Z. Angew. Math. Phys.c© 2014 Springer BaselDOI 10.1007/s00033-013-0394-1

Zeitschrift fur angewandteMathematik und Physik ZAMP

Contribution of higher order terms in electron-acoustic solitary waveswith vortex electron distribution

Hilmi Demiray

Abstract. The basic equations describing the nonlinear electron-acoustic waves in a plasma composed of a cold electronfluid, hot electrons obeying a trapped/vortex-like distribution, and stationary ions, in the long-wave limit, are re-examinedthrough the use of the modified PLK method. Introducing the concept of strained coordinates and expanding the fieldvariables into a power series of the smallness parameter ε, a set of evolution equations is obtained for various order terms inthe perturbation expansion. The evolution equation for the lowest order term in the perturbation expansion is characterizedby the conventional modified Korteweg–deVries (mKdV) equation, whereas the evolution equations for the higher orderterms in the expansion are described by the degenerate(linearized) mKdV equation. By studying the localized traveling wavesolution to the evolution equations, the strained coordinate for this order is determined so as to remove possible secularitiesthat might occur in the solution. It is observed that the coefficient of the strained coordinate for this order corresponds tothe correction term in the wave speed. The numerical results reveal that the contribution of second order term to the waveamplitude is about 20 %, which cannot be ignored.

Mathematics Subject Classification. 35Q53 · 35Q35.

Keywords. Electron-acoustic waves · Solitary waves · Modified KdV equations · Modified PLK method.

1. Introduction

The idea of electron-acoustic mode had been conceived by Fried and Gould [1] during numerical solutionsof the linear electrostatic Vlassov dispersion equation in an unmagnetized, homogeneous plasma. Besidesthe well-known Langmuir and ion-acoustic waves, they noticed the existence of a heavily damped acoustic-like solution of the dispersion equation. It was later shown that in the presence of two distinct groups(cold and hot) of electrons and immobile ions, one indeed obtains a weakly damped electron-acousticmode [2], the properties of which significantly differ from those of the Langmuir waves.

To study the properties of electron-acoustic solitary structures Dubouloz et al. [3] considered a one-dimensional, unmagnetized collisionless plasma consisting of cold electrons, Maxwellian hot electrons,and stationary ions. However, in practice, the hot electrons may not follow a Maxwellian distributiondue to the formation of phase space holes caused by the trapping of hot electrons in a wave potential.Accordingly, in most space plasma the hot electrons follow the trapped/vortex-like distribution [4–6].Therefore, in the present work, we shall consider a plasma model consisting of a cold electron fluid, hotelectrons obeying a non-isothermal (trapped/vortex-like) distribution, and stationary ions.

The propagation of small-but-finite amplitude waves in a plasma with one-dimensional ion-acousticmodel had been studied by several scientists (see, for instance, Washimi and Taniuti [7] and with one-dimensional electron-acoustic by Schamel [4,5] and Mamun and Shukla [8] by use of the classical reductiveperturbation method [9]. Due to nonlinearity of the governing equations, in the long-wave limit, oneobtains the KdV or modified KdV equations for the lowest order term in the perturbation expansion,the solution of which may be described by solitons. To study the higher order terms in the perturbationexpansion, the use of reductive perturbation method [9] lead to the secularity in the solution that can

H. Demiray ZAMP

be removed by introducing some additional slow scale variables [10] or by the re-normalization procedureby Kodama and Taniuti [11]). The re-normalization method is rather heuristic and criticized by severalscientists (see, [12–14]). To eliminate such secularities in the solution, Demiray [13] presented a methodso called “the modified reductive perturbation method,” which is based on the idea that the higher ordernonlinearities should be balanced by higher order dispersive effects. The detail of the method can befound in the above references.

In the present work, we consider the nonlinear propagation of electron-acoustic waves in a plasmacomposed of a cold electron fluid, hot electrons obeying a trapped/vortex-like distribution, and stationaryions. The basic equations of above described plasma, in the long-wave limit, are re-examined through theuse of the modified PLK method. We introduce the concept of strained coordinates and expand the fieldvariables into a power series of the smallness parameter ε, and a set of evolution equations is obtainedfor various order terms in the perturbation expansion. The evolution equation for the lowest order termin the perturbation expansion is characterized by the conventional modified Korteweg–deVries (mKdV)equation, whereas the evolution equations for the higher order terms in the expansion are described by thedegenerate(linearized) mKdV equation. By studying the localized traveling wave solution to the evolutionequations, the strained coordinate for this order is determined so as to remove possible secularities thatmight occur in the solution. It is shown that the coefficient of the strained coordinate for this ordercorresponds to the correction term in the speed of propagation.

