Direct Perturbation Method Research Articles

Direct Perturbation Method for Derivative Nonlinear Schrödinger Equation

We extend Lou's direct perturbation method for solving the nonlinear Schrödinger equation to the case of the derivative nonlinear Schrödinger equation (DNLSE). By applying this method, different types of perturbation solutions are obtained. Based on these approximate solutions, the analytical forms of soliton parameters, such as the velocity, the width and the initial position, are carried out and the effects of perturbation on solitons are analyzed at the same time. A numerical simulation of perturbed DNLSE finally verifies the results of the perturbation method.

Asymptotical solutions of coupled nonlinear Schrödinger equations with perturbations

In this paper Lou's direct perturbation method is applied to the perturbed coupled nonlinear Schrodinger equations to obtain their asymptotical solutions, which include not only the zero-order solutions but also the first-order modifications. Based on the asymptotical solutions, the effects of perturbations on soliton parameters and the collision between two solitons are then discussed in brief. Furthermore, we directly simulate the perturbed coupled nonlinear Schrodinger equations by split-step Fourier method to check the validity of the direct perturbation method. It turns out that our analytical results are well supported by the numerical calculations.

Approximate Solutions of Perturbed Nonlinear SchrödingerEquations

By applying Lou's direct perturbation method toperturbed nonlinear Schrödinger equation and the criticalnonlinear Schrödinger equation with a small dispersion, theirapproximate analytical solutions including the zero-order and thefirst-order solutions are obtained. Based on these approximatesolutions, the analytical forms of parameters of solitons areexpressed and the effects of perturbations on solitons are brieflyanalyzed at the same time. In addition, the perturbed nonlinearSchrödinger equations is directly simulated by split-stepFourier method to check the validity of the direct perturbationmethod. It turns out that the analytical results given by thedirect perturbation method are well supported by numerical calculations.

Generation and control of chaos in a Bose–Einstein condensate

For a Bose-Einstein condensate (BEC) confined in a double lattice consisting of two weak laser standing waves we find the Melnikov chaotic solution and chaotic region of parameter space by using the direct perturbation method. In the chaotic region, spatial evolutions of the chaotic solution and the corresponding distribution of particle number density are bounded but unpredictable between their superior and inferior limits. It is illustrated that when the relation k1 ≈ k2 between the two laser wave vectors is kept, the adjustment from k2 < k1 to k2 ≥ k1 can transform the chaotic region into regular one or the other way round. This suggests a feasible scheme for generating and controlling chaos, which could lead to an experimental observation in the near future.

Stationary Classical Chaos of Trapped Two-ComponentBose–Einstein Condensates in 1-D OpticalLattice Potentials

We study the nonlinear dynamics of two-component Bose–Einsteincondensates in one-dimensional periodic optical lattice potentials. The stationary state perturbation solutions of the coupled two-component nonlinear Schrödinger/Gross–Pitaevskii equations are constructed by using the direct perturbation method. Theoretical analysis revels that the perturbation solution is the chaotic one, which indicates the existence of chaos and chaotic region in parameter space. The corresponding numerical calculation results agree well with the analytical results. By applying the chaotic perturbation solution, we demonstrate the atomic spatial population and the energy distribution of the system are chaotic generally.

Adjoint Sensitivity of an Ocean General Circulation Model to Bottom Topography

Abstract Bottom topography, or more generally the geometry of the ocean basins, is an important ingredient in numerical ocean modeling. With the help of an adjoint model, it is shown that scalar diagnostics or objective functions in a coarse-resolution model, such as the transport through Drake Passage, the strength of the Atlantic Ocean meridional overturning circulation, the Deacon cell, and the meridional heat transport across 32°S, are sensitive to bottom topography as much as they are to surface boundary conditions. For example, adjoint topography sensitivities of the transport through Drake Passage are large in choke-point areas such as the Crozet–Kerguélen Plateau and south of New Zealand; the Atlantic meridional overturning circulation is sensitive to topography in the western boundary region of the North Atlantic Ocean and along the Scotland–Iceland Ridge. Many sensitivities are connected to steep topography and can be interpreted in terms of bottom form stress, that is, the product of bottom pressure and topography gradient. The adjoint sensitivities are found to agree with direct perturbation methods with deviations smaller than 30% for significant perturbations on time scales of 100 yr, so that the assumption of quasi linearity that is implicit in the adjoint method holds. The horizontal resolution of the numerical model affects the sensitivities to bottom topography, but large-scale patterns and the overall impact of changes in topography appear to be robust. The relative impact of changes in topography and surface boundary conditions on the model circulation is estimated by multiplying the adjoint sensitivities with assumed uncertainties. If the uncertainties are correlated in space, changing the surface boundary conditions has a larger impact on the scalar diagnostics than topography does, but the effects can locally be on the same order of magnitude if uncorrelated uncertainties are assumed. In either case, bottom topography variations within their prior uncertainties affect the solution of an ocean circulation model. To this extent, including topography in the control vector can be expected to compensate for identifiable model errors and, thus, to improve the solutions of estimation problems.

