Solitary Wave Structures Research Articles

Simulation of zonal flow excitation by drift mode turbulence: applications to tokamaks and the magnetopause

Recently, we investigated the interaction between broadband drift mode turbulence and zonal flows near the edge of a region of magnetized plasma (Trines et al 2005 Phys. Rev. Lett. 94 165002). Our simulation results showed the development of a zonal flow through the modulational instability of the drift wave distribution, as well as the existence of solitary zonal flow structures about an ion gyroradius wide, drifting towards steeper relative density gradients. Both the growth rate of the turbulence and the particle/energy transport across the plasma boundary can be stabilized by adjusting the plasma density gradient. This spontaneous formation of solitary wave structures has also been found in Cluster satellite observations (Trines et al 2007 Phys. Rev. Lett. 99 205006), confirming our earlier theoretical predictions. We will discuss the consequences of our results for our understanding of the Earth's magnetopause, as well as for the study of zonal flows in tokamaks.

Solitary wave propagation in quantum electron-positron plasmas

Existence of large amplitude stationary solitary wave structures in an unmagnetized electron-positron (e-p) plasma is studied using a quantum hydrodynamic (QHD) model that includes the quantum force (tunnelling) associated with the Bohm potential and the Fermi-dirac pressure law. It is found that in a quasi-neutral pair (e-p) plasma, where the dispersion is only due to the the quantum tunnelling effects, the large amplitude stationary solitary structure exists only when the normalized Mach speed,M <√2. Such solitary structures do not exist in absence of the Bohm potential term in an unmagnetized quasineutral pair (e-p) plasma. The system is shown to support only rarefactive stationary solitary waves. For such waves the amplitude, being independent of the quantum parameter H (the ratio of the electron plasmon to electron Fermi energy), decreases with the Mach number M, whereas the width increases with both M and H. The present theory is applicable to analyze the formation of localized coherent solitary structures at quantum scales in dense astrophysical objects as well as in intense laser fields.

Nonlinear adiabatic models of ion-acoustic waves in dust plasma

Nonlinear adiabatic models of ion-acoustic waves in a dust plasma are developed. The problem of the structure of subsonic periodic and supersonic solitary ion-acoustic waves is exactly solved analytically under the assumption of a constant charge of dust particles; the critical Mach numbers for the solitary wave are determined. The problem of the wave structure is solved numerically for the case when the charge of dust particles was assumed to be variable.

Dust-ion-acoustic solitary waves in a two-temperature electrons with charge fluctuations in a dusty plasma

The nonlinear structure of small amplitude dust-ion-acoustic (DIA) solitary waves is investigated in a hot magnetized dusty plasma consisting of negative dust grains, positive ions, and two-temperature isothermal electrons. The charge fluctuation dynamics of the dust grains has also been considered. A Korteweg–de Vries (KdV) equation, Modified Korteweg–de Vries (MKdV) equation and Korteweg–de Vries (KdV) type equation at critical densities of ions are derived using a reductive perturbation theory. The effects of two-temperature electrons and external oblique magnetic field on the properties of solitary waves are studied. Such study showed that the presence of a second component of electrons modifies the nature of DIA solitary waves structures. This has an important consequence for the reinterpretation of several astrophysical phenomena involving two-temperature electrons. The present investigation can be relevance to electrostatic solitary structures observed in many cosmic dust–laden plasmas, such as Saturn’s rings.

Effects of two-temperature electrons, external oblique magnetic field, and higher-order nonlinearity on dust acoustic solitary waves in a dusty plasma with vortex-like ion distribution

Nonlinear properties of the dust acoustic (DA) solitary waves in a dusty plasma consisting of negatively variable-charged dust particles, vortex-like distributed ions and two-temperature isothermal electrons are reported. A reductive perturbation theory has been used to derive a modified Korteweg-de Vries (mKdV) equation for the first-order perturbed potential and a linear inhomogeneous mKdV-type equation for the second-order perturbed potential. The renormalization method is used to obtain stationary solutions of these coupled equations. The modifications in the amplitude and width of the solitary wave structure due to the inclusion of two different types of isothermal electrons, external oblique magnetic field, higher-order nonlinearity, and vortex-like distributed ions are investigated. Also a method based on energy consideration was used to obtain the stability condition. Moreover, the numerical results are applied to investigate some nonlinear characteristics of the DA solitary waves.

