Analytical solution for nonplanar waves in a plasma with q-nonextensive nonthermal velocity distribution: Weighted residual method

Chaos, Solitons and Fractals 130 (2020) 109448

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Analytical solution for nonplanar waves in a plasma with q-nonextensive nonthermal velocity distribution:Weighted residual method Hilmi Demiray Isik University, Faculty of Arts and Sciences, Department of Mathematics, Sile 34980 Istanbul, Turkey

a r t i c l e

i n f o

Article history: Received 8 March 2019 Revised 29 July 2019 Accepted 13 September 2019

Keywords: Nonplanar solitary waves Cairns-Tsallis distribution q-nonextensive nonthermal distribution

a b s t r a c t The basic nonlinear equations describing the dynamics of a two component plasma consisting of cold positive ions and electrons obeying hybrid q- nonextensive nonthermal velocity distribution are examined in the cylindrical(spherical) coordinates through the use of reductive perturbation method and the cylindrical(spherical) KdV and the modified KdV equations are obtained. An approximate analytical method for the progressive wave solution is presented for these evolution equation in the sense of weighted residual method. It is observed that both amplitudes and the wave speeds decrease with the time parameter τ . Since the wave profiles change with τ , the waves cannot be treated as solitons. It is further observed that the amplitudes of spherical waves are larger than those of the cylindrical waves; and the wave amplitudes of modified KdV equation are much larger than those of the KdV equation. The effects of physical parameters (α , q) on the wave characteristics are also discussed. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Solitons are localized pulse shaped stable nonlinear entities which arise as a manifestation of balance between the nonlinearity and dispersion. Washimi and Taniuti [1] were the first to use the reductive perturbation method to derive the KortewegdeVries(KdV) equation for ion-acoustic solitons(IASs) in plasma. Early investigations on IASs were based on the Maxwellian distribution, which are believed to be universally valid. However, the recent studies on space and laboratory plasmas indicate the presence of energetic particles in tailed- particle distribution. Moreover, the observations made by Viking spacecraft [2] and Freja satellite [3] showed the importance of electrostatic solitary structures. Cairns et al. [4] introduced a distribution model in terms of a parameter α , which measures the deviation from the Maxwellian distribution function. In another development, Tribeche et al. [5] extended the work of Cairns et al. [4] and introduced a hybrid CairnsTsallis distribution. Using this model Wang et al. [6], Saha et al. [7], Saha and Tamang [8], and Tamang et al. [9] showed the existence of electron-acoustic solitary waves in such a plasma. It was further shown that the regions of existence and the amplitude of the solitary waves are effected by the nonextensive parameter q and the nonthermal parameter α . More recently Bouzit et al. [10,11] have

E-mail address: [email protected] https://doi.org/10.1016/j.chaos.2019.109448 0960-0779/© 2019 Elsevier Ltd. All rights reserved.

investigated the effect of an interplay between nonthermality and nonextensivity on ion-acoustic solitons. They further observed that, ion-acoustic solitons exhibit compression or rarefaction, depending on the nonextensivity and nonthermality of modulational instability of ion acoustic waves. Moreover, it is shown that both nonthermal and nonextensive parameters affect the domains of instabilities. In all these works the planar field equations of plasmas in one dimension are investigated. The propagation of nonlinear waves in cylindrically and spherically symmetric plasmas had been studied before by several researchers (see, for instance, Maxon and Viecelli [12,13], Mamun and Shukla [14], Sahu and Roychoudhury [15] and Xue [16]) and obtained the cylindrical and the spherical KdV equations. Presently, there is no analytical solutions available for these evolution equations but some numerical solutions exist. Demiray and Bayindir [17] and Demiray and El-Zahar [18] presented some approximate analytical solutions based on the weighted residual methods and the results are compared with the results of numerical solutions. It is observed that the analytical results agree well with the numerical solutions. In the present work, employing the basic nonlinear equations describing the dynamics of a two component plasma consisting of cold positive ions and electrons obeying hybrid q- nonextensive nonthermal velocity distribution are examined in the cylindrical(spherical) coordinates through the use of reductive perturbation method and the cylindrical(spherical) KdV and modified KdV

