Intrinsic Localized Spin Wave Modes and Modulational Instability in a Two-Dimensional Heisenberg Ferromagnet

An analytical study on the properties of intrinsic localized modes and modulational instability in a quantum two-dimensional ferromagnet with single-ion uniaxial anisotropy is completed in the semiclassical limit. By making use of the semidiscrete multiple-scale method, we obtain a line localized solution and a radially symmetric localized solution, and analyze their existence conditions. Taking into account that the existence of bright localized solutions is closely linked to the modulational instability of plane waves, we analytically study the discrete modulational instability of plane spin waves. The result of the modulational instability analysis show that the uniaxial anisotropy plays a key role in the appearance of our intrinsic localized spin wave modes.     

Modulational instability and discrete breathers in the discrete cubic-quintic nonlinear Schrödinger equation

We investigate the properties of modulational instability and discrete breathers in the cubic-quintic discrete nonlinear Schrödinger equation. We analyze the regions of modulational instabilities of nonlinear plane waves. Using the Page approach [J.B. Page, Phys. Rev. B 41 (1990) 7835], we derive the conditions for the existence and stability for bright discrete breather solutions. It is shown that the quintic nonlinearity brings qualitatively new conditions for stability of strongly localized modes. The application to the existence of localized modes in the Bose-Einstein condensate (BEC) with three-body interactions in an optical lattice is discussed. The numerical simulations agree with the analytical predictions.     

Discrete modulational instability and bright localized spin wave modes in easy-axis weak ferromagnetic spin chains involving the next-nearest-neighbor coupling

We report a theoretical work on the properties of modulational instability and bright type nonlinear localized modes in one-dimensional easy-axis weak ferromagnetic spin lattices involving next-nearest-neighbor couplings.With a linear stability analysis, we calculate the growth rates of the modulational instability, and plot the instability regions.When the strength of the next-nearest-neighbor coupling is large enough, two new asymmetric modulational instability regions appear near the boundary of the first Brillouin zone.Furthermore, analytical forms of the bright nonlinear localized modes are constructed by means of a quasi-discreteness approach.The influence of the next-nearest-neighbor coupling on the Brillouin zone center mode and boundary mode are discussed.In particular, we discover a reversal phenomenon of the propagation direction of the Brillouin zone boundary mode.     

Quantum Solitons and Breathers in an Anisotropic Ferromagnet with Octupole-Dipole Interaction

By using a full quantum approach based on the time-dependent Hartree approximation and the semidiscrete multiple-scale method, we study quantum nonlinear excitations in a one-dimensional ferromagnetic chain with octupole-dipole interaction and on-site uniaxial anisotropy. We find that quantum solitons and breathers can exist in the ferromagnetic chain, and analyze existence conditions of these excitations. Since the system states corresponding to quantum breathers are stationary states, we can get the energy level formula of such quantum breathers.     

Quantum Two-breathers Formed by Ultracold Bosonic Atoms in Optical Lattices

Two-discrete breathers are the bound states of two localized modes that can appear in classical nonlinear lattices. I investigate the quantum signature of two-discrete breathers in the system of ultracold bosonic atoms in optical lattices, which is modeled as Bose–Hubbard model containing n bosons. When the number of bosons is small, I find numerically quantum two-breathers by making use of numerical diagonalization and perturbation theory. For the cases of a large number of bosons, I can successfully construct quantum two-breather states in the Hartree approximation.     

Modulational instability and gap solitons in periodic ferromagnetic films

The modulational instability and gap solitons are theoretically studied in the ferromagnetic films under a periodic magnetic field. By multiple scale expansion, the envelope soliton solutions are obtained naturally. Due to the periodic modulation of dispersion, the solitons may be pushed into the gap region. For the easy-axis magnetic film, the red-shift of frequency leads to a modulational instability in the bottom of band and generates a bright gap soliton. For the easy-plane case, the blue-shift leads to an instability in the top of band and a dark gap soliton emerges. The weak damping produces an attenuation factor and a small oscillation.     

Classical breathers and quantum coherent states in discrete NLS systems

We present a comparison of quantum and “semiclassical” trajectories of coherent states that correspond to classical breather solutions of finite discrete nonlinear Schrödinger (DNLS) lattices. The main goal is to explain earlier numerical observations of recurrent return to the vicinity of initial coherent states corresponding to stable breathers that are also spatially localized. This effect can be considered as a quantum manifestation of classical spatial localization. We show that these phenomena are encoded in a simple expression for the distance between the quantum and semiclassical states that involves the basic frequencies of the classical and quantum systems, as well as the breather amplitude and quantum spectral decomposition of the system. A corollary is that recurrence phenomena are robust under perturbation of the initial conditions for stable breathers.     

