Exact solutions to the

Alexandria Engineering Journal (2016) 55, 1635–1645

H O S T E D BY

Alexandria University

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ORIGINAL ARTICLE

Exact solutions to the (3+1)-dimensional coupled Klein–Gordon–Zakharov equation using exp(U(n))-expansion method M.G. Hafez Department of Mathematics, Chittagong University of Engineering & Technology, Bangladesh Received 23 August 2014; revised 20 January 2016; accepted 20 February 2016 Available online 5 March 2016

KEYWORDS The nonlinear coupled Klein–Gordon–Zakharov equation; Traveling wave solutions; Solitary wave solutions; The expðUðnÞÞ-expansion method

Abstract In this article, the expðUðnÞÞ-expansion method is modified for (3+1)-dimensional space–time coordinate system and successfully implemented to construct the new exact traveling wave solutions of the (3+1)-dimensional coupled Klein–Gordon–Zakharov equation. The solutions of this equation are expressed in terms of hyperbolic, trigonometric, exponential and rational functions. The results illustrate its effectiveness for solving nonlinear coupled partial differential equations arises in mathematical physics and engineering. The annihilation phenomena of the wave propagation in the x–y plane are also investigated. Furthermore, the three-dimensional surface plots due to the obtained solutions are also given to make the dynamics of the equation visible. Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction It is well known that partial differential equations (PDEs) are special kind of nonlinear evolution equations. Already a large number of authors [1–3] have derived the nonlinear PDEs considering different theoretical model equations. Many issues of physical phenomena in nature may be described by solving the nonlinear PDEs. The explicit solutions of these equations play a vital role to study such physical phenomena in mathematical physics. Thus, the nonlinear equations may become most exciting and enormously active research areas for mathematicians, physicists and engineers. On the other hand, the traveling wave solutions play an essential role to study the models arising from various natural and complex phenomena in the field of E-mail address: [email protected] Peer review under responsibility of Faculty of Engineering, Alexandria University.

applied sciences and engineering. For instance, the wave propagation phenomena were observed in optical fibers, elastic media, quantum mechanics, fluids dynamics, chemical physics, solid mechanics, biophysics, Higgs mechanism, quantum field theory, plasma physics and so on. To obtain the exact solutions of the nonlinear PDEs, the initial dominant mathematical tool was the inverse scattering method [4]. Some of other most efficient methods for finding analytical solutions of the nonlinear PDEs are the three-wave method [5], the improved F-expansion method [6], the generalized method [7], the Weis s–Tabor–Carnevale method [8], the tanh-function method [9], the extended tanh-method [10,11], the modified extended tanh-function method [12,13], the Jacobi elliptic function expansion method [14], the extended Jacobi elliptic function expansion rational method [15], the Exp-function method [16–18], the truncated Painleve expansion method [19], the homogeneous balance method [20–22] and so on. Recently, the expðUðnÞÞ-expansion method [23–27] has also been used

http://dx.doi.org/10.1016/j.aej.2016.02.010 1110-0168 Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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M.G. Hafez

Figure 1

Shape of ju1 ðx; yÞj in x–y plane at (a) t ¼ 0, (b) t ¼ 10 and (c) t ¼ 30.

Figure 2

Shape of m1 in x–y plane at (a) t ¼ 0, (b) t ¼ 10 and (c) t ¼ 30.

to find the exact traveling wave solutions of single nonlinear PDEs. Moreover, many researchers [28–32] have investigated the exact solutions of the coupled nonlinear PDEs, that is,

the Maccari system, Higgs field equation, (1+1)-dimensional nonlinear coupled Klein–Gordon–Zakharov equation, etc. employing the Exp-function method [28], truncated expansion

Exact solutions to the (3+1)-dimensional coupled Klein–Gordon–Zakharov equation using

Figure 3

Figure 4

Shape of ju2 j in x–y plane at (a) t ¼ 0, (b) t ¼ 10 and (c) t ¼ 1000.