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H. Demiray / Chaos, Solitons and Fractals 130 (2020) 109448

equations are obtained. Using the weighted residual method developed in [19], approximate analytical solutions for the progressive wave are obtained for these evolution equations. It is observed that both the wave amplitudes and the wave speeds decrease with the time parameter τ . Since the wave profiles change with τ , the waves cannot be treated as solitons (see the reference [23]). It is further observed that the amplitudes of spherical waves are larger than those of the cylindrical waves; and the wave amplitudes of modified KdV equation are much larger than those of the KdV equation. The effects of the physical parameters α and q on the wave speeds and the steepening characteristic of the waves are also discussed. 2. Basic field equations We consider a homogeneous, collisionless, unmagnetized plasma consisting of cold positive ions with electrons obeying qnonextensive nonthermal velocity distribution. The dynamics of such a system may be described by the set of normalized equations (Williams et al. [20]):

∂n + ∇ . ( nv ) = 0, ∂t

(1)

∂v + (v.∇ )v + ∇φ = 0, ∂t

(2)

∇ 2 φ − ne + n = 0,

(3)

where n and ne are the normalized cold ion and electron number densities, v is the velocity vector of the ion fluid and φ is the electrostatic potential, respectively. The normalized q-nonextensive nonthermal electron density profile is given by q+1





ne = [1 + (q − 1 )φ ] 2(q−1) 1 + G1 φ + G2 φ 2 ,

(4)

where the parameter q, called the nonextensive parameter, stands for the strength of nonextensive system and the coefficients G1 and G2 are defined by





G1 = −16qα / 3 − 14q + 15q2 + 12α ,

G2 = − ( 2q − 1 )G1 .

(5)

Here α is a parameter determining the number of nonthermal electrons in the model. The range and the validity of (q, α ) for solitons are discussed by Williams et al. [20]. In the extensive limiting case (q → 1) and α = 0, the above distribution reduces to the wellknown Maxwell- Boltzmann velocity distribution. For (q → 1) and α = 0, the above distribution reduces to Cairns distribution [4]. In this work, we shall study the cylindrically (spherically) symmetric plasma, in which case the field equations take the following form

∂n ∂ m + ( nv ) + ( nv ) = 0, ∂t ∂ r r

(6)

∂v ∂v ∂φ +v + = 0, ∂t ∂r ∂r

(7)

∂ 2 φ m ∂φ + − ne + n = 0, r ∂r ∂ r2

(8)

where the coefficients d1 , d2 , d3 , are defined by



The Eqs. (6)–(9) will be used as we study the propagation of cylindrical(spherical) solitary waves of small- but finite amplitude. 3. Derivation of cylindrical(spherical) KdV and mKdV Equations For the derivation of the cylindrical(spherical) KdV and mKdV evolution equations we shall employ the classical reductive perturbation method (Taniuti [21]). For that purpose we introduce the following stretched coordinates

ξ =  1/2 (r − λt ), τ =  3/2t,

(11)

where  is the smallness parameter measuring the weakness of nonlinearity and dispersion and λ is a parameter to be determined from the solution. We should point out that the transformation (11) is presented on the basis of the linear dispersion relation for planar case. The limitations of such a transformation for nonplanar case had been discussed by several researchers (see, for instance, the references [22,23]). But, for our future studies, in this work we shall adopt the transformation given in (11). Introducing (11) into the Eqs. (6)-(9) the following equations are obtained

−λ

∂ m ∂n ∂n + + ( nv ) + ( nv ) = 0, ∂ξ ∂τ ∂ξ λτ + ξ

(12)

−λ

∂v ∂φ ∂v ∂v + +v + = 0, ∂t ∂τ ∂ξ ∂ξ

(13)



m 2 ∂φ ∂ 2φ + − 1 − d1 φ − d2 φ 2 − d3 φ 3 + n = 0. 2 λτ + ξ ∂ξ ∂ξ

(14)

We shall assume that the field variables n, v and φ can be expanded into a power series in  as:





n = 1 +  n1 +  n2 +  2 n3 + . . . . ,

  v =  v1 + v2 +  2 v3 + . . . ,   φ =  φ1 + φ2 +  2 φ3 + . . . . ,

(15)

Introducing the expansion (15) into the field Eqs. (12)-(14) and setting the coefficients of like powers of  equal to zero we obtain some sets of differential equations. From the solution of these differential equations the following evolution equation governing the lowest order term in the perturbation expansion is obtained

