Nonlinear dusty magnetosonic waves in a strongly coupled dusty plasma

Proton firehose instability: An interplay of thermal core and protons Calculation of anharmonic effect of the reactions related to NO2 in fuel combustion mechanism ABSTRACT The nonlinear propagation of magnetosonic waves in a magnetized strongly coupled dusty plasma consisting of inertialess electrons and ions as well as strongly coupled inertial charged dust particles is presented. A generalized viscoelastic hydrodynamic model for the strongly coupled dust particles and a quantum hydrodynamic model for electrons and ions are considered. In the kinetic regime, we derive a modified Kadomstev-Petviashvili (KP) equation for nonlinear magnetosonic waves of which the amplitude changes slowly with time due to the effect of a small amount of dust viscosity. The approximate analytical solutions of the modified KP equations are obtained with the help of a steady state line-soliton solution of the second type KP equation in a frame with a constant velocity. The dispersion relationship in the kinetic regime shows that the viscosity is no longer a dissipative effect.


I. INTRODUCTION
Magnetosonic waves propagating perpendicular to the applied magnetic field in a dusty plasma differ from those in an electron-ion plasma because of not only the unusual time and space scale but also the correlations of the dust grains. 1,2 Self-gravitation due to heavy dust grains could also modify the traditional Jeans instability of magnetosonic modes, 3 which is generalized to multiple dust species with the effects of mass distribution. 4 For magnetosonic waves propagating parallel to the magnetic field, the presence of dust grains can greatly cause nonresonant firehose instability. 5 When the drift velocity of the dust beam parallel to the applied magnetic field is larger than that of the phase velocity of magnetosonic waves, the instability of the sheared flow grows, and the amplitude of magnetosonic waves also grows. 6 Accordingly, the linear magnetosonic waves and nonlinear magnetosonic waves in a dusty magnetoplasma 7 were well studied by linear dispersion relations or nonlinear equations in a different regime. [8][9][10] Quantum effects on magnetosonic waves 11,12 or dusty acoustic waves 13 have attracted much attention recently. As a plasma is cooled to a relatively low-temperature or if its number density is relatively high, the de Broglie wavelength of the particles can be comparable to the distance between the charged particles. Accordingly, the quantum effects should be considered. [14][15][16][17] Quantum effects can greatly affect the collective nonlinear excitation in dusty quantum plasmas. [18][19][20][21][22][23] The electrostatic or electromagnetic waves in a dusty degenerate or nondegenerate plasma have been well studied in recent times. [24][25][26][27][28] As the mass of dust particles is much larger than the mass of electrons and ions, the quantum effects on dust particles are usually less important than those on electrons and ions. [29][30][31][32][33] Accordingly, many authors have only considered quantum effects on electrons or ions by Bohm potential. The dusty grains was described by a fluid equation [34][35][36] or taken as a static background. [37][38][39] In magnetized quantum dusty plasmas, the propagating properties of magnetosonic waves were already well studied for both slow and fast modes, which shows that all kinds of quantum effects, such as diffraction effects, statistic effects, and spin-1/2 effects, can have great effects on magnetosonic solitary waves. [40][41][42][43][44] In this paper, we will use the Coulomb coupling parameter Γ d = q 2 d /kBT d r d0 in a classical regime for dusty species, where q d is the equilibrium charge of the dust grains, T d is the temperature, kB is the Boltzmann constant, and r d0 is the Wigner -Seitz radius. We will use the coupling parameter Γj = q 2 j /kBTjr j0 for electrons (j = e) and ions (j = i), respectively. For a dusty plasma with a relatively ARTICLE scitation.org/journal/adv low-temperature and high number density, we will consider Γe,i < 1 and Γ d > 1 for typical parameters in an astrophysical object. In fact, the linear or nonlinear waves in strongly coupled dusty plasmas are quite different from those in weakly coupled systems. [45][46][47][48][49][50] In astrophysical objects, such as the interior of heavy planets, white dwarfs, and neutron stars, the dusty plasma is in a strongly coupled state. 51 Thus, in this paper, we consider the dusty grains to be strongly coupled. Accordingly, our aim in this work is to investigate magnetosonic waves in a strongly coupled magnetized dusty plasma. Here, we focused on a kinetic regime in which magnetosonic solitary waves can be excited. In fact, the nonlinear waves in a strongly coupled nonmagnetized dusty plasma 32,52 were already studied in a kinetic regime. 52,53 In this paper, we will study the quantum effects and strongly coupled effects on nonlinear magnetosonic waves in a magnetized dusty plasma. Then, one may expect that dust viscosity together with the quantum effects on electrons and ions has a strong effect on the properties of magnetosonic waves, which is important for astrophysical objects, such as the interior of heavy planets, white dwarfs, and neutron stars. 51

