Linear Coupling Terms Research Articles

Periodic solutions, breathers and rogue waves in a generalized coupled Hirota system

In this paper, a generalized coupled Hirota system with higher-order and linear coupling terms has been investigated. Based on the Lax integrable property, the Darboux transformation (DT) algorithm has been constructed. Some new solutions under the vanishing and non-vanishing backgrounds are derived, including periodic solutions, breathers and rogue wave solutions. By some graphic simulations the propagation characteristics of those solutions influenced by higher-order and linear coupling terms are analyzed.

Spatial, temporal, and spatio-temporal modulational instabilities in a planar dual-core waveguide

We investigate modulational instability (MI) in a planar dual-core waveguide (DWG), with a Kerr and non-Kerr polarizations based on coupled nonlinear Schrödinger equations in the presence of linear coupling term, coupling coefficient dispersion (CCD) and other higher order effects such as third order dispersion (TOD), fourth order dispersion (FOD), and self-steepening (ss). By employing a standard linear stability analysis, we obtain analytically, an explicit expression for the MI growth rate as a function of spatial and temporal frequencies of the perturbation and the material response time. Pertinently, we explicate three different types of MI—spatial, temporal, and spatio-temporal MI for symmetric/antisymmetric continuous wave (cw), and spatial MI for asymmetric cw, and emphasize that the earlier studies on MI in DWG do not account for this physics. Essentially, we discuss two cases: (i) the case for which the two waveguides are linearly coupled and the CCD term plays no role and (ii) the case for which the linear coupling term is zero and the CCD term is nonzero. In the former case, we find that the MI growth rate in the three different types of MI, seriously depends on the coupling term, quintic nonlinearity, FOD, and ss. In the later case, the presence of quintic nonlinearity, CCD, FOD, and ss seriously enhances the formation of MI sidebands, both in normal as well as anomalous dispersion regimes. For asymmetric cw, spatial MI is dependent on linear coupling term and quintic nonlinearity.

Nonlinear switching in multicore versus multimode waveguide junctions for mode-locked laser applications

The main differences in nonlinear switching behavior between multicore versus multimode waveguide couplers are highlighted. By gradually decreasing the separation between the two cores of a dual-core waveguide and interpolating from a multicore to a multimode scenario, the role of the linear coupling, self-phase modulation, cross-phase modulation, and four-wave mixing terms are explored, and the key reasons are identified behind higher switching power requirements and lower switching quality in multimode nonlinear couplers.

Non-Gaussian Test Models for Prediction and State Estimation with Model Errors

Turbulent dynamical systems involve dynamics with both a large dimensional phase space and a large number of positive Lyapunov exponents. Such systems are ubiquitous in applications in contemporary science and engineering where the statistical ensemble prediction and the real time filtering/state estimation are needed despite the underlying complexity of the system. Statistically exactly solvable test models have a crucial role to provide firm mathematical underpinning or new algorithms for vastly more complex scientific phenomena. Here, a class of statistically exactly solvable non-Gaussian test models is introduced, where a generalized Feynman-Kac formulation reduces the exact behavior of conditional statistical moments to the solution to inhomogeneous Fokker-Planck equations modified by linear lower order coupling and source terms. This procedure is applied to a test model with hidden instabilities and is combined with information theory to address two important issues in the contemporary statistical prediction of turbulent dynamical systems: the coarse-grained ensemble prediction in a perfect model and the improving long range forecasting in imperfect models. The models discussed here should be useful for many other applications and algorithms for the real time prediction and the state estimation.

Coupling of In-Plane Flexural, Tangential, and Shear Wave Modes of a Curved Beam

In this paper, the coupling effects among three elastic wave modes, flexural, tangential, and radial shear, on the dynamics of a planar curved beam are assessed. Two sets of equations of motion governing the in-plane motion of a curved beam are derived, in a consistent manner, based on the theory of elasticity and Hamilton’s principle. The first set of equations retains all resulting linear coupling terms that includes both static and dynamic coupling among the three wave modes. In the derivation of the second set of equations, the effects of Coriolis acceleration and high-order elastic coupling terms are neglected, which leads to a set of equations without dynamic coupling terms between the tangential and shear wave modes. This second set of equations of motion is the one most commonly used in studies on thick curved beams that include the effects of centerline extensibility, rotary inertia, and shear deformation. The assessment is carried out by comparing the dynamic behavior predicted by each curved beam model in terms of the dispersion relations, frequency spectra, cutoff frequencies, natural frequencies and mode shapes, and frequency responses. In order to ensure the comparison is based on accurate results, the wave propagation technique is applied to obtain exact wave solutions. The results suggest that the contributions of the dynamic and high-order elastic coupling terms to the response of a thick curved beam are quite significant and that these coupling terms should not be neglected for an accurate analysis of a thick curved beam with a large curvature parameter.