Shape of m2 in x–y plane at (a) t ¼ 0, (b) t ¼ 10 and (c) t ¼ 100.

1637

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M.G. Hafez

Figure 5


method [29], first integral method [30], and conservative difference methods [31]. Many types of coupled Klein–Gordon–Zakharov equation [27,31,33] have obtained to study the nonlinear propagation of physical phenomena in physics, especially in plasma physics and fluid dynamics. In order to get the plane wave solutions for the coupled nonlinear evolution equations in three dimensional space and time, we have modified the expðUðnÞÞexpansion method for solving (3+1)-dimensional coupled nonlinear evolution equations. The aim of the present work was to find the exact traveling wave solutions of (3+1)dimensional nonlinear coupled Klein–Gordon–Zakharov equation. Thus, the paper is arranged as follows: Modification of the expðUðnÞÞ expansion method is given in Section 2. The traveling wave solutions of (3+1)-dimensional coupled Klein–Gordon–Zakharov equation are depicted in Section 3. The physical significances along with relevant discussion are mentioned in Section 4. Finally, the conclusion is drawn in Section 5. 2. Description of the modified exp(U(n))-expansion method The general form of the nonlinear PDE can be written as follows: Fðu; ut ; ux ; uy ; uz ; uxx ; uyy ; uzz ; utt ; utx ; . . .Þ ¼ 0;

ð1Þ

where u ¼ uð~ r; tÞ is an unknown function of x; y; z and t, F is a polynomial in uð~ r; tÞ and its derivative in which higher order derivatives and nonlinear terms involved and the subscripts stand for the partial derivatives.

Now, combining the real variables x; y; z and t by a compound variable n is as follows: uð~ r; tÞ ¼ uðnÞ;

~ ~ n¼P r xt;

ð2Þ

^ ~ ^ where p; q; r are the ~ ¼ p^ r ¼ x^ i þ y^j þ zk, Here P i þ q^ j þ rk, constant magnitude along the axes of x; y; z respectively and x is the speed of the traveling wave. Using the traveling wave variables transformation (2), Eq. (1) can be reduced to an ordinary differential equation (ODE) for u ¼ uðnÞ: Rðu; u0 ; u00 ; u000 ; . . .Þ ¼ 0;

ð3Þ

where R is a polynomial of u and its derivatives and primes indicate the ordinary derivatives with respect to n. The traveling wave solution of Eq. (3) can be expressed as follows: uðnÞ ¼

N X Ai ðexpðUðnÞÞÞi ;

ð4Þ

i¼0

where Ai ð0 6 i 6 NÞ are constants to be evaluated later, such that AN – 0 and U ¼ UðnÞ satisfy the following ODE: U0 ðnÞ ¼ expðUðnÞÞ þ l expðUðnÞÞ þ k;

ð5Þ

where k and l are arbitrary constants. It is noted that Eq. (5) gives the following general solutions: Type 1: When l – 0; H ¼ k2 4l > 0, ! pffiffiffiffi pffiffiffiffi H tanh 0:5 Hðn þ EÞ k UðnÞ ¼ ln 2l

ð6Þ


Figure 6

Shape of v3 in x–y plane at (a) t ¼ 0, (b) t ¼ 10 and (c) t ¼ 30.

Type 2: When l – 0; H ¼ k2 4l < 0, ! pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi H tan 0:5 Hðn þ EÞ k UðnÞ ¼ ln 2l Type 3: When l ¼ 0; k – 0, and H ¼ k2 4l > 0, k UðnÞ ¼ ln expðkðn þ EÞÞ 1 Type 4: When l – 0; k – 0, and H ¼ k2 4l ¼ 0, 2ðkðn þ EÞ þ 2Þ UðnÞ ¼ ln k2 ðn þ EÞ

