Publicly Available Published by De Gruyter February 1, 2018

Painlevé Test, Bäcklund Transformation and Consistent Riccati Expansion Solvability for two Generalised Cylindrical Korteweg-de Vries Equations with Variable Coefficients

Rehab M. El-Shiekh

From the journal Zeitschrift für Naturforschung A

https://doi.org/10.1515/zna-2017-0349

Abstract

In this paper, the integrability of the (2+1)-dimensional cylindrical modified Korteweg-de Vries equation and the (3+1)-dimensional cylindrical Korteweg-de Vries equation with variable coefficients arising in dusty plasmas in its generalised form was studied by two different techniques: the Painlevé test and the consistent Riccati expansion solvability. The integrability conditions and Bäcklund transformations are constructed. By using Bäcklund transformations and the solutions of the Riccati equation many new exact solutions are found for the two equations in this study. Finally, the application of the obtained solutions in dusty plasmas is investigated.

Keywords: Bäcklund Transformation; Consistent Riccati Expansion Solvability; Cylindrical Korteweg-de Vries Equations with Variable Coefficients; Dust Ion Acoustic Waves; Painléve Test

1 Introduction

Dust acoustic waves (DAW) are very important in plasma physics because they are affected by a wide range of phenomena such as basic plasma physics. They can be studied at the single particle level and in dusty plasma DAWs are possible mechanisms for fluctuations observed in many systems like the Earth’s mesosphere, planetary rings, comet tails and the moon. Moreover, DAWs and shocks accelerate dust to enable like-charged dust grains to stick together even in the presence of Coulomb repulsion [1], [2 and references therein].

Additionally, there are two types of acoustic waves, the low-frequency DAWs involving mobile dust grains and the high-frequency dust ion acoustic waves (DIAWs) which are the usual ion acoustic waves modified by the presence of dust grains [3]. In most studies the model describes DAWs and finishes with the study of the Korteweg-de Vries (KdV) equation or its variants like the Korteweg-de Vries Burger equation (KdVB) and the Kadomstev-Petviashvili (KP) equation [4], [5]. Due to its importance we are going to study two important models of acoustic waves given using KdV equations with variable coefficients in two and three dimensions as follows:

The cylindrical modified Korteweg-de Vries (cmKdV) or KP equation with variable coefficients representing the DIAWs waves with azimuthal perturbation in a dusty plasma [6]

(1)(ψτ+Aψ2ψζ+Bψζζζ+ψ2τ)ζ+μ2τ2ψηη=0,

where A=15μ2−14μ,B=12μ32 and ψ present the first approximation of the electrostatic wave potential of the DIAWs. Moreover, by combining the effects of bounded cylindrical geometry, azimuthal and axial perturbations, the non-linear DAWs in an unmagnetised plasma consists of negatively charged dust grains, non-extensive ions and non-extensive electrons. A (3+1)-dimensional variable-coefficient cylindrical KdV (cKdV) equation is derived [7]

(2)(ψτ+Aψψζ+V22ψζζζ+ψ2τ)ζ+12V T2ψηη+V2ψζζ=0,

where B=V32, C=V2. To make our study for (1) and (2) deal with many models in physics, not only in DAWs, we use (1) and (2) in the following generalised form [8]

(3)(ut+Au2ux+Buxxx+α(t)u)x+β(t)uyy=0,

(4)(ut+Auux+Buxxx+α(t)u)x+β(t)uyy+Cuzz=0,

where α(t) and β(t) are arbitrary functions of t.

2 Integrability and Exact Solutions for (3)

In this section we study the integrability of the variable coefficients (2+1)-dimensional cmKdV equation by using the classical Painlevé test [9], [10] and the new consistent Riccati expansion (CRE) solvability [11] as follows:

2.1 Painlevé Test for (3)

Assume that the solution of (3) can be expressed in the Laurent series as

(5)u(x, u, t)=∑j=0∞uj(x, y, t)ϕ(x, y, t)j+α,

by substituting (5) in (3) and by balancing the non-linear term with the most dispersive term the leading order α is determined as

(6)α=−1 and u0(x, y, t)=−6BAϕx,

then, to find the resonance point j, where the arbitrary functions exist we substituted

(7)u(x, y, t)=u0ϕ−1+ujϕj−1, j≥1

in (3) using (6), the following characteristic equation for resonances is obtained

(8)(j+1)(j−3)(j−4)(j−4)=0.

Therefore, the resonances j occur at j=−1, 3 and 4. The resonance at j=−1 corresponds to the arbitrariness of the singular manifold ϕ(x, y, t)=0.