∂ϕ1 m ∂ϕ ∂ 3ϕ + ϕ1 + μ1 ϕ1 1 + μ2 31 = 0, ∂τ 2τ ∂ξ ∂ξ

(16)

where μ1 and μ2 are defined by

where v is the radial velocity of the ion fluid for the cylindrical (m = 1) and the spherical (m = 2) cases, respectively. For small φ , by expanding the expression (4) into a power series of φ , the electron number density ne may be approximated by

ne = 1 + d1 φ + d2 φ 2 + d3 φ 3 + . . .



q+1 q+1 (q + 1 )(3 − q ) , d2 = G2 + G1 + , 2 2 8 (q + 1 )(3 − q )(5 − 3q ) (q + 1 )(3 − q ) (q + 1 ) d3 = + G1 + G2 . 48 8 2 (10) d1 = G1 +

(9)

3 2

μ1 = d11/2 −

d2 d13/2

,

μ2 =

1 2d13/2

.

(17)

The evolution Eq. (16) is known as the cylindrical(spherical) KdV equation. The details of the derivation are given in Appendix A.1 As is seen from the expression of the μ1 , this coefficient is a function of the physical parameters α and q. For the region where μ1 > 0, the solitary wave is called compressive and for μ1 < 0, it is called rarefactive (see, references [10,11]). For certain group of values (see Fig. 1) of these parameters this coefficient μ1 vanishes. In this case the evolution Eq. (16) degenerates into the linear KdV equation. Physically, this means that the order of nonlinearity is weaker as compared to the order of dispersive effects. In order to

H. Demiray / Chaos, Solitons and Fractals 130 (2020) 109448

1

The evolution equations we have derived here may be expressed by

μ1 = 0

0.95

∂u m ∂u ∂ 3u + u + P uk +Q = 0, ∂τ 2τ ∂ξ ∂ξ 3

μ1 > 0

0.9

q

0.8 0.75

∂ u0 ∂ u0 ∂ 3 u0 + P uk0 +Q = 0. ∂τ ∂ξ ∂ξ 3

0.7 μ1 < 0

0.65 0.6

0

0.02 0.04 0.06 0.08

0.1

0.12 0.14 0.16

Fig. 1. The variation of nonextensvity (q) with nonthermal parameter (α ) for μ1 = 0.

balance the nonlinearity with dispersion one has to introduce new stretched coordinates as:

ξ =  (r − λt ), τ =  3t.

(18)

Introducing the transformation (18) into the Eqs. (6)–(9) the following equations are obtained

∂n ∂ m 2 ∂n + 2 + ( nv ) + ( nv ) = 0, ∂ξ ∂τ ∂ξ λτ +  2 ξ

∂v ∂v ∂φ ∂v −λ + 2 +v + = 0, ∂t ∂τ ∂ξ ∂ξ ∂φ ∂ 2φ m 4 + − 1 − d1 φ − d2 φ 2 − d3 φ 3 + n = 0. 2 ∂ξ λτ +  2 ξ ∂ξ

(19)

(21)

(22)

3 2d11/2



d12 + d2 −



d3 . d1

(23)

ζ0 , ζ0 = c0 (ξ − v0 τ ),

(26)

where a0 is the constant wave amplitude and the other quantities are defined by

c02 =

Pak0 k2 , 2Q (k + 1 )(k + 2 )

v0 =

2Pak0 . (k + 1 )(k + 2 )

(27)

Motivated with the solution given in (26) we shall propose a solution to the Eq. (24) of the following form 2/k

u(ξ , τ ) = a(τ ) sech

ζ , ζ = c ( τ )[ ξ − v ( τ )] ,

(28)

where a(τ ), c(τ ) and v(τ ) are some unknown functions and related to each other by

(20)

where the coefficient μˆ1 is defined by

μˆ1 =

2/k

u0 (ξ , τ ) = a0 sech

c ( τ )2 =

Introducing the expansion (15) into the field Eqs. (19)–(21) and setting the coefficients of like powers of  equal to zero we obtain some sets of differential equations. From the solution of these differential equations the following evolution equation governing the lowest order term in the perturbation expansion is obtained

