A quantum field theory of localized and resonant modes in 3D nonlinear lattices at finite temperatures I

The effective interaction Hamiltonian in 3D nonlinear lattices is established taking into account the repetitions of the up and down transition of an atom between two levels at the same site. The effective interaction Hamiltonian leads to the Heisenberg equation for phonon operators, which yields the conventional dynamical equation for displacements of atoms in 3D nonlinear lattices in the tree approximation by the boson transformation method. Making the one-loop approximation to the nonlinear potential in the Heisenberg equation, we obtain a dynamical equation with a self-consistent potential created by a localized or a resonant mode. In paper II, we show that the dynamical equation yields solutions for localized and resonant modes at finite temperatures.     

Localized modes in a one-dimensional sphalerite-structure lattice with anharmonicity

The nonlinear localized vibrational modes of a one-dimensional atomic chain with two periodically alternating masses and force constants are analytically investigated using a discrete multiple-scale expansion method. This model simulates a row of atoms in the <1 1 1>-direction of sphalerite, or zinc blende, crystals. Owing to the structural asymmetry, the vibrational amplitude is governed by a perturbed nonlinear Schr?dinger equation instead of the standard one found in one-dimensional lattices with two alternating masses but uniform force constant. Although the stationary localized modes with carrier wavevector at the Brillouin-zone boundary are similar to those of ionic lattices, the moving localized modes with wavevectors within the zone are different owing to the perturbation. The calculation shows that the height of the moving localized modes in this lattice dampens with time. Received 14 May 2001 and Received in final form 12 July 2001     

Beam propagation in gain–loss balanced waveguides

Linear and nonlinear localized modes of a slab waveguide with distributed gain and loss are studied. The structure is an optical analog of parity-time symmetric potentials in quantum mechanics. Such waveguide structures support stable localized modes. The explicit equation for the propagation constant of linear modes is obtained. The existence of stable nonlinear modes in such waveguides is demonstrated. Bend losses in such structures are analyzed.     

Two-dimensional dissipative solitons supported by localized gain

We show that the balance between localized gain and nonlinear cubic dissipation in the two-dimensional nonlinear Schr?dinger equation allows for the existence of stable localized modes that we identify as solitons. Such modes exist only when the gain is strong enough and the energy flow exceeds certain threshold value. Above the critical value of the gain, symmetry breaking occurs and asymmetric dissipative solitons emerge.     

Dynamical barrier for the formation of solitary waves in discrete lattices

We consider the problem of the existence of a dynamical barrier of “mass” that needs to be excited on a lattice site to lead to the formation and subsequent persistence of localized modes for a nonlinear Schrödinger lattice. We contrast the existence of a dynamical barrier with its absence in the static theory of localized modes in one spatial dimension. We suggest an energetic criterion that provides a sufficient, but not necessary, condition on the amplitude of a single-site initial condition required to form a solitary wave. We show that this effect is not one-dimensional by considering its two-dimensional analog. The existence of a sufficient condition for the excitation of localized modes in the non-integrable, discrete, nonlinear Schrödinger equation is compared to the dynamics of excitations in the integrable, both discrete and continuum, version of the nonlinear Schrödinger equation.     

Localization in physical systems described by discrete nonlinear Schrodinger-type equations

Following a short introduction on localized modes in a model system, namely the discrete nonlinear Schrodinger equation, we present explicit results pertaining to three different physical systems described by similar equations. The applications range from the Raman scattering spectra of a complex electronic material through intrinsic localized vibrational modes, to the manifestation of an abrupt and irreversible delocalizing transition of Bose-Einstein condensates trapped in two-dimensional optical lattices, and to the instabilities of localized modes in coupled arrays of optical waveguides.     

Quantum Solitons and Localized Modes in a One-Dimensional Lattice Chain with Nonlinear Substrate Potential

In the classical lattice theory, solitons and localized modes can exist in many one-dimensional nonlinear lattice chains, however, in the quantum lattice theory, whether quantum solitons and localized modes can exist or not in the one-dimensional lattice chains is an interesting problem. By using the number state method and the Hartree approximation combined with the method of multiple scales, we investigate quantum solitons and localized modes in a one-dimensional lattice chain with the nonlinear substrate potential. It is shown that quantum solitons do exist in this nonlinear lattice chain, and at the boundary of the phonon Brillouin zone, quantum solitons become quantum localized modes, phonons are pinned to the lattice of the vicinity at the central position j=j₀.     

