-type model, describing a one-dimensional waveguide array in an active Kerr medium with intrinsic, saturable gain and damping. It is shown that exponentially localized, traveling discrete dissipative breather-solitons may exist as stable attractors supported only by intrinsic properties of the medium, i.e., in the absence of any external field or symmetry-breaking perturbations. Through an interplay by the gain and damping effects, the moving soliton may overcome the Peierls-Nabarro barrier, present in the corresponding conservative system, by self-induced time-periodic oscillations of its power (norm) and energy (Hamiltonian), yielding exponential decays to zero with different rates in the forward and backward directions. In certain parameter windows, bistability appears between fast modes with small oscillations and slower, large-oscillation modes. The velocities and the oscillation periods are typically related by lattice commensurability and exhibit period-doubling bifurcations to chaotically walking modes under parameter variations. If the model is augmented by intersite Kerr nonlinearity, thereby reducing the Peierls-Nabarro barrier of the conservative system, the existence regime formoving solitons increases considerably, and a richer scenario appears including Hopf bifurcations to incommensurately moving solutions and phase-locking intervals. Stable moving breathers also survive in the presence of weak disorder.

  • 44.
    Johansson, Magnus
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics .
    Rasmussen, K.O.
    Rasmussen, K.Ø., Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, United States.
    Statistical mechanics of general discrete nonlinear Schrödinger models: Localization transition and its relevance for Klein-Gordon lattices2004In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, Vol. 70, no 6 2Article in journal (Refereed)
    Abstract [en]

    A statistical-mechanics description of a general class of discrete nonlinear Schrödinger (DNLS) models, was presented. Simple analytical conditions for the transition into the statistical localization regime, were obtained. Numerical simulation was performed to show the nature of the localization dynamics outside the 'normal' Gibbsian regime for various cases. It is concluded that the results from the DNLS model can be transferred into approximate conditions for statistical formation of long-lived breathers in weakly coupled Klein-Gordon chains.

  • 45.
    Johansson, Magnus
    et al.
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics . Linköping University, The Institute of Technology.
    Sukhorukov, Andrey A
    Australian National University.
    Kivshar, Yuri S
    Australian National University.
    Discrete reduced-symmetry solitons and second-band vortices in two-dimensional nonlinear waveguide arrays2009In: PHYSICAL REVIEW E, ISSN 1539-3755, Vol. 80, no 4, p. 046604-Article in journal (Refereed)
    Abstract [en]

    Considering a two-dimensional lattice of weakly coupled waveguides, where each waveguide may carry two orthogonal modes of dipolar character, we present a nonlinear discrete vector model for the study of Kerr optical solitons with profiles having a reduced symmetry relative to the underlying lattice. We describe analytically and numerically existence and stability properties of such states in square and triangular lattices and also reveal directional mobility properties of two-dimensional gap solitons which were recently observed in experiment. The model also describes one-site peaked discrete vortices corresponding to experimentally observed "second-band" vortex lattice solitons, for which oscillatory instabilities are predicted. We also introduce a concept of "rotational Peierls-Nabarro barrier" characterizing the minimum energy needed for rotation of stable dipole modes and compare numerically translational and rotational energy barriers in regimes of good mobility.

  • 46.
    Johansson, Magnus
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics .
    Öster, Michael
    Linköping University, The Institute of Technology. Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics .
    Gaididei, Yu.B.
    Christiansen, P.L.
    Eriksson, A.
    Some properties of a physically motivated generalized DNLS equation with inter-site nonlinearities2007In: Hamiltonian Lattice Dynamical Systems,2007, 2007Conference paper (Refereed)
  • 47.
    Kroon, Lars
    et al.
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics . Linköping University, The Institute of Technology.
    Johansson, Magnus
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics . Linköping University, The Institute of Technology.
    Kovalev, A S
    National Acadamy of Science Ukraine.
    Yu Malyuta, E
    National Acadamy of Science Ukraine.
    The appearance of gap solitons in a nonlinear Schrodinger lattice2010In: PHYSICA D-NONLINEAR PHENOMENA, ISSN 0167-2789, Vol. 239, no 5, p. 269-278Article in journal (Refereed)
    Abstract [en]

    We study the appearance of discrete gap solitons in a nonlinear Schrodinger model with a periodic on-site potential that possesses a gap evacuated of plane-wave solutions in the linear limit. For finite lattices supporting an anti-phase (q = pi/2) gap edge phonon as an anharmonic standing wave in the nonlinear regime, gap solitons are numerically found to emerge via pitchfork bifurcations from the gap edge. Analytically, modulational instabilities between pairs of bifurcation points on this "nonlinear gap boundary" are found in terms of critical gap widths, turning to zero in the infinite-size limit, which are associated with the birth of the localized soliton as well as discrete multisolitons in the gap. Such tunable instabilities can be of relevance in exciting soliton states in modulated arrays of nonlinear optical waveguides or Bose-Einstein condensates in periodic potentials. For lattices whose gal) edge phonon only asymptotically approaches the anti-phase solution, the nonlinear gap boundary splits in a bifurcation scenario leading to the birth of the discrete gap soliton as a continuable orbit to the gap edge in the linear limit. The instability-induced dynamics of the localized soliton in the gap regime is found to thermalize according to the Gibbsian equilibrium distribution, while the spontaneous formation of persisting intrinsically localized modes (discrete breathers) from the extended out-gap soliton reveals a phase transition of the solution.

