Physics Letters A 366 (2007) 606–610

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Fully nonlinear ion-sound waves in a dense Fermi magnetoplasma

S. Ali ∗, W.M. Moslem 1, P.K. Shukla 2, I. Kourakis

Institut für Theoretische Physik IV, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germany

Received 7 May 2007; accepted 15 May 2007

Available online 2 June 2007

Communicated by V.M. Agranovich

Abstract

The nonlinear propagation of ion-sound waves in a collisionless dense electron-ion magnetoplasma is investigated. The inertialess electrons areassumed to follow a non-Boltzmann distribution due to the pressure for the Fermi plasma and the ions are described by the hydrodynamic (HD)equations. An energy balance-like equation involving a new Sagdeev-type pseudo-potential is derived in the presence of the quantum statisticaleffects. Numerical calculations reveal that the profiles of the Sagdeev-like potential and the ion-sound density excitations are significantly affectedby the wave direction cosine and the Mach number. The present studies might be helpful to understand the excitation of nonlinear ion-sound wavesin dense plasmas such as those in superdense white dwarfs and neutron stars as well as in intense laser-solid density plasma experiments.© 2007 Elsevier B.V. All rights reserved.

PACS: 52.35.Fp; 52.35.Mw; 52.27.-h

Quantum plasma has recently attracted much attentionamongst the plasma physics community because of its poten-tial applications in different scientific areas such as quantumcomputers, semiconductor devices [1], quantum dots, quan-tum wires [2], and quantum wells, carbon nanotubes, quantumdiodes [3], ultra-cold plasmas [4], microplasmas [5], biopho-tonic [6], superdense giant planets [7], and intense laser-soliddensity plasma experiments [8]. Quantum plasmas which canbe modeled with the help of the Schrödinger–Poisson and the

* Corresponding author at: Department of Physics, Government College Uni-versity, Lahore 54000, Pakistan.

E-mail addresses: shahid_gc@yahoo.com (S. Ali),wmmoslem@hotmail.com, wmm@tp4.rub.de (W.M. Moslem), ps@tp4.rub.de(P.K. Shukla), ioannis@tp4.rub.de (I. Kourakis).

1 Present address: Department of Physics, Faculty of Education-Port Said,Suez Canal University, 42111, Egypt.

2 Also at the Nonlinear Physics Centre & Center for Plasma Scienceand Astrophysics, Ruhr-Universität Bochum, D-44780 Bochum, Germany;Department of Physics, Umeå University, SE-90187 Umeå, Sweden; Max-Planck-Institut für extraterrestrische Physik, D-85741 Garching, Germany;GoLP/Instituto Superior Técnico, 1049-001 Lisbon, Portugal; CCLRC Cen-tre for Fundamental Physics, Rutherford Appleton Laboratory, Chilton, Didcot,Oxon 0X11 0QX, UK; SUPA Department of Physics, University of Strathclyde,Glasgow G 40NG, UK; School of Physics, Faculty of Science & Agriculture,University of Kwazulu-Natal, Durban 4000, South Africa.

0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2007.05.073

Wigner–Poisson equations, consist of the electrons and ionswhose densities (temperatures) are extremely high (low), con-trary to the classical plasma. Quantum mechanical effects comeinto play when the de-Broglie wavelength exceeds the De-bye wavelength and approaches the Fermi wavelength (viz.λBe > λDe and λBe ∼ λFe), where λBe (λFe) is the de-Broglie(Fermi) wavelength, and λDe is the Debye wavelength. Numer-ous analytical and computational efforts [9–15] have been madeto investigate the collective processes involving the quantummechanical effects. By using the Schrödinger–Poisson equa-tions, Haas et al. [16] investigated a multistreaming instabilityinvolving a new quantum mode in an unmagnetized quantumplasma. This is reminiscent of an earlier study, which em-ployed the Wigner–Poisson system [17], where it was pointedout that Landau damping suppresses the streaming instabili-ties. Recently, Haas et al. [18] studied the linear and nonlinearproperties of the IAWs in quantum plasmas taking into accountthe Bohm potential and the Fermi–Dirac distribution. Haas [18]also developed a quantum HD model for a magnetized plasma.More recently, Shukla and Eliasson [19] have presented a nu-merical study of the dark solitons and vortices in quantumelectron plasmas.

