Exact solutions to the (3+1)-dimensional coupled Klein–Gordon–Zakharov equation using exp(−Φ(ξ))-expansion method

==Abstract==

In this article, the <math display="inline">exp(-\Phi (\xi ))</math>-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 <math display="inline">x</math>–<math display="inline">y</math> 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.

==Keywords==

The nonlinear coupled Klein–Gordon–Zakharov equation; Traveling wave solutions; Solitary wave solutions; The exp(-Φ(ξ))exp(-Φ(ξ))-expansion method

==1. Introduction==

It is well known that partial differential equations (PDEs) are special kind of nonlinear evolution equations. Already a large number of authors [[#b0005|[1]]], [[#b0010|[2]]] and [[#b0015|[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 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 [[#b0020|[4]]]. Some of other most efficient methods for finding analytical solutions of the nonlinear PDEs are the three-wave method [[#b0025|[5]]], the improved F-expansion method [[#b0030|[6]]], the generalized method [[#b0035|[7]]], the Weiss–Tabor–Carnevale method [[#b0040|[8]]], the tanh-function method [[#b0045|[9]]], the extended tanh-method [[#b0050|[10]]] and [[#b0055|[11]]], the modified extended tanh-function method [[#b0060|[12]]] and [[#b0065|[13]]], the Jacobi elliptic function expansion method [[#b0070|[14]]], the extended Jacobi elliptic function expansion rational method [[#b0075|[15]]], the Exp-function method [[#b0080|[16]]], [[#b0085|[17]]] and [[#b0090|[18]]], the truncated Painleve expansion method [[#b0095|[19]]], the homogeneous balance method [[#b0100|[20]]], [[#b0105|[21]]] and [[#b0110|[22]]] and so on. Recently, the <math display="inline">exp(-\Phi (\xi ))</math>-expansion method [[#b0115|[23]]], [[#b0120|[24]]], [[#b0125|[25]]], [[#b0130|[26]]] and [[#b0135|[27]]] has also been used to find the exact traveling wave solutions of single nonlinear PDEs. Moreover, many researchers [[#b0140|[28]]], [[#b0145|[29]]], [[#b0150|[30]]], [[#b0155|[31]]] and [[#b0160|[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 [[#b0140|[28]]], truncated expansion method [[#b0145|[29]]], first integral method [[#b0150|[30]]], and conservative difference methods [[#b0155|[31]]].

Many types of coupled Klein–Gordon–Zakharov equation [[#b0135|[27]]], [[#b0155|[31]]] and [[#b0165|[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 <math display="inline">exp(-\Phi (\xi ))</math>-expansion 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 <math display="inline">exp(-\Phi (\xi ))</math> expansion method is given in Section [[#s0010|2]]. The traveling wave solutions of (3+1)-dimensional coupled Klein–Gordon–Zakharov equation are depicted in Section [[#s0015|3]]. The physical significances along with relevant discussion are mentioned in Section [[#s0020|4]]. Finally, the conclusion is drawn in Section [[#s0025|5]].

<span id='s0010'></span>
==2. Description of the modified exp(−''Φ''(''ξ''))-expansion method==

The general form of the nonlinear PDE can be written as follows:

<span id='e0005'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>F(u\mbox{,}u_t\mbox{,}u_x\mbox{,}u_y\mbox{,}u_z\mbox{,}u_{xx}\mbox{,}u_{yy}\mbox{,}u_{zz}\mbox{,}u_{tt}\mbox{,}u_{tx}\mbox{,}\ldots )=</math><math>0\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (1)
|}

where <math display="inline">u=u(\overset{\rightarrow}{r}\mbox{,}t)</math> is an unknown function of <math display="inline">x\mbox{,}y\mbox{,}z</math> and <math display="inline">t</math>, <math display="inline">F</math> is a polynomial in <math display="inline">u(\overset{\rightarrow}{r}\mbox{,}t)</math> 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 <math display="inline">x\mbox{,}y\mbox{,}z</math> and ''t  '' by a compound variable <math display="inline">\xi </math> is as follows:

<span id='e0010'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u(\overset{\rightarrow}{r}\mbox{,}t)=u(\xi )\mbox{,}\quad \xi =</math><math>\overset{\rightarrow}{P}\cdot \overset{\rightarrow}{r}\pm \omega t\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
|}