∂ϕ1 m ∂ϕ ∂ 3ϕ + ϕ1 + μˆ1 ϕ12 1 + μ2 31 = 0, ∂τ 2τ ∂ξ ∂ξ

(25)

For very large τ (say, τ > τ 0 ), the evolution Eq. (25) may be considered as the limiting case of the Eq. (24), in which case the ratio (mu/2τ ) may be negligibly small. Then, the evolution Eq. (25) may assume the solitary wave solution of the form

α

2

(24)

where k = 1 stands for the KdV and k = 2 for the modified KdV equations and the coefficients P and Q characterize the nonlinearity and dispersion, respectively. For planar case (m=0) the evolution equation reduces to

0.85

−λ

3

P a ( τ )k k2 , 2Q (k + 1 )(k + 2 )

v ( τ ) =

2P a ( τ )k . (k + 1 )(k + 2 )

(29)

We note that the relations (29) are formally the same with those of (27). Introducing (28) into (24) and considering the relations (29) the following residue term is obtained



R (τ , ζ ) =

a m + a 2τ





a 2/k − ζ tanhζ sech ζ . a

(30)

Since the expression given in (28) is not the exact solution of (24) the residue term R(τ , ζ ) will not be zero. Motivated with the weighted residual method, which is extensively used in applied mathematics, we shall multiply the residue term R(τ , ζ ) by a weighing function and integrate the result with respect to ζ , from ζ = −∞ to ζ = ∞ and set the result equal to zero. In the present work, by selecting the weighing function as sech2/k ζ and performing the above described operation the following equation is obtained



1−

k 4



a m + a 2τ



∞

4/k

sech −∞

ζ d ζ = 0.

(31)

The evolution Eq. (22) is known as the modified cylindrical(spherical) KdV equation. It should be pointed out that this evolution equation is valid for the group of values of (q, α ) for which μ1 = 0 (see Fig. 1). The details of the derivation are given in Appendix A.2.

∞ 4/k Since the integral −∞ sech ζ dζ = 0, one gets the following ordinary differential equation for a(τ )

4. Progressive wave solutions

The solution of this ordinary differential equation is given by

In this section we shall try to present an approximate analytical solution for the cylindrical(spherical) KdV and the modified KdV equations given in (16) and (22) by use of the modified form of weighted residual method (see, Appendix B). Some numerical solutions for these equations had been presented in the literature (see, for instance, Maxon and Viecelli [12,13] and Mamun and Shukla [14]). To our knowledge, there is no analytical solution exists in the current literature.



k 1− 4



a m + = 0. a 2τ

 τ −( 42−km )  τ −( 4mk−k ) , c ( τ ) = c0 , τ0 τ0    1− 2mk 4−k τ ( 4−k ) v ( τ ) = v 0 τ0 1 + −1 , 4 − k − 2mk τ0

(32)

a ( τ ) = a0

(33)

where a0 is a constant and τ 0 corresponds to the time parameter τ such that for τ > τ 0 the ratio (mu/2τ ) may be negligibly small. Hence, the approximate analytical solution for the generalized

4

H. Demiray / Chaos, Solitons and Fractals 130 (2020) 109448

m=1 3

τ =3 τ =6 τ =9

2.5 2 1.5

ϕ1

ϕ1

m=2

1 0.5 0

−10

−5

0

5

10

8 7 6 5 4 3 2 1 0

τ =3 τ =6 τ =9

−10

−5

0

ξ

5

10

ξ

Fig. 2. The variation of wave profiles for the cylindrical(m=1) and the spherical(m=2) KdV equations.

cylindrical(spherical) KdV equation is given by

purpose we shall investigate the solution of the KdV and the modified KdV equations, separately.

 τ −( 42−km )

2 sech ζ , τ0   1− 2mk  τ −( 4mk−k ) 4−k τ 4−k ξ −v0 τ0 1 + ζ = c0 −1 , (34) τ0 4 − k − 2mk τ0

u = a0

The speed of the propagation may be defined by

v p = v ( τ ) = v0

 τ − 24mk −k τ0

.