II. BASIC PLASMA THEORY
We consider the nonlinear propagation of magnetosonic waves in a magnetized dusty plasma consisting of inertial strongly coupled classical dusts and inertialess quantum electrons and ions with weak interparticle interactions. We also consider the applied magnetic field along the Z axis. The nondegenerated electrons and ions obey the pressure law Pj = P j0 (nj/n j0 ) 5 3 , where Pj is the pressure of electrons/ions, and P j0 = n j0 kBTj. The number density nj is normalized by its equilibrium density n j0 of j-species particles, where j = e, and i stands for electrons and ions, respectively.
Then, the momentum equation for electrons and ions can be given by the quantum hydrodynamic equation, 54 On the other hand, the dynamics of strongly coupled dusts can be described by a generalized hydrodynamic model as whereτm, which is normalized by the inverse dusty plasma frequency ω pd , denotes the viscoelastic relaxation time, 46 whereη = ξ * + 4η * /3 is the coefficient of effective dust viscosity, in which η * and ξ * are, respectively, the shear and bulk viscosities. The viscosityη stands for the correlation effects, which gives the additional corrections of the dispersion relation and changes the phase velocity of the waves throughη terms. If one considers a weak coupling limit, the viscosityη reduces to the Navier-Stokes viscosity, which comes from the collision of the dusty particles. 55 For a strongly coupling limit, a transverse shearlike mode can appear in a dusty plasma. 46 Again, it has been found that the viscosity coefficient η/n d0 kBT d in Eq. (5) has a wide minima (∼1) in 1 < Γ d < 10, and it tends to increase as ∝ Γ 32,56 Thus, τm (∝η/n d0 kBT d ) becomes high in weak coupling (Γ d < 1) as well as in strong coupling (Γ d > 1) regimes. So, the kinetic modes exist only for (Γ d < 1) or (Γ d ≫ 1) where the condition ωτm ≫ 1 is satisfied.
Here, the parameter γ d is the dust adiabatic index. u(Γ d ) is a measure of the excess internal dust energy and can be given as 33,57,58 ). Furthermore, the compressibility parameter μ d appearing in Eq. (5) is given by 46 Since u(Γ d ) is negative for increasing values of Γ d , μ d can change its sign. It has been shown that for values of Γ d in 1 < Γ d < 10, this change of sign can cause the dispersion curve to turn over with the group velocity going to zero and then to negative values. 46 We consider dust pressure to include the thermodynamic contribution as well as the pressure due to mutual electrostatic repulsion of like charged dust particles as P d = γ d kBT df n d , 33,59,60 where T df = T * + μ d T d is the effective dust temperature in which T * appears due to electrostatic interactions between strongly coupled dusts and is given by T * = Nnn * Γ d * T d . 59,60 Here, Nnn is determined by the dust structure and corresponds to the number of nearest neighbors (e.g., in the crystalline state, Nnn = 12 for the fcc and hcp lattices, and Nnn = 8 for the bcc lattice). Although the parameter μ d can be negative for increasing values of Γ d , T * may be comparable or even dominate μ d T d for Γ d ≫ 1, so the effective temperature T df (>0) in the limit of κ → 0 is most likely due to the strong coupling of dust grains. Then, the system is enclosed by the following Maxwell's equations: Equations (1)-(4), (7), and (8) have been dimensionless. The number density nj is normalized by the equilibrium number density n j0 for electrons, ions, and dusty particles. The equilibrium number densities of different species satisfy the electrical neutrality conditions as n e0 + Z d0 n d0 = n i0 , where Z d0 is the number of charges of the dust grains, which is determined in units of electron charge. The parameter α = n e0 /Z d0 n d0 is used to satisfy the electrical neutrality condition. The fluid velocity vj is also normalized by the effective thermal speed V T = √ γ d kBT df /m d , where m d is the mass of the dust grains. The space and time coordinates are also normalized by the effective Debye length λD = √ γ d kBT df /4πn d0 Z 2 d0 e 2 and the dust plasma oscillation frequency ω pd = V T /λD, respectively.