Now, substitute the Laurent series in (3) as

(9)u(x, y, t)=∑j=04uj(x, y, t)ϕ(x, y, t)j−1.

To test the existence of a sufficient number of arbitrary functions, collect the coefficients of ϕ^−j−1 as follows:

The coefficient of ϕ⁻⁵ gives u0(x, y, t)=−6BAϕx, which corresponds to the resonance j=−1
The coefficient of ϕ⁻⁴ gives u1(x, y, t)=12−6BAϕxxϕx,
The coefficient of ϕ⁻³ gives u2(x, y, t)=112Bϕx3−6BA(2β(t)ϕy2+33Bϕxx2+26Bϕxϕxxx+2ϕxϕt),
From the coefficients of ϕ⁻² and ϕ⁻¹ we find equations do not include u₃(x, y, t) and u₄(x, y, t), this means that those functions are arbitrary, we can conclude that the (2+1)-dimensional cmKdV equation with variable coefficients has passed the Painlevé test.

2.2 Bäcklund Transformation for (3)

Now, to construct Bäcklund transformation for (3) [12], [13], [14], [15], [16], [17], [18], assume that the arbitrary functions u₃ and u₄ are zeros and also u₂=0,

(10)u(x, y, t)=−6BAϕxϕ+u1(x, y, t),

where u₁(x, y, t) is a solution of (3).

Let ϕ(x, y, t)=Φ(w) where w=w(t, x, y) is an undetermined function given by Kruskal’s simplification method as

(11)w(x, y, t)=kx+χ(y, t),

then,

(12)u(x, y, t)=−6BA∂∂xln(Φ(w))+u1(x, y, t).

So that, we have the following relations

(13)Φ″Φ‴=−112Φ‴″, (Φ″)2=−16Φ″″, Φ′Φ‴=−13Φ″″Φ′Φ″=−12Φ‴, Φ′2=−Φ″

Substituting (11, 12) into (3) using relation (13) and equating the linear coefficients Φ‴″, Φ″″, …, Φ′ by zero, a partial differential system is obtained by assuming that

(14)Φ=M+exp(w),

where M is the arbitrary constant; the following integrability condition between the variable coefficients α(t) and β(t) is obtained and u₁(x, y, t)

(15)α(t)=−β(t)2∫β(t)dt, u1(x, y, t)=−k2−6BA

Also

(16)w(x, y, t)=kx−ky24∫β(t)dt+y4∫β(t)dt +132∫−2β(t)+16Bk4(∫β(t)dt)2k(∫β(t)dt)2 dt+c1

where c₁ is an arbitrary constant. By substituting (14–16) into (12), the following soliton solution is obtained

(17)u(x, y, t)=k−6BAewM+ew−k2−6BA

where w is given by (16).

By using the following two important formulas [14]

(18)exp(w)M+exp(w)={1for M=012[tanh12(w−lnM)+1]for M>012[coth12(w−ln(−M))+1] for M<0

the following kink-type solitary wave solutions are obtained

(19)u11=k2−6BAtanh[12(w−lnM)], for M>0

(20)u12=k2−6BAcoth[12(w−ln(−M))], for M<0

where w is given by (16).

2.3 Consistent Riccati Expansion Solvability for (3)

According to the CRE solvability [11], the solution of (3) can be put in the form

(21)u(x, y, t)=u0(x, y, t)+u1(x, y, t)R(w(x, y, t)),

where u₀(x, y, t) and u₁(x, y, t) are arbitrary functions and R(w) is a solution of the Riccati equation

(22)Rw(w)=a0+a1R(w)+a2R2(w),

where a₀, a₁ and a₂ are arbitrary constants to be determined. By back substitution from (21) into (3) using (22), by collecting the coefficients of R and its power, then equating it with zero, a partial differential system is obtained. By solving it, we get

(23)u0=12−6BA(wxx+a1wx2wx), u1=a2−6BAwx

with the variable coefficients cmKdV−w equation

(24)Ba12wx4−4a0a2Bwx4−2β(t)wy2−2Bwxwxxx−2wtwx +3Bwxx2=0.

Theorem 1:If w is a solution of the variable coefficients cmKdV−w equation (24), then

u=12−6BA(wxx+a1wx2wx)+a2−6BAwxR(w)

is a solution of the (2+1)-dimensional cmKdV equation (3) with R(w) being a solution of the Riccati equation (22).

From the previous theorem we conclude that (3) is integrable according to the CRE solvability.

Remark 1: By using the same w described by (16) we have found that it satisfies the cmKdV−w equation with the same integrability condition (15). Unfortunately, by using the solutions of the Riccati equation (22), we get the same solutions (19) and (20).

