Whispering gallery like modes along pinned vortices

We study sound propagation in stationary and locally irrotational vortex flows where the circulation is wound around a long (rotating) cylinder, using Unruh’s formalism of acoustic space-times. Apart from the usual scattering solutions, we find anomalous modes which are bound to the vicinity of the cylinder and propagate along its axis—similar to whispering gallery modes. These modes exist for subsonic and supersonic flow velocities. In the supersonic case (corresponding to an effective ergoregion in the acoustic space-time), they can even have zero frequency ω = 0 and thus the associated quasiparticles with E = ?ω = 0 are easy to excite from an energetic point of view. Hence they should be relevant for the question of stability or instability of this setup.     

On stability of solutions of theU(N) chiral model in two dimensions

We discuss the stability properties of classical solutions of theU(N) sigma models in two Euclidean dimensions. We show that all nontrivial solutions are unstable. For a general case we exhibit one mode of instability; in some special cases (corresponding to a grassmannian solution and an instantonic grassannian embedding) we exhibit two such independent modes.     

Azimuthal modulational instability of vortices in the nonlinear Schrödinger equation

We study the azimuthal modulational instability of vortices with different topological charges, in the focusing two-dimensional nonlinear Schrödinger (NLS) equation. The method of studying the stability relies on freezing the radial direction in the Lagrangian functional of the NLS in order to form a quasi-one-dimensional azimuthal equation of motion, and then applying a stability analysis in Fourier space of the azimuthal modes. We formulate predictions of growth rates of individual modes and find that vortices are unstable below a critical azimuthal wave number. Steady-state vortex solutions are found by first using a variational approach to obtain an asymptotic analytical ansatz, and then using it as an initial condition to a numerical optimization routine. The stability analysis predictions are corroborated by direct numerical simulations of the NLS. We briefly show how to extend the method to encompass nonlocal nonlinearities that tend to stabilize such solutions.     

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.     

Disk-shaped Bose-Einstein condensates in the presence of an harmonic trap and an optical lattice

We study the existence and stability of solutions of the two-dimensional nonlinear Schrodinger equation in the combined presence of a parabolic and a periodic potential. The motivating physical example consists of Bose-Einstein condensates confined in an harmonic (e.g., magnetic) trap and an optical lattice. By connecting the nonlinear problem with the underlying linear spectrum, we examine the bifurcation of nonlinear modes out of the linear ones for both focusing and defocusing nonlinearities. In particular, we find real-valued solutions (such as multipoles) and complex-valued ones (such as vortices). A primary motivation of the present work is to develop "rules of thumb" about what waveforms to expect emerging in the nonlinear problem and about the stability of those modes. As a case example of the latter, we find that among the real-valued solutions, the one with larger norm for a fixed value of the chemical potential is expected to be unstable.     

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.     

(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.     

On utilizing delayed feedback for active-multimode vibration control of cantilever beams

We present a single-input single-output multimode delayed-feedback control methodology to mitigate the free vibrations of a flexible cantilever beam. For the purpose of controller design and stability analysis, we consider a reduced-order model consisting of the first n vibration modes. The temporal variation of these modes is represented by a set of nonlinearly coupled ordinary-differential equations that capture the evolving dynamics of the beam. Considering a linearized version of these equations, we derive a set of analytical conditions that are solved numerically to assess the stability of the closed-loop system. To verify these conditions, we characterize the stability boundaries using the first two vibration modes and compare them to damping contours obtained by long-time integration of the full nonlinear equations of motion. Simulations show excellent agreement between both approaches. We analyze the effect of the size and location of the piezoelectric patch and the location of the sensor on the stability of the response. We show that the stability boundaries are highly dependent on these parameters. Finally, we implement the controller on a cantilever beam for different controller gain-delay combinations and assess the performance using time histories of the beam response. Numerical simulations clearly demonstrate the controller ability to mitigate vibrations emanating from multiple modes simultaneously.     