Several years ago, Shukla and Yu [20] presented exact local-ized solutions for nonlinear ion-acoustic waves (IAWs) propa-

S. Ali et al. / Physics Letters A 366 (2007) 606–610 607

gating obliquely to the external magnetic field in the classicalelectron–ion plasma. The results of Ref. [20] was generalizedby Yu et al. [21] who presented a fully nonlinear theory forIAWs, accounting for three-dimensional ions and Boltzmanndistributed electrons. Bharuthram and Shukla [22] provided aone-dimensional theory for finite amplitude monotonic doublelayers associated with IAWs in an unmagnetized plasma con-taining cold ions and a double Maxwellian distribution for theelectrons.

In this Letter, we study fully nonlinear ion-sound waves(ISWs) propagating obliquely to an external magnetic field ina collisionless dense magnetoplasma appropriate pressure lawfor a dense Fermi plasma and three-dimensional ion responseto the nonlinear ISWs, we derive an energy equation. The latteris numerically solved to study the profiles of the Sagdeev-likepotential [23] and the fully nonlinear ion-acoustic solitary pulseproperties.

We consider a dense magnetoplasma whose constituents arethe electrons and singly charged ions. The plasma is confinedin an external magnetic field B0 = zB0, where z is the unit vec-tor along the z-axis and B0 is the strength of the magnetic field.The electrostatic ion-sound waves are supposed to propagateobliquely against the external magnetic field in the x–z plane.We also assume that the quantum plasma satisfies the conditionTFe � TFi and follows a one-dimensional electron pressure law[18] pe = men

3eV

2Fe/3n2

e0, where VFe = (2KBTFe/me)1/2 is the

Fermi electron thermal speed, KB is the Boltzmann constant,TFe (TFi) is the Fermi electron (ion) temperature, me is the elec-tron mass, ne is the electron number density, with the equilib-rium value ne0. The dynamics of the low-frequency (ω � ωci )nonlinear ISWs in a dense magnetoplasma is governed by

(1)∂ni

∂ t+ ∇ · (niUi ) = 0,

(2)

(∂

∂t+ Ui · ∇

)Ui = −∇ϕ + Ui × z,

(3)0 = ∂ϕ

∂Z− 1

2

∂n2e

∂Z+ H 2

e

2

∂

∂Z

( ∇2√ne√ne

),

and

(4)ne = ni ,

where ∇ = x∂x + z∂z, Ui is the normalized ion fluid velocity,and ϕ is the normalized scalar potential. The quantum diffrac-tion effects are denoted by a parameter He = hωLH/2KBTFe,h is the Planck constant divided by 2π , ωLH = (ωciωce)

1/2 isthe lower-hybrid resonance frequency, and ωci (ωce) is the ion(electron) gyrofrequency. Furthermore, mi is the mass of ion,and ni is the ion number density. It is noted that the dynamics ofthe inertialess electrons rapidly thermalized along the magneticfield direction [24]. However, they do not follow the Boltzmannlaw, as in the case for the classical plasma.

