Fifth-order Dispersion Research Articles

Soliton solutions of an integrable nonlinear Schrödinger equation with quintic terms.

We present the fifth-order equation of the nonlinear Schrödinger hierarchy. This integrable partial differential equation contains fifth-order dispersion and nonlinear terms related to it. We present the Lax pair and use Darboux transformations to derive exact expressions for the most representative soliton solutions. This set includes two-soliton collisions and the degenerate case of the two-soliton solution, as well as beating structures composed of two or three solitons. Ultimately, the new quintic operator and the terms it adds to the standard nonlinear Schrödinger equation (NLSE) are found to primarily affect the velocity of solutions, with complicated flow-on effects. Furthermore, we present a new structure, composed of coincident equal-amplitude solitons, which cannot exist for the standard NLSE.

Approximate solutions of nonlinear PDEs by the invariant expansion

It is difficult to obtain exact solutions of the nonlinear partial differential equations (PDEs) due to their complexity and nonlinearity, especially for non-integrable systems. In this paper, some reasonable approximations of real physics are considered, and the invariant expansion is proposed to solve real nonlinear systems. A simple invariant expansion with quite a universal pseudopotential is used for some nonlinear PDEs such as the Korteweg-de Vries (KdV) equation with a fifth-order dispersion term, the perturbed fourth-order KdV equation, the KdV—Burgers equation, and a Boussinesq-type equation.

Improved XPM-induced crosstalk with higher order dispersion in SCM–WDM optical transmission link

In this paper, the XPM-induced crosstalk has been evaluated with higher order dispersion in SCM–WDM optical transmission link at different modulation frequencies, transmission lengths and optical powers. It has been observed that there is exponent increase in XPM-induced crosstalk with the increase in modulation frequencies, transmission lengths and optical powers, respectively. The impact of 3OD, 4OD and 5OD is small as compared to 2OD but still contributes when the combined terms are considered. The combined effect of second, third, fourth and fifth-order dispersion parameters, the induced crosstalk introduced by XPM little increase.

XPM-induced crosstalk with higher order dispersion in SCM–WDM optical transmission link

In this paper, the XPM-induced crosstalk has been evaluated with higher order dispersion in SCM–WDM optical transmission link at different modulation frequencies. It has been observed that there is exponent increase in XPM-induced crosstalk with the increase in modulation frequency from 0 to 3GHz. The impact of 3OD, 4OD and 5OD is small as compared to 2OD but still contributes when the combined terms are considered. The combined effect of second, third, fourth and fifth-order dispersion parameters is that the induced crosstalk introduced by XPM increases.

Numerical Method Based on the Lattice Boltzmann Model for the Kuramoto-Sivashinsky Equation

In this paper, we proposed a lattice Boltzmann model based on the higher-order moment method for the Kuramoto-Sivashinsky equation. A series of partial differential equations obtained by using multi-scale technique and Chapman-Enskog expansion. According to Hirt's heuristic stability theory, the stability of the scheme can be controlled by modulating some special moments to design the fifth-order dispersion term and the sixth-order dissipation term. As results, the Kuramoto-Sivashinsky equation is recovered with higher-order truncation error. The numerical examples show the higher-order moment method can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the Kuramoto-Sivashinsky equation.

Small-signal theory of a grating-based free-electron laser in three dimensions

We present an analytic theory for the small-signal operation of a grating-based free-electron laser that includes the effects of transverse diffraction on the evanescent wave. In this device, the electron beam interacts with an evanescent wave of the grating that bunches the beam and creates superradiant Smith-Purcell radiation. We find that the evanescent wave is guided by the electron beam, giving an optical-mode width that depends on the gain. We consider the cases of very wide and very narrow electron beams. For a wide electron beam, the cubic dispersion relation previously found for slow-wave structures is recovered. When the electron beam is narrow, so that gain guiding is important, a fifth-order dispersion relation is found instead. Diffraction in a system where the group velocity is very different (sometimes negative) from the phase velocity leads to unexpected results. The Brillouin zone subdivides into four regions; only two physically allowed (gain-guided) roots are obtained in the regions near the center of the Brillouin zone, but three are found in the regions away from the center. In the left half of the Brillouin zone, corresponding to high electron energy, the device operates on a convective instability, as an amplifier. In the right half of the Brillouin zone, where the group velocity is negative, the device operates on an absolute instability, as an oscillator. In the region where only two guided modes exist, oscillator operation will be more difficult.