ð7Þ

ð8Þ

ð9Þ

Type 5: When l ¼ 0; k ¼ 0, and H ¼ k2 4l ¼ 0, UðnÞ ¼ ln ðn þ EÞ

1639

ð10Þ

where E is the integration constant. The value of the positive integer N can be determined by balancing the higher order derivative linear terms with the nonlinear terms of the highest order that appeared in Eq. (3). If the degree of uðnÞ is D½uðnÞ ¼ N, then the degree of

the other expressions can be evaluated by the following formulae: p q s d uðnÞ p d uðnÞ D ¼ N þ p; D u ¼ Np þ sðN þ qÞ: dnp dnq By substituting Eq. (4) into Eq. (3) and using (5) rapidly, one can obtain a system of algebraic equations by equating all the coefficients of expðUðnÞÞ to zero. The obtaining system can be solved to find the value of A1 ; A2 . . . ; x; p; q; r with the help of symbolic computation software such as Maple. Substituting the values of A1 ; A2 . . . ; x; p; q; r into Eq. (4) along with general solutions of Eq. (5) completes the determination of the solution of Eq. (1). Advantages: The advantage of the proposed method over other existing method is that it is provided the new exact traveling wave solutions along with additional free parameters. The exact solutions have its great importance to reveal the inner mechanism of the natural physical phenomena. The obtaining solutions of nonlinear evolution equations may be helpful for the numerical solvers to compare the correctness of their results and stability analysis. Various types of traveling wave solutions are obtained when the related physical

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M.G. Hafez

Figure 7


parameters received their particular values. Algebraic manipulation of the proposed scheme is much easier rather than the other methods with the help of symbolic computation, such as, Maple. The effectiveness of this method is provided in the next section. 3. Application of the method This section presents the application of the modified expðUðnÞÞ-expansion method for finding the exact traveling wave solutions of the (3+1)-dimensional coupled Klein– Gordon Zakharov equation [33]. Let us consider the coupled Klein–Gordon–Zakharov equation ) utt c20 r2 u þ f20 u þ duv ¼ 0 : ð11Þ vtt c20 r2 v br2 juj2 ¼ 0 Here uð~ r; tÞ is a complex function describing the fast timescale component of electric field raised by electrons. vð~ r; tÞ is real function indicating the deviation of ion density form its equilibrium. The system of Eqs. (11) described the interaction of the Langmuir wave and the ion acoustic wave in high frequency plasma in plasma physics (more details in Ref. [30]). To find out the wave packet solution, one can introduce the following transformation: n

o uð~ r; tÞ ¼ exp i k~ ~ r Xt UðnÞ; vð~ r; tÞ ¼ VðnÞ; ð12Þ

~~ with k~ ~ r ¼ k1 x þ k2 y þ k3 z, n ¼ P r xt ¼ px þ qy þ rz xt, where k1 ; k2 ; k3 are the constant magnitudes along the axes of x; y; z respectively. By using Eq. (12) into Eq. (11), one obtains the following coupled nonlinear ordinary differential equations: 2 9 x c20 P2 U00 þ 2i xX c20 ðk1 p þ k2 q þ k3 rÞ U0 > = 2 x c20 K2 f20 U þ dUV ¼ 0 ; ð13Þ > 2 00 ; 2 2 00 2 2 x c0 P V bP ðU Þ ¼ 0 where K2 ¼ k21 þ k22 þ k23 , P2 ¼ p2 þ q2 þ r2 and primes denote the differentiation with respect to n. c2 ~ then Eq. (13) becomes If we set x ¼ X0 k~ P, ) 2 x c20 P2 U00 x2 c20 K2 f20 U þ dUV ¼ 0 : ð14Þ 2 00 x c20 P2 V00 bK2 ðU2 Þ ¼ 0 Integrating the second equation of (14) twice with respect to n, we obtain V¼

x2

C bK2 þ U2 2 2 2 c0 P x c20 P2

ð15Þ

where C is the integrating constant. By using (15) into the first equation of (14), we obtain 2 2 x c20 P2 U00 þ x2 c20 P2 x2 þ c20 K2 þ f20 þ dC U þ dbK2 U3 ¼ 0