Intrinsic Localized Spin-Wave Modes in an Anisotropic Ferromagnet with Dzyaloshinsky-Moriya Interaction

The existence and properties of intrinsic localized spin-wave modes in a ferromagnetic XXZ spin chain with Dzyaloshinsky-Moriya interaction are investigated analytically in the semiclassical limit. The model Hamiltonian is quantized by introducing the Dyson-Maleev transformation and the coherent state representation is chosen as the basic representation of the system. By making use of the method of multiple scales combined with a quasidiscreteness approximation, the equation of motion for the coherent-state amplitude is reduced to the nonlinear Schrödinger equation. It is shown that a bright intrinsic localized spin-wave mode whose eigenfrequency lies below the bottom of the magnon frequency band can exist in the ferromagnetic system. We also show that the system can produce a dark intrinsic localized spin-wave mode, i.e., nonpropagating kink, whose eigenfrequency is below the upper of the magnon frequency band. In addition, we find that the introduction of the Dzyaloshinsky-Moriya interaction changes wave numbers in the Brillouin-zone corresponding to the appearance of intrinsic localized spin-wave modes.     

Two-dimensional matter-wave solitons and vortices in competing cubic-quintic nonlinear lattices

The nonlinear lattice — a new and nonlinear class of periodic potentials — was recently introduced to generate various nonlinear localized modes. Several attempts failed to stabilize two-dimensional (2D) solitons against their intrinsic critical collapse in Kerr media. Here, we provide a possibility for supporting 2D matter-wave solitons and vortices in an extended setting — the cubic and quintic model — by introducing another nonlinear lattice whose period is controllable and can be different from its cubic counterpart, to its quintic nonlinearity, therefore making a fully “nonlinear quasi-crystal”.A variational approximation based on Gaussian ansatz is developed for the fundamental solitons and in particular, their stability exactly follows the inverted Vakhitov–Kolokolov stability criterion, whereas the vortex solitons are only studied by means of numerical methods. Stability regions for two types of localized mode — the fundamental and vortex solitons — are provided. A noteworthy feature of the localized solutions is that the vortex solitons are stable only when the period of the quintic nonlinear lattice is the same as the cubic one or when the quintic nonlinearity is constant, while the stable fundamental solitons can be created under looser conditions. Our physical setting (cubic-quintic model) is in the framework of the Gross–Pitaevskii equation or nonlinear Schrödinger equation, the predicted localized modes thus may be implemented in Bose–Einstein condensates and nonlinear optical media with tunable cubic and quintic nonlinearities.     

Localized and periodic wave patterns for a nonic nonlinear Schrödinger equation

Propagating modes in a class of ‘nonic’ derivative nonlinear Schrödinger equations incorporating ninth order nonlinearity are investigated by application of two key invariants of motion. A nonlinear equation for the squared wave amplitude is derived thereby which allows the exact representation of periodic patterns as well as localized bright and dark pulses in terms of elliptic and their classical hyperbolic limits. These modes represent a balance among cubic, quintic and nonic nonlinearities.     

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.     

Nonlinear localized modes in bandgap microcavities

We study experimentally an electrically pumped GaAs-based bandgap structure based on a vertical cavity surface emitting laser (VCSEL). We demonstrate that a microcavity embedded into this bandgap VCSEL structure supports localized optical modes without any holding beam. We propose a model of surface-structured VCSELs based on a reduced dissipative wave equation for describing electromagnetic modes in such semiconductor cavities and analyze a crossover between linear and nonlinear solitonlike cavity modes.     

(2+1)-Dimensional Spatial Localized Modes in Cubic-Quintic Nonlinear Media with the PT-Symmetric Potentials

We obtain exact spatial localized mode solutions of a (2+1)-dimensional nonlinear Schrödinger equation with constant diffraction and cubic-quintic nonlinearity in PT-symmetric potential, and study the linear stability of these solutions. Based on these results, we further derive exact spatial localized mode solutions in a cubic-quintic medium with harmonic and PT-symmetric potentials. Moreover, the dynamical behaviors of spatial localized modes in the exponential diffraction decreasing waveguide and the periodic distributed amplification system are investigated.     