Contribution of Higher-Order Dispersion to Nonlinear Dust Ion Acoustic Waves in Inhomogeneous Mesospheric Dusty Plasma with Dust Charge Fluctuation

The propagation of dust ion acoustic waves (DIAWs) in a weakly inhomogeneous, weakly coupled, collisionless, and unmagnetized four components dusty plasma are examined. The fluid system considered in this work consists of cold positive ions, cold negatively and positively charged dust particles associated with isothermal electrons. For nonlinear (DIAW) waves, a reductive perturbation method was employed to obtain the variable coefficients Kortewege-de Vries (KdV) equation for the first-order potential. For local inhomogenity, the present system admits the coexistence of rarefactive and compressive solitons. As a matter of fact, when the wave amplitude enlarged, the width and velocity of the wave deviate from the prediction of the KdV equation. It means that we have to extend our analysis to obtain the variable coefficients Kortewege-de Vries (KdV) equation with fifth-order dispersion term. For locally constant parameters, the higher-order solution for the resulting equation has been achieved via what is called perturbation technique. The effects of positive and negative dust charge fluctuations on the higher-order soliton amplitude and width of electrostatic solitary structures are outlined.

Shock waves and compactons for fifth-order non-linear dispersion equations

The followingfirst problemis posed:is a correct ‘entropy solution’ of the Cauchy problem for the fifth-order degenerate non-linear dispersion equations (NDEs), same as for the classic Euler oneut+uux= 0,These two quasi-linear degenerate partial differential equations (PDEs) are chosen as typical representatives; so other (2m+ 1)th-order NDEs of non-divergent form admit such shocks waves. As a relatedsecond problem, the opposite initial shockS+(x) = −S−(x) = signxis shown to be a non-entropy solution creating ararefaction wave, which becomesC∞for anyt> 0. Formation of shocks leads to non-uniqueness of any ‘entropy solutions’. Similar phenomena are studied for afifth-order in timeNDEuttttt= (uux)xxxxinnormal form.On the other hand, related NDEs, such asare shown to admit smoothcompactons, as oscillatorytravelling wavesolutions with compact support. The well-known non-negative compactons, which appeared in various applications (first examples by Dey, 1998,Phys. Rev.E, vol. 57, pp. 4733–4738, and Rosenau and Levy, 1999,Phys. Lett.A, vol. 252, pp. 297–306), are non-existent in general and are not robust relative to small perturbations of parameters of the PDE.

Contribution of higher order dispersion to nonlinear dust-acoustic solitary waves in dusty plasma with different sized dust grains and nonthermal ions

The propagation of nonlinear dust-acoustic waves (DAWs) in an unmagnetized, collisionless dusty plasma consisting of dust grains obeying the power-law dust size distribution and nonthermal ions are investigated. For nonlinear DAWs, a reductive perturbation method was employed to obtain a Korteweg–de Vries (KdV) equation for the first-order potential. As the wave amplitude increases, the width and the velocity of the soliton deviate from the prediction of the KdV equation, i.e. the breakdown of the KdV approximation occurs. To overcome this weakness, we extended our analysis to obtain the KdV equation with the fifth-order dispersion term. After that, the higher order solution for the resulting equation has been achieved via what is called the perturbation technique. The effects of dust size distribution, dust radius and nonthermal distribution of ions on the higher order soliton amplitude, width and energy of electrostatic solitary structures are presented.

Four-wave mixing analysis in WDM optical communication systems with higher-order dispersion

Four-wave mixing (FWM) is one of the major limiting factors in WDM optical fiber communication systems. In this paper, we analyze the individual and combined effect of second-, third-, fourth- and fifth-order dispersion parameters on FWM at different input channel powers and core effective areas, which have not been calculated earlier. FWM power versus channel power graphs for individual and combined effects of dispersion parameters have been presented, and it has been observed that FWM reduces for combined effect of dispersion parameter.

Effects of parallel ion motion on zonal flow generation in ion-temperature-gradient mode turbulence

The role of parallel ion motion for zonal flow generation in ion-temperature-gradient (ITG) mode turbulence is investigated with focus on the effects of acoustic modes and toroidicity on the zonal flow. One possible reason for the weak suppression of ITG turbulence by zonal flows found in experiments in the Columbia Linear Machine [Phys. Plasmas 13, 055905 (2006)] might be due to the small toroidicity (ϵn=2Ln∕R) in the experiment. The zonal flow is often directly dependent on the ITG mode and the coupling of zonal flow to acoustic modes and hence is directly affected by any change of the relevant parameters. The model consists of the continuity, temperature, and parallel ion momentum equations for the ITG turbulence. The zonal flow time evolution is described by a Hasegawa-Mima-like equation, and a fifth-order zonal flow dispersion relation is derived. The results are interpreted in terms of quality of zonal flows, i.e., the ratio of growth rate and real frequency (Q=ΩIM∕ΩRE). It is found that the quality of the zonal flow rapidly decreases with decreasing toroidicity.