Numerical simulation on the interaction of an internal solitary wave with a step-type topography

Based on a nonhydrostatic numerical ocean model developed by one of the authors, the interaction of an internal solitary wave with a step-type topography was investigated. Over the step topography, the flow pattern could be classified into three categories: 1) the propagation and spatial structure of the internal solitary wave was little influenced by the bottom topography, 2) the internal solitary wave was significantly distorted by the blocking effect of the topography without the occurrence of wave breaking and 3) the internal solitary wave was broken as it encountered and passed over the bottom topography. A detailed description of the processes leading to wave breaking is given in this paper together with energy budget analysis. The results revealed that the maximum of the energy dissipation rate is no more than 40%, which is consistent with available experimental data.

Cluster observations of electrostatic solitary waves near the Earth's bow shock

Using a period of internal burst mode data from the Cluster Electric Field and Wave instrument a number of electrostatic solitary structures have been identified in the foot region of Earth's quasi‐perpendicular bow shock. The four individual probe potential measurements are utilized to investigate the fundamental characteristics of the solitary wave structures such as wave propagation vector, propagation velocity, scale‐size and potential amplitude. Two classes of waves are observed. Bipolar solitary waves typically propagate in the solar wind direction toward the shock but at a significant angle from the ambient magnetic field. Unipolar/tripolar solitary waves tend to propagate along the ambient magnetic field. The wave amplitude‐scale size relation is similar to that obtained for similar structures observed in the auroral zone. The structures lie in the theoretically allowed region in width‐amplitude space to be consistent with the BGK ion holes. Using a period of internal burst mode data from the Cluster Electric Field and Wave instrument a number of electrostatic solitary structures have been identified in the foot region of Earth's quasi‐perpendicular bow shock. The four individual probe potential measurements are utilized to investigate the fundamental characteristics of the solitary wave structures such as wave propagation vector, propagation velocity, scale‐size and potential amplitude. Two classes of waves are observed. Bipolar solitary waves typically propagate in the solar wind direction toward the shock but at a significant angle to the ambient magnetic field in contrast to most previous studies which assume parallel propagation to the ambient magnetic field. In contrast, unipolar/tripolar solitary waves tend to propagate along the ambient magnetic field. The wave amplitude‐scale size relation is similar to that obtained for structures observed in the auroral zone. The structures lie in the theoretically allowed region in width‐amplitude space to be consistent with the BGK (Bernstein‐Greene‐Kruskal) ion holes. The two classes of observed solitary waves may greatly influence the ambient plasma dynamics around the shock. The bipolar solitary waves do not exhibit a large net potential difference but may still play an important role in plasma thermalisation by particle scattering. Unipolar/tripolar solitary waves exhibit a remarkable net potential difference that may be responsible for the plasma energisation along the ambient magnetic field.

Soliton-like excitation in a nonlinear model of DNA dynamics with viscosity

The study of solitary wave solutions is of prime significance for nonlinear physical systems. The Peyrard-Bishop model for DNA dynamics is generalized specifically to include the difference among bases pairs and viscosity. The small amplitude dynamics of the model is studied analytically and reduced to a discrete complex Ginzburg-Landau (DCGL) equation. Exact solutions of the obtained wave equation are obtained by the mean of the extended Jacobian elliptic function approach. These amplitude solutions are made of bubble solitons. The propagation of a soliton-like excitation in a DNA is then investigated through numerical integration of the motion equations. We show that discreteness can drastically change the soliton shape. The impact of viscosity as well as elasticity on DNA dynamic is also presented. The profile of solitary wave structures as well as the energy which is initially evenly distributed over the lattice are displayed for some fixed parameters.

Spontaneous Generation of Self-Organized Solitary Wave Structures at Earth’s Magnetopause

Spontaneous formation of solitary wave structures has been observed in Earth's magnetopause, and is shown to be caused by the breakup of a zonal flow by the action of drift wave turbulence. Here we show matched observations and modeling of coherent, large-scale solitary electrostatic structures, generated during the interaction of short-scale drift wave turbulence and zonal flows at the Earth's magnetopause. The observations were made by the Cluster spacecraft and the numerical modeling was performed using the wave-kinetic approach to drift wave-zonal flow interactions. Good agreement between observations and simulations has been found, thus explaining the emergence of the observed solitary structures as well as confirming earlier theoretical predictions of their existence.

