Stability of two-dimensional ion-acoustic wave packets in quantum plasmas
ABSTRACT The nonlinear propagation of two-dimensional (2D) quantum ion-acoustic waves (QIAWs) is studied in a quantum electron–ion plasma. By using a 2D quantum hydrodynamic model and the method of multiple scales, a new set of coupled nonlinear partial differential equations is derived which governs the slow modulation of the 2D QIAW packets. The oblique modulational instability (MI) is then studied by means of a corresponding nonlinear Schrödinger equation derived from the coupled nonlinear partial differential equations. It is shown that the quantum parameter H (ratio of the plasmon energy density to Fermi energy) shifts the MI domains around the kθ -plane, where k is the carrier wave number and θ is the angle of modulation. In particular, the ion-acoustic wave (IAW), previously known to be stable under parallel modulation in classical plasmas, is shown to be unstable in quantum plasmas. The growth rate of the MI is found to be quenched by the obliqueness of modulation. The modulation of 2D QIAW packets along the wave vector k is shown to be described by a set of Davey–Stewartson-like equations. The latter can be studied for the 2D wave collapse in dense plasmas. The predicted results, which could be important to look for stable wave propagation in laboratory experiments as well as in dense astrophysical plasmas, thus generalize the theory of MI of IAW propagations both in classical and quantum electron–ion plasmas.
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arXiv:0912.4439v1 [physics.plasm-ph] 22 Dec 2009
Stability of two-dimensional ion-acoustic wave packets in quantum plasmas
Amar P. Misra,1, ∗Mattias Marklund,1, †Gert Brodin,1, ‡and Padma K. Shukla2, §
1Department of Physics, Ume˚ a University, SE-901 87 Ume˚ a, Sweden
2Institut f¨ ur Theoretische Physik IV, Fakult¨ at f¨ ur Physik and Astronomie,
Ruhr-Universit¨ at Bochum, D-44780 Bochum, Germany
(Dated: 22 Dec., 2009)
The nonlinear propagation of two-dimensional (2D) quantum ion-acoustic waves (QIAWs) is studied in a
quantum electron-ion plasma. By using a 2D quantum hydrodynamic model and the method of multiple scales,
a new set of coupled nonlinear partial differential equations is derived which governs the slow modulation of
the 2D QIAW packets. The oblique modulational instability (MI) is then studied by means of a corresponding
nonlinear Schr¨ odinger equation derived fromthecoupled nonlinear partial differential equations. Itisshownthat
the quantum parameter H ∝ ¯ h, associated with the Bohm potential, shifts the MI domains around the kθ-plane,
where k is the carrier wave number and θ is the angle of modulation. In particular, the ion-acoustic wave (IAW),
previously known tobe stableunder parallel modulation inclassical plasmas, isshown tobe unstable inquantum
plasmas. The growth rate of the MI is found to be quenched by the obliqueness of modulation. The modulation
of 2D QIAW packets along k is shown to be described by a set of Davey-Stewartson-like equations. The latter
can be studied for the 2D wave collapse in dense plasmas. The predicted results, which could be important
to look for stable wave propagation in laboratory experiments as well as in dense astrophysical plasmas, thus
generalize the theory of MI of IAW propagations both in classical and quantum electron-ion plasmas.
PACS numbers: 52.27.Aj; 52.30.Ex; 52.35.Fp; 52.35.Mw; 52.35.Sb
The importance of quantum effects has been recognized
over the last few years in view of its remarkable applications
in metallic and semiconductor nanostructures (e.g., metal-
lic nanoparticles, metal clusters, thin metal films, nanotubes,
quantumwell andquantumdots, nano-plasmonicdevicesetc.)
[1, 2, 3, 4, 5, 6, 7, 8] as well as in dense astrophysical envi-
ronments (e.g., white dwarfs, neutron stars, supernovae etc.)
It is well known that the nonlinear propagation of wave
packets in a dispersive plasma medium is generically sub-
ject to amplitude modulations due to the carrier wave self-
interaction, i.e., a slow variation of the wave packet’s enve-
lope due to nonlinearities. Under certain conditions, the sys-
tem’s evolution may thus undergo a modulational instability
(MI), leading to energy localization via the formation of en-
velope solitons. Such solitons are governed by a nonlinear
Schr¨ odinger(NLS) equationwhere the nonlinearityat the first
stage of the amplitude is balanced by the group dispersion.