2. Governing equations

We consider a one-dimensional, collisionless plasma consisting of a cold electron fluid, hot electronsobeying a trapped/vortex-like distribution, and stationary ions. The dynamics of electron-acoustic wavesis governed by the following equations

∂nc

∂t+

∂

∂x(ucnc) = 0, (1)

∂uc

∂t+ uc

∂uc

∂x− α

∂φ

∂x= 0, (2)

∂2φ

∂x2=

1αnc + nh −

(1 +

1α

), (3)

where nc is the normalized cold electron number density, nh is the normalized hot electron number density,uc is the cold electron fluid velocity, φ is the electrostatic potential and the coefficient α is defined byα = nh0/nc0, where nc0 and nh0 are the equilibrium values of the cold and hot electron number densities,respectively. The hot electron number density nh (for β < 0) can be expressed by Schamel [4]

nh = I(φ) +2√−πβWD(

√−βφ), (4)

where

I(x) = [1 − erf(√x)] exp(x), WD(x) = exp(−x2)

x∫0

exp(y2)dy, (5)

where erf(x) is the error function. For φ � 1, the Eq. (4) gives

nh = 1 + φ− 43√π

(1 − β)φ3/2 +φ2

2− 8

15√π

(1 − β2)φ5/2 +φ3

6+ · · · . (6)

Denoting the fluctuation of the cold electron number density from its equilibrium value by n, i.e., nc =1 + n, the Eqs. (1)–(3) can be written as

Contribution of higher order terms in electron-acoustic

∂n

∂t+∂uc

∂x+

∂

∂x(ucn) = 0, (7)

∂uc

∂t+ uc

∂uc

∂x− α

∂φ

∂x= 0, (8)

∂2φ

∂x2=

1αn+ φ− 4

3√π

(1 − β)φ3/2 +φ2

2− 8

15√π

(1 − β2)φ5/2 +φ3

6+ · · · . (9)

These equations will be used as we study the solitary waves of small-but-finite amplitude.

3. Modified PLK formalism

When the classical reductive perturbation method [7] is applied to study the higher order perturbationexpansion, it leads to the secularity in the progressive wave solution [10]. To remove such secularitiesDemiray [13] presented the modified reductive perturbation method, which is mainly based on the balanceof higher order nonlinearities with higher order dispersive effects.

In the present work, we shall show that the modified form of extended PLK method can efficientlybe used to study the higher order perturbation expansion. For that purpose, we consider the followingstrained coordinates

ε1/2(x− t) = ξ +∞∑

n=1

εnPn(τ), τ = ε3/2t, (10)

where ε is the smallness parameter measuring the weakness of nonlinearity and Pn(τ) (n = 1, 2, 3, . . .)are some unknown functions to be determined from the solution. When we introduce (10) into (7)–(9),the following equations are obtained

−∂n

∂ξ+ ε

∂n

∂τ+∂uc

∂ξ−

∞∑n=1

εn+1 dPn(τ)dτ

∂n

∂ξ+

∂

∂ξ(ucn) = 0, (11)

−∂uc

∂ξ+ ε

∂uc

∂τ−

∞∑n=1

εn+1 dPn(τ)dτ

∂uc

∂ξ+ uc

∂uc

∂ξ− α

∂φ

∂ξ= 0, (12)

−ε∂2φ

∂ξ2+

1αn+ φ− 4

3√π

(1 − β)φ3/2

+φ2

2− 8

15√π

(1 − β2)φ5/2 +φ3

6+ · · · = 0. (13)

We shall assume the field quantities can be expanded into power series in ε as:

n = ε2[n(1) + εn(2) + ε2n(3) + · · · ],uc = ε2[u(1)

c + εu(2)c + ε2u(3)

c + · · · ],φ = ε2[φ(1) + εφ(2) + ε2φ(3) + · · · ]. (14)

When we are introducing the expansion (14) into Eqs. (11)–(13) and set the coefficient of like powers ofε equal to zero, the following sets of differential equations are obtained.O(ε2) equations:

− ∂n(1)

∂ξ+∂u

(1)c

∂ξ= 0,

∂u(1)c

∂ξ+ α

∂φ(1)

∂ξ= 0,

n(1)

α+ φ(1) = 0. (15)

H. Demiray ZAMP

O(ε3) equations:

−∂n(2)