A numerical study of compressible turbulent boundary layers

Compressible turbulent boundary layers with free-stream Mach number ranging from 2.5 up to 20 are analyzed by means of direct numerical simulation of the Navier–Stokes equations. The fluid is assumed to be an ideal gas with constant specific heats. The simulation generates its inflow condition using the rescaling-recycling method. The main objective is to study the effect of Mach number on turbulence statistics and near-wall turbulence structures. The present study shows that supersonic/hypersonic boundary layers at zero pressure gradient exhibit close similarities to incompressible boundary layers and that the main turbulence statistics can be correctly described as variable-density extensions of incompressible results. The study also shows that the spanwise streak’s spacing of 100 wall units in the inner region (y+≈15) still holds for the considered high Mach numbers. The probability density function of the velocity dilatation shows significant variations as the Mach number is increased, but it can also be normalized by accounting for the variable-density effect. The compressible boundary layer also shows an additional similarity to the incompressible boundary layer in the sense that without the linear coupling term, near-wall turbulence cannot be sustained.

Effect of coupled ferroelectric and antiferromagnetic fluctuations on dielectric anomalies inspin induced multiferroics

Dielectric and magnetodielectric peaks have been evidenced byε(T, H) measurements in spininduced ferroelectrics: CuCrO2 and AgCrO2 delafossites. Such behaviour, also found in several other improper ferroelectrics,can be explained in the frame of Landau analysis of phase transitionswith two coupled order parameters: antiferromagnetic ordered moment,L, andpolarization, P. The existence of such anomalies in the dielectric constant observed atTN is very general. The existence of this peak is not due to any linear coupling term betweenP andL in this system, butrather due to the L2P2 term which always exists in every compound, whatever the symmetry/space group is.

ChemInform Abstract: The Jahn-Teller Effect in Triptycene.

The irregular vibronic structure resolved in the S1←S0 resonant two‐photon ionization (R2PI) spectrum of supersonically cooled triptycene (9,10‐dihydro‐9,10[1’2’]benzenoanthracene) is assigned in terms of a single‐mode E’⊗e’ Jahn–Teller vibronic Hamiltonian for the excited state, with linear and quadratic coupling terms. The Jahn–Teller active vibrational mode is a benzene wagging framework mode. To fit to the observed vibronic levels yields a very low frequency νe’ =47.83 cm−1 and linear and quadratic terms are k=1.65 and g=0.426. This fit accounts for ≊98% of the observed absorption band intensities over the observable range 0–350 cm−1. The quadratic term is unusually large, leading to localization of the lowest vibronic levels in the three symmetry‐equivalent minima. Emission spectra from 13 vibronic levels in the excited E’ state show extended vibrational progressions with up to 25 members in the analogous e’ ground state vibration, which is highly harmonic in the electronic ground state. The Franck–C...

Three dimensional classical theory of rainbow scattering of atoms from surfaces

In this work, we extend to three dimensions our previous stochastic classical theory on surface rainbow scattering. The stochastic phonon bath is modeled in terms of linear coupling of the phonon modes to the motion of the scattered particle. We take into account the three polarizations of the phonons. Closed formulae are derived for the angular and energy loss distributions. They are readily implemented when assuming that the vertical interaction with the surface is described by a Morse potential. The hard wall limit of the theory is derived and applied to some model corrugated potentials. We find that rainbow structure of the scattered angular distribution reflects the underlying symmetries of the surface. We also distinguish between “normal rainbows” and “super rainbows”. The latter occur when the two eigenvalues of the Hessian of the corrugation function vanish simultaneously.

Relativistic Jahn–Teller effects in the quartet states of K3 and Rb3: A vibronic analysis of the 2 E4′←1 A42′ electronic transitions based on ab initio calculations

We apply second-order multireference Rayleigh-Schrodinger perturbation theory to obtain the adiabatic potential energy surface of the 1 (4)A(2)(') lowest quartet state and the 2 (4)E(') excited state of K(3) and Rb(3). Both trimers show a typical Emultiply sign in circlee Jahn-Teller distortion in their 2 (4)E(') state, which is analyzed in terms of relativistic Jahn-Teller effect theory. Linear, quadratic, and spin-orbit coupling terms are extracted from the ab initio results and used to generate simulated spectra for a direct comparison with laser-induced fluorescence and magnetic circular dichroism spectra of alkali-doped helium nanodroplets [Aubock et al., J. Chem. Phys. 129, 114501 (2008)].