This mechanism is encountered in various physical contexts,
including pulse formation in nonlinear optics, in material sci-
ence, as well as in plasma physics. A number of works can
be found in the literature for the investigation of MI in clas-
sical (see, e.g., Refs. [11, 12, 13, 14, 15, 16, 17]) as well as
quantum plasmas (see e.g., Refs. [18, 19, 20, 21]). The MI
of ion-acoustic waves (IAWs) has been shown to be a general
Department of Mathematics, Siksha Bhavana, Visva-Bharati University,
Santiniketan-731 235, India
†Electronic address: firstname.lastname@example.org
‡Electronic address: email@example.com
§Electronic address: firstname.lastname@example.org
property in a nonlinear dispersive plasma medium, when the
modulation is considered obliquely to the direction of propa-
gation of the wave vector . Experimental observations of
the MI of IAWs have been reported by Watanabe . Sta-
ble wave propagation from modulational obliqueness, in both
classical [13, 14, 16, 17] and quantum plasmas , has also
been investigated by a number of authors.
On the otherhand, in a wide varietyof scientific fields (e.g.,
nonlinear optics, plasma physics, fluid dynamics etc.), certain
nonlinear governing equations often exhibit important phe-
nomena other than solitons, such as shocks (wave singular-
ity), self-similar structures, wave collapse (i.e., blow up with
growing amplitude or amplitude decay at finite time or finite
distance of propagation), as well as the wave radiation emis-
sion leading to the onset of chaos. One such importantsystem
in this context is the Davey-Stewartson (DS)-like equations
, where the system has both quadratic as well as cubic
nonlinearities. Such equations can appear not only in the field
of fluid dynamics as in the case of water wave propagation
[22, 23], or in optical communications and information pro-
cessing in nonlinear optical media [24, 25, 26], but also in
plasma physics community [27, 28]. The DS description of
collective excitations in Bose-Einstein condenstates has also
been studied in the context of matter-wave solitons [29, 30].
The purpose of the present work is to investigate, in more
detail, the stability and instability criteria for the modula-
tion of quantum IAW (QIAW) packets using two-dimensional
(2D) quantum fluid model, and to generalize the previous in-
vestigation , both from classical and quantum points of
view. Here we show that the nonlineardynamicsof QIAWs as
well as the static zeroth harmonic field is governed by a new
set of coupled nonlinear partial differential equations, which,
in particular, reduces to a set of Davey-Stewartson (DS)-like
equations for surface water waves . We show that the
quantum parameter H ∝ ¯ h, associated with the Bohm poten-
tial, shifts the MI domains in the kθ-plane in 2D quantum
plasmas. Here k is the carrier wave number and θ is the an-
gle of modulation with the propagation vector. We also show
that the maximum growth rate of MI can be reduced by the
obliquenes parameter θ rather than H as in one-dimensional
(1D) quantum plasmas [18, 19]. Moreover, a criterion for the
existence of 2D wave collapse is presented from the DS-like
The paper is organized as follows. In Sec. II, we describe
the 2D quantum hydrodynamicmodel, and derive the govern-
ing system of equations using the method of multiple scales.
In Sec. III the MI of the QIAWs is studied and the condition
for the existence of 2D wave collapse is derived. Finally, the
Sec. IV is left for concluding the results.
II.FLUID MODEL AND DERIVATION OF THE
We consider the nonlinear propagation of QIAWs in a 2D
quantum electron-ion plasma. The basic normalized set of
equations in a two-componentunmagnetizedquantumplasma
∂t+∇·(niV) = 0,
∂t+(V·∇)V = −∇φ,
∇2φ = ne−ni,
where ∇ ≡ (∂/∂x,∂/∂y), ne(i)is the electron (ion) number
density normalized by their equilibrium value n0, V ≡ (u,v)
is the ion velocity normalized by the ion-acoustic speed cs=
?kBTFe/miwith kBdenoting the Boltzmann constant, mithe
perature, and ¯ h the scaled Planck’s constant.
¯ hωpe/kBTFeis denoting the ratio of the ‘plasmon energy den-
sity’ to the Fermi thermal energy, where ωpj=?n0e2/ε0mj
the electrostatic potential normalized by kBTFe/e.The space
and time variables are respectively normalized by cs/ωpiand
the inverse of ωpi. The electron pressure gradient (∇pe) and
quantum force in Eq. (3) appear due to the electron degen-
eracy in a dense plasma with the Fermi distribution function.