Linear and nonlinear waves in surface and wedge index potentials

We study optical beams that are supported at the surface of a medium with a linear index potential and by a piecewise linear wedge-type potential. In the linear limit the modes are described by Airy functions. In the nonlinear regime we find families of solutions that bifurcate from the linear modes and study their stability for both self-focusing and self-defocusing Kerr nonlinearity. The total power of such nonlinear waves is finite without the need for apodization.     

Continuation of normal modes in finite NLS lattices

We study the continuation of breather solutions of the discrete NLS equation as the intersite coupling parameter is varied. Considering the case of a finite one-dimensional lattice of N sites, we show the existence of N branches of breathers that persist for arbitrary coupling, thus connecting normal modes of the linear system to breathers of the uncoupled, anticontinuous limit system. The proof is based on global bifurcation theory, applied to the continuation from the weakly nonlinear regime. As the coupling parameter varies these solutions generally change their stability, and we detect parameter regions where trajectories starting near unstable breathers appear to reach equipartition of power.     

Complex dynamics in delay-differential equations with large delay

We investigate the dynamical properties of delay differential equations with large delay. Starting from a mathematical discussion of the singular limit τ → ∞, we present a novel theoretical approach to the stability properties of stationary solutions in such systems. We introduce the notion of strong and weak instabilities and describe a method that allows us to calculate asymptotic approximations of the corresponding parts of the spectrum. The theoretical results are illustrated by several examples, including the control of unstable steady states of focus type by time delayed feedback control and the stability of external cavity modes in the Lang-Kobayashi system for semiconductor lasers with optical feedback.     

固体激光器的稳模式热稳条件分析

对传统的稳模式热稳条件进行了推广，使之也可适用于非球面驻波稳定腔的其它谐振腔，如环形腔等；引入了稳模式弱热稳条件的概念，进而提出了一种折衷的优化设计方法——使谐振腔同时满足稳功率热稳条件和稳模式弱热稳条件，从而使得激光器的输出既稳功率又具有一定的模式热稳定性，如发散角的热稳定性；此外，还给出了球面驻波稳定腔、球面环形稳定腔的稳模式强、弱热稳条件的解析解     

Stability of rod-shaped nanoparticles embedded in an elastic matrix

In many biological tissues as well as in some technical materials we find nano-sized rod-shaped particles embedded in a relatively soft matrix. Loss of stability of equilibrium, i.e. buckling, is one of the possible failure modes of such materials. In the present paper different kinds of load transfer between matrix and reinforcing particles, which are typical for rod-shaped nanostructures in biological tissues, are considered with respect to stability of equilibrium. Two regimes of matrix stiffnesses leading to different modes of buckling, and a transition regime in between, have been found: soft matrix materials leading to the so-called ‘flip mode’ (also called ‘tilt mode’) and hard matrix materials resulting in ‘bending mode’ buckling. The transition regime is of particular interest for biological tissues. Numerical and semi-analytical as well as asymptotic concepts are employed leading to results for estimating the critical load intensities both in the form of closed form solutions and diagrams. The analytical solutions are compared with results of finite element analyses. From these comparisons indications are gained for deciding which of the different analytical approaches should be chosen for a particular nanostructure configuration in terms of the associated buckling modes.     

Fermion Localization on the <Emphasis Type="Italic">orbifold</Emphasis><Emphasis Type="Italic">S</Emphasis><Subscript>1</Subscript>/<Emphasis Type="Italic">Z</Emphasis><Subscript>2</Subscript>

The hierarchical structure of fermion masses of the Standard Model is explained in split fermion models by localizing the fermions at different points in an extra dimension. We consider split fermion models with two bulk scalars compactified on an orbifold. In the static case we find analytical expression for the localizer. We also address the issue of stability of the localizer. We also find exact solutions for the fermion zero modes. We explore the parameter space of the model. We find ample opportunity for construction of phenomenologically viable theories exist.     

