Modified Zakharov Equations Research Articles

Exact periodic wave solutions for the modified Zakharov equations with a quantum correction

In this paper, exact solutions for the modified Zakharov equations with a quantum correction are studied. We construct the periodic wave solutions for the quantum Zakharov equations by applying the dynamical systems method and F-expansion method. Moreover, when the modulus h of the Jacobian elliptic functions converges to 1, the periodic solutions will converge to the corresponding solitary wave solutions.

Alfvén solitary waves with effect of arbitrary temperature degeneracy in spin quantum plasma

Nonlinear Alfvén waves are studied in a fluid model for nonrelativistic, magnetized spin-1/2 quantum plasmas with an arbitrary degeneracy effect. Following a local Fermi-Dirac distribution function, a modified equation of state is utilized which is applicable to both classical and degenerate limits. Using the fluid equations for Hall magnetohydrodynamics with quantum corrections due to statistical effects, Bohm potential, spin magnetization energy, and temperature degeneracy, a set of modified Zakharov equations are derived for circularly polarized nonlinear Alfvén waves. Ions are assumed to be cold, and the spin effects of electrons are incorporated through spin force along with spin magnetization current. A linear dispersion relation for finite amplitude Alfvén waves duly modified by spin magnetization and arbitrary temperature degeneracy effects is also obtained. Employing the Sagdeev potential approach, the properties of Alfvén solitary profiles in quantum plasmas with arbitrary degeneracy effects of electrons are analyzed. The amplitude of Sagdeev potential and of the associated soliton structure for both right and left-hand circularly polarized Alfvén waves is observed to decrease with the decrease in the value of the arbitrary temperature degeneracy factor G for the case of the nearly degenerate limit. Similarly, it is found that the amplitude of Sagdeev potential and of the related solitary profile increases for both kinds of circular polarized Alfvén waves with the increasing value of G in the case of the nearly non-degenerate limit.

Nonlinear coupling of Langmuir and electron acoustic waves in a viscous plasma

A nonlinear analysis of finite amplitude electron acoustic waves is considered in a viscous plasma. The two fluid two time scale model is used to describe the two temperature electron species in a fixed ion background. We have obtained two sets of modified Zakharov equations where the modification comes due to the presence of viscosity in the plasma system. We have shown that, for very low frequency, these viscosity modified Zakharov equations reduce to a modified nonlinear Schrödinger's equation where viscosity introduces a new term via collective effects. Perturbative analysis shows the formation of soliton structures with an oscillating tail. The relevance of the results is important in the context of astrophysical and laboratory plasma.

The Time-Periodic Solutions to the Modified Zakharov Equations with a Quantum Correction

This paper investigates the existence and uniqueness of time-periodic solutions of the periodic initial value problem for the modified Zakharov equations with a quantum correction. By combining a priori estimates with the Galerkin method and Leray–Schauder fixed point theorem, we prove that there exist a unique strong time-periodic solution and a unique classical time-periodic solution under some conditions on the forcing terms f and g.

On global generalized solutions for a two-dimensional modified Zakharov equations

This paper considers the existence of the generalized solution to the Cauchy problem for a class of modified Zakharov equation in (2+1) dimensions. by a priori integral estimates and Galerkin method, the author establish global in time existence of the global generalized solution to the problem.

Long time behavior of the solutions for the dissipative modified Zakharov equations for plasmas with a quantum correction

In this paper, we study the long time behavior of the solutions for the dissipative modified Zakharov equations for plasmas with a quantum correction. Subsequent to our previous work (L.Y. Jin et al. 2012), where the existence of the global attractor A⊂L2(Ω)×(H2(Ω)∩H01(Ω))×(H3(Ω)∩H01(Ω)) is obtained via delicate a prior estimates, we are further going to investigate and obtain the estimates of the upper bounds of Hausdorff and fractal dimensions for the global attractor.

Modified nonlinear evolution of Langmuir waves

Zakharov equations describe the nonlinear coupling between the electron and ion motion in a plasma mediated by the ponderomotive effect. It has been recently shown that the conventional ponderomotive theory has several limitations and under certain conditions, the plasma density depends on the square of the electric potential and not the electric field [K. Shah, Phys. Plasmas 17, 112301 (2010)]. In this paper, the modified Zakharov equations are derived using this modified ponderomotive density. This modified equation does not admit any spatially localized solution and leads to a spatial broadening of the periodic wave solutions of the Zakharov equations. It is also shown that in this modified description of the nonlinear evolution of Langmuir waves, the high frequency electron oscillations must have a frequency slightly higher than the electron plasma frequency.

Existence of attractors for modified Zakharov equations for plasmas with a quantum correction

In this paper, we consider the asymptotic behavior of the solutions for the quantum Zakharov system in one dimension, with the inclusion of phenomenological damping terms in both equations for the envelope electric field and for the density fluctuation. The existence of solutions and attractors are obtained, respectively, in different Sobolev spaces via a prior estimates and the Galerkin method.

Initial boundary value problem for modified Zakharov equations

In this paper the authors consider the existence and uniqueness of the solution to the initial boundary value problem for a class of modified Zakharov equations, prove the global existence of the solution to the problem by a priori integral estimates and Galerkin method.

Exact traveling wave solutions of modified zakharov equations for plasmas with a quantum correction

In this article, the authors study the exact traveling wave solutions of modified Zakharov equations for plasmas with a quantum correction by hyperbolic tangent function expansion method, hyperbolic secant expansion method, and Jacobi elliptic function expansion method. They obtain more exact traveling wave solutions including trigonometric function solutions, rational function solutions, and more generally solitary waves, which are called classical bright soliton, W-shaped soliton, and M-shaped soliton.