The former can be given (since the equilibrium pressure is
truly three-dimensional) by the following equation of state
ion mass, TFe≡ ¯ h2(3π2n0)2/3/2kBmethe electron Fermi tem-
Also, H =
is the plasma frequency for the j-th particle. Moreover, φ is
where VFe≡?kBTFe/meis the Fermi thermal speed of elec-
In order to obtain the evloution equations for QIAW pack-
ets, we employ the standard multiple-scale technique (MST)
 in which the space and the time variables are stretched
ξ = ε(x−vgxt),η = ε(y−vgyt),τ = ε2t,
where ε is a small parameter representing the strength of the
wave amplitude and vg≡ (vgx,vgy) is the normalized (by cs)
group velocity, to be determined later by the compatibility
condition. The dynamical variables are expanded as
of the corresponding quantity. Notice that in Eqs. (7)-(9)
the perturbed states depend on the fast scales via the phase
(k·r−ωt) (where k and ω respectively denote the wave vec-
tor and wave frequency), whereas the slow scales only enter
the l-th harmonic amplitude. We suppose that k makes an
angle θ with the x-axis, and the modulation is along any di-
rection in the xy-plane. In a general way, the wave vector k
is then k ≡ (kx,ky) = (kcosθ,ksinθ) and the group velocity,
vg≡ (vgx,vgy) = (vgcosθ,vgsinθ).
Substituting the expressions from Eqs. (7)-(9) into the Eqs.
(1)-(4) and collecting the terms in different powers of ε we
obtain for n = 1,l = 1 the linear dispersion relation for the
normalized wave frequency (ω → ω/ωpi) and the wave num-
ber (k → kcs/ωpi) as
with asterisk denoting the complex conjugate
satisfy the reality condition
where Λ ≡ 1/3+H2k2/4. Note that the wave number k in
Eq. (10) is not very small such that the wave frequency ω
is sufficiently large to prevent the appearance of harmonic
modes of kx,ky as proper modes. These harmonic modes
will virtually appear in higher orders of perturbations. Also,
since k is normalizedby the inverse of Fermi screening length
(λF≡ cs/ωpi), the values of k greater than unity is inadmis-
sible, otherwise the wavelength would become smaller than
the screening length. As a result, the collective behaviors of
the plasma will disappear. Moreover, we consider the quan-
tum parameter H, to vary in the range 0.1 < H ? 0.45 such
that the ratio VFe/c ≪ 1 [an approximate condition for the
nonrelativistic quantum hydrodynamicmodel to be valid] and
the coupling parameter, gQ≡ 2mee2/ε0¯ h2(3π2√n0)2/3? 1
(which corresponds to the density region where the quantum
collective and mean field effects become important).
On the other hand, for the second order reduced equations
with n = 2,l = 1, we obtain the following compatibility con-
ditions for the group velocity components
Next, proceeding in the same way as in Refs. [18, 19, 21]
and finally considering the equations for l = 1 and n = 3, we
obtain the following coupled equations for the propagation of
3˜ ψφ +Qq
∂ ¯ ψ
4¯ ψφ +Qq
∂ ¯ ψ
∂ ˜ ψ
6˜ χφ = 0, (13)
∂ ˜ χ
j = 1,...,6 for Q’s, j = 1,2,3 for P’s and j = 1,...,4 for
others.The corresponding coefficients Pc
the propagation of 2D classical IAWs can also be obtained
by the similar method as discussed in Appendix B. The
superscripts ‘q’ and ‘c’ are used to denote the coefficients
corresponding to 2D quantum and 2D classical electron-ion
plasmas. Thus, we have obtained a new system of nonlocal
nonlinear equations, which describe the slow (and general)
modulation of the QIAW packets in 2D quantum plasmas.