A new application of the Korteweg–de Vries Benjamin-Ono equation in interfacial electrohydrodynamics

We consider waves on a layer of finite depth governed by the Euler equations in the presence of gravity, surface tension, and vertical electric fields. We use perturbation theory to identify canonical scalings and to derive a Korteweg–de Vries Benjamin-Ono equation arising in interfacial electrohydrodynamics. When the Bond number is equal to 1∕3, dispersion disappears and the equation reduces to the Benjamin-Ono equation. In the additional limit of vanishing electric fields, we show how to obtain a new evolution equation that contains third- and fifth-order dispersion as well as a nonlocal electric field term.

Viscous Damping of Non-Adiabatic MHD Waves in an Unbounded Solar Coronal Plasma

The damping of MHD waves in solar coronal magnetic field is studied taking into account thermal conduction and compressive viscosity as dissipative mechanisms. We consider viscous homogeneous unbounded solar coronal plasma permeated by a uniform magnetic field. A general fifth-order dispersion relation for MHD waves has been derived and solved numerically for different solar coronal regimes. The dispersion relation results three wave modes: slow, fast, and thermal modes. Damping time and damping per periods for slow- and fast-mode waves determined from dispersion relation show that the slow-mode waves are heavily damped in comparison with fast-mode waves in prominences, prominence–corona transition regions (PCTR), and corona. In PCTRs and coronal active regions, wave instabilities appear for considered heating mechanisms. For same heating mechanisms in different prominences the behavior of damping time and damping per period changes significantly from small to large wavenumbers. In all PCTRs and corona, damping time always decreases linearly with increase in wavenumber indicate sharp damping of slow- and fast-mode waves.

Exact solutions and conservation laws for coupled generalized Korteweg–de Vries and quintic regularized long wave equations

A new model equation is proposed for simulating unidirectional wave propagation in nonlinear media with dispersion. This model equation is a one-dimensional evolutionary partial differential equation that is obtained by coupling the generalized Korteweg–de Vries (gKdV) equation, which includes a nonlinear term of any order and cubic dispersion, with the quintic regularized long wave (qRLW) equation, which includes fifth-order dispersion. Exact solitary wave solutions are derived for this model equation. These analytical solutions are obtained for any order of the nonlinear term and for any given values of the coefficients of the cubic and quintic dispersive terms. Analytical expressions for three conservation laws and for three invariants of motion for solitary wave solutions of this new equation are also derived.

On the propagation of solitary pulses in microstructured materials

Abstract KdV-type evolution equation, including the third- and the fifth-order dispersive and the fourth-order nonlinear terms, is used for modelling the wave propagation in microstructured solids like martensitic–austenitic alloys. The character of the dispersion depends on the signs of the third- and the fifth-order dispersion parameters. In the present paper the model equation is solved numerically under localised initial conditions in the case of mixed dispersion, i.e., the character of dispersion is normal for some wavenumbers and anomalous for others. Two types of solution are defined and discussed. Relatively small solitary waves result in irregular solution. However, if the amplitude exceeds a certain threshold a solution having regular time–space behaviour emerges. The latter has tree sub-types: “plaited” solitons, two solitary waves and single solitary wave. Depending on the value of the amplitude of the initial pulse these sub-types can appear alone or in a certain sequence.

Higher-order nonlinearity of electron-acoustic solitary waves with vortex-like electron distribution and electron beam

The nonlinear wave structure of small-amplitude electron-acoustic solitary waves (EASWs) is investigated in a four-component plasma consisting of cold electron fluid, hot electrons obeying vortex-like distribution traversed by a warm electron beam and stationary ions. The streaming velocity of the beam, uo, plays the dominant role in determining the roots of the linear dispersion relation associated with the system. Using the reductive perturbation theory, the basic set of equations is reduced to a modified Korteweg–de Vries (mKdV) equation. With the inclusion of higher-order nonlinearity, a linear inhomogeneous mKdV type equation with fifth-order dispersion term is derived and the higher-order solution is obtained using a renormalization method. However, both mKdV and mKdV-type solutions present a positive potential, which corresponds to a hole (hump) in the cold (hot) electron number density. The mKdV-type solution has a smaller energy amplitude and a wider width than that of mKdV solution. The dependence of the energy amplitude, the width, and the velocity on the system parameters is investigated. The findings of this investigation are used to interpret the electrostatic solitary waves observed by the Geotail spacecraft in the plasma sheet boundary layer of the Earth’s magnetosphere.