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Here, the amplitude of the electric field vector E was normalized by m d ω pd V T /Z d0 e. The amplitude of the magnetic field B was normalized by the applied magnetic field vector B 0 . The dust gyrofrequency Ω d = Z d eB 0 /m d ω pd c is normalized by the dusty plasma frequency ω pd . In Eqs. (1) and (2), the coefficient of pressure term is given by σj = (3Z d0 Tj)/(2γ d T df ) for electrons and ions. The coefficient The normalized shear and bulk viscosities are η * = ηω pd /n d0 γ d kBT df and ξ * = ξω pd /n d0 γ d kBT df .
We will derive a general linear dispersion relation for strongly coupled plasmas to identify different collective modes. In this paper, we will focus on the modes propagating perpendicular to the applied magnetic field. Here, the transverse perturbations are very weak and can be regarded as higher order perturbations. Accordingly, we have v dy1 ≪ v dx1 , E y1 and ky, and kz ≪ kx. By linearizing Eqs. (1)-(4), (7), and (8), we obtain the following dispersion relation: Dispersion relation (9) only describes the fast magnetosonic waves propagating perpendicular to the applied magnetic field. We find that dispersion relation (9) for fast magnetosonic waves is greatly modified by the quantum diffraction effects, nondegenerate pressure, and dust fluid viscosity (∝τm orη) associated with the strong coupling of the dust particles. As mentioned before, the memory function τm (proportional to the dust fluid viscosity) defines two characteristic time scales to distinguish between two classes of wave modes, namely, "hydrodynamic modes" (ωτm ≪ 1) and "kinetic modes" (ωτm ≫ 1).
In the hydrodynamic regime (ωτm ≪ 1), dispersion relation (9) reduces to Assuming that the wave frequency has real and imaginary parts, i.e., ω = ωr + iωi, and the wave number k is real, separating the real and imaginary parts, we obtain the following from Eq. (10): Consequently, the magnetosonic waves suffer viscous effects in the hydrodynamic regime, which were also observed in strongly coupled dusty plasmas with Boltzmann distributed electrons and ions. 46 In the long-wavelength limit, the phase velocity of magnetosonic waves (obtained from the real part of ω) assumes a constant value, i.e., the wave becomes dispersionless. From Eq. (11), we also find that the magnetosonic wave frequency increases with increasing values of k.
In the upper panel of Fig. 1, we give a plot of the real part of ω with respect to k. It is found that as the pressure parameter σe increases, the wave frequency increases with a cut-off at a larger wave number. From Eq. (11), one can also find that the wave has cut-off frequencies at k = kc as follows: From Eq. (13), one can clearly see that this zero-frequency mode occurs due to the existence of dust viscosity, which is related to the first term of the denominator on the right side of Eq. (13). In the lower panel of Fig. 1, we also find that as the pressure parameter of ions σi increases, the wave frequency increases with a cut-off at a larger wave number. In Fig. 1, we used the physical parameters me = 9.1 × 10 −28 g, mi = 1836me, m d = 10 9 mi, Z d0 = 300, n e0 = 2.0 × 10 22 cm −3 , n d0 = 5.6 × 10 19 cm −3 , and n i0 = n e0 + Z d0 n d0 ; Ti ∼ 10 6 K and T d ∼ In Fig. 2, we also give the plot of the real part of ω with respect to k to consider the quantum diffraction effects on the dispersion relation. One can find that as the quantum diffraction parameter of electrons He increases, the wave frequency decreases with a cutoff at a lower wave number. The cut-off frequencies at k = kc can also be obtained from Eq. (11). The quantum diffraction parameters are given as He = 0.0019 (solid line), He = 0.0021 (dashed line), He = 0.0023 (dotted line), and He = 0.0024 (dashed-dotted line), which correspond to the number density n d0 = 5.6 × 10 19 cm −3 , n d0 = 9.6 × 10 19 cm −3 , n d0 = 15.6 × 10 19 cm −3 , and n d0 = 20.6 × 10 19 cm −3 , respectively. These dispersion curves in Figs. 1 and 2 show that the quantum diffraction and the weakly coupled pressure lead to dispersive correction and the changes in the phase velocity. We can also find that the wave modes have a smooth evolution trend from kinetic modes to the hydrodynamic mode with the increasing of the quantum effects although the physical parameters is quite different for the two wave modes.