∂ξ+∂u

(2)c

∂ξ+∂n(1)

∂τ= 0,

∂u(2)c

∂ξ+ α

∂φ(2)

∂ξ− ∂u

(1)c

∂τ= 0,

∂2φ(1)

∂ξ2=n(2)

α+ φ(2) − 4

3√π

(1 − β)φ(1)3/2. (16)

O(ε4)equations:

−∂n(3)

∂ξ+∂u

(3)c

∂ξ+∂n(2)

∂τ− dP1

dτ∂n(1)

∂ξ+

∂

∂ξ[n(1)u(1)

c ] = 0,

−∂u(3)c

∂ξ− α

∂φ(3)

∂ξ+∂u

(2)c

∂τ− dP1

dτ∂u

(1)c

∂ξ+ u(1)

c

∂u(1)c

∂ξ= 0,

−∂2φ(2)

∂ξ2+n(3)

α+ φ(3) − 2√

π(1 − β)φ(1)1/2

φ(2) +12φ(1)2 = 0. (17)

3.1. Solution of the field equations

From the solution of the set (15) one obtains

φ(1) = ϕ1(ξ, τ), n(1) = −αϕ1, u(1)c = −αϕ1, (18)

where ϕ1(ξ, τ) is an unknown function of its argument and the evolution equation of it will be obtainedlater.

When we are introducing the solution (18) into the Eq. (16), the following differential equations areobtained

−∂n(2)

∂ξ+∂u

(2)c

∂ξ− α

∂ϕ1

∂τ= 0,

∂u(2)c

∂ξ+ α

∂φ(2)

∂ξ+ α

∂ϕ1

∂τ= 0,

n(2) = −αφ(2) + α

[∂2ϕ1

∂ξ2+

43√π

(1 − β)ϕ13/2

]. (19)

The solution of (19) gives

n(2) = −αϕ2 + α

[∂2ϕ1

∂ξ2+

43√π

(1 − β)ϕ13/2

],

u(2)c = −αϕ2 +

α

2

[∂2ϕ1

∂ξ2+

43√π

(1 − β)ϕ13/2

], (20)

where ϕ2(ξ, τ) is another unknown function whose governing equation will be obtained later.Eliminating n(2), u

(2)c and ϕ2 between the Eqs. (19) and (20), we obtain the following evolution

equation

∂ϕ1

∂τ+

1√π

(1 − β)ϕ11/2 ∂ϕ1

∂ξ+

12∂3ϕ1

∂ξ3= 0. (21)

This evolution equation is known as the modified Korteweg–deVries (mKdV) equation.To obtain the solution for O(ε4) equations, we first add (17)1,2 side by side and eliminate φ(3) through

the use of (17)3, which gives

Contribution of higher order terms in electron-acoustic

−α∂3ϕ2

∂ξ3− 2

α√π

(1 − β)∂

∂ξ[ϕ1

1/2ϕ2] − 2αc1∂ϕ1

∂τ

+∂

∂τ[u(2)

c + n(2)] + (α+ 3α2)ϕ1∂ϕ1

∂ξ= 0. (22)

Introducing the expressions of u(2)c and n(2) into the Eq. (22), we have the following evolution equation

∂ϕ2

∂τ+

1√π

(1 − β)∂

∂ξ[ϕ1

1/2ϕ2] +12∂3ϕ2

∂ξ3= R2(ϕ1), (23)

where R1(ϕ1) is defined by

R2(ϕ1) =dP1

dτ∂ϕ1

∂ξ+

34∂3ϕ1

∂ξ2∂τ+

32√π

(1 − β)ϕ11/2 ∂ϕ1

∂τ+

(12

+32α

)ϕ1∂ϕ1

∂ξ. (24)

The evolution Eq. (23) is the degenerate(linearized) form of the modified KdV equation with the non-homogeneous term R2(ϕ1). Here, we note that the function R2(ϕ1) contains an unknown function dP1/dτ ,which is to be determined so as to eliminate the possible secularities that might occur in the progressivewave solution.

If the higher order contribution is studied, the evolution equation for the nth equation order term inthe perturbation expansion will have the following form

∂ϕn

∂τ+

1√π

(1 − β)∂

∂ξ[ϕ1

1/2ϕn] +12∂3ϕn

∂ξ3= Rnϕ1, . . . , ϕn−1) n = 2, 3, . . . , (25)

where the non-homogeneous term Rn(ϕ1, ϕ2, . . . , ϕn−1) contains the unknown functions dPn/dτ n =1, 2, 3, . . . , n− 1. To save the space, we will not give the detailed calculation here.