Here <math display="inline">\overset{\rightarrow}{P}=p\hat{i}+q\hat{j}+r\hat{k}</math>, <math display="inline">\overset{\rightarrow}{r}=x\hat{i}+y\hat{j}+z\hat{k}</math>, where <math display="inline">p\mbox{,}q\mbox{,}r</math> are the constant magnitude along the axes of <math display="inline">x\mbox{,}y\mbox{,}z</math> respectively and <math display="inline">\omega </math> is the speed of the traveling wave. Using the traveling wave variables transformation [[#e0010|(2)]], Eq. [[#e0005|(1)]] can be reduced to an ordinary differential equation (ODE) for <math display="inline">u=u(\xi )</math>:

<span id='e0015'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>R(u\mbox{,}u^{{'}}\mbox{,}u^{{''}}\mbox{,}u^{{'''}}\mbox{,}\ldots )=</math><math>0\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3)
|}

where <math display="inline">R</math> is a polynomial of ''u  '' and its derivatives and primes indicate the ordinary derivatives with respect to <math display="inline">\xi </math>.

The traveling wave solution of Eq. [[#e0015|(3)]] can be expressed as follows:

<span id='e0020'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u(\xi )=\sum_{i=0}^NA_i{\left(exp(-\Phi (\xi ))\right)}^i\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (4)
|}

where <math display="inline">A_i\quad (0\leqslant i\leqslant N)</math> are constants to be evaluated later, such that <math display="inline">A_N\quad \not =\quad 0</math> and <math display="inline">\Phi =\Phi (\xi )</math> satisfy the following ODE:

<span id='e0025'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>{\Phi }^{{'}}(\xi )=exp(-\Phi (\xi ))+\mu exp(\Phi (\xi ))+</math><math>\lambda \mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (5)
|}

where <math display="inline">\lambda </math> and <math display="inline">\mu </math> are arbitrary constants.

It is noted that Eq. [[#e0025|(5)]] gives the following general solutions:
* '''Type 1''': When <math display="inline">\mu \quad \not =\quad 0\mbox{,}\quad \Theta ={\lambda }^2-</math><math>4\mu >0</math>,

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\Phi (\xi )=ln\left(\frac{-\sqrt{\Theta }tanh\left\{0.5\sqrt{\Theta }(\xi +E)\right\}-\lambda }{2\mu }\right)</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (6)
|}
* '''Type 2:''' When <math display="inline">\mu \quad \not =\quad 0\mbox{,}\quad \Theta ={\lambda }^2-</math><math>4\mu <0</math>,

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\Phi (\xi )=ln\left(\frac{\sqrt{-\Theta }tan\left\{0.5\sqrt{-\Theta }(\xi +E)\right\}-\lambda }{2\mu }\right)</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (7)
|}
* '''Type 3''': When <math display="inline">\mu =0\mbox{,}\quad \lambda \quad \not =\quad 0</math>, and <math display="inline">\Theta ={\lambda }^2-4\mu >0</math>,

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\Phi (\xi )=-ln\left(\frac{\lambda }{exp(\lambda (\xi +E))-1}\right)</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (8)
|}
* '''Type 4''': When <math display="inline">\mu \quad \not =\quad 0\mbox{,}\quad \lambda \quad \not =</math><math>\quad 0</math>, and <math display="inline">\Theta ={\lambda }^2-4\mu =0</math>,

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\Phi (\xi )=ln\left(-\frac{2(\lambda (\xi +E)+2)}{{\lambda }^2(\xi +E)}\right)</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (9)
|}
* '''Type 5''': When <math display="inline">\mu =0\mbox{,}\quad \lambda =0</math>, and <math display="inline">\Theta ={\lambda }^2-4\mu =0</math>,

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\Phi (\xi )=ln\left(\xi +E\right)</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (10)
|}

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.  [[#e0015|(3)]]. If the degree of <math display="inline">u(\xi )</math> is <math display="inline">D[u(\xi )]=N</math>, then the degree of the other expressions can be evaluated by the following formulae:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>D\left[\frac{d^pu(\xi )}{d{\xi }^p}\right]=N+p\mbox{,}\quad D\left[u^p{\left(\frac{d^qu(\xi )}{d{\xi }^q}\right)}^s\right]=</math><math>Np+s(N+q)\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

By substituting Eq. [[#e0020|(4)]] into Eq. [[#e0015|(3)]] and using [[#e0025|(5)]] rapidly, one can obtain a system of algebraic equations by equating all the coefficients of <math display="inline">exp(-\Phi (\xi ))</math> to zero. The obtaining system can be solved to find the value of <math display="inline">A_1\mbox{,}A_2\ldots \mbox{,}\omega \mbox{,}p\mbox{,}q\mbox{,}r</math> with the help of symbolic computation software such as Maple. Substituting the values of <math display="inline">A_1\mbox{,}A_2\ldots \mbox{,}\omega \mbox{,}p\mbox{,}q\mbox{,}r</math> into Eq. [[#e0020|(4)]] along with general solutions of Eq. [[#e0025|(5)]] completes the determination of the solution of Eq. [[#e0005|(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 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.