Dissipative discrete breathers: periodic,quasiperiodic, chaotic,and mobile

The properties of discrete breathers in dissipative one-dimensional lattices of nonlinear oscillators subject to periodic driving forces are reviewed. We focus on oscillobreathers in the Frenkel-Kontorova chain and rotobreathers in a ladder of Josephson junctions. Both types of exponentially localized solutions are easily obtained numerically using adiabatic continuation from the anticontinuous limit. Linear stability (Floquet) analysis allows the characterization of different types of bifurcations experienced by periodic discrete breathers. Some of these bifurcations produce nonperiodic localized solutions, namely, quasiperiodic and chaotic discrete breathers, which are generally impossible as exact solutions in Hamiltonian systems. Within a certain range of parameters, propagating breathers occur as attractors of the dissipative dynamics. General features of these excitations are discussed and the Peierls-Nabarro barrier is addressed. Numerical scattering experiments with mobile breathers reveal the existence of two-breather bound states and allow a first glimpse at the intricate phenomenology of these special multibreather configurations.     

Discrete breathers

Nonlinear classical Hamiltonian lattices exhibit generic solutions in the form of discrete breathers. These solutions are time-periodic and (typically exponentially) localized in space. The lattices exhibit discrete translational symmetry. Discrete breathers are not confined to certain lattice dimensions. Necessary ingredients for their occurrence are the existence of upper bounds on the phonon spectrum (of small fluctuations around the groundstate) of the system as well as the nonlinearity in the differential equations. We will present existence proofs, formulate necessary existence conditions, and discuss structural stability of discrete breathers. The following results will be also discussed: the creation of breathers through tangent bifurcation of band edge plane waves; dynamical stability; details of the spatial decay; numerical methods of obtaining breathers; interaction of breathers with phonons and electrons; movability; influence of the lattice dimension on discrete breather properties; quantum lattices — quantum breathers.Finally we will formulate a new conceptual approach capable of predicting whether discrete breathers exist for a given system or not, without actually solving for the breather. We discuss potential applications in lattice dynamics of solids (especially molecular crystals), selective bond excitations in large molecules, dynamical properties of coupled arrays of Josephson junctions, and localization of electromagnetic waves in photonic crystals with nonlinear response.     

Localization of Bose-Einstein condensates in optical lattices

The dynamics of repulsive bosons condensed in an optical lattice is effectively described by the Bose-Hubbard model. The classical limit of this model, reproduces the dynamics of Bose-Einstein condensates, in a periodic potential, and in the superfluid regime. Such dynamics is governed by a discrete nonlinear Schrödinger equation. Several papers, addressing the study of the discrete nonlinear Schrödinger dynamics, have predicted the spontaneous generation of (classical) breathers in coupled condensates. In the present contribute, we shall focus on localized solutions (quantum breathers) of the full Bose-Hubbard model. We will show that solutions exponentially localized in space and periodic in time exist also in absence of randomness. Thus, this kind of states, reproduce a novel quantum localization phenomenon due to the interplay between bounded energy spectrum and non-linearity.     

Modulational instability and localized breather in discrete Schrödinger equation with helicoidal hopping and a power-law nonlinearity

We investigate the discrete nonlinear Schrödinger model with helicoidal hopping and a power-law nonlinearity, motivated by the tunable nonlinearity in the model of DNA chain and ultra-cold atoms trapped in a helix-shaped optical trap. In the study of modulational instability, we find a successive destabilization along with increasing nonlinear-power. In particular, the critical amplitudes of second-stage instability decrease as nonlinear-power increases. Furthermore, it is shown that information on the stability properties of weakly localized solutions can be inferred from the plane-wave modulational instability results. This link enable us to analytically estimate the critical parameters at which the breather solutions turn unstable, and find these parameters are dramatically influenced by the nonlinear-power. The stability properties of localized breathers perform an obvious change when the nonlinear power crosses a critical value ${\gamma }_{cr}$. It is reflected that at weak nonlinearity the breathers exhibit monostability, while exceeding ${\gamma }_{cr}$ the bistability and instability will set in. The interplay between nonlinear-power and long-range hopping on the stability properties of breathers is also discussed in detail.     

Rogue waves in the basin of intermediate depth and the possibility of their formation due to the modulational instability

The properties of rogue waves in the basin of intermediate depth are discussed in comparison with known properties of rogue waves in deep waters. Based on observations of rogue waves in the ocean of intermediate depth we demonstrate that the modulational instability can still play a significant role in their formation for basins of 20 m and larger depth. For basins of smaller depth, the influence of modulational instability is less probable. By using the rational solutions of the nonlinear Schrodinger equation (breathers), it is shown that the rogue wave packet becomes wider and contains more individual waves in intermediate rather than in deep waters, which is also confirmed by observations.     

