Brazilian Journal of Physics

https://doi.org/10.1007/s13538-019-00693-2

GENERAL AND APPLIED PHYSICS

Compressive and Rarefactive Ion Acoustic Nonlinear Periodic

Waves in Nonthermal Plasmas
Ata-ur-Rahman1 · Muhammad Khalid1 · Aurang Zeb1

Received: 11 March 2019

© Sociedade Brasileira de Fı́sica 2019

Abstract
The ion acoustic nonlinear periodic (cnoidal) waves are studied in an unmagnetized plasma consisting of cold ions and
nonthermal electrons following Cairns distribution. By employing reductive perturbation method, the nonlinear Korteweg-
de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations are derived, and their periodic wave solutions are
obtained and analyzed numerically. It has been pointed that the KdV equation fails at plasma critical composition, and we
have thus taken into account the higher order nonlinearity and derived the mKdV equation. In the latter case, the coexistence
of compressive and rarefactive periodic wave structures is pointed out in the critical case. So far, this aspect has not been
tackled at all in the nonthermal plasma literature on periodic waves. In the present work, it is the degree of nonthermality of
electrons that is responsible for the rarefactive solutions. The present plasma model accounts for the cnoidal wave structures
(Jovanovic and Shukla Phys. Rev. Lett. 84, 4373 2000) in the magnetosphere observed via FAST and POLAR spacecrafts.

Keywords Ion acoustic nonlinear periodic waves · Reductive perturbation method · Nonthermal plasma

1 Introduction The investigation of nonlinear periodic (cnoidal) waves

has engendered renewed interest in recent past or so
Power law-like velocity distribution functions showing [8–11] due to its wide ranging applications in diverse areas
enhanced high-energy tails are often observed in various of physics. One of the application includes the nonlinear
space and astrophysical environments. These are usually transport phenomena. The main characteristic properties of
fitted by κ-distributions, as was initially suggested by nonlinear periodic waves are manifested through sharper
Vasyliunas [1] and was later on employed in various phys- crests and flatter troughs. The cnoidal waves are worked
ical contexts. However, in 1995, motivated by the obser- out in terms of Jacobi elliptical functions such as sn,
vations of cavitons (rarefaction of ion number density) cn, and dn. These solutions of KdV/mKdV equations are
with the help of Freja satellite and Viking spacecraft, stable (due to small perturbations) and are exact periodic
Cairns et al. [2] introduced an alternative nonthermal dis- solutions linked with proper boundary conditions, while in
tribution function for electron species to investigate for a weakly dispersive media give the nonlinear periodic wave
electrostatic solitary structures with density depressions morphologies [9–11]. Kauschke and Schlüter [12] noticed
observed in space. The Maxwellian distributed electrons that the nonlinear periodic signals detected at the edge of
were unable to explain this behavior. The Cairns distribution, plasma in their experiment can be described in a befitting
in contrast to the power law-like distribution, has wings, manner through the cnoidal waves. Experimentally, the
superimposed on a Maxwellian-like low-energy component. cnoidal waves have also been observed in LiNbO3 -(SiO
Plasmas with nonthermal components (electrons and/or film) structure [13], in shallow water [14, 15], and in
ions) have received a great interest recently; too many to photorefractive bismuth titanate crystal [16]. It was initially
cite, a few are given here, e.g., see Refs. [3–7]. due to Konno et al. [17] who explored the ion acoustic
cnoidal wave solution of the KdV equation in electron-ion
 Ata-ur-Rahman plasma. Following this, Yadav et al. [18] investigated ion
ata797@yahoo.com acoustic nonlinear periodic waves in a magnetized plasma,
using a reductive perturbation approach involving warm
1 Department of Physics, Islamia College Peshawar (Public adiabatic ions and two electron species in one dimension.
Sector University), Peshawar 25120, Pakistan The dynamics therein are modeled by a KdV equation. We
 Braz J Phys