Asymptotic solutions of perturbed N-component nonlinear Schr?dinger equations

We apply the direct perturbation method to an integrable nonlinear Schrdinger equation with a correction term to obtain its asymptotic solutions. It is shown that there is a good qualitative agreement between the asymptotic and the exact solutions when ε is small enough. Then the direct perturbation method is applied to the perturbed N-component nonlinear Schrdinger equations and their asymptotical solutions are obtained.

Effect of Higher-Order Terms on Nonlinear Schrödinger DarkSolitons in Optical Fibres

We solve the nonlinear Schrödinger equation with higher order terms,which describes the propagation of femtosecond pulses in optical fibres,by means of a direct perturbation method for dark solitons. An exactdark-soliton solution of the higher order nonlinear Schrödingerequation is found, provided that parameters satisfy certain conditions.The soliton's velocity with the high order effects as well as theformula for calculation of the first-order correction representing theperturbation-induced radiative field is presented.

Buckling of non-uniform beams by a direct functional perturbation method

This study is divided into two parts. In the first, the buckling load ( P ) of heterogeneous columns is found by applying the Functional Perturbation Method (FPM) directly to the Buckling (eigenvalue) Differential Equation (BDE). The FPM is based on considering P and the transverse deflection ( W ) as functionals of heterogeneity, i.e., the elastic bending stiffness “ K ” (or the compliance S = 1 / K ). The BDE is expanded functionally, yielding a set of successive differential equations for each order of the (Fréchet) functional derivatives of P and W . The obtained differential equations differ only in their RHS, and therefore a single modified Green function is needed for solving all orders. Consequently, an approximated value for the buckling load is obtained for any given morphology. Both deterministic and stochastic examples of simply supported columns are solved and discussed. Results are compared with solutions found in the literature for validation. In the second part, the Optimized DFPM (ODFPM) is presented. It is based on finding a new material property (which is a function of K or S ) around which the DFPM solution is more accurate. The new material property is found by requiring that the second order perturbation term in the Fréchet series is minimized. As a result, a non-linear differential equation is obtained which relates the new material property with K through morphology. An exact solution for this equation is found, in a power form K f , where f depends on morphology. Calculating P with respect to this new property yields more accurate results for the statistical characteristics of P .

Chaotic phase oscillation of a proton beam in a synchrotron

We investigate the chaotic phase oscillation of a proton beam in a cooler synchrotron. By using direct perturbation method, we construct the general solution of the 1st-order equation. It is demonstrated that the general solution is bounded under some initial and parameter conditions. From these conditions, we get a Melnikov function which predicts the existence of Smale-horseshoe chaos iff it has simple zeros. Our result under the 1st-order approximation is in good agreement with that in [H. Huang et al., Phys. Rev. E 48 (1993) 4678]. When the perturbation method is not suitable for the system, numerical simulation shows the system may present transient chaos before it goes into periodical oscillation; changing the damping parameter can result in or suppress stationary chaos.

A direct perturbation method investigation of the interplay between self-frequency shift and cross-phase modulation for solitons

The self-frequency shift of solitons, mainly observed in femtosecond pulses and arising from the Raman effect, can be counteracted by the cross-phase modulation (XPM) arising from collisions between pulses of different frequency. The same could be said for the time displacement that follows the frequency shift. The current work is an analytical approach to the interplay between the Raman self-frequency shift and XPM with third-order dispersion (TOD) effects taken into account. By using coupled NLS equations, the effect of XPM between two channels is considered for the cases of complete and incomplete collision. The analysis is based on the direct perturbation method and provides quantitative and qualitative insight of the spectral and temporal evolution of ultra short soliton pulses.

Conditional stability of diatomic molecule driven by a weak laser field

Using a direct perturbation method, we investigate the stability of a diatomic molecule modelled by a weakly laser-driven Morse oscillator. It is shown that stationary state solution of the system is stable in the sense of Lyapunov and the periodical one possesses conditional stability, namely its stability depends on the initial conditions and system parameters. The corresponding sufficient and necessary conditions are established that indicate the stable states associated with some discrete energies. The results reveal how a diatomic molecule can be stabilized or dissociated with a weak laser and demonstrate that the mathematical conditional stability works in the considered physical system.