ð16Þ


Figure 8


This is the elliptic-like equation with cubic nonlinearity. The pole of this equation is N ¼ 1. Therefore, the expðUðnÞÞexpansion method allows us to use the solution in the following form: UðnÞ ¼ A0 þ A1 expðUðnÞÞ;

1641

A1 – 0

ð17Þ

where A0 and A1 are unknown constants. By means of Eq. (17), Eq. (16) turns out into a polynomial of expðUðnÞÞ and equating the coefficient of ðexpðUðnÞÞÞi ; ði ¼ 0; 1; 2; 3Þ to zero, we obtain a system of algebraic equations as follows:

Solving the system algebraic Eqs. (18)–(21), we obtain the following set of solutions: C¼

1 4x4 l 8x2 c20 P2 l x4 k2 l þ 2x2 c20 P2 2c20 P2 f20 2d

c40 P4 k2 þ 2x2 f20 þ 2x2 c20 P2 k2 2c40 P2 K2 þ 2x2 K2 c20 i þ 4c40 P4 l 2x4 A0 ¼ pffiffiffiffiffiffiffiffi c20 P2 x2 k; K 2db i A1 ¼ pffiffiffiffiffiffiffiffi c20 P2 x2 : K 2db

ð22Þ

dbK2 A31 þ 2A1 c40 P4 þ 2x4 A1 4x2 c20 P2 A1 ¼ 0:

ð18Þ

3A1 x4 k 6A1 x2 c20 P2 k þ 3dbK2 A0 A21 þ 3A1 c40 P4 k ¼ 0

ð19Þ

where B0 ; k and l are arbitrary constants. Using (12), (15), (17) and (22), the traveling wave solutions of the coupled Klein–Gordon–Zakharov equation are obtained as follows: When H ¼ k2 4l > 0; l – 0, we have

ð20Þ

~ u1 ð~ r; tÞ ¼ eiðk~rXtÞ c20 P2 x2

2A1 x4 l 4A1 x2 c20 P2 l þ 3dbK2 A20 A1 þ dCA1 þ A1 x4 k2 þ x2 c20 P2 A1 c20 P2 f20 A1 þ A1 c40 P4 k2 þ x2 f20 A1 2A1 x2 c20 P2 k2 c40 P2 K2 A1 þ x2 K2 þ c20 A1 þ 2A1 c40 P4 l x4 A1 ¼ 0

c40 P2 K2 A0 2A1 x2 c20 P2 kl þ A1 x4 kl þ x2 c20 P2 A0 þ x2 K2 c20 A0 þ dCA0 þ dbK

2

A30 c20 P2 f20 A0 þ x2 f20 A0 x4 A0

þ A1 c40 P4 kl ¼ 0

ð21Þ

ik pffiffiffiffiffiffiffiffi K 2db

!) i 2l pffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffiffi K 2db H tanh 0:5 Hð~ p ~ r xt þ EÞ þ k ð23Þ

1642

M.G. Hafez

Figure 9


v1 ð~ r; tÞ ¼

C k bK2 x2 c20 P2 pffiffiffiffiffiffiffiffi 2 2 2 K 2db x c0 P

!)2 1 2l pffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffi K 2db H tanh 0:5 Hð~ p ~ r xt þ EÞ þ k ð24Þ When H ¼ k2 4l < 0, we have ~ u2 ð~ r; tÞ ¼ eiðk~rXtÞ c20 P2 x2

ik i pffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffi K 2db K 2db !)