Observation of Slowly Varying Envelope Approximation Breakdown by Comparison between the Extended Finite-Difference Time-Domain Method and the Beam Propagation Method for Ultrashort-Laser-Pulse Propagation in a Silica Fiber

Conventionally, the beam propagation method (BPM) for solving the generalized nonlinear Schrödinger equation (GNLSE) including the slowly varying envelope approximation (SVEA) has been used to describe ultrashort-laser-pulse propagation in an optical fiber. However, if the pulse duration approaches the optical cycle regime (<10 fs), this approximation becomes invalid. Consequently, it becomes necessary to use the finite difference time domain method (FDTD method) for solving the Maxwell equation with the least approximation. In order to both experimentally and numerically investigate nonlinear femtosecond ultrabroad-band-pulse propagation in a silica fiber, we have extended the FDTD calculation of Maxwell's equations with nonlinear terms to that including all exact Sellmeier-fitting values. We compared the results of this extended FDTD method with the solution of the BPM that includes the Raman response function, which is the same as in the extended FDTD method, up to fifth-order dispersion with the SVEA, as well as with the experimental results for nonlinear propagation of a 12 fs laser pulse in a silica fiber. Furthermore, in only the calculation, pulse width was gradually shortened from 12 fs to 7 fs to 4 fs to observe the breakdown of the SVEA in detail. Moreover, the soliton number N was established as 1 or 2. To the best of our knowledge, this is the first observation of the breakdown of the SVEA by comparison between the results of the extended FDTD and the BPM calculations for the nonlinear propagation of an ultrashort (<12 fs) laser pulse in a silica fiber.

Higher order solution of the dust ion acoustic solitons in a warm dusty plasma with vortex-like electron distribution

Abstract The ratios of dust to free electron and free to trapped electron temperatures are examined in warm dusty plasmas with vortex-like electron distribution through the derivation of a modified Korteweg–de Vries (MKdV) equation using a reductive perturbation theory. As the wave amplitude increases, the width and velocity of the soliton deviate from the prediction of the MKdV equation, i.e., the breakdown of the MKdV approximation. To describe the soliton of larger amplitude, the MKdV equation with the fifth-order dispersion term is employed and its higher-order solutions are obtained.

Local Axisymmetric Diffusive Stability of Weakly Magnetized, Differentially Rotating, Stratified Fluids

We study the local stability of stratified, differentially rotating fluids to axisymmetric perturbations in the presence of a weak magnetic field and of finite resistivity, viscosity, and heat conductivity. This is a generalization of the Goldreich-Schubert-Fricke (GSF) double-diffusive analysis to the magnetized and resistive, triple-diffusive case. Our fifth-order dispersion relation admits a novel branch that describes a magnetized version of multidiffusive modes. We derive necessary conditions for axisymmetric stability in the inviscid and perfect-conductor (double-diffusive) limits. In each case, rotation must be constant on cylinders and angular velocity must not decrease with distance from the rotation axis for stability, irrespective of the relative strength of viscous, resistive, and heat diffusion. Therefore, in both double-diffusive limits, solid-body rotation marginally satisfies our stability criteria. The role of weak magnetic fields is essential to reach these conclusions. The triple-diffusive situation is more complex, and its stability criteria are not easily stated. Numerical analysis of our general dispersion relation confirms our analytic double-diffusive criteria but also shows that an unstable double-diffusive situation can be significantly stabilized by the addition of a third, ostensibly weaker, diffusion process. We describe a numerical application to the Sun's upper radiative zone and establish that it would be subject to unstable multidiffusive modes if moderate or strong radial gradients of angular velocity were present.

Dispersion management for solitons in a Korteweg-de Vries system.

The existence of "dispersion-managed solitons," i.e., stable pulsating solitary-wave solutions to the nonlinear Schrodinger equation with periodically modulated and sign-variable dispersion is now well known in nonlinear optics. Our purpose here is to investigate whether similar structures exist for other well-known nonlinear wave models. Hence, here we consider as a basic model the variable-coefficient Korteweg-de Vries equation; this has the form of a Korteweg-de Vries equation with a periodically varying third-order dispersion coefficient, that can take both positive and negative values. More generally, this model may be extended to include fifth-order dispersion. Such models may describe, for instance, periodically modulated waveguides for long gravity-capillary waves. We develop an analytical approximation for solitary waves in the weakly nonlinear case, from which it is possible to obtain a reduction to a relatively simple integral equation, which is readily solved numerically. Then, we describe some systematic direct simulations of the full equation, which use the soliton shape produced by the integral equation as an initial condition. These simulations reveal regions of stable and unstable pulsating solitary waves in the corresponding parametric space. Finally, we consider the effects of fifth-order dispersion. (c) 2002 American Institute of Physics.