also observe that Maxwellian conditions were considered in Here, the dimensionless variables ni and ui are the ion
that study. density and ion fluid velocity, respectively, with charge
For the sake of rigor, it may be mentioned that for some com- e, Boltzmann’s constant kB , and mass mi , normalized
 by
plex plasma compositions, the KdV picture no longer holds, equilibrium density and by ion acoustic speed cs = Te
mi .
and one can encounter plasma parameter values which allow
The electrostatic potential φ is normalized by eφ Te . Space
the nonlinearity coefficient in the KdV equation to vanish. For
those critical values, the mKdV equation having cubic non- 
and time variables have been normalized in terms of the
Te
plasma screening length λD = 2 and inverse of
linearity is derived (e.g., see Ref. [19]). We will encounter a  4πni0 e
−1 me
similar situation in the present paper. ion plasma frequency ωpi = 4πn e2 , respectively. We
i0
In the present paper, we search for the propagation of ion note that ne0 = ni0 is assumed to establish quasi neutrality
acoustic cnoidal waves in a plasma composed of cold ions and at equilibrium (the index “0” stands for the equilibrium
nonthermal electrons. With the help of reductive perturbation quantities). The ionic charge state Zi = 1 was assumed and
technique, KdV equation is derived and its cnoidal wave solu- ionic thermal effects are ignored, for simplicity.
tion is determined and discussed at length. However, at some The electron distribution is assumed to be non-
plasma compositional parameters, the coefficient of the Maxwellian modeled via a nonthermal Cairns distribution,
quadratic nonlinearity in the KdV equation vanishes. In this and hence, the (reduced) electron density in (3) is given by
special case, we then derive the mKdV equation with cubic  
nonlinearity and its periodic wave solution is determined. ne = 1 − βφ + βφ 2 eφ , (4)
The plan of the paper is organized as follows. In Section 2,
the normalized fluid equations for the system are presented. where the macroscopic nonthermality parameter β is
Using reductive perturbation technique, the KdV equation is defined as β = 1+3δ 4δ
, wherein δ represents the population
derived in Section 3. In Section 4, cnoidal wave solution of of nonthermal electrons. More details concerned to the
the KdV equation is obtained and the effect of relevant Cairns distribution can be seen in the original paper which
plasma parameters is discussed at length. Section 5 is dedi- introduced the nonthermal Cairns distribution [2]. It is to
cated to the particular case when the plasma compositional be noted that β = 0 (i.e., δ = 0) corresponds to the usual
parameters take on values such that the nonlinear coeffi- Boltzmann density distribution, i.e., ne = eφ (in normalized
cient in the KdV equation vanishes. In this critical case, units). Allowing δ to vary in the range 0 to ∞, one notices
we consider the higher order nonlinearity and derive the that β is limited to 0 < β < 4/3.
mKdV equation. The periodic wave solution of the mKdV
equation is determined in Section 6. Section 7 then briefly
summarizes the main conclusions. 3 Derivation of KdV Equation

To model ion acoustic nonlinear periodic waves, namely,

2 Ion Fluid Model ion acoustic cnoidal waves, and inspired by methods in the
literature [17], the following stretching of the spatial and
We are modeling ion acoustic waves propagating in an temporal coordinates in terms of an infinitesimal parameter
unmagnetized electron-ion plasma where the electrons are is introduced:
assumed to follow the Cairns distribution, while the ions
ζ = 1/2
(x − λt) and τ= 3/2
t, (5)
are considered a cold (Ti = 0) population. In case of ion
acoustic waves, it is required that the wave phase speed must where λ is the ion excitation speed interpreted as the phase
lie between ion and electron thermal speeds. In other words, velocity of the nonlinear wave. The dependent variables are
the wave phase speed is much higher than ion thermal speed expanded about their equilibrium values in a power series in
and yet in turn much smaller than electron thermal speed to as:
avoid Landau damping. Adopting the one-dimensional fluid ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
⎡ ⎤ ⎡ ⎤ (1) (2) (3)
description, the dynamics of positive ions may be described ne,i 1 ne,i ne,i ne,i
by the following set of normalized fluid equations ⎣ ui ⎦ = ⎣ 0 ⎦ + ⎢
⎣ u(1)
⎥
⎦+
2 ⎢ (2) ⎥
⎣ ui ⎦ +
3 ⎢ (3) ⎥
⎣ ui ⎦ + · · ·. (6)
i
φ 0 φ (1) φ (2) φ (3)
∂ni ∂
+ (ni ui ) = 0 (1)
∂t ∂x Substituting (5) and (6) into (1)–(4), we obtain the first-
order perturbations and the ion acoustic phase velocity as
∂ui ∂ui ∂φ
+ ui =− (2) follows
∂t ∂x ∂x
∂ 2φ (1) φ (1)
= ne − ni . (3) ni = (7)
∂x 2 λ2
Braz J Phys