Modulational instability for a self-attractive two-component Bose--Einstein condensate

By means of the multiple-scale expansion method, the coupled nonlinear Schr?dinger equations without an explicit external potential are obtained in two-dimensional geometry for a self-attractive Bose-Einstein condensate composed of different hyperfine states. The modulational instability of two-component condensate is investigated by using a simple technique. Based on the discussion about two typical cases, the explicit expression of the growth rate for a purely growing modulational instability and the optimum stable conditions are given and analysed analytically. The results show that the modulational instability of this two-dimensional system is quite different from that in a one-dimensional system.     

Modulational instability and spatial structures of the Ablowitz-Ladik equation

Spatial structures as a result of a modulational instability are obtained in the integrable discrete nonlinear Schrödinger equation (Ablowitz-Ladik equation). Discrete slow space variables are used in a general setting and the related finite differences are constructed. Analyzing the ensuing equation, we derive the modulational instability criterion from the discrete multiple scales approach. Numerical simulations in agreement with analytical studies lead to the disintegrations of the initial modulated waves into a train of pulses.     

Quantum Breathers in an Anisotropic Ferromagnetic Heisenberg Chain with Biquadratic Exchange Interaction

Based on the Hartree approximation and the semidiscrete multiple-scale method, quantum breathers in an anisotropic ferromagnetic Heisenberg chain with biquadratic exchange interaction are predicted analytically. The existence condition of quantum breathers is obtained, and the properties of quantum breathers are analyzed. Moreover, it is shown that the energy of such quantum breathers is quantized.     

Exact results for strongly correlated fermions in 2 + 1 dimensions

We derive exact results for a model of strongly interacting spinless fermions hopping on a two-dimensional lattice. By exploiting supersymmetry, we find the number and type of ground states exactly. Exploring various lattices and limits, we show how the ground states can be frustrated, quantum critical, or combine frustration with a Wigner crystal. We show that on generic lattices the model is in an exotic "superfrustrated" state characterized by an extensive ground-state entropy.     

Interacting signal packets in a lossless nonlinear transmission network with linear dispersion

In the small amplitude limit, we use the reductive perturbation method and the continuum limit approximation to derive a coupled nonlinear Schrö dinger (CNLS) equation describing the dynamics of two interacting signal packets in a discrete nonlinear electrical transmission line (NLTL) with linear dispersion. With the help of the derived CNLS equations, we present and analyze explicit expressions for the instability growth rate of a purely growing modulational instability (MI). We establish that the phenomenon of the MI can be observed only for “small” nonzero modulation wavenumbers. Also, we point out the effects of the linear dispersive element, as well as of the frequencies of the signal packets, on the instability growth rate. It is shown that the linear dispersion and the frequencies of signal packets can be well used to control the instability domain. Through the CNLS equations, we analytically investigate the propagation of solitary waves in the network. Our analytical studies show four types of interaction of signal packets propagating in the network: bright–bright, dark–dark, bright–dark and dark–bright soliton interactions.     

Modulational Instability and Energy Localization in a Chain of Interacting Frenkel Excitons

The modulational instability in one-dimensional molecule chain of interacting Frenkel excitons is investigated. The formation of localized modes via modulational instability is predicted and the previous numerical and analytical results are explained. The Peierls-Nabarro (PN) potential barrier for the localized modes is also discussed.     

Exact solutions for bright and dark solitons in spatially inhomogeneous nonlinearity

We present exact analytical results for bright and dark solitons in a type of one-dimensional spatially inhomogeneous nonlinearity. We show that the competition between a homogeneous self-defocusing (SDF) nonlinearity and a localized self-focusing (SF) nonlinearity supports stable fundamental bright solitons. For a specific choice of the nonlinear parameters, exact analytical solutions for fundamental bright solitons have been obtained. By applying both variational approximation and Vakhitov-Kolokolov stability criterion, it is found that exact fundamental bright solitons are stable. Our analytical results are also confirmed numerically. Additionally, we show that a homogeneous SF nonlinearity modulated by a localized SF nonlinearity allows the existence of exact dark solitons, for certain special cases of nonlinear parameters. By making use of linear stability analysis and direct numerical simulation, it is found that these exact dark solitons are linearly unstable.     

Discrete quantum breathers: what do we know about them?

The knowledge about discrete quantum breathers, accumulated during the last two decades, is reviewed. "Prehistory" of the problem is described and some important properties differentiating localized and extended vibrational modes are outlined. The state of art of our understanding of the principal features of the quantum discrete breathers is presented.