Exploring the connection between quasistationary and squared eigenfunction expansion techniques in soliton perturbation theory

We give a short review of the techniques that have been used to study soliton perturbations for nonlinear integrable systems in 1 + 1 dimensions. In particular, we discuss the connection between direct perturbation methods using squared eigenfunction expansions and quasistationary perturbation theory. We show for the KdV (Korteweg–de Vries) how the results for quasistationary approximations are related to those for the standard adiabatic perturbation theory.

Green's Function Method for Perturbed sine-Gordon Equation**The project supported by National Natural Science Foundation of China under Grant Nos. 10375041 and 10174054

The usual Green's function method is introduced to show the completeness of the squared Jost solutions of the multi-soliton case in this work. Since sine-Gordon equation contains second derivative in time, the known procedure with generalized Marchenko equation to show completeness relation of the eigenfunctions of linearized equation is unavailable. And the explicit expressions of Jost solutions are not necessary here. Thus a general method of direct perturbation method for the perturbed sine-Gordon equation is developed.

Numerical Simulations of the Fluxon Dynamics of a Two-Dimensional Josephson Junction System

A direct perturbation method applied to a long Josephson junction molded by one-dimensional sine-Gordon equation is presented and the modulation of perturbations on fluxon velocity and stable state is studied. Simulations of a two-dimensional Josephson junction with dc bias current in an external magnetic field provide an I–V characteristic curve for the system and determine the various dynamic behaviors, including the periodic, quasiperiodic, and chaotic motions. Physical explanations for the behaviors are presented. The numerical results are in good agreement with the energetic analysis.

Stochastic Sensitivity of 3D-Elastodynamic Response Subjected to Random Ground Excitation

The present study deals with structural sensitivity of dynamic response having uncertainties in design parameters subjected to random earthquake loading. Earthquake is modeled as stationary random process defined by Kanai–Tajimi power spectral density. The uncertain design parameters are modeled as homogeneous Gaussian process and discretized through 3D local averaging. Subsequently the Cholesky decomposition of respective co-variance matrix is used to simulate random values of design parameters. The Neumann expansion blended with Monte Carlo simulation (NE-MCS) is explored for computing response sensitivity in frequency domain. Application examples related to a building frame and a gravity dam are presented serving to validate the NE-MCS technique in terms of its accuracy and effectiveness compared to direct Monte Carlo simulation and perturbation method.

Chaotic Solutions of a Typical Nonlinear Oscillator in a Double Potential Trap

We have obtained a general unstable chaotic solution of a typical nonlinear oscillator in a double potential trap with weak periodic perturbations by using the direct perturbation method. Theoretical analysis reveals that the stable periodic orbits are embedded in the Melnikov chaotic attractors. The corresponding chaotic region and orbits in parameter space are described by numerical simulations.

Nonlinear Excitations in Hydrogen-Bonded Chains with Anharmonic Interatomic Interactions

We discuss the nonlinear excitations and the motion of a kink soliton pair in hydrogen-bonded chains with anharmonic interatomic interactions, based on a two-component soliton model, using a direct perturbation method. The expression for the asymmetric solutions of the kink soliton pair is found because of anharmonicity, and the energy, the momentum and the effective mass of a kink pair for cubic and quartic anharmonicity are calculated, which are in good agreement with the experimental data.

Direct perturbation analysis on cables with concentrated loads

The dynamics of cables with concentrated loads is studied using a direct perturbation method (DPM) combined with multiple time scales. A general cable model is assumed to have complex loads, including nonlinear aerodynamic force and concentrated masses. An early developed perturbation method––DPM––is applied to investigate the cable’s motion. This approach does not need to transform the original partial differential equations to ordinary differential equations, and the solution procedure is systematic. Static profile is found and the frequency equation is explicitly derived using linear analysis. The direct perturbation approach is employed to find nonlinear solutions, and governing equations are obtained which can be used to determine the stabilities of the steady-state solutions.

Sensitivity Statistics of 3D Structures under Parametric Uncertainty

Determination of sensitivity gradient is a major prerequisite for structural optimization, reliability assessment, and parameter identification. As the conventional deterministic sensitivity analysis cannot provide complete information, stochastic analysis is needed to tackle the uncertainties in structural parameters. This study focuses on the utility of the stochastic finite-element method for random response sensitivity analysis. The stochastic modeling of a random parameter is based on a commonly used 2D local averaging method generalized for a 3D case. The Choleski decomposition technique is then employed for digital simulation. The Neumann expansion based finite-element simulation method is extended for stochastic sensitivity analysis. This technique leads to a considerable saving of computational time. Example problems are used to compare the accuracy of this method to the direct Monte Carlo simulation and perturbation method in terms of varying stochasticity and efficiency in CPU time.