New extension of the tanh-function method and application to the Whitham–Broer–Kaup shallow water model with symbolic computation

Abstract In this Letter, the tanh-function method is further extended by introducing a more generalized expansion form. Compared with other existing tanh-function methods, our algorithm provides the possibility for constructing more types of travelling-wave solutions (especially the solitary-wave solutions) for a given system of nonlinear evolution equations (NLEEs). Applying such an algorithm to the Whitham–Broer–Kaup shallow water model with symbolic computation, abundant solitary-wave solutions are obtained and some novel solitary-wave structures are also revealed under certain parametric conditions. After suitable improvements, this algorithm can also be applicable to the construction of nontravelling solitary-wave solutions for variable-coefficient and higher-dimensional NLEEs.

Solitary waves bifurcated from Bloch-band edges in two-dimensional periodic media

Solitary waves bifurcated from edges of Bloch bands in two-dimensional periodic media are determined both analytically and numerically in the context of a two-dimensional nonlinear Schrödinger equation with a periodic potential. Using multiscale perturbation methods, the envelope equations of solitary waves near Bloch bands are analytically derived. These envelope equations reveal that solitary waves can bifurcate from edges of Bloch bands under either focusing or defocusing nonlinearity, depending on the signs of the second-order dispersion coefficients at the edge points. Interestingly, at edge points with two linearly independent Bloch modes, the envelope equations lead to a host of solitary wave structures, including reduced-symmetry solitons, dipole-array solitons, vortex-cell solitons, and so on-many of which have not been reported before to our knowledge. It is also shown analytically that the centers of envelope solutions can be positioned at only four possible locations at or between potential peaks. Numerically, families of these solitary waves are directly computed both near and far away from the band edges. Near the band edges, the numerical solutions spread over many lattice sites, and they fully agree with the analytical solutions obtained from the envelope equations. Far away from the band edges, solitary waves are strongly localized, with intensity and phase profiles characteristic of individual families.

Modulational instability and exact solutions of the nonlinear Schrödinger equation coupled with the nonlinear Klein–Gordon equation

The modulational instability (MI) of the coupled nonlinear Schrödinger and nonlinear Klein–Gordon equations is investigated. It is found that there are a number of possibilities for the MI regions due to the generalized dispersion relation, which relates the frequency and wavenumber of the modulating perturbations. Some exact travelling wave solutions are constructed via the solutions of a ϕ4 model through a simple mapping relation. Furthermore, we present five different types of solutions representing possible final states of modulationally unstable perturbations. The profiles of solitary wave structures are displayed for some fixed parameters.

Bäcklund Transformations, Solitary Waves, Conoid Waves and Bessel Waves of the (2+1)-Dimensional Euler equation

Some simple special Backlund transformation theorems are proposed and utilized to obtain exact solutions for the (2+1)-dimensional Euler equation. It is found that the (2+1)-dimensional Euler equation possesses abundant soliton or solitary wave structures, conoid periodic wave structures and the quasi-periodic Bessel wave structures on account of the arbitrary functions in its solutions. Moreover, all solutions of the arbitrary two dimensional nonlinear Poisson equation can be used to construct exact solutions of the (2+1)-dimensional Euler equation.

Nonlinear 1-D stationary flows in multi-ion plasmas – sonic and critical loci – solitary and &amp;quot;oscillatory&amp;quot; waves