All physical quantities appearing in Eqs. (1)–(4) have beenappropriately normalized: nj = nj/nj0, Ui = Ui/Cs , Uez =Uez/Cs , ∇ = ∇ρs , Z = Z/ρs , t = tωci , ϕ = eϕ/2KBTFe,where ρs = Cs/ωci is the Fermi ion-sound gyroradius, andCs = √

(2KBTFe)/mi is the Fermi ion-sound speed. Dropping

the “hat” notation over reduced variables, we express Eqs. (1)and (2) into their scalar components as

(5)∂ni

∂t+ ∂

∂X(niUix) + ∂

∂Z(niUiz) = 0,

(∂

∂t+ Uix

∂

∂X+ Uiz

∂

∂Z

)Uix

(6)= −1

2

∂n2e

∂X+ H 2

e

2

∂

∂X

(∇2√ne√ne

)+ Uiy,

(7)

(∂

∂t+ Uix

∂

∂X+ Uiz

∂

∂Z

)Uiy = −Uix,

and(∂

∂t+ Uix

∂

∂X+ Uiz

∂

∂Z

)Uiz

(8)= −1

2

∂n2e

∂Z+ H 2

e

2

∂

∂Z

(∇2√ne√ne

),

where we have used (3) to eliminate ϕ.To obtain localized solutions, we consider a new moving

frame of single variable ξ = lxX + lzZ − Mt , normalizedwith ρs , in which the localized ion-sound pulse is moving withspeed M , lx and lz are the directional cosines of the wavevec-tor k along the x- and z-axes, so that l2

x + l2z = 1. Using the

charge-neutrality condition (viz. ne = ni = n) and combiningEqs. (5)–(8), we obtain an equation in terms of a single inde-pendent variable as

∂

∂ξ

{(3an − 1

n3

)∂n

∂ξ− P

n

{−1

2

∂

∂ξ

[1

n

(∂n

∂ξ

)2]

+ 1

2

∂3n

∂ξ3

}− Q

∂

∂ξ

( ∂2

∂ξ2

√n

√n

)}

= −bn4 + n

M2

(1 − 1

n

)

(9)+ P

M2

[− 1

2n

(∂n

∂ξ

)2

+ 1

2

∂2n

∂ξ2

]+ bn,

where P = 3al2zH

2e /2, Q = 3al2

xH2e /2, a = 1/3M2, and b =

l2z /3M4. Eq. (9) is our main result, which incorporates the

quantum mechanical effects in a dense electron-ion quantummagnetoplasma. It is obvious that the Bohr quantum diffractioneffects are embedded in (9) through P and Q while the quan-tum statistical effect produces a different type of nonlinearityfrom that appearing for the classical plasma [20,21].

An analytical solution of Eq. (9) is difficult to obtain. For thesake of analytical tractability, we shall here consider the spe-cial case P = Q = 0, obtained by formally setting He → 0. Anenergy-balance-like equation incorporating the quantum statis-tical effect, then turns out to be

(10)

(∂n

∂ξ

)2

+ V (n;M) = 0,

where

608 S. Ali et al. / Physics Letters A 366 (2007) 606–610

(a) (b)

Fig. 1. (Color online.) The Sagdeev potential V (n) against the number density n for (a) different values of wave directional cosine lz , lz = 0.3 (thick curve), lz = 0.32(thin curve), and lz = 0.35 (dashed curve), while the Mach number is M = 0.4 everywhere. (b) Here, M = 0.4 (thick curve), M = 0.42 (thin curve), and M = 0.5(dashed curve) with lz = 0.3 everywhere.

(a) (b)

Fig. 2. (Color online.) The maximum amplitude nmax against lz and M with fixed values of M = 0.7 and lz = 0.3, respectively.

V (n;M) = n6

(3an4 − 1)2

[(a + 3)b + 1 − a

M2+ 1 − 2n

n2M2

(11)

− bn2 + abn6 − a(2n − 3)n2

M2− 2b

(an3 + 1

n

)].

In deriving (10), we have imposed the boundary conditions:n → 1, ∂n/∂ξ → 0, at |ξ | → ∞. Eq. (11) which represents theSagdeev pseudo-potential for a quantum magnetoplasma and issignificantly different from that in Ref. [21] due to the new elec-tron pressure law for the dense Fermi plasma. Furthermore, werecall that the pseudo-potential V is essentially a function of n,since the Mach number M enters as a free parameter (whoseallowed range of values has to be determined).