  • 48.
    Makela, H
    et al.
    Umeå University.
    Johansson, Magnus
    Linköping University, The Institute of Technology. Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics .
    Zelan, M
    Umeå University.
    Lundh, E
    Umeå University.
    Stability of nonstationary states of spin-1 Bose-Einstein condensates2011In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, Vol. 84, no 4, p. 043646-Article in journal (Refereed)
    Abstract [en]

    The stability of nonstationary states of homogeneous spin-1 Bose-Einstein condensates is studied by performing Bogoliubov analysis in a frame of reference where the state is stationary. In particular, the effect of an external magnetic field is examined. It is found that a nonzero magnetic field introduces instability in a (23)Na condensate. The wavelengths of this instability can be controlled by tuning the strength of the magnetic field. In a (87)Rb condensate this instability is present already at zero magnetic field. Furthermore, an analytical bound for the size of a stable condensate is found, and a condition for the validity of the single-mode approximation is presented. Realization of the system in a toroidal trap is discussed, and the full time development is simulated.

  • 49.
    Morgante, A.M.
    et al.
    Lab. Leon Brillouin (CEA-CNRS), CEA Saclay, F-91191 Gif-sur-Yvette Cedex, France.
    Johansson, Magnus
    Linköping University, The Institute of Technology. Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics .
    Aubry, S.
    Lab. Leon Brillouin (CEA-CNRS), CEA Saclay, F-91191 Gif-sur-Yvette Cedex, France.
    Kopidakis, G.
    Department of Physics, University of Crete, PO Box 2208, GR-71003 Heraklion, Crete, Greece.
    Breather-phonon resonances in finite-size lattices: 'Phantom breathers'?2002In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 35, no 24, p. 4999-5021Article in journal (Refereed)
    Abstract [en]

    We investigate the resonance mechanisms for discrete breathers in finite-size Klein-Gordon lattices, when some harmonic of the breather frequency enters the linear phonon band. For soft on-site potentials, the second-harmonic resonances typically result in the appearance of solutions with non-zero tails, phonobreathers. However, these tails may be very weak, and for small systems where the phonon frequencies are sparsely distributed, we identify 'phantom breathers' as being practically localized solutions, existing with frequencies in-between the phonon frequencies. For particular parameter values the tails completely vanish, and the phantom breathers decay exponentially over the whole system. We also describe briefly a first-harmonic resonance with a constant-amplitude wave and the generation of phonobreathers for hard potentials.

  • 50.
    Morgante, A.M.
    et al.
    Laboratoire Léon Brillouin (CEA-CNRS), CEA Saclay, F-91191 Gif-sur-Yvette Cedex, France.
    Johansson, Magnus
    Linköping University, The Institute of Technology. Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics .
    Kopidakis, G.
    Laboratoire Léon Brillouin (CEA-CNRS), CEA Saclay, F-91191 Gif-sur-Yvette Cedex, France, Department of Physics, University of Crete, P.O. Box 2208, GR-71003 Heraklion, Crete, Greece.
    Aubry, S.
    Laboratoire Léon Brillouin (CEA-CNRS), CEA Saclay, F-91191 Gif-sur-Yvette Cedex, France.
    Standing wave instabilities in a chain of nonlinear coupled oscillators2002In: Physica D: Non-linear phenomena, ISSN 0167-2789, E-ISSN 1872-8022, Vol. 162, no 1-2, p. 53-94Article in journal (Refereed)
    Abstract [en]

    We consider existence and stability properties of nonlinear spatially periodic or quasiperiodic standing waves (SWs) in one-dimensional lattices of coupled anharmonic oscillators. Specifically, we consider Klein-Gordon (KG) chains with either soft (e.g., Morse) or hard (e.g., quartic) on-site potentials, as well as discrete nonlinear Schrödinger (DNLS) chains approximating the small-amplitude dynamics of KG chains with weak inter-site coupling. The SWs are constructed as exact time-periodic multibreather solutions from the anticontinuous limit of uncoupled oscillators. In the validity regime of the DNLS approximation these solutions can be continued into the linear phonon band, where they merge into standard harmonic SWs. For SWs with incommensurate wave vectors, this continuation is associated with an inverse transition by breaking of analyticity. When the DNLS approximation is not valid, the continuation may be interrupted by bifurcations associated with resonances with higher harmonics of the SW. Concerning the stability, we identify one class of SWs which are always linearly stable close to the anticontinuous limit. However, approaching the linear limit all SWs with non-trivial wave vectors become unstable through oscillatory instabilities, persisting for arbitrarily small amplitudes in infinite lattices. Investigating the dynamics resulting from these instabilities, we find two qualitatively different regimes for wave vectors smaller than or larger than p/2, respectively. In one regime persisting breathers are found, while in the other regime the system rapidly thermalizes. © 2002 Elsevier Science B.V. All rights reserved.

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