Nonlinear distribution function for a plasma obeying Vlasov-Maxwell system of equations

The nonlinear distribution function of Allis, generalised to include the transverse electromagnetic waves in a plasma, is used to set up the coupled wave equations for the longitudinal and the transverse modes. These are solved, keeping terms up to the cubic order of nonlinearity, by using the method of multiple scales. The equations of wave modulation are derived, which are solved to discuss the nature of the modulational instability and solitary wave propagation. It is found that the solutions so obtained satisfy conditions which are very similar to the well known Lighthill criterion for stability, appropriately modified due to the coupling of the two modes. The role of the average constant current due to any flow of the resonant and trapped electrons in determining the stability, is also discussed.     

Three-dimensional absolute and convective instabilities in mixed convection of a viscoelastic fluid through a porous medium

By using the mathematical formalism of absolute and convective instabilities we study the nature of unstable three-dimensional disturbances of viscoelastic flow convection in a porous medium with horizontal through-flow and vertical temperature gradient. Temporal stability analysis reveals that among three-dimensional (3D) modes the pure down-stream transverse rolls are favored for the onset of convection. In addition, by considering a spatiotemporal stability approach we found that all unstable 3D modes are convectively unstable except the transverse rolls which may experience a transition to absolute instability. The combined influence of through-flow and elastic parameters on the absolute instability threshold, wave number and frequency is then determined, and results are compared to those of a Newtonian fluid.     

Modal stability of inclined cables subjected to vertical support excitation

In this paper the out-of-plane dynamic stability of inclined cables subjected to in-plane vertical support excitation is investigated. We compute stability boundaries for the out-of-plane modes using rescaling and averaging methods. Our study focuses on the 2:1 internal resonance phenomenon between modes that occurs when the excitation frequency is twice the first out-of-plane natural frequency of the cable. The second in-plane mode is excited directly, while the out-of-plane modes can be excited parametrically. An analytical model is developed in order to study the stability regions in parameter space. In this model we include nonlinear coupling effects with other modes, which have thus far been omitted from previous models of parametric excitation of inclined cables. Our study reflects the importance of such effects. Unstable parameter regions are defined for the selected cable configuration. The validity of the proposed stability model was tested experimentally using a small-scale cable actuator rig. A comparison between experimental and analytical results is presented in which very good agreement with model predictions was obtained.     

On the quasinormal modes of the de Sitter spacetime

Modifying a method by Horowitz and Hubeny for asymptotically anti-de Sitter black holes, we establish the classical stability of the quasinormal modes of the de Sitter spacetime. Furthermore using a straightforward method we calculate the de Sitter quasinormal frequencies of the gravitational perturbations and discuss some properties of the radial functions of these quasinormal modes.     

Nondegenerate soliton solutions in certain coupled nonlinear Schrödinger systems

In this paper, we report a more general class of nondegenerate soliton solutions, associated with two distinct wave numbers in different modes, for a certain class of physically important integrable two component nonlinear Schrödinger type equations through bilinearization procedure. In particular, we consider coupled nonlinear Schrödinger (CNLS) equations (both focusing as well as mixed type nonlinearities), coherently coupled nonlinear Schrödinger (CCNLS) equations and long-wave-short-wave resonance interaction (LSRI) system. We point out that the obtained general form of soliton solutions exhibit novel profile structures than the previously known degenerate soliton solutions corresponding to identical wave numbers in both the modes. We show that such degenerate soliton solutions can be recovered from the newly derived nondegenerate soliton solutions as limiting cases.     

Hydrodynamical model for bulk and surface plasmons in cylindrical wires

In this paper, we study the electrostatic surface and volume modes of a cylindrical wire using the hydrodynamical model of plasmon excitation, which allows an analytical study of dispersion effects. We solve the hydrodynamical equations for a cylindrical wire geometry, obtaining new analytical expressions for the bulk and surface modes. New dispersion relations are obtained for each type of mode and numerical solutions are given. We analyze in detail the characteristics of the solutions and their differences with previous treatments based on non-dispersive models. These differences become important for wires of small radii, particularly in the range of few nanometers.