(1) φ (1) in respect of which the wave is stationary, where v0 is the

ui = + C1 (τ ) (8) velocity of the cnoidal wave. Using this transformation, (11)
λ
takes the following form
ne(1) = (1 − β) φ (1) (9)
d d d3
− (v0 − C1 ) φ + aφ φ + b 3 φ = 0 (14)
λ = 1/ 1 − β (10) dη dη dη
where C1 (τ ) is the constant of integration and is the Integrating twice (14) with respect to η, we obtain the
function of τ only. In our model for λ to be real, it is required following equation
that β < 1 must hold. Besides, as illustrated in Fig. 1, if 2
1 dφ
β → 0, then λ → 1, which is the scaled ion sound speed + W (φ) = 0 (15)
2 dη
in Maxwellian conditions [17]. Further, λ is seen to increase
with increasing β, and hence, the ion acoustic phase speed In (15), W (φ) is the Sagdeev potential given by
is smaller in Maxwellian plasma as compared with that in a a 3 u 1
W (φ) = φ − φ 2 + ρφ − E 2 (16)
nonthermal plasma. 6b 2b 2
Proceeding to the the next highest orders of (i.e., to the where ρ and E are the charge density and electric field,
second order of perturbation), we get the KdV equation for respectively, when the electrostatic potential φ vanishes and
the present plasma system in the following form u = v0 − C1 . To find the nonlinear periodic wave solution
∂φ ∂φ ∂φ ∂ 3φ using initial conditions φ (0) = α1 and dφ(0)
dη = 0, we can
+ aφ + C1 +b 3 =0 (11) find
∂τ ∂ζ ∂ζ ∂ζ a 3 u 2
where the nonlinear coefficient a and dispersion coefficient E2 = α − α + 2ρα1 (17)
3b 1 b 1
b have been defined as Substituting (16) and (17) into (15), after factorization, we
2 + 3β 2 − 6β obtain the following equation,
a= 3
(12)
2 (1 − β) 2 dφ 2
a
1 = (α1 − φ) (φ − α2 ) (φ − α3 ) (18)
b= (13) dη 3b
3
2 (1 − β) 2 where α2 and α3 are defined as
At this stage, the special case of a plasma with Boltzmann   
electrons, as treated in the literature [17], is obtained at 3 u α1 1
α2,3 = − ± (b1 − α1 ) (α1 − b2 ) (19)
β = 0, i.e., a = 1 and b = 1/2. It is clear that the nonlinear 2 a 3 3
coefficient (12) and dispersion coefficient (13) depend upon In (19), we have defined
nonthermality parameter β. In (11), φ (1) is replaced by φ. 
u u2 2bρ
b1,2 = ± 2 2 − . (20)
a a a
4 Periodic Wave Solution of KdV Equation The following inequalities must hold in order to find the
nonlinear periodic wave solution, i.e., b2 ≤ α1 ≤ b1 or
In order to find the steady-state solution of (11), we define
b1 ≤ α1 ≤ b2 . From (15) and (18), the velocity of the
a traveling wave transformation of the form η = ζ − v0 τ
cnoidal wave in terms of roots α1 , α2 , and α3 is
a
u = (α1 + α2 + α3 ) (21)
3
The nonlinear periodic (cnoidal) wave solution in terms of
Jacobi elliptic function of (15) is given as
φ (η) = α2 + (α1 − α2 ) cn2 (Dη, s) (22)
Here, cn is the Jacobian elliptical function. The modulus
s (0 < s ≤ 1) is a measure of the nonlinearity and is defined
as
α1 − α2
s2 = (23)
α1 − α3
while the quantity D is found as

a
Fig. 1 Variation of λ as β is increased, indicating how λ approaches D= (α1 − α3 ) (24)
the Maxwellian limit as β → 0 12b
 Braz J Phys

From solution of (15), we get the initial condition φ (0) =

α1 for η = 0. Further, for ab > 0, the real numbers α1 , α2 ,
and α3 are regulated as α1 > α2 ≥ α3 and α2 ≤ φ ≤ α1 ,
and for ab < 0, the condition must be α1 < α2 ≤ α3 and
α2 ≥ φ ≥ α1 . The amplitude A and wavelength λ0 of the
ion acoustic cnoidal waves are given by
A = α1 − α2 (25)
and