3.2. Solitary waves

In this sub-section, we shall study the propagation of solitary waves in such a medium. For that purpose,we shall seek a progressive wave solution to the evolution Eqs. (21) and (23) of the following form

ϕi(ξ, τ) = ψi(ζ), ζ = χ(ξ − u0τ), i = 1, 2, (26)

where u0 is the speed of propagation and χ is a constant to be determined from the solution. Inserting(26) for i = 1 into the evolution Eq. (21), we obtain

− u0ψ′1 +

1√π

(1 − β)ψ1/21 ψ′

1 +χ2

2ψ′′′

1 = 0, (27)

where the prime denotes the differentiation of the corresponding quantity with respect to ζ. Integrating(27) with respect to ζ and utilizing the localization condition, i. e., ψ1 and its various derivatives vanishas ζ → ±∞, we obtain

− u0ψ1 +2

3√π

(1 − β)ψ3/21 +

χ2

2ψ′′

1 = 0. (28)

This nonlinear ordinary differential equation assumes a solitary wave solution of the form

ψ1 = a sech4ζ, (29)

where a is the constant amplitude of the wave, u0 and χ are defined by

u0 =8(1 − β)a1/2

15√π

, χ2 =(1 − β)a1/2

15√π

. (30)

Here it is seen that the speed of the solitary wave is proportional to the square root of the amplitude.

H. Demiray ZAMP

To obtain progressive wave solution to the evolution Eq. (23), we insert (26) for i = 2 into (23),integrate the result with respect to ζ and utilize the localization condition yields the following non-homogeneous differential equation

ψ′′2 +

(30a1/2

ψ1/21 − 16

)ψ2 =

30√π

(1 − β)a1/2

dP1

dτψ1

−4(1 − β)a1/2

5√π

ψ′′1 − 16

(1 − β)√π

ψ3/21 +

(12

+32α

)15

√π

(1 − β)a1/2ψ2

1 . (31)

Inserting the solution (29) into (31), we have

ψ′′2 +

(30sech2ζ − 16

)ψ2 = a

(1)0 sech4ζ + a

(1)1 sech8ζ, (32)

where the coefficients a(1)0 and a(1)

1 are defined by

a(1)0 =

[30a1/2

√π

1 − β

dP1

dτ− 64(1 − β)a3/2

5√π

], a

(1)1 =

(12

+32α

)15

√π

(1 − β)a3/2. (33)

Here it is easily seen that sech4ζtanhζ is one of the fundamental solutions of the homogeneous differentialequation derived from (32). Thus it is convenient to introduce the following change of dependent variable

ψ2 = f(ζ)sech4ζtanhζ, (34)

where f(ζ) is an unknown function to be determined from the solution. Inserting (34) into (32), we have

f ′′tanhζ + (−8 + 10sech2ζ)f ′ = a(1)0 + a

(1)1 sech4ζ. (35)

When we make the substitution f ′ = U , the Eq. (35) reduces to the first-order, non-homogeneous differ-ential equation

U ′tanhζ + (−8 + 10sech2ζ)U = a(1)0 + a

(1)1 sech4ζ. (36)

The solution of the homogeneous equation is given by

Uh =cosh10ζ

sinh2ζ. (37)

The general solution of (36) may be given by

U = d2cosh10ζ

sinh2ζ− a

(1)0

8coth2ζ − a

(1)1

31

sinh22ζ, (38)

where d2 is an integration constant. Integrating (38) with respect to ζ, one obtains the auxiliary functionf(ζ) as

f(ζ) = d1 +

(63256

d2 − a(1)0

8

)ζ + d2

[105512

sinh(2ζ) +15256

sinh(4ζ) +15

1024sinh(6ζ)

+5

2048sinh(8ζ) +

15120

sinh(10ζ)]

+

(a(1)0

8+

a(1)1

12

)cothζ +

a(1)1

12tanhζ, (39)

Contribution of higher order terms in electron-acoustic

where d1 is another integration constant. Thus, the general solution for ψ2 may be given by

ψ2 = d1sech4ζtanhζ +

(a(1)0

8+

a(1)1

6

)sech4ζ − a(1)

1

12sech6ζ

+

(63256

d2 − a(1)0

8

)ζsech4ζtanhζ + d2sech4ζtanhζ

[105512

sinh(2ζ)

+15256

sinh(4ζ) +5

1024sinh(6ζ) +

52048

sinh(8ζ) +1

5120sinh(10ζ)