2l pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi H tan 0:5 Hð~ p ~ r xt þ EÞ k

2 C k 1 2 2 2 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi bK x c P 0 K 2db K 2db x2 c20 P2 2 k ð28Þ exp fkð~ p ~ r xt þ EÞg 1

v3 ð~ r; tÞ ¼

When l – 0; k – 0, and k2 4l ¼ 0, we have ~ u4 ð~ r; tÞ ¼ eiðk~rXtÞ c20 P2 x2

ik i k2 fkð~ p ~ r þ EÞg pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 2fkð~ p ~ r xt þ EÞg þ 4 K 2db K 2db ð29Þ

ð25Þ

2 C k 1 2 2 2 pffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffi bK x c P 0 2 2 2 K 2db K 2db x c0 P !)2 2l ð26Þ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi H tan 0:5 Hð~ p ~ r xt þ EÞ k

r; tÞ ¼ v2 ð~

When k2 4l > 0; l ¼ 0, we have ~ u3 ð~ r; tÞ ¼ eiðk~rXtÞ c20 P2 x2

ik i k pffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffi p ~ r xt þ EÞg 1 K 2db K 2db exp fkð~ ð27Þ

2 C k 1 2 2 2 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi bK x c P 0 K 2db K 2db x2 c20 P2 k2 ð~ p ~ r þ EÞ ð30Þ 2fkð~ p ~ r xt þ EÞg þ 4

r; tÞ ¼ v4 ð~

When l ¼ 0; k ¼ 0, and k2 4l ¼ 0, we have ~ u5 ð~ r; tÞ ¼ eiðk~rXtÞ c20 P2 x2

ik i 1 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi p ~ r xt þ E K 2db K 2db ~ C bK2 x2 c20 P2 2 2 c0 P

2 k 1 1 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi p ~ r xt þ E K 2db K 2db ~

v5 ð~ r; tÞ ¼

ð31Þ

x2

ð32Þ


Figure 10

1643


The above determined solutions may be useful for researchers to understand the interaction of the Langmuir wave and the ion acoustic wave in high frequency plasma in plasma physics. The physical significances of these solutions are also described in the next section. 4. Physical significance This section presents the physical explanation of traveling wave solutions to the coupled Klein–Gordon–Zakharov equation in the x–y plane for different values of time t. The coupled Kl ein–Gordon–Zakharov equation are given various types of traveling wave propagation in the x–y plane, such as, kink type, singular kink type, soliton type, cupson type and bell type solitary waves as well as periodic traveling waves. It is noted the delicate balance between linear and nonlinear effects on the solitons solution which are fully interacting with others and come back with the same speed and shape. That is, the amplitude, velocity and wave shape of solitons remain unchanged after nonlinear interaction and showing completely elastic collisions. However, some solutions of coupled Klein–Gordon–Z akharov equation in the x–y plane for different values of time among solitonic excitations are not completely elastic, because their shapes are changed after their interactions. Thus,

the traveling wave solutions are described physically with elastic property and presented graphically in Figs. 1–10. The shape of the solutions u1 and v1 of the coupled Klein–Gordon–Zakharov equation is presented in Figs. 1 and 2 respectively with some fixed parametric values k1 ¼ 2, k2 ¼ 1, k3 ¼ 2, p ¼ 1, q ¼ 1, r ¼ 1, c0 ¼ 0:1, E ¼ 1, k ¼ 8, l ¼ 2, f0 ¼ 1, X ¼ 5, d ¼ 5, and b ¼ 3 at times t ¼ 0, 10, 25. From Figs. 1 and 2, it is observed that the traveling wave solutions u1 and v1 initially form kink type soliton in the x–y plane. Kink waves are special type of traveling waves which come up from one asymptotic state to another. It is also observed that the amplitude and shape of these waves become smaller and smaller after interactions. At the final state, they reduce to zero, which shows that the solitonic excitations are not completely elastic. Figs. 3 and 4 show the wave profiles corresponding to the solutions u2 and v2 considering the fixed values of the parameters k1 ¼ 2, k2 ¼ 1, k3 ¼ 2, p ¼ 1, q ¼ 1, r ¼ 1, c0 ¼ 0:1, E ¼ 1, k ¼ 4, l ¼ 6, f0 ¼ 1, X ¼ 0:5, d ¼ 5, and b ¼ 3 for different values of t ¼ 0, 10, 1000. The exact solutions u2 and v2 of Eq. (11) are represented periodic wave solutions in the x–y plane for different values of t. It is shown that the periodic wave solutions remain unchanged. That is, the periodic wave solution is completely elastic solution after nonlinear interaction.