The coefficients Pq
persion and 2D motion, and Pq
of the carrier wave propagation as well as the modulation of
the QIAW packets. The nonlinear coefficients Qq
the carrier wave self-interaction originating from the zeroth
harmonic modes (or slow modes), i.e., the ponderomotive
force, and Qq
and self-interaction.The nonlinear-nonlocal coefficients
field associated with the first harmonic (with a ‘cascaded’
effect from the second harmonic) and a static field generated
due to the mean motion (zeroth harmonic) in the plasma. The
appearance of the coefficients Rq
Equations (13) and (14) can be studied, as for example, for
the modulation of Stokes wave train (plane wave) with con-
stant amplitude to a small perturbation as well as to look for
envelope solitons, and wave collapse, if any. There are, of
course, many other physical insights (e.g., dromion solution),
that can also be recovered from this system. In particular,
looking for the modulation of QIAW packets parallel to the
carrier wave vector, one can obtain the DS-like equations 
1, ¯ ψ
?∂ξ|φ|2∂η, ˜ χ ≡
0∂η, ˜ ψ
jare given in Appendix A, where
2appear due to the wave group dis-
3for the arbitrary orientations
1is due to
6are for the combined effects of 2D motion
4arise due to the coupling between the dynamical
jcan also be explained
in 2D quantum plasmas
22ψφ = 0,(15)
where the coefficients are those obtained at θ = 0 from the
general expressions. Also, Qq
be obtained by transforming the x-axis through an angle θ, so
that the wave vector k and the modulation direction coincide.
Now, our aim is to investigate the MI of QIAWs by consid-
ering its modulation along the x-axis through an angle θ with
the carrier wave vector k. To this end, we disregard the group
velocity component of the modulated wave along the y-axis,
as well as the y- or η- dependence of the physical variables.
Thus, we obtain fromEqs. (13) and (14) the NLS equationfor
the oblique modulation of QIAW packets
2. Similar forms like Eqs. (15) and (16) can also
∂ξ2+Q|φ|2φ = 0,
where P ≡ Pq
1and Q ≡ Qq
III.ANALYSIS OF MODULATION INSTABILITY
We consider the MI of a plane wave solution of Eq. (17)
for φ with constant amplitude φ0. The boundary conditions
that φ → 0 as ξ → ∞ must now be relaxed, because the plane
wave train is still unmodulated and the solution is not unique.
Thus, we can represent the solution as the monochromatic
solution φ = φ0exp(iQ|φ0|2τ), where ∆(τ) = −Q|φ0|2is the
nonlinear frequency shift. To study the stability of this so-
lution we modulate the amplitude against linear perturbation
as φ =[φ0+φmcos(Kξ −Ωτ)]exp?iQ|φ0|2τ?, where K and Ω
ulation. We then readily obtain from Eq. (17) the dispersion
are, respectively, the wave number and the frequency of mod-
that the MI sets in for a wave number K < Kc, i.e. for all
wavelengths above the threshold, λc= 2π/Kc. The instability
growth rate (letting Ω = iΓ) is then given by
?2Q/P|φ0| is the critical wave number such
Γ = PK2
The maximum growth rate, Γmax= |Q||φ0|2as equals to the
amount of the nonlinear frequency shift |∆(τ)|, is achieved at
K = Kc/√2. Clearly, the instability condition depends only
on the sign of the product PQ, which, in turn, depends on
the angle of modulation θ as well as the quantum parameter
H. These may be studied numerically, relying on the exact
expressions of P and Q.
H = 0.3 (i.e., as one enters the density region n0= 1.5295×
1033m−3), P > 0 in 0.6958? θ < π/2 and Q < 0 in 0 < θ ?
0.5281 and 1.138? θ < π/2 for 0 < k < 1. Other domains of
k and θ where P and Q change sign depend on the P = 0 and
Q = 0 curves in the kθ-plane. Similarly, considering another
value of H = 0.45 (where n0= 1.34×1032m−3) one can find
P > 0 in 0.6748 ? θ < π/2 and Q < 0 in 0 < θ ? 0.6 and
1.137 ? θ < π/2 for 0 < k < 1. As above, P,Q also change
their sign in other regions in the kθ-plane.