(35)

In the limit as m → 0 the solutions given in (33)–(35) reduce to the one given in (26) and (27). Special cases: The approximate analytical progressive wave solution for the cylindrical(spherical)KdV and modified KdV equations may be obtained from the general formulation by choosing the parameters k, P and Q in a suitable way. (i) Solution for the cylindrical(spherical) KdV equation: From the general formulation by setting k = 1, P = μ1 and Q = μ2 the analytical solution may be given by

 τ −( 23m ) 2 ϕ1 = a0 sech ζ , τ0  μ a 1/2  τ −( m3 ) 1 0 ζ = 12μ2 τ0  μ a τ  1 0 0 × ξ− 1+ 3

3 3 − 2m



τ  ( 1− 3 ) −1 τ0 2m

. (36)

(ii) Solution for the modified cylindrical(spherical) KdV equation: From the general formulation by choosing k = 2, P = μˆ1 and Q = μ2 , the analytical solution may be given by

 τ −m ϕ1 = a0 sechζ , τ0  1/2  −m τ μˆ1 ζ = a0 6 μ2 τ0  μˆ1 a20 τ0 × ξ− 1+ 6

5.1. Numerical evaluation of the KdV solution For this case the character of the solitary wave depends on the sign of the coefficient μ1 , i. e., when μ1 > 0 (μ1 < 0) the wave is called compressive (rarefactive) solitary wave. Here we shall only study the compressive solitary waves. For the numerical evaluation in the compressive region (see Fig. 1) we shall select q = 0.9 and α = 0.04 and the values of μ1 and μ2 are found to be μ1 = 0.601 and μ2 = 0.755. In this case by choosing a0 = 1 the solution (35) takes the following form

ϕ1 =

 τ −( 23m ) τ0

2

sech

ζ,

 τ − m3

ζ = 0.258 τ0  × ξ − 0 . 2 τ0 1 +

3 3 − 2m



τ  ( 1− 3 ) −1 τ0 2m

.

(38)

The variations of the wave profiles for the cylindrical(m=1) and the spherical(m=2) waves for τ0 = 14 and various values of the time parameter τ and the spatial coordinate ξ are depicted on Fig. 2. As is seen from these figures, the wave amplitudes decrease with increasing time parameter τ . Since the wave amplitudes change with the time parameter τ , the progressive waves in the cylindrical and spherical geometries cannot be treated as solitons, which preserve their shapes [23]. It is further observed that the distortion in the wave profiles for spherical waves is much stronger than that of the cylindrical waves. As a final remark, the amplitude of the spherical wave is larger that that of the cylindrical wave. 5.2. Numerical evaluation of modified KdV solutions

1 1 − 2m



τ (1−2m) −1 . τ0

(37)

As might be seen from Eqs. (36) and (37) for non-planar case the amplitude and the speed of the wave change with the time parameter τ . 5. Numerical results and discussion In this section, for illustrative purposes of the analytical results, we shall present a numerical evaluation of the solutions. For that

The modified KdV equation we have derived here is valid for the set of values of (α , q) such that μ1 = 0 (or, equivalently d2 = 1.5d12 ). In this case the coefficient μˆ1 becomes

μˆ1 =

3 2d11/2





2.5d12 −

d3 . d1

(39)

Choosing α = 0.04, q = 0.73, which roughly corresponds to μ1 = 0, the coefficient μˆ1 and μ2 are found to be: μˆ1 = 0.441, μ2 = 1.45. Hence, the solution function ϕ 1 becomes

ϕ1 =

 τ −m τ0

sechζ ,

H. Demiray / Chaos, Solitons and Fractals 130 (2020) 109448

m=2 25

τ =3 τ =6 τ =9

τ =3 τ =6 τ =9

20 15 ϕ1

ϕ1

m=1 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 −20 −15 −10 −5

5

10 5

0

5

10 15 20

0 −40 −30 −20 −10 0

ξ

10

20

30

40

ξ

Fig. 3. The variations of wave profiles for the modified cylindrical(m=1) and the spherical(m=2) KdV equations.

1 0.5 0 μ1

−0.5 −1 −1.5 −2

α = 0.02 α = 0.06 α = 0.10

−2.5

−3 0.65 0.7 0.75 0.8 0.85 0.9 0.95

1

q Fig. 4. The variation of nonlinearity coefficient μ1 with q for various values of α .