3 Integrability and Exact Solutions for (4)

According to the same procedures used before in the previous section the Painlevé property, Bäcklund transformation and CRE solvability can be applied to the (3+1)-dimensional cKdV equation with variable coefficients (4) and the following results are obtained.

3.1 Painlevé Test for (4)

By applying the test on (4), the leading order is obtained as α=−2 and the solution of equation (4) can be assumed as

(25)u(x, y, z, t)=u0ϕ−2+u1ϕ−1+ujϕj−2, j≥2

by substitution from the previous equation into (4), the characteristic equation for resonances is given by

(26)(j+1)(j−4)(j−5)(j−6)=0.

So that, the resonances j occur at j=−1, 4, 5 and 6. Now, we can put u in the Laurent series as follows

(27)u(x, y, z, t)=∑j=06uj(x, y, z, t)ϕ(x, y, z, t)j−2.

By collecting the coefficients of ϕ^−j we get,

the coefficient of ϕ⁻⁶ gives u0(x, y, z, t)=−12BAϕx2, corresponds to the resonance j=−1
the coefficient of ϕ⁻⁵ gives u1(x, y, t)=12BAϕxx,
the coefficient of ϕ⁻⁴ gives u2(x, y, t)=1Aϕx2(3Bϕxx2−4Bϕxϕxxx−Cϕz2−ϕxϕt−β(t)ϕy2),
from the coefficients of ϕ⁻³, ϕ⁻² and ϕ⁻¹ we find equations do not include u₃(x, y, t), u₄(x, y, t), and u₅(x, y, t), this means that those functions are arbitrary and we can conclude that the (3+1)-dimensional cKdV equation with variable coefficients satisfy the Painlevé property.

3.2 Bäcklund Transformation for (4)

From the Painlevé property, we can obtain the following auto-Bäcklund transformation for (4)

(28)u(x, y, t)=−12BA∂2∂x2ln(Φ(w))+u2(x, y, z, t),

where u₂(x, y, z, t) is a solution of equation (4), ϕ(x, y, z, t)=Φ(w) and w=w(t, x, y, z) is given by Kruskal’s simplification method. Additionally, by using the same assumptions (13) and (14) in subsection 2.2, the following new soliton solution for the (3+1)-dimensional variable-coefficient cylindrical KdV (cKdV) equation is obtained

(29)u(x, y, z, t)=−12Bk2e2wA(M+ew)2+12Bk2ewA(M+ew)+F(t)

with

(30)w=kx−k(12y2+c2y+1)e∫2α(t)dt+sz−∫(Bk3+Cks2+F(t)kA+ k(c22−2)α(t)e∫2α(t)dt)dt+c3

(31)β(t)=α(t)e−∫2α(t)dt

where s, c₂, c₃ are arbitrary constants and F(t) is the arbitrary function of t.

From formula (18), the following shock wave and periodic wave solutions are obtained for the (3+1)-dimensional variable-coefficient cylindrical KdV equation (cKdV).

(32)u21=−3Bk2A(tanh212(w−ln(M))−1)+F(t) for M>0

(33)u22=−3Bk2A(coth212(w−ln(M))−1)+F(t) for M<0

3.3 CRE Method for (4)

By applying the CRE method to (4), u can be presented as

(34)u(x, y, z, t)=u0(x, y, z, t)+u1(x, y, z, t)R(w(x, y, z, t)) +u2(x, y, z, t)R2(w(x, y, z, t)),

where u₀(x, y, z, t), u₁(x, y, z, t), and u₂(x, y, z, t) are arbitrary functions and R(w) is a solution of the Riccati equation (22). By back substitution from (34) into (4) using (22), then collecting the coefficients of various powers of R and then equating it by zero, a partial differential system is obtained by solving it, and so we get

(35)u0=−1Awx2(Ba12wx4+6Bwx2wxxa1+4Bwxwxxx−3Bwxx2+8Ba0a2wx4+Cwz2+wxwt+β(t)wy2)u1=−12BAa2(a1wx2+wxx), u2=−12BAa2wx2

and the variable coefficients (3+1)-dimensional cKdV − w equation

(36)wx2wxt+3Bwxx2+β(t)wx2wyy+Cwx2wzz+α(t)wx3−4Bwxwxxwxxx −Ba12wx4wxx+Bwx2wxxxx−wxwtwxx+4a0a2Bwx4wxx−Cwz2wxx −β(t)wy2wxx=0.