We find that for a fixed value of
Figure 1 displays the contour plots of PQ = 0 boundary
curves against the normalized wave number k and the mod-
ulation angle θ. The upper panel corresponds to the quan-
tum case, whereas the lower one is for the classical electron-
ion plasmas. The stable (PQ < 0) and unstable (PQ > 0) re-
gions are separated in the kθ-plane by the boundary curves of
PQ = 0 as indicated by the white and the gray regions respec-
tively. We have allowed k to vary in a range 0 < k < 1 (as
explained in Sec. II) and θ to vary between θ = 0 and θ = π,
so that all plots seem to be π/2 periodic, and, in fact, sym-
metric uponreflection with respect to either θ =0 or θ =π/2
lines. We also consider H, in the range 0.1 < H ? 0.45 such
that gQ? 1 (as explained before). It is clear from Fig. 1 that
the stability and instability domains are quite differentin clas-
sical andquantumplasmas. Evidently,the quantumparameter
H shifts the MI domains in the kθ-plane. It is, of course, hard
to find the common instability regions for both classical and
quantum cases. Increasing the value of H (? 0.45) gives no
effective change, except in reducing the stable regions around
k ∼ 1.
In particular, considering the modulation of the QIAWs
along the carrier wave vector k (i.e., the case of θ = 0), the
QIAW is shown to be modulational unstable (see Fig. 2).
This is in contrast to the classical case where the IAWs are
knownto be stable underparallelmodulation. Physically,
the QIAW under parallel modulation becomes unstable due to
the nonlinearself-interactionsoriginatingfromthe zerothhar-
monic modes or slow modes as well as the second harmonic
modes(since P andQ are always negative). However,thecase
of modulation perpendicular to the wave vector k (θ = π/2)
gives rise stable wave propagation, an agreement with clas-
sical plasmas . Here the wave is stable due to the second
Thus, for the MI to occur the second harmonic mode is very
essential in order to maintain the same sign of P and Q.
On the other hand, the MI growth rate as given by Eq. (19)
can be calulated as depicted in Fig. 3. The upper panel (cor-
responding to QIAWs) shows that the growth rate can be re-
duced by increasingthe obliqueness of modulationgivingrise
cut-offs at lower wave numbers of modulation. The quantum
parameter H has no significant role in reducing such growth
rate. In the classical case, the maximum growth rate reduces
slightly with a small change of the angle of modulation (see
lower panel of Fig. 3)
FIG. 1: The equation PQ = 0 is contour plotted in the kθ-plane to
show the stable and unstable regions in quantum (upper panel) and
classical (lower panel) plasmas [Eq.(17)] for H = 0.3. The white or
stable (gray or unstable) areas correspond to the regions in the kθ-
plane where PQ < 0 (PQ > 0).
FIG. 2: The case of parallel modulation (θ = 0): PQ is plotted
against k to show that the QIAW is unstable (PQ > 0). The solid
and dashed lines respectively correspond to the cases where H = 0.3
and H = 0.45.
IV.VARIATIONAL PRINCIPLE AND 2D-EVOLUTION
The DS-like equations (15) and (16) describes the case of
general 2D-evolutionof slowly varyingweakly nonlinearion-
acousticwaves inthe quantumregime. Herewe start bypoint-
ing out a variational principle for the DS-like system. Intro-
FIG. 3: The MI growth rate Γ is shown with respect to the wave
number of modulation, K. The growth rate can be quenched by the
obliqueness parameter θ in quantum plasmas (upper panel). In the
classical case (lower panel), the maximum growth rate is reduced
slightly with a small change of θ. The solid, dashed and dotted lines
correspond to the values θ = 0.99,1.0,1.05 respectively.
ducing the Lagrangian density
where the action functional is A (φ,φ∗,u) =?Ldξdτ, and
(15) and (16) by varying φ, φ∗and u and minimizing the ac-
tion as usual. It is straightforward to show that (15) and (16)
posses constants of motion, representing the conservations of
the number of high frequency quanta
∂2u/∂ξ2≡ ψ plays the role of a potential. We obtain eqs.
−∞|φ|2dηdξ = 0(21)
and of longitudinal momentum
dηdξ = 0(22)
as well as transverse momentum
dηdξ = 0.
Furthermore, since no explicit time dependence occurs in the
Lagrangian, the Hamiltonian?∞
−∞H dηdξ, derived from
L in (20), is also conserved. The existence of a Variational
principle makes it possible to use trial functions as a means to
obtain approximate solutions. A good review on this topic for
wave equations similar to (15) and (16) is given by Ref. .
A thorough variational study of (15) and (16) for the general
case is beyond the scope of the present paper. However, in or-
der to illustrate the usefulness of the variational technique,we
will state a variational result that applies when a cylindrically
symmetric collapse is possible. For the general case, Eqs.