Theoretical Study of the Jahn−Teller Effect in CH3CN+(X̃2E) and CD3CN+(X̃2E): Multimode Spin-Vibronic Energy Level Calculations

The spin-vibronic energy levels for CH(3)CN(+)(X(2)E) and CD(3)CN(+)(X(2)E) have been calculated using a diabatic model including multimode vibronic couplings and spin-orbit interaction without adjusting any parameter. The diabatic potential energy surfaces are represented by the Taylor expansions including linear, quadratic and bilinear vibronic coupling terms. The normal coordinates used in the Taylor expansion were expressed by the mass-weighted Cartesian coordinates. The adiabatic potential energy surfaces for CH(3)CN(+) and CD(3)CN(+) were calculated at the level of CASPT2/cc-pvtz, and the spin-orbit coupling constant was calculated at the level of MRCI/CAS/cc-pvtz. The spin-orbit energy splittings for the ground vibrational states of CH(3)CN(+)(X(2)E) and CD(3)CN(+)(X(2)E) are 20 and 16 cm(-1), respectively, which are resulted from the quenching of the spin-orbit coupling strength of 51 cm(-1). The calculated spin-vibronic levels are in good agreement with the experimental data. The calculation results show that the Jahn-Teller effects in CH(3)CN(+)(X(2)E) and CD(3)CN(+)(X(2)E) are essential to understand their spin-vibronic energy structure.

Soliton dynamics in linearly coupled discrete nonlinear Schrödinger equations

We study soliton dynamics in a system of two linearly coupled discrete nonlinear Schrödinger equations, which describe the dynamics of a two-component Bose gas, coupled by an electromagnetic field, and confined in a strong optical lattice. When the nonlinear coupling strengths are equal, we use a unitary transformation to remove the linear coupling terms, and show that the existing soliton solutions oscillate from one species to the other. When the nonlinear coupling strengths are different, the soliton dynamics is numerically investigated and the findings are compared to the results of an effective two-mode model. The case of two linearly coupled Ablowitz–Ladik equations is also briefly discussed.

Impact of Linear Coupling on Thermoacoustic Instabilities

There have been conflicting opinions in literature, about the consequences of ignoring the linear coupling terms in finite-dimensional thermoacoustic models. We try to clarify this issue in this paper. We study consequences of ignoring the linear coupling terms on the non-normal thermoacoustic interactions in a horizontal Rijke tube. In the Galerkin formulation, we use pseudospectra and Henrici index of non-normality to study the impact of linear coupling between the Galerkin modes on the non-normality and the ensuing transient growth. This linear coupling is important for the energy transfer between the eigenmodes of the system (which are not the same as the natural acoustic modes of the duct). Although ignoring this linear coupling results in negligible shift in the eigenfrequencies, it may have serious consequences on the non-normality of the thermoacoustic system and its dynamics. Nonlinear phenomena such as triggering, boot-strapping and limit cycle oscillations may be completely missed out when this linear coupling is ignored. Ignoring the linear coupling between the Galerkin modes is justified only when the non-normality and the nonlinear dynamics of the system are not altered significantly.

Averaging Method Analysis of Synchronization Characteristics of a Large Number of Nonlinearly Coupled van der Pol Oscillators

The averaged equation for an arbitrary number of oscillators coupled by nonlinear coupling scheme invented by S. Nagano, is derived. This system is invented as a model of uni-cellular slime amoeba. By using the averaged equation, we investigate the synchronization characteristics of five coupled oscillators and a large number of coupled oscillators. In particular, we present the statistical property of coupled oscillators in terms of coupling factor γ. We also investigate the effect of linear and nonlinear coupling terms for achieving synchronization, and confirm that the nonlinear coupling term plays an important role for strong synchronization than linear coupling term does.

Effects of nonlinear wave coupling: Accelerated solitons

Boomerons are described as accelerated solitons for special integrable systems of coupled wave equations. A general formalism based on the Lax pair method is set up to introduce such systems which look of Nonlinear Schrödinger-type with linear, quadratic and cubic coupling terms. The one-soliton solution of such general systems is also briefly discussed. We display special instances of wave systems which are of potential interest for applications, including dispersion-less models of resonating waves. Among these, special attention and details are given to the celebrated equations describing the resonant interaction of three waves, in view of their application to optical pulse propagation in quadratic nonlinear media. For this particular case, we present exact solutions of the three-wave resonant interaction system, in the form of triplets moving with a common nonlinear velocity (simultons). The simultons have nontrivial phase-fronts and exist for different velocities and energy flows. We studied simulton stability upon propagation, and found that solitons with a velocity greater than a certain critical value are stable. We explore a novel consequence of the particle-like nature of three-wave simultons, namely their inelastic scattering with particular linear waves. Such phenomenon is associated with the excitation (decay) of stable (unstable) simultons by means of the absorption (emission) of the energy carried by a particular isolated pulse. Inelastic processes are exactly described in terms of boomerons. We also briefly consider collisions between different three-wave simultons.