 τ −m ζ = 0.226 τ0  × ξ − 0.074τ0 1 +

Fig. 5. The variations of the steepening parameter μ1 /μ2 with q for various values of α .

6. Conclusions

1 ( 1 − 2m )



τ (1−2m) −1 τ0

.

(40)

The variations of the wave profiles for the cylindrical(m=1) and spherical(m=2) waves are shown in the Fig. 3. Here, also, the wave profiles change with the time (τ ) and the space (ξ ) variables. These waves cannot be treated as solitons. The wave amplitudes for modified KdV equations are very large as compared to the amplitudes in KdV equation. Finally, the distortion in spherical waves is much stronger than that of the cylindrical waves. It might be instructive to study the effects of the physical parameters α and q on some wave characteristics. As pointed out before, the coefficients μ1 and μ2 are functions of these physical parameters. The Eq. (35) shows that the speed of propagation vp is proprtional to the speed v0 of the planar case, which is proportional to the nonlinearity coefficient μ1 . The variations μ1 with the physical parameter α and q are depicted on Fig. 4. As is seen from this figure, the coefficient μ1 increases with q but decreases with the parameter α . In other words, for the fixed time parameter, the speed of propagation increases with q but decreases with α . Another important characteristics of solitary waves is the bandwidth (or, steepening effect), which is proportional to the ratio of μ1 /μ2 . The variations of this ratio with q for various values of α are shown on Fig. 5. The numerical results indicate that the waves get steepened with increasing parameter q but get flattened with increasing α .

Utilizing the basic nonlinear equations describing the dynamics of a two component plasma consisting of cold positive ions and electrons obeying hybrid q- nonextensive nonthermal velocity distribution are examined in the cylindrical(spherical) coordinates through the use of reductive perturbation method and obtained the cylindrical(spherical) KdV and modified KdV equations. Being aware of that it is not possible to obtain an exact solution for these evolution equations, an approximate anaytical solution method, so called the weighted residual method, is presented. It is observed that both the amplitude and the wave speed vary with the time parameter τ . The numerical studies indicate that the amplitudes of these waves decrease with increasing time parameter τ . Since the wave profiles change with τ , the waves cannot be treated as solitons. It is further observed that the amplitudes of spherical waves are larger than those of the cylindrical waves. Moreover, the wave amplitude of modified KdV equation is much larger than that of the KdV equation. As a final remark, it should be stated that the speed of the propagation increases with the parameter q and decreases with α and the time parameter τ . We should also mention that the wave profiles get flattened with increasing time and the parameter α but get steepened with increasing values of the parameter q. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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H. Demiray / Chaos, Solitons and Fractals 130 (2020) 109448

Appendix A. Derivations of the Evolution Equations In this Section we shall present the derivation of KdV and modified KdV equations.

Appendix B. Weighted residual method

A1. Derivation of KdV equation Introducing the expansion (15) into the field Eqs. (12)–(14) and setting the coefficients of like powers of  equal to zero, the following sets of differential equations are obtained O( ) equations:

−λ

∂ n1 ∂v1 ∂v1 ∂φ1 + = 0, −λ + = 0 , n 1 = d 1 φ1 . ∂ξ ∂ξ ∂ξ ∂ξ

(A.1)

O( 2 ) equations:

−λ

∂ ∂ n2 ∂v2 ∂ n1 m + + + ( n v ) = 0, v + ∂ξ ∂ξ ∂τ λτ 1 ∂ξ 1 1 ∂v1 ∂v2 ∂φ2 ∂v1 −λ + + + v1 = 0, ∂ξ ∂ξ ∂τ ∂ξ ∂ 2 φ1 − + d1 φ2 + d2 φ12 − n2 = 0. ∂ξ 2

(A.2)

From the solution of the set (A.1) one obtains

φ1 = ϕ1 (ξ , τ ), v1 = d11/2 ϕ1 , n1 = d1 ϕ1 , λ = (d1 )−1/2 , (A.3) where ϕ 1 (ξ , τ ) is an unknown function of its argument and its evolution equation will be obtained later. Introducing (A.3) into the set (A.2) one obtains