Stability and instability of nonlinear defect states in the coupled mode equations—Analytical and numerical study

Coupled backward and forward wave amplitudes of an electromagnetic field propagating in a periodic and nonlinear medium at Bragg resonance are governed by the nonlinear coupled mode equations (NLCME). This system of PDEs, similar in structure to the Dirac equations, has gap soliton solutions that travel at any speed between 0 and the speed of light. A recently considered strategy for spatial trapping or capture of gap optical soliton light pulses is based on the appropriate design of localized defects in the periodic structure. Localized defects in the periodic structure give rise to defect modes, which persist as nonlinear defect modes as the amplitude is increased. Soliton trapping is the transfer of incoming soliton energy to nonlinear defect modes. To serve as targets for such energy transfer, nonlinear defect modes must be stable. We therefore investigate the stability of nonlinear defect modes. Resonance among discrete localized modes and radiation modes plays a role in the mechanism for stability and instability, in a manner analogous to the nonlinear Schrödinger/Gross-Pitaevskii (NLS/GP) equation. However, the nature of instabilities and how energy is exchanged among modes is considerably more complicated than for NLS/GP due, in part, to a continuous spectrum of radiation modes which is unbounded above and below. In this paper we (a) establish the instability of branches of nonlinear defect states which, for vanishing amplitude, have a linearization with eigenvalues embedded within the continuous spectrum, (b) numerically compute, using Evans function, the linearized spectrum of nonlinear defect states of an interesting multiparameter family of defects, and (c) perform direct time-dependent numerical simulations in which we observe the exchange of energy among discrete and continuum modes.     

在一维均匀铁磁链中磁振动的内禀局域模

利用多标度方法和准离散近似,我们考察了在一维均匀铁磁链中磁振动的内禀局域模; 结果表明磁振动的内禀局域模在许多方面都与晶格振动的内禀局域模相类似;它们是近邻自旋之间非线性相互作用的结果.这种内禀局域模的存在并没有破坏系统的平移对称性,它们能在任何晶格位被激发.它们的量子本征频率在简谐磁振动频带的上方.     

Self focusing of acoustically excited Faraday ripples on a bubble wall

A theoretical explanation is presented to explain pattern formation during the generation of Faraday waves on a bubble wall. The theory derives the Hamiltonian formulation of the nonlinear bubble dynamics. The nonlinear Schrödinger equation for the envelope of surface modes on the bubble wall has been obtained. The solitary wave solution predicts that the shape distortions should be localized near the equator of the bubble.     

Internal Modes of Relativistic Solitons

We apply a linear perturbation analysis to investigate the relationship between soliton oscillations and the integrability of nonlinear PDEs in bi-dimensional spacetime. For this purpose, we consider a localized solution of the nonlinear differential equation, and study small amplitude fluctuations around it. The linearized equation is a Schrödinger-like, eigenvalue problem. By considering several nonlinear PDEs, which are known to have soliton and solitary wave solutions, we find that in systems which are integrable, this eigenvalue equation has one and only one bound state with zero frequency. Non-integrable equations—in contrast—show extra bound states. The time evolution of the oscillations are also calculated, using a numerical program to integrate the time-dependent equation. The behavior of the modes are studied, using the Fourier transform of the evolving solutions.     

TWO-DIMENSIONAL LOCALIZED MODES IN CONJUGATED POLYMERS: THE NONLINEAR ELECTRON-PHONON COUPLING EFFECT

The two-dimensional localized modes around a soliton have been investigated by using an extension of Su-Shrieffer-Heeger model, in which is included the nonlinear term of electron-phonon interaction. The results show that there appears an additional localized mode, and the two modes obtained in the previous work without the nonlinear term disappear. Furthermore, the frequencies of the modes are shifted and their localizations are changed after turning on the nonlinear term.     

Stationary dark localized modes: discrete nonlinear Schrödinger equations

Various kinds of stationary dark localized modes in discrete nonlinear Schr?dinger equations are considered. A criterion for the existence of such excitations is introduced and an estimation of a localization region is provided. The results are illustrated in examples of the deformable discrete nonlinear Schr?dinger equation, of the model of Frenkel excitons in a chain of two-level atoms, and of the model of a one-dimensional Heisenberg ferromagnetic in the stationary phase approximation. The three models display essentially different properties. It is shown that at an arbitrary amplitude of the background it is impossible to reach strong localization of dark modes. In the meantime, in the model of Frenkel excitons, exact dark compacton solutions are found.     

Stability of strongly localized excitations in discrete media with cubic nonlinearity

By using a linear analysis it is analytically shown that the stability of strongly localized modes depends on their symmetry, the sign of nonlinearity, and the degree of localization. The existence of a stable, bright, even mode of the discrete nonlinear Schrödinger equation is demonstrated and confirmed by direct numerical simulations. Possible applications to all-optical switching are discussed.