Double layers in strong turbulent plasma with suprathermal electrons

Distributions of plasma with excess numbers of superthermal electrons are common in space environments and double layer (DL) is one of the very important electrostatic nonlinear wave structures ubiquitous in plasma systems. Based on the modified Zakharov equations, the DLs are studied in the strong turbulent plasmas with Kappa distributed electrons. It appears that in the strong turbulence regime, the presence of additional superthermal particles does not make qualitative changes on the DLs behavior, but modify the thicknesses of the DLs.

Strong Langmuir turbulence in Kappa distributed plasmas

Superthermal electrons are often observed in space and astrophysics and can be appropriate modeled by the family of Kappa distribution functions. Taking the nonlinear wave-wave, wave-particle interactions and the effect of superthermal electrons into account, the strong Langmuir turbulence is investigated in kinetic regime. The modified Zakharov equations are obtained for the case of no damping or driving terms. On the basis of these equations, dynamics of collapse have been studied by the means of the general virial theorem, and the collapse thresholds which are strong modified by superthermal index κe are given.

Modified Zakharov equations for plasmas with a quantum correction

Quantum Zakharov equations are obtained to describe the nonlinear interaction between quantum Langmuir waves and quantum ion-acoustic waves. These quantum Zakharov equations are applied to two model cases, namely, the four-wave interaction and the decay instability. In the case of the four-wave instability, sufficiently large quantum effects tend to suppress the instability. For the decay instability, the quantum Zakharov equations lead to results similar to those of the classical decay instability except for quantum correction terms in the dispersion relations. Some considerations regarding the nonlinear aspects of the quantum Zakharov equations are also offered.

Response to “Comment on ‘Modified Zakharov equations in dusty plasmas with variable charges on dust particles’ ” [Phys. Plasmas 11, 4151 (2004)

The comments [R. Annou, Phys. Plasmas 11, 4151 (2004)] are unfounded and are shown to be based on the complete misunderstanding of our article by the author. The claim of an incorrect analysis is strongly refuted and firmly rejected.

Comment on “Modified Zakharov equations in dusty plasmas with variable charges on dust particles” [Phys. Plasmas 10, 984 (2003)

Abstract

Modified Zakharov equations in dusty plasmas with variable charges on dust particles

The nonlinear coupling between large amplitude Langmuir waves and dust ion-acoustic waves in a charge-varying dusty plasma is considered. By employing a two-time-scale three fluid approach, modified Zakharov equations are derived. A normal mode analysis of these nonlinearly coupled equations is performed to obtain a general dispersion relation that is useful for studying parametric instabilities. The results show that the impurities can significantly alter the characteristics and damping of these instabilities by reducing their phase velocity and causing charging related damping.

Langmuir turbulence in moderately magnetized space plasmas*

Beam-driven Langmuir turbulence is studied in two moderately magnetized (Ωe≊ωe) space-plasma regimes: regions of the lower solar corona and the Earth’s auroral ionosphere. The turbulence is modeled using modified Zakharov equations, which are employed in two-dimensional numerical simulations. For coronal parameters, highly anisotropic coherent wave packets form and collapse when Ωe<ωe. By contrast, the turbulence is phase incoherent when Ωe≳ωe, as a result of change in the topology of the Langmuir dispersion relation. In the auroral ionosphere, intense Langmuir waves (up to 500 mV/m) have been measured, in conjunction with field-aligned electron streams and nonthermal electron tails. Approximate agreement with high-time-resolution electric-field measurements, is found in the simulations. However, because of strong damping on nonthermal electrons, wave collapse is inhibited, irrespective of the ordering of Ωe and ωe.

Energy computations for evolution of class I and II instabilities of Stokes waves

The modified Zakharov equation is used to study the coupled evolution of class I and class II instabilities of surface gravity waves on infinitely deep water. In contrast to single class (I or II) evolution, the coupled behaviour is non-periodic. Except for the very steep waves a dominance of class I modes over those of class II is observed. Energy calculations show that the Hamiltonian of the wave field considered is nearly constant. Thus the Zakharov and the modified Zakharov equations represent consistent approximations of the original water-wave problem.

Effect of random density irregularities on nonlinear evolution of Langmuir waves in interplanetary medium

The evolution of nonlinear Langmuir waves in the interplanetary medium is investigated by appropriately accounting for the random density irregularities of the medium. A pair of modified Zakharov equations, which describe these waves, is solved numerically as an initial value problem for large scale (≫ 102 km) initial pertubations. For an ion acoustic-Langmuir solitary wave, the random irregularities damp the Langmuir wave by way of scattering and let the ion density perturbation radiate away in a few days. However an initial solitary or shock-like Langmuir wave excites the ion density perturbations within a fraction of a second, and then itself gets damped. These effects will strongly decelerate the collapse of large scale Langmuir waves. The possibility of detecting these processes, by means of interplanetary scintillation, is discussed.

Modulational instability and soliton formation during ionospheric heating

The most intense electric fields during ionospheric heating occur a fraction of a kilometer below the classical reflection point. At this location, the nonlinear evolution of Langmuir waves is studied within the context of the modified Zakharov equations. It is found that the modulational instability (oscillating two‐stream instability) is more important than the three‐wave parametric decay instability, leading to the rapid formation of solitons.