On the other hand, for relatively high-frequency magnetosonic waves, we obtain the following dispersion relation for the kinetic wave modes (ωτm ≫ 1): Clearly, viscosity is no longer a dissipative effect; however, it modifies the dispersion curve. Furthermore, Eq. (14) shows that the phase speed of the wave remains greater than the effective dust thermal speed and assumes a constant value in the long-wavelength limit k → 0. The kinetic wave modes have no cut-off frequencies. The kinetic magnetosonic wave frequency increases with increasing values of k. As the nonlinear hydrodynamic modes are well known already, we focus on the kinetic wave modes in a nonlinear regime.

III. NONLINEAR MAGNETOSONIC WAVES
We note that the normalized relaxation time τm in Eq. (5) represents two characteristic time scales, which describe two classes of wave modes, namely, the hydrodynamic modes and kinetic modes. In this section, we focused on the kinetic modes. We will use the reductive perturbation method, which was already used to investigate the nonlinear waves in a nonmagnetized dusty plasma, 52,53 to investigate the nonlinear magnetosonic waves in a kinetic regime. As we mainly consider the effects of transverse perturbation on nonlinear magnetosonic solitary waves, we introduce stretched coordinates X = ϵ(x − vpt), Y = ϵ 2 y, and T = ϵ 3 t, where ϵ is an arbitrary small parameter, and vp is the phase velocity of solitary waves. The dependent variables, which are expanded in powers of ϵ, have the form for ne, ni, n d , Bz, Ey, and vjx, where f (0) = 1 for ne, ni, n d , and Bz, and f (0) = 0 for vjx and Ey. The other dependent variables Ex and vjy are expanded in a different way in powers of ϵ as In what follows, we substitute the stretched coordinates and the expansions given in (15) and (16) into Eqs. (1)-(4), (7), and (8).
In the lowest order of ϵ, we obtain from Eqs. (1)-(4), (7), and (8) the following first-order quantities: n j1 = B z1 , v jx1 = vpB z1 , and E y1 = Ω d vpB z1 . One can find that quasineutrality is satisfied from the first order quantities. The other relationships for the first order physical quantities are given as where we have introduced the parameterη = ξ * + 4η * /3. By using the first order results, one can obtain the dispersion relation as One can combine the second order expansion equations and use the first order physical quantities to derive a modified Kadomstev-Petviashvili (mKP) equation as where ϕ stands for B z1 . Here, the coefficients of nonlinearity, dispersion, transverse perturbation, and viscosity effects, i.e., A, B, δ, and γ, are given as ARTICLE scitation.org/journal/adv From Eq. (20), one can see that the nonlinear properties, dispersive relationship, transverse perturbation, and viscosity effects are all related to the strongly coupled effects and quantum effects. The KP equation can be used to investigate a soliton in a weak two dimensional nonlinear excitation, which stands for transverse perturbations effects. 61 If we neglect the dust viscosity effect (η = 0), the coefficient γ vanishes, and Eq. (20) reduces to the usual KP equation as The KP equation can have an analytical line soliton solution and lump solution, which depend on the sign of the transverse perturbation coefficient δ. For our work, the KP equation is the KPII equation as the transverse perturbation δ is always positive, which has a steady line-soliton solution.