(15) and (16) do not admit a cylindrically symmetrical col-
lapse. However, for geometries and parameter regimes where
the coefficients fulfill either R2≪ R1, or Qq
our system reduces to a 2D NLS equation for which cylindri-
cally symmetrical solutions are possible.
least one of the above strong inequalities apply, and that we
have cylindrically symmetric initial conditions (to be defined
below), the evolution equation for our system can be written
Assuming that at
+Qeff|φ|2φ = 0 (24)
where we have introducedthe radial coordinate r2= ξ2/Pq
spatial dependence is only through this variable . The coef-
ficient Qeffis either Qq
(16) was due to the inequality Qq
For the case of Qeff> 0, collapse is possible. A good es-
timate for the collapse threshold can be obtained by using
the variational formulation, with a trial function of the form
φ(τ,r) = F(τ)sech(r/f(τ))exp(ib(τ)r2), see e.g. Ref. 
for mathematical details. The result is that there will be a
collapse towards a zero radius of the pulse profile, f(τ) → 0
within a finite time, provided the initial profile is sufficiently
intense and well localized, as expressed by the collapse con-
2, and assumed, cylindrical symmetry, i.e. that the
11(in case the reduction from (15) and
22S1/R1(in case the reduction was due to R2≪ R1).
11) or Qeff=
Qefff2(τ = 0)??F2(τ = 0)??? 1.35
numerical calculations. Naturally, for cases where cylindrical
symmetry does not apply, the evolution will be much more
complicated than indicated by our simple example.
The accuracy of this collapse condition is well supported by
We have investigated the nonlinear propagation of QIAW
packets in a 2D quantum plasma. A multiple scale technique
is used to derive a coupled set of nonlinear partial differential
equations, which governs the dynamics of modulated QIAW
packets. The set of equations, in particular (i.e., modulation
along the direction of the pump carrier wave), is shown to be
reducible to a well-known Davey-Stewartson (DS)-like equa-
tions . The latter can be studied for the 2D wave collapse
in dense plasmas. The oblique MI of QIAWs is then stud-
ied by means of a corresponding NLS equation. It is found
that the quantum parameter H has the significant role in shift-
ing the instability domains around the kθ-plane. The case of
parallel modulation, which is known to be stable in classical
plasmas , is shown to be unstable in quantum plasmas. In
contrast to the classical case, the MI growth rate in quantum
plasmas can significantly be reduced by increasing the angle
It is to be mentioned that the numerical simulation of the
DS equations could furnish more convincing evidence for the
process of soliton collapse, where the quantum parameter H
may play a crucial role in accelerating or decelarating the col-
lapse process. However, it needs extra effort, and could be an
open issue to be explored in future studies. The integrability
as well as the dromion solution, if there be any, of the system
will also be an another investigation, but beyond the scope of
the present work.
A. P. M. gratefully acknowledges support from the Kempe
Foundations. This research is supported by the European Re-
search Council under Contract No. 204059-QPQV and the
Swedish Research Council under Contract No. 2007-4422
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The coefficients Pq
(14) are given as follows.
jappearing in Eqs. (13) and
In the expression for Qq
brackets is due to the first and second harmonic modes (the
first and zeroth harmonic modes).
1the first (second) term in the square
The coefficients corresponding to classical electron-ion
plasmas can be obtained by considering a 2D fluid model
with contininuity and momentum equations for cold ions and
Boltzmann distributed electrons [see, e.g., the Eqs. (5)-(8) in
Ref. ]. The normalizations and the corresponding fluid
equations for the variables can be recoveredby simply replac-
ing the Fermi temperature TFeby the classical temperature Te
with considering the electron pressure as pe= kBTeneinstead
of the Fermi pressure, and disregarding the Bohm potential
term proportional to ¯ h. These coefficients are presented for
a general interest of the readers to study the dynamics of ion-
(14) or (15), (16) in 2D classical electron-ion plasmas.
Thus, the coefficients Pc
tions like (13) and (14) in the case of classical plasmas can be
given as follows:
jto be appeared in equa-
where, in the expression for Qc
square brackets is due to the first and second harmonic modes
(the first and zeroth harmonic modes). Moreover,
1, the first (second) term in the
where ω2= k2/(1+k2) is the normalized classical disper-
sion relation (ω → ω/ωpiand k → kcs/ωpi). We note that the
difference in the dispersion relation as compared to Eq. (10)
stems from the use of a classical (i.e. not a Fermi) equation of