The existence of the ion-sound soliton is possible, if theSagdeev potential satisfies the following conditions [25]:

(12)V ′′(n)|n=1 = M2(M2 − l2z

)(M2 − 1

)< 0,

(13)V ′(n)|n=1 = 0 and V (n)|n=1 = 0,

where the prime denotes the differentiation with respect to n.Eq. (12) implies that the limits of the Mach number mustbe l2

z < M2 < 1. It is clear that only subsonic localized ion-acoustic waves exist in a quantum magnetoplasma. We have

displayed the Sagdeev potential V (n) profiles, from Eq. (11), inFig. 1. It is seen that the region of the existence of the ion-soundsoliton (i.e. the negative V area in Fig. 1) becomes narrower bydecreasing the directional cosine (lz), so localized ISW exci-tations will be shorter and wider for larger angles (viz. smallercosine); see Fig. 1(a). On the other hand, the permitted region ofthe density values associated with the localized excitations ex-pand as the Mach number (M) acquires higher values, implyingthat faster pulse excitations will be taller and narrower.

It is straightforward to show that a maximum V (nm) = 0may exist at some density values, say n = nm (representingthe maximum amplitude of the localized pulse excitation), pro-vided that

n6m

(3an4m − 1)2

(a + 3)b + 1 − a

M2+ 1 − 2nm

n2mM2

− bn2m

(14)= a(2nm − 3)n2m

M2+ 2b

(an3

m + 1

nm

)− abn6

m.

Eq. (14) has been numerically solved and nm is plotted againstthe directional cosine lz and the Mach number M in Fig. 2.It is found that the maximum density nm decreases by increas-ing lz (see Fig. 2(a)), while it is an increasing function of M (see

S. Ali et al. / Physics Letters A 366 (2007) 606–610 609

(a) (b)

Fig. 3. (Color online.) The number density n against ξ for: (a) values of the direction cosine lz = 0.3 (thick curve), lz = 0.32 (thin curve), and lz = 0.35 (dashedcurve) with fixed M = 0.4; (b) Mach number values M = 0.4 (thick curve), M = 0.42 (thin curve), and M = 0.5 (dashed curve) with fixed lz = 0.3.

Fig. 2(b)). This confirms the qualitative discussion above, basedon Fig. 1, leading to the conclusion that the pulse amplitudewill be larger for higher Mach number values, and/or for higherobliqueness angles (hence smaller cosine values).

We have numerically solved the energy equation (10) forvarious sets of parameter values. We have thus obtained thedensity pulse profiles depicted in Fig. 3. We note that higherpulses are narrower, while shorter are wider, in agreement withthe existing (e.g. Korteweg–de Vries equation related) solitonphenomenology. It is obvious that the density profile becomesshorter and wider by increasing lz, while varying M yields theopposite effect.

Concluding, we have presented a study of the nonlinear ion-acoustic waves propagating obliquely to an external magneticfield in a collisionless quantum electron–ion magnetoplasma.An energy-balance-like expression involving a Sagdeev-likepseudo-potential has been derived by employing the QHDequations. Analytical and numerical calculations reveal thatonly subsonic ion-sound solitary waves may exist. The depen-dence of the pseudo-potential profiles and of the density pulseexcitation characteristics on the Fermi distribution, the wavedirectional cosine and the Mach number, have been investi-gated.

Our results should elucidate the excitation of nonlinear ion-acoustic waves in quantum plasmas, particularly in superdenseastrophysical objects (e.g. in the interior white dwarfs and neu-tron stars), microplasmas, as well as in intense laser-solid den-sity plasma experiments.

Acknowledgements

S.A. acknowledges the partial financial support from theDeutscher Akademischer Austausch Dienst (Bonn, Germany).W.M.M. thanks the Alexander von Humboldt-Stiftung (Bonn,Germany) for financial support. I.K. acknowledges supportfrom the German Research Society (Deutsche Forschungsge-meinschaft, DFG) under the Emmy-Noether Programme (grantSH 93/3-1).

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