Here, we neglect the strongly coupled effects to find the analytical line-soliton solution of (25). Then, we consider the physical quantity to be ϕ(ζ), where ζ is defined as ζ = X + Y − VT with V being the normalized velocity of steady magnetosonic solitary waves. Using the above relationship, the KP equation without the strongly coupled effects can be written as With the help of boundary conditions ϕ → 0 and dϕ/dζ → 0 at ζ → ∞, the solution of the steady state KP equation is calculated as 62 The steady solution (27) is the well known line-soliton profile. We mention that an exact analytic solution of Eq. (20) is much complicated. However, we can seek for an approximate timedependent soliton solution of Eq. (20) with a small effect of γ and high-order transverse effects of δ, which is confirmed from the stretched transverse coordinate Y = ϵ 2 y. To this end, we rewrite Eq. (20) after integrating with respect to X as where we have used the following boundary conditions: ϕ → 0, ∂ϕ/∂X → 0 as X → ±∞. In order to study the effects of viscosity on the character of magnetosonic solitary waves, we consider the normalized solitary speed V to be dependent on time T, To determine the time dependent speed V(T), we use the momentum conservation law in the presence of viscosity as 63 where the momentum of this system I = (1/2) ∫ +∞ −∞ ϕ 2 dX. Substituting Eq. (29) into Eq. (30), we obtain the analytical expression for time dependent speed as where the parameters T 0 = 5A √ B/(V 0 −δ) 3/2 γ and V 0 is the velocity of the magnetosonic waves when T = 0. From the expression of the speed V(T), one can see that the amplitude of magnetosonic solitary waves will change slowly with time, which is due to the small amount of dust viscosity. From Eq. (31), it can also be seen that as the wave amplitude increases, the propagation speed also increases in widening the pulse width.
To obtain the typical parameter for the strongly coupled parameter, we use the dense plasmas in the interiors of heavy planets, white dwarfs and neutron stars. 26,64,65 In such dense astrophysical scenarios, the plasma density lies in the range 10 22 -10 29 cm −3 , the temperature lies in the range 10 5 -10 7 K, and the magnetic field is estimated to lie in the range 10 4 -10 10 G. 66 Here, we use the following parameters: me = 9.1 × 10 −28 g, mi = 1836 * me, m d = 10 9 mi, Z d0 = 200, n e0 = 2.0 × 10 22 cm −3 , n d0 = 5.6 × 10 19 cm −3 , and n i0 = n e0 + Z d0 n d0 ; Te ≈ Ti ∼ 10 6 K and T d ∼ 10 6 K, where Te, Ti, and T d are

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scitation.org/journal/adv the electron, ion, and dust grain temperatures, respectively. From the above physical parameters, one can obtain the strongly coupled parameter of the dust grain as Γ d ∼ 125. In Fig. 3, we give the propagation of the magnetosonic solitary waves described by Eq. (29). In Fig. 3, the magnetosonic solitary waves were illustrated for different times: T = 0, T = 200, T = 400, and T = 600. The amplitude and velocity of magnetosonic solitary waves both changed with time.

IV. SUMMARY AND CONCLUSION
To summarize, we have investigated the linear and nonlinear propagation of magnetosonic waves in a strongly coupled dusty plasma in a kinetic regime, which can illustrate the viscoelastic relaxation effects coming from the strongly coupled dust grains. As the number density of a dusty plasma is relatively high in astrophysical objects, such as the interior of heavy planets, white dwarfs, and neutron stars, the quantum diffraction effects for electrons and ions are included in our model. We analytically investigated the steady line soliton solutions of the KP equation without the viscoelastic effects. With the help of a steady line soliton solution, we obtain an approximate solitary wave analytical solution of the nonlinear magnetosonic waves in the presence of viscoelastic effects. It is shown that the amplitude of the nonlinear magnetosonic solitary waves changes with time with minimal influence of the dust viscosity and weak transverse perturbation.