3b
λ0 = 4 K(s) (26)
a(α1 − α3 )
where K(s) is the complete elliptical integral of the first
kind. Now when ρ = 0 and E = 0, then α2 = α3 = 0, so Fig. 2 Variation of a and b as β is increased, indicating how a shifts
s → 1, the cnoidal wave solution may approach to solitary from positive to negative values at βc and how b remains positive and
wave solution. Therefore we obtain increases monotonically from 1/2
3u
A = α1 = = φm (27)
a the combination of negative (rarefactive) and positive
and (compressive) cnoidal wave structures are possible in the
  present plasma system. In Fig. 2, we have given a graph
a u 1
D= α1 = = (28) of how the nonlinearity coefficient a and the dispersion
12b 4b w
coefficient b vary as the nonthermality parameter β is
In the limiting case s → 1, the Jacobian elliptical increased from 0 to 0.6. It is seen that a changes sign at
function transforms to secant hyperbolic function, that is, some critical value of β. It is observed that a is positive
cn (Dη, 1) = sech (Dη), therefore, (22) takes the form, for 0 ≤ β < βc which defines the existence region for
η compressive cnoidal wave structures while a is negative
φ (η) = φm sech2 (29) for βc < β < 1 which shows the existence domain for
w
 rarefactive cnoidal wave structures.
where φm = 3u a and w = 4b
u are, respectively, the peak Figure 3a depicts the variation of Sagdeev potential
amplitude and width of ion acoustic solitary pulses. W (φ) versus φ for different values of nonthermality para-
In order to study ion acoustic nonlinear periodic wave meter β (considering β < βc and keeping all other parame-
structures, we recall that the nonlinear coefficient a ters fixed). It is seen that W (φ)  = 0, at φ = 0, for
and dispersion coefficient b appearing in the KdV (11) ion acoustic nonlinear periodic waves, represented by solid
are functions of various relevant plasma configuration (thick), solid, dashed, dotdashed curves (i.e., orange, red,
parameters. The value of these coefficients strongly affects green, and blue colors). It appears that the amplitude increa-
the structural characteristics of the pulses and may therefore ses as β is increased (i.e., the nonthermal behavior of the
lead to the generation of different kinds of cnoidal wave plasma becomes more significant). It is also found that if we
structures in the given plasma system. To obtain some increase β, the depth of Sagdeev potential well increases.
insight into the behavior of the ion acoustic cnoidal waves, The black dotted curve with ρ = 0 and E = 0, however,
the real zeros α1 , α2 , and α3 of Sagdeev potential (with represents the Sagdeev potential W (φ) corresponding to
nonzero finite values of the integration constants ρ and solitary waves which becomes zero at φ = 0. For the same
E) are determined. We may remark with no surprise that set of parameters as used in Fig. 3a, we have illustrated the
W (φ) = 0, at φ = 0, for solitary waves and W (φ)  = 0, at phase curves of compressive ion acoustic nonlinear periodic
φ = 0, for cnoidal waves. The Sagdeev potential illustrates waves (bounded inner curves) and solitary pulses separatrix
the fact that the particle oscillates to and fro periodically (outer black dotted curve) in Fig. 3b for positive potential.
between α1 and α2 , which is formally analogous to the main It is noted that the phase separatrix for positive potential
characteristic behavior of nonlinear periodic waves. increases with the increase of β. Furthermore, the orange
It is instructive to mention here that the cnoidal wave solid thick curve appearing in Fig. 3 corresponds to β = 0
solution may give either positive (compressive) or negative (i.e., Maxwellian conditions) and shows that the amplitude
(rarefactive) pulse, depending on the sign of the nonlinearity is small as compared with other values (viz., nonzero
coefficient a. Interestingly, the nonlinear coefficient a values) of β. The deviation from a thermal (Maxwellian)
possesses positive as well as negative values, and thus, distribution thus provides enhanced amplitude ion acoustic
Braz J Phys

(a)
(a)