1644 Figs. 5 and 6 present the wave profiles corresponding to u3 and v3 with fixed parametric values k1 ¼ 2, k2 ¼ 1, k3 ¼ 2, p ¼ 1, q ¼ 1, r ¼ 1, c0 ¼ 0:1, E ¼ 1, k ¼ 0:1, l ¼ 0, f0 ¼ 1, X ¼ 5, d ¼ 5, and b ¼ 3 for t ¼ 0, 10, 30. The exact solutions u3 and v3 of Eq. (11) initially formed bell type solitary wave in the x–y plane. The amplitude and width of these waves also become smaller and smaller after interaction according to increase of time t. It is investigated that the shapes of the bell type solitary wave are changed, but the wave propagation of plasma does not reduce to zero. Figs. 7 and 8 exhibit the wave profiles of u4 and v4 with fixed parametric values k1 ¼ 2, k2 ¼ 1, k3 ¼ 2, p ¼ 1, q ¼ 1, r ¼ 1, c0 ¼ 0:1, E ¼ 1, k ¼ 1, l ¼ 2, f0 ¼ 1, X ¼ 5, d ¼ 5; and b ¼ 3 at times t ¼ 0, 10, 30. The exact solutions u4 and v4 of Eq. (11) initially represented singular kink traveling wave solutions in the x–y plane. Figs. 9 and 10 present the wave profiles of the solutions u5 and v5 taking the fixed values of the parameters k1 ¼ 2, k2 ¼ 1, k3 ¼ 2, p ¼ 1, q ¼ 1, r ¼ 1, c0 ¼ 0:1, E ¼ 1, k ¼ 0, l ¼ 0, f0 ¼ 1, X ¼ 5, d ¼ 5, and b ¼ 3 for t ¼ 0, 10, 1000. The exact solutions u5 and v5 of Eq. (11) are initially represented cupson and soliton wave propagation respectively in the x–y plane. Cupson is a special type of soliton wave propagation. 5. Conclusion The modified expð/ðnÞÞ-expansion method has been implemented to construct the new explicit solutions of the coupled (3+1) dimensional Klein–Gordon-Zakharov equation. The hyperbolic, trigonometric, exponential and rational function type traveling wave solutions are obtained considering the fixed values of the related parameters. The annihilation phenomena of the wave propagation in the x–y plane are also described. It is observed that the soliton solutions of kink type are not completely elastic, but the periodic traveling wave solutions are completely elastic. It can be concluded that the method presented here is very effective to solve many other nonlinear evolution equations. In future, this method may be applied to find the analytical solutions of the (3+1)dimensional KdVB equation for investigating the oblique propagation of wave dynamics in plasma physics. Those results will be reported in future publications. References [1] M.G. Hafez, M.R. Talukder, Ion acoustic solitary waves in plasmas with nonextensive electrons, Boltzmann positrons and relativistic thermal ions, Astrophys. Space Sci. 359 (2015) 27. [2] M.G. Hafez, M.R. Talukder, M.H. Ali, Ion acoustic shock and solitary waves in highly relativistic plasmas with nonextensive electrons and positrons, Phys. Plasmas 23 (2016) 012902. [3] M.G. Hafez, M.R. Talukder, M.H. Ali, New analytical solutions for propagation of small but finite amplitude ionacoustic waves in a dense plasma, Waves Random Complex Media 26 (2016) 68–79. [4] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform, Cambridge University Press, Cambridge, 1991. [5] M.T. Darvishi, M. Najafi, Some exact solutions of the (2+1)dimensional breaking soliton equation using the three-wave method, World Acad. Sci. Eng. Technol. 87 (2012) 31–34.

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