Theorem 2:If w is a solution of the variable coefficients (3+1)-dimensional cKdV − w equation (36), then

u(x, y, z, t)=−1Awx2(Ba12wx4+6Bwx2wxxa1+4Bwxwxxx−3Bwxx2+8Ba0a2wx4+Cwz2+wxwt+β(t)wy2)−12BAa2(a1wx2+wxx)R(w)−12BAa2wx2R2(w)

is a solution of the (3+1)-dimensional cKdV equation (4) with R(w) being a solution of the Riccati equation (22).

Therefore, (4) is integrable according to the CRE solvability.

Remark 2: By substituting (30) into (36) we found it is satisfied with respect to the integrability condition (31). Additionally, by using the solutions of the Riccati equation (22) [19], [20], [21], we obtained more new types of solutions for the variable coefficients (3+1)-dimensional cKdV equation given as

(37)u23=F(t)−Bk2A(1+3(tan(w)±sec(w))2),where a0=a2=12, a1=0,u24=F(t)−Bk2A(1+3tan2(w2)),where a0=a2=12, a1=0,u25=F(t)−Bk2A(1+3cot2(w2)),where a0=a2=12, a1=0,u26=F(t)+3Bk2A(3−4tanh2(w)),where a0=1, a2=−1, a1=0,u27=F(t)+3Bk2A(3−4coth2(w)),where a0=1, a2=−1, a1=0,u28=F(t)−Bk2A(7+12tan2(w)),where a0=a2=−1, a1=0,u29=F(t)−Bk2A(7+12cot2(w)),where a0=a2=−1, a1=0,u210=F(t)+Bk2A(33−48tanh2(2w)),where a0=1, a2=−4, a1=0,u211=F(t)+Bk2A(33−48coth2(2w)),where a0=1, a2=−4, a1=0,u212=F(t)−Bk2A(31+48tan2(2w)),where a0=1, a2=4, a1=0,u213=F(t)−Bk2A(31+48cot2(2w)),where a0=1, a2=4, a1=0,u214=F(t)+Bk2A(1−12w2),where a0=0, a2=1, a1=0,

where w is given by (30) with the integrability condition (31).

4 Application in Dusty Plasmas

From the importance of DAWs mentioned before in the introduction we plotted here the physical situation given in [6], [7] . From (1) α(t)=12τ and β(t)=μ2τ2 by comparing with (15) the integrability condition is satisfied. By taking the parameters as given in [6] μ=14, then A=4 and B=−132, moreover, we assumed that c₁=0 and M=1 in solutions (17), (19) and (20) and Figures 1–3 are given.

$Figure 1: The dark soliton DIAW u given by (17) with k=−12,$k = - {1 \over 2},$t=35.$

Figure 1:

The dark soliton DIAW u given by (17) with k=−12,t=35.

$Figure 2: The bright soliton DIAW u11 with k=12,$k = {1 \over 2},$t=35.$

Figure 2:

The bright soliton DIAW u₁₁ with k=12,t=35.

$Figure 3: The DIAW u12 with k=12,$k = {1 \over 2},$t=35.$

Figure 3:

The DIAW u₁₂ with k=12,t=35.

Additionally, by comparing (2) with (4), we obtain α(t)=12τ and β(t)=12τ2 also the integrability condition (31) is satisfied, therefore, all solutions obtained for the (3+1)-dimensional KdV equation are satisfied for this physical case. Taking the same physical parameters as in [7] where V=−2.8, A=−32.9, B=−10.98, C=−1.4 and the integration constants given as c₂=c₃=1, F(t)=0, s=0 and M=1, see Figures 4–6.

$Figure 4: The soliton DAW given by (29) with k=12$k = {1 \over 2}$ at t=15.$

Figure 4:

The soliton DAW given by (29) with k=12 at t=15.

$Figure 5: The DAW given by u22 with k=12$k = {1 \over 2}$ and t=15.$

Figure 5:

The DAW given by u₂₂ with k=12 and t=15.

$Figure 6: The DAW solution u24 where k=12$k = {1 \over 2}$ at t=15.$

Figure 6:

The DAW solution u₂₄ where k=12 at t=15.

5 Conclusion

In this paper, the (2+1)-dimensional cmKdV equation with variable coefficients and the (3+1)-dimensional variable coefficients cKdV in general their forms (3) and (4) were investigated using two integrability techniques, the Painlevé test and CRE solvability. Using those methods we can conclude that (3) and (4) are integrable. Moreover, many new exact solutions for (3) and (4) are obtained using both Bäcklund transformation and the solutions of the Riccati equation (22). The studied Kdv-type equations govern many physical situations and one of them is the DAWs in dusty plasma given by (1) and (2). From the importance of that DAWs we have applied it on our solutions and plotted them by using the same physical values for the different parameters as given in [6], [7].