Abstract. One-dimensional stationary flows of a plasma consisting of two ion populations and electrons streaming against a heavy ion cloud are studied. The flow structure is critically governed by the position of sonic and critical points, at which the flow is shocked or choked. The concept of sonic and critical points is suitably generalized to the case of multi-ion plasmas to include a differential ion streaming. For magnetic field free flows, the sonic and critical loci in the (upx, uhx) space coincide. Amongst the different flow patterns for the protons and heavy ions, there is a possible configuration composed of a "heavy ion shock" accompanied by a proton rarefaction. The magnetic field introduces a "stiffness" for the differential ion streaming transverse to the magnetic field. In general, both ion fluids respond similarly in the presence of "ion obstacle"; the superfast (subfast) flows are decelerated (accelerated). The collective flow is choked when the dynamic trajectory (upx, uhx) crosses the critical loci. In specific regimes the flow contains a sequence of solitary structures and as a result, the flow is strongly bunched. In each such substructure the protons are almost completely replaced by the heavies. A differential ion streaming is more accessible in the collective flows oblique to the magnetic field. Such a flexibility of the ion motion is determined by the properties of energy integrals and the Bernoulli energy functions of each ion species. The structure of flows, oblique to the magnetic field, depends critically on the velocity regime and demonstrates a rich variety of solitary and oscillatory nonlinear wave structures. The results of the paper are relevant to the plasma and field environments at comets and planets through the interaction with the solar wind.

Nonlinear Optics of Intense Attosecond Light Pulses

The interaction of an intense light pulse of "subatomic" duration with a system of multiple discrete quantum states is analyzed. The nonperturbative character of the response to the pulse field leading to an efficient conversion into high order harmonics is predicted. The spatial-temporal evolution of the field is shown to obey a generalized nonlinear wave equation of the double-sine-Gordon type. In addition to the solitary wave structures, it predicts a nontrivial regime of pulse amplification accompanied by extreme temporal self-contraction of the amplified field.

Modulational instability of dust acoustic waves in a dusty plasma with nonthermal electrons and ions

Effects of nonadiabaticity of variable dust charge, dust fluid temperature, trapped electrons as well as nonisothermality of ions on the amplitude modulation of dust acoustic waves in an unmagnetized dusty plasma are investigated. A modified nonlinear Schrodinger equation (NLSE) is obtained by the standard reductive perturbation technique and is solved numerically by the split-step Fourier method. The modulational instability and the envelope solitary wave structure are found to be modified somewhat by the effects of nonthermally distributed ions and trapped electrons.

Variable separation solutions and new solitary wave structures to the (1+1)-dimensional Ito system

A variable separation approach is proposed and extended to the (1+1)-dimensional physics system. The variable separation solution of (1+1)-dimensional Ito system is obtained. Some special types of solutions such as non-propagating solitary wave solution, propagating solitary wave solution and looped soliton solution are found by selecting the arbitrary function appropriately.

Trapped-electron solitary wave structures in a magnetized plasma

The classic one-dimensional (1D) BGK-mode (trapped-particle) analysis is extended to 3D plasmas in which a background magnetic field is present. A theoretical analysis is made of such nonlinear states from the coupled plasma equations for a magnetized plasma, which yields a generalized BGK-like equation for trapped-particle distributions that produce this structure in the guiding-center approximation, and the resulting conditional requirement for electron trapping as a function of the electric-field components, and magnetic gradient and polarization drifts. This condition shows that the amount of trapping decreases as the angle of the electric field with respect to the background magnetic field increases, often ceasing at a critical finite angle. The full solution to the generalized BGK-type equation for the trapped-electron distribution is obtained and reduced to the extent possible, thereby extending the classic 1D solution for electron holes to that for the 3D magnetized plasmas for cylindrical symmetry. This solution is very easily generalized for the inclusion of any cross-tail drift for the case of interest. For the case that the E×B drift dominates, the solution shows that these trapped-particle modes usually are restricted to angles θ&amp;lt;sin−1−2B2W∕meE2, where −W is the available trapping energy for the electrons, although in rarer cases slightly higher angles for these modes may be allowed. In no case can θ be as large as sin−12−B2W∕meE2.

Critical density solitary waves structures in a hot magnetized dusty plasma with vortexlike ion distribution in phase space

The nonlinear properties of solitary waves structures in a hot magnetized dusty plasma consisting of isothermal hot electrons, nonisothermal ions, and high negatively charged massive dust grains are reported. A modified Korteweg–de Vries (modified KdV) equation, which admits a solitary waves solution, for small but finite amplitude, is derived using a reductive perturbation theory. A nonisothermal ion distribution provides the possibility of existence of rarefactive solitary waves. On the other hand, the dynamics of solitary waves at a critical ion density is governed by KdV equation. The modification in the amplitude and width of the solitary waves structures due to the inclusion of obliqueness and external magnetic field are also investigated.

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.