Low-Temperature Phase Transformations ofPbZr1−xTixO3in the Morphotropic Phase-Boundary Region

We present anelastic and dielectric spectroscopy measurements of PbZr(1-x)Ti(x)O(3) with 0.455 < or = x < or = 0.53, which provide new information on the low-temperature phase transitions. The tetragonal-to-monoclinic transformation is first order for x < 0.48 and causes a softening of the polycrystal Young's modulus whose amplitude may exceed the one at the cubic-to-tetragonal transformation; this is explainable in terms of linear coupling between shear strain components and tilting angle of polarization in the monoclinic phase. The transition involving rotations of the octahedra below 200 K is visible both in the dielectric and anelastic losses, and it extends within the tetragonal phase, as predicted by recent first-principle calculations.

Foundations of nonlinear gyrokinetic theory

Nonlinear gyrokinetic equations play a fundamental role in our understanding of the long-time behavior of strongly magnetized plasmas. The foundations of modern nonlinear gyrokinetic the- ory are based on three important pillars: (1) a gyrokinetic Vlasov equation written in terms of a gyrocenter Hamiltonian with quadratic low-frequency ponderomotive-like terms; (2) a set of gyrokinetic Maxwell (Poisson-Ampere) equations written in terms of the gyrocenter Vlasov dis- tribution that contain low-frequency polarization (Poisson) and magnetization (Ampere) terms derived from the quadratic nonlinearities in the gyrocenter Hamiltonian; and (3) an exact energy conservationlaw for the gyrokineticVlasov-Maxwell equations that includes all the relevant linear and nonlinear coupling terms. The foundations of nonlinear gyrokinetic theory are reviewed with an emphasis on the rigorous applications of Lagrangian and Hamiltonian Lie-transform perturba- tion methods used in the variationalderivationof nonlineargyrokineticVlasov-Maxwell equations. The physical motivations and applications of the nonlinear gyrokinetic equations, which describe the turbulent evolution of low-frequency electromagnetic fluctuations in a nonuniform magnetized plasmas with arbitrary magnetic geometry, are also discussed.

Jahn-Teller and related effects in the silver trimer. II: Vibrational analysis of the ÃE″2-X̃E′2 electronic transition

The laser-excited, jet-cooled A 2E"-X 2E' electronic spectrum of the silver trimer yields detailed information about its A- and X-state vibronic structure. Following extensive parameter fitting, the absorption and emission spectra are simulated and the bands are assigned. The Jahn-Teller analysis includes both linear and quadratic coupling terms, considered simultaneously with spin-orbit coupling. The spin-orbit splitting is shown to be largely quenched in both the A and X electronic states. The Jahn-Teller analysis of the A and X vibronic structures reveals the distortion of their corresponding potential energy surfaces.

Vibration of the Translation and the Inclination Motions Coupled System Under the Periodic Base Motion (Auto-parametric Resonance and Influences of Height of Center of Mass, Unbalance of Mass and Difference between Support Stiffness)

This is a study on the vibration characteristics of the translation and the inclination motions-coupled systems, which is the dynamical model of a train, a vehicle, and so on. It mainly investigates the nonlinear vibration of such a system which is subject to the base periodic displacement. The study observes the occurrence of the auto-parametric resonance at the resonance point of the translational motion, when the ratio between the natural frequencies of the inclinational motion and the translational motion is close to one-to-two. Moreover, the occurrence of an almost periodic motion is observed when this ratio is slightly different from one-to-two. The main discussion is based on the occurrence regions of the auto-parametric resonance and the almost periodic motion. This paper clarifies that the occurrence region of the auto-parametric resonance widely exists from about the ratio two-to-three to about the ratio one-to-three. It also notes on the influence of unbalance of the center of mass which produces slight linear coupling terms between the translational motion and the inclinational motion. The results of this paper indicate that the auto-parametric resonance is not the special phenomenon and it easily occurs. These theoretical results are confirmed by the experiments.

Coupling of structure and wake oscillators in vortex-induced vibrations

A class of low-order models for vortex-induced vibrations is analyzed. A classical van der Pol equation models the near wake dynamics describing the fluctuating nature of vortex shedding. This wake oscillator interacts with the equation of motion of a one degree-of-freedom structural oscillator and several types of linear coupling terms modelling the fluid–structure interaction are considered. The model dynamics is investigated analytically and discussed with regard to the choice of the coupling terms and the values of model parameters. Closed-form relations of the model response are derived and compared to experimental results on forced and free vortex-induced vibrations. This allows us to set the values of all model parameters, then leads to the choice of the most appropriate coupling model. A linear inertia force acting on the fluid is thus found to describe most of the features of vortex-induced vibration phenomenology, such as Griffin plots and lock-in domains.