Finally, we have some concluding remarks on the application of both Painlevé test and CRE solvability as follows:

The new CRE method can be used to construct Bäcklund transformations given by (10) and (28) from the solutions given by theorems 1 and 2, respectively, if Φ=1R and a₀=0, a₁=1, therefore, we can conclude that the CRE is applicable and is straightforward for non-linear variable coefficients partial differential equations.
The values of resonances and the results given in subsection 2.1 for (3) cover the results obtained before as a special case of this equation. For details see [22].

References

[1] S. A. Elwakil, M. A. Zahrana, E. K. El-Shewy, and A. E. Mowafya, Adv. Space Res. 48, 1067 (2011).10.1016/j.asr.2011.04.034Search in Google Scholar

[2] R. L. Merlino, J. Plasma Phys. 80, 773 (2014).10.1017/S0022377814000312Search in Google Scholar

[3] O. H. EL-Kalaawy, Comput. Math. Appl. 72, 1031 (2016).10.1016/j.camwa.2016.06.013Search in Google Scholar

[4] H. R. Pakzad, Chaos, Solitons Fractals 42, 874 (2009).10.1016/j.chaos.2009.02.016Search in Google Scholar

[5] N. S. Saini, N. Kaur, and T. S. Gill, Adv. Space Res. 55, 2873 (2015).10.1016/j.asr.2015.03.013Search in Google Scholar

[6] B. Tian and Y.-T. Gao, Phys. Lett. A 362, 283 (2007).10.1016/j.physleta.2006.10.094Search in Google Scholar

[7] S. Guo, H. Wang, and L. Mei, Phys. Plasma 19, 063701 (2012).10.1063/1.4729682Search in Google Scholar

[8] M. R. El-Shiekh, Comput. Math. Appl. In press doi: org/10.1016/j.camwa.2017.11.031.org/10.1016/j.camwa.2017.11.031Search in Google Scholar

[9] J. D. Gibbon and M. Tabor, Dynamical Problems Soliton Syst. 55 (1985). pp. 55–61.10.1007/978-3-662-02449-2_9Search in Google Scholar

[10] M. F. El-Sayed, G. M. Moatimid, M. H. M. Moussa, R. M. El-Shiekh, F. A. H. El-Shiekh, et al., Math. Methods Appl. Sci. 38, 3670 (2015).10.1002/mma.3307Search in Google Scholar

[11] S. Y. Lou, Stud. Appl. Math. 134, 372 (2015).10.1111/sapm.12072Search in Google Scholar

[12] M. H. M. Moussa, M. R. El-Shiekh, Phys. Lett. A 372, 1429 (2008).10.1016/j.physleta.2007.09.056Search in Google Scholar

[13] M. H. M. Moussa, M. R. El-Shiekh, Int. J. Nonlinear Sci. 7, 29 (2009).Search in Google Scholar

[14] M. R. El-Shiekh, Am. J. Comput. Math. 5, 215 (2015).10.4236/ajcm.2015.52018Search in Google Scholar

[15] X. Lü, W.-X. Ma, Y. Zhou, and C. M. Khalique, Comput. Math. Appl. 71, 1560 (2016).10.1016/j.camwa.2016.02.017Search in Google Scholar

[16] X. Lü, W.-X. Ma, S. Chen, and C. M. Khalique, Appl. Math. Lett. 58, 13 (2016).10.1016/j.aml.2015.12.019Search in Google Scholar

[17] X. Lü, W.-X. Ma, J. Yu and C. M. Khalique, Commun. Nonlinear Sci. Numer. Simul. 31, 40 (2016).10.1016/j.cnsns.2015.07.007Search in Google Scholar

[18] X. Lü and F. Lin, Commun. Nonlinear Sci. Numer. Simul. 32, 241 (2016).10.1016/j.cnsns.2015.08.008Search in Google Scholar

[19] M. R. El-Shiekh, Z. Naturforsch. A 68, 255 (2013).10.5560/ZNA.2012-0108Search in Google Scholar

[20] M. R. El-Shiekh, Z. Naturforsch. A 70, 445 (2015).10.1515/zna-2015-0057Search in Google Scholar

[21] M. R. El-Shiekh, Comput. Math. Appl. 73, 1414 (2017).10.1016/j.camwa.2017.01.008Search in Google Scholar

[22] X.-L. Gai, Y.-T. Gao, L. Wang, De-X. Meng, X. Lü, et al., Commun. Nonlinear Sci. Numer. Simul. 16, 1776 (2011).10.1016/j.cnsns.2010.07.021Search in Google Scholar

Received: 2017-09-28

Accepted: 2017-12-22

Published Online: 2018-02-01

Published in Print: 2018-02-23