]. (40)

It is seen that the last two terms in Eq. (40) cause to secularities (see [10]). In order to avoid the secularity,the coefficients of these expressions must vanish, i. e.,

d2 = 0, a(1)0 = 0. (41)

This result shows that, in order to avoid the secularity, the coefficient of sech4ζ, which is a(1)0 for this

order, in Eq. (32)must vanish one must have a(1)0 = 0. This restriction gives

dP1

dτ− 32(1 − β)2a

75π= 0, or P1 =

32(1 − β)2a75π

τ. (42)

The remaining part of the solution takes the following form

ψ2 = d1sech4ζtanhζ +(

18

+38α

)5√πa3/2

(1 − β)(2sech4ζ − sech6ζ). (43)

Here the first term corresponds to the solution of the homogeneous differential equation in (31)and thesecond term to the particular solution. The, the total solution up to O(ε3) may be given by

φ = ε2a

{[1 + εa1/2

(14

+34α

)5√π

(1 − β)

]sech4ζ

−εa1/2

(18

+38α

)5√π

(1 − β)sech6ζ + ...

}(44)

where in terms of real space and time variables, ζ is defined by

ζ =[(1 − β)a1/2

15√π

]1/2

ε1/2

(x− t− ε

8(1 − β)a1/2

15√π

t− ε232(1 − β)2a

75πt− · · ·

). (45)

Here [32(1 − β)2a/75π] corresponds to the speed correction term for this order.If the localized progressive wave solution to the evolution (25) is studied, the governing ordinary

differential equation will have the form

ψ′′n + (30sech2ζ − 16)ψn = a

(n−1)0 sech4ζ +

∑j>1

a(n−1)j sech4+2jζ, (46)

where the coefficient a(n−1)0 , which causes to secularity in the solution, contains dPn−1/dτ as unknown

term. In order to avoid the secularity the coefficient a(n−1)0 must vanish. This requirement provides a

relation to determine dPn−1/dτ , which is nth order speed correction term, may be obtained. To save thespace, we shall not give them here.

H. Demiray ZAMP

3.3. Numerical results and discussion

It might be instructive to illustrate the analytical results obtained in the previous sub-section by somenumerical graphs. For numerical purposes, following Mamun and Shukla [8], we shall select the parametersas α = 5.0, β = −0.5, α = 1.0 and ε = 0.1. The analytical results for ϕ1, given in (29), and for ϕ1 + εϕ2,given in (44) and (45), are evaluated numerically for various values of time parameter t, and the resultsare depicted in Figs. 1 and 2, respectively. The Fig. 1 shows the progress of bell-shaped solitary wave intime. The variation of the wave profile for the sum of the first and the second order terms is shown inFig. 2. As is seen from this figure, the contribution of second order term to the wave amplitude is about20 %, which cannot be ignored.

4. Conclusion

We have considered a plasma consisting of a cold electron fluid, hot electrons obeying a trapped/vortex-like distribution, and stationary ions. By employing the modified PLK perturbation method, we studiedthe propagation of small-but-finite amplitude waves and obtained a set of evolution equations governing

Fig. 1. The variation of solitary wave profile ϕ1 for various values of the time parameter t

Fig. 2. The variation of the solitary wave profile ϕ1 + εϕ2 for various values of the time parameter t

Contribution of higher order terms in electron-acoustic

various order terms in the perturbation expansion. It is observed that the lowest order term in the per-turbation expansion is governed by conventional modified Korteweg–deVries (mKdV) equation, whereasthe higher order terms in the expansion are governed by the degenerate(linearized) mKdV equation withnon-homogeneous term. Seeking a localized traveling wave solution to the evolution equations, we havedetermined the strained coordinate so as to remove the possible secularities that might occur in thesolution. It is observed that the coefficient of the strained coordinate for this order corresponds to thecorrection term to the wave speed. Although we studied the evolution equation for the second order termsin the perturbation expansion and determined the strained coordinates for this order, the higher orderterms can also be studied without any principal difficulty.

The analytical results obtained in the Sect. 3.2 are evaluated numerically, and the results are depictedon some figures. The results of numerical evaluation reveal that the contribution of second order term inthe perturbation expansion to the wave amplitude is about 20 %, which might not be neglected.

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Hilmi DemirayFaculty of Arts and SciencesDepartment of MathematicsIsik University34980 SileIstanbulTurkeye-mail: demiray@isikun.edu.tr

(Received: July 19, 2013; revised: December 18, 2013)