∂ϕ1 md1 ∂ ∂ n2 ∂v2 + + d1 + ϕ + d13/2 (ϕ12 ) = 0, ∂ξ ∂ξ ∂τ τ 1 ∂ξ ∂ϕ1 ∂ϕ1 ∂v2 ∂ϕ2 −λ + + d11/2 + d1 ϕ1 = 0, ∂ξ ∂ξ ∂τ ∂ξ ∂ 2 ϕ1 n2 = − + d1 ϕ2 + d2 ϕ12 . ∂ξ 2 −λ

(A.4)

where ϕ 2 (ξ , τ ) is another unknown function of its argument. Eliminating ϕ 2 , v2 and n2 between the Eqs. (A.4) the evolution equation given in (16) can be obtained. A2. Derivation of modified KdV equation Introducing the expansion (15) into the field Eqs. (19)-(21) and setting the coefficients of like powers of  equal to zero the following sets of differential equations are obtained O( ) equations:

−λ

∂ n1 ∂v1 ∂v1 ∂φ1 + = 0, −λ + = 0 , n 1 = d 1 φ1 . ∂ξ ∂ξ ∂ξ ∂ξ

O( 2 )

O

(A.5)

equations:

∂ ∂ n2 ∂v2 −λ + + ( n v ) = 0, ∂ξ ∂ξ ∂ξ 1 1 ∂v1 ∂v2 ∂φ2 −λ + + v1 = 0, ∂ξ ∂ξ ∂ξ n2 − d1 φ2 − d2 φ12 = 0. ( 3 )

(A.6)

equations:

∂ ∂ n3 ∂v3 ∂ n1 m + + + ( n v + n2 v1 ) = 0, v + ∂ξ ∂ξ ∂τ λτ 1 ∂ξ 1 2 ∂ ∂v3 ∂φ3 ∂v1 −λ + + + ( v v ) = 0, ∂ξ ∂ξ ∂τ ∂ξ 1 2 ∂ 2 φ1 n3 + − d1 φ3 − 2d2 φ1 φ2 − d3 φ13 = 0. ∂ξ 2

Following the same procedure, from the solution of the differential Eqs. (A.5)–(A.7) the evolution equation given in (22) is obtained.

−λ

(A.7)

Weighted residual method is a generic class of methods developed to obtain approximate solutions to the differential equation of the form

L ( φ ) + f = 0,

in

D,

(B.1)

where φ (x) is the unknown dependent variable, f(x) is a known function and L denotes the differential operator involving the spatial derivatives. Weighted residual method involves two major steps. In the first step, based on the general behavior of the dependent variable, an approximate solution is assumed. The assumed solution is often selected so as to satisfy the boundary conditions for φ . When this solution is substituted into the differential equation, in general, it does not satisfy the differential equation and the resulting error is called the residual. This residual is then made to vanish in some average sense over the domain of definition, which produces a system of algebraic equations. From the solution of these algebraic equations the approximate solution is completely determined. If we have an evolution equation of the form

∂φ + L (φ ) + f = 0 ∂t

in

D,

(B.2)

where t is the time parameter, the variables φ and f are functions of x and t and the differential operator L involves the spatial derivatives. The same approach can be applied to such evolutionary problems and the resulting system of equations will be a system of ordinary differential equations in terms of the parameter t, rather than the algebraic equations. The solution of these differential equations give the time evolution of the approximate solution. For more information about the weighted residual method the readers may consul to the references [26,27]. This idea of approximate analytical solution will be used as we study progressive wave solution for the cylindrical(spherical) KdV and modified KdV equations. As a matter of fact, this method was used before to obtain approximate analytical solution for perturbed KdV and the dissipative nonlinear Schrödinger equation [19,24,25]. The obtained results are exactly the same with those of found by inverse scattering method. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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H. Demiray / Chaos, Solitons and Fractals 130 (2020) 109448 [23] Sheridan TE. Phys Plasmas 2017;24:092303. [24] Demiray H. Appl Math Comput 2003;145:179. [25] Demiray H. An approximate wave solution for perturbed kdv and NLS equations:weighted residual method. TWMS J Appl Engr Math 2019;9. doi:10. 26837/jaem.590449. (to appear).

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[26] Finlayson A. The method of weighted residuals and variational principles. New York: Academic Press; 1972. [27] Fletcher AJ. Weighted residual methods:in computational techniques for fluid dynamics. Springer series in computational physics. Berlin, Heidelberg: Springer; 1981.