(b)
(b)
Fig. 3 Plot of a variation in the Sagdeev potential W (φ) with φ and
b phase curves of positive potential ion acoustic cnoidal waves (inner Fig. 4 Plot of a variation in the Sagdeev potential W (φ) with φ and
curves) for u = 0.3, ρ = 0.02, E = 0.0007, and different values b phase curves of negative potential ion acoustic cnoidal waves (inner
of β. The black dotted curve having β = 0.25, ρ = 0, and E = 0 curves) for u = 0.3, ρ = − 0.02, E = 0.0007, and different values
corresponds to the solitary pulses of β. The black dotted curve having β = 0.52, ρ = 0, and E = 0
corresponds to the rarefactive solitary pulses

cnoidal waves. It is also remarkable to note that in the limit

β → 0, the nonlinear coefficient a (viz., a = 1 for β → 0) The above results on the characteristics of ion acoustic
has a positive sign and thus prescribing the positive polarity cnoidal waves have been graphically summarized in Figs. 5
(i.e., compressive in nature) for ion acoustic cnoidal waves and 6. It is very clear from Fig. 5 that higher values of β lead
in a Maxwellian plasma [17]. to larger amplitude positive potential ion acoustic cnoidal
Figure 4a displays the variation of Sagdeev potential waves, while in the context of the rarefactive region, the
W (φ) versus φ for different values of β. Note that we have amplitude of the ion acoustic cnoidal waves decreases as β
considered β > βc and ρ = − 0.02 which corresponds to is increased as shown in Fig. 6.
negative potential cnoidal wave structures. Note that these It is important to note that the nonlinear coefficient a
negative potential cnoidal waves have not been reported becomes zero at β = βc and the KdV equation no longer
previously. In this case, when we assume ρ = 0, and remains valid. It is therefore important to consider higher
E = 0, we obtain negative potential solitary pulses. order nonlinearity to derive the mKdV equation.
The ion acoustic cnoidal waves have been represented
by solid, dashed, and dotdashed curves while solitary
pulses correspond to a black dotted curve. In Fig. 4b, the 5 Derivation of the mKdV Equation
rarefactive phase curves dφdη versus φ for different values of
β are shown for fixed values of all other parameters. From It has been pointed out that the KdV equation fails at
Fig. 4b, one might conclude that lower values of β imply a = 0. For example, in the considered plasma model, we
stronger rarefactive ion acoustic cnoidal waves. may notice that for critical composition, the nonlinearity
 Braz J Phys

In the next highest order of , the second-order momentum

equation results in the following
(2) φ (2) 1  2
ui = + 3 φ (1) + Ci (31)
λ 2λ
With the first- and second-order solutions, the next higher
order equations result the mKdV equation as
∂φ ∂φ ∂φ ∂ 3φ
+ φ 2 + Ci +b 3 =0 (32)
∂τ ∂ζ ∂ζ ∂ζ
where
3λ 3 3 (3β + 1) λ3
= + 3− (33)
4 2λ 4
Again in (32), φ is used instead of φ (1) .

Fig. 5 Plot of compressive ion acoustic cnoidal waves versus η for

different values of β. Here, u = 0.15, ρ = 0.023, and E = 0.0007 6 Periodic Wave Solution of mKdV Equation

coefficient a becomes zero. In this case, the expression To find the steady state solution of (32), we assume η =
a = 2+3β −6β
2
reduces to 2 + 3β 2 − 6β = 0, and thus, ζ − v0 τ where v0 is the velocity of the nonlinear structure
3
2(1−β) 2 in the comoving frame. Using this transformation into (32)
the√required expression for critical value of β, i.e., β = and integrating twice, we obtain the following equation
3± 3
3 , is obtained.
√
As seen from (10), the phase velocity 2
is λ = 1/ 1 − β requiring that√β < 1 for real λ, and 1 dφ
+ W (φ) = 0 (34)
a single critical value, βc = 3−3 3 = 0.423, is obtained 2 dη
within the allowed range. To investigate ion acoustic cnoidal where the Sagdeev potential W (φ) is given by
waves in such a situation, we take into account the higher  4 u1 2 1 2
order nonlinearity and derive the mKdV equation. Again we W (φ) = φ − φ − E (35)
12b 2b 2
use the reductive perturbation technique and introduce the
modified stretching of coordinates as Here u1 = v0 − Ci and 12 E 2 is the constant of integration,
when the potential φ vanishes. To find the nonlinear
ζ = (x − λt) and τ= 3
t (30) periodic solution, substitute (35) in (34), after factorization,
we obtain the following equation,
Substituting (6) and (30) into (1)–(4), we get the same
equations in the lowest order of as in the KdV derivation. dφ 2   
= δ1 − φ 2 φ 2 − δ2 (36)
dη 6b
where δ1 and δ2 are defined as

3u1 9u21 6bE 2
δ1,2 = ± 2
+ (37)
  
From (34) and (36), we obtain

u1 = (δ1 + δ2 ) (38)
6
The cnoidal wave solution in terms of Jacobian elliptic
function of (34) is given as
φ (η) = δ1 cn (D1 η, s1 ) (39)
Here, cn is the Jacobian elliptic function. The modulus s1 is
a measure of the nonlinearity and is defined as
δ1
Fig. 6 Plot of rarefactive ion acoustic cnoidal waves versus η for s12 = (40)
different values of β. Here u = 0.15, ρ = − 0.023, and E = 0.0007 δ1 − δ2
Braz J Phys

While quantity D1 is defined as



D1 = (δ1 − δ2 ) (41)
6b
with δ1 > 0 and δ2 ≤ 0. Now when E = 0, then δ2 = 0, so
s1 → 1, the cnoidal wave solution may approach to mKdV
solitary wave solution. Therefore, we obtain
η
φ (η) = φ0 sech (42)
w1
where the amplitude of ion acoustic solitary waves is

6u1
φ0 = ± (43)

and the width of ion acoustic solitary waves is given by

b Fig. 8 Plot of φ versus η corresponding to the critical plasma
w1 = (44) composition with u = 0.2. Here, the solid curve corresponds to
u1 E = 0.07 and for the dotted curve E = 0
The appearance of plus and minus signs in (43),
respectively, represents the occurrence of both compressive
and rarefactive solitons. It is clearly seen from (43) that compressive mKdV solitons. However, for E  = 0, the
compressive and rarefactive mKdV solitons can coexist in phase curve is repeated on the same path and the mKdV
the present plasma system. The experimental verification of cnoidal waves are formed. While in mechanical analogy,
the coexistence of the compressive and rarefactive mKdV whenever the velocity of the pseudoparticle becomes zero
solitons in a plasma with negative ions has been carried (i.e., dφ/dη = 0), it still feels the restoring force (viz.,
out in Ref. [20]. The phase curves using (34) have been the potential force, since dW (φ) /dφ  = 0) and reflects
√ it
plotted in Fig. 7 for the critical case (viz., a = 0). back and thus oscillates between two points, ± δ1 , and
When E = 0, the continuous phase curve reduces to two resultantly, we have a symmetric phase curve around both
symmetric contours. The left (right) side contour represents of the axes. In other words, the corresponding potential of
that the phase curve begins from the origin, revolving in the mKdV√nonlinear periodic wave oscillates between two
clockwise direction about the negative (positive) φ axis, and values ± δ1 .
at origin, it again stops entering from the upper (lower) It is to be pointed out that the most interesting result
side. In physical space, the left and right side contours, reported here is that, in the critical case (i.e., for which a =
respectively, represent the formation of rarefactive and 0), the mKdV (32) admits a combination of compressive and
rarefactive modes (see Fig. 8).
We may stress here that this result appears particularly in
the context of plasma physics. The criticality resulting in the
mKdV structure cannot exist for surface solitons on shallow
water [21], which are always compressive.

7 Summary

To summarize, we have studied the propagation charac-

teristics of ion acoustic periodic (cnoidal) waves in an
unmagnetized nonthermal plasma consisting of fluid ions
and inertialess electrons obeying Cairns distribution. Using
the reductive perturbation technique, KdV and mKdV equa-
tions have been derived and the nonlinear periodic wave
Fig. 7 Phase curves for (34) corresponding to the critical plasma solution of these equations was determined.
composition with u = 0.2 and so here the mKdV equation is needed.
Due to the variation of the critical value of β, the
Solid curve corresponds to E = 0.07 while the dotted curve is for E =
0. The coexistence of compressive and rarefactive mKdV nonlinear KdV equation admits a combination of compressive and
periodic waves is noted rarefactive nonlinear periodic structures in the given plasma
 Braz J Phys

system. It was found that nonlinear coefficient a becomes structures [23] in the magnetosphere observed via FAST and
zero at β = βc and the KdV equation no longer POLAR spacecrafts.
remains valid. We, therefore, take into account the higher
Acknowledgments Authors gratefully acknowledges the constructive
order nonlinearity and derive the mKdV equation. In suggestions of an anonymous referee which significantly improved the
conditions of higher nonthermality, the dispersive term quality of the manuscript.
of the KdV equation is enhanced and correspondingly
larger (in amplitude) and wider (in width) compressive
ion acoustic nonlinear periodic waves are obtained. On the References
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explanation for the coherent electric field of cnoidal wave jurisdictional claims in published maps and institutional affiliations.