<span id='s0015'></span>
==3. Application of the method==

This section presents the application of the modified <math display="inline">exp(-\Phi (\xi ))</math>-expansion method for finding the exact traveling wave solutions of the (3+1)-dimensional coupled Klein–Gordon Zakharov equation [[#b0165|[33]]].

Let us consider the coupled Klein–Gordon–Zakharov equation

<span id='e0055'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}
u_{tt}-c_0^2{\nabla }^2u+f_0^2u+\delta uv=0\\
v_{tt}-c_0^2{\nabla }^2v-\beta {\nabla }^2\left|u\right|^2=0
\end{array}\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (11)
|}

Here <math display="inline">u(\overset{\rightarrow}{r}\mbox{,}t)</math> is a complex function describing the fast timescale component of electric field raised by electrons. <math display="inline">v(\overset{\rightarrow}{r}\mbox{,}t)</math> is real function indicating the deviation of ion density form its equilibrium. The system of Eqs. [[#e0055|(11)]] described the interaction of the Langmuir wave and the ion acoustic wave in high frequency plasma in plasma physics (more details in Ref. [[#b0150|[30]]]).

To find out the wave packet solution, one can introduce the following transformation:

<span id='e0060'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u(\overset{\rightarrow}{r}\mbox{,}t)=exp\left\{i\left(\overset{\rightarrow}{k}\cdot \overset{\rightarrow}{r}-\right. \right. </math><math>\left. \left. \Omega t\right)\right\}U(\xi )\mbox{,}\quad v(\overset{\rightarrow}{r}\mbox{,}t)=</math><math>V(\xi )\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (12)
|}

with <math display="inline">\overset{\rightarrow}{k}\cdot \overset{\rightarrow}{r}=</math><math>k_1x+k_2y+k_3z</math>, <math display="inline">\xi =\overset{\rightarrow}{P}\cdot \overset{\rightarrow}{r}-</math><math>\omega t=px+qy+rz-\omega t</math>, where <math display="inline">k_1\mbox{,}k_2\mbox{,}k_3</math> are the constant magnitudes along the axes of <math display="inline">x\mbox{,}y\mbox{,}z</math> respectively. By using Eq. [[#e0060|(12)]] into Eq. [[#e0055|(11)]], one obtains the following coupled nonlinear ordinary differential equations:

<span id='e0065'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}
\left({\omega }^2-c_0^2P^2\right)U^{{''}}+2i\left\{\omega \Omega -c_0^2\left(k_1p+k_2q+k_3r\right)\right\}U^{{'}}\\
\quad -\left({\omega }^2-c_0^2K^2-f_0^2\right)U+\delta UV=0\\
\left({\omega }^2-c_0^2P^2\right)V^{{''}}-\beta P^2{\left(U^2\right)}^{{''}}=0
\end{array}\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (13)
|}

where <math display="inline">K^2=k_1^2+k_2^2+k_3^2</math>, <math display="inline">P^2=p^2+q^2+r^2</math> and primes denote the differentiation with respect to <math display="inline">\xi </math>.

If we set <math display="inline">\omega =\frac{c_0^2}{\Omega }\overset{\rightarrow}{k}\cdot \overset{\rightarrow}{P}</math>, then Eq. [[#e0065|(13)]] becomes

<span id='e0070'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}
\left({\omega }^2-c_0^2P^2\right)U^{{''}}-\left({\omega }^2-c_0^2K^2-f_0^2\right)U+\delta UV=0\\
\left({\omega }^2-c_0^2P^2\right)V^{{''}}-\beta K^2{\left(U^2\right)}^{{''}}=0
\end{array}\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (14)
|}

Integrating the second equation of [[#e0070|(14)]] twice with respect to <math display="inline">\xi </math>, we obtain

<span id='e0075'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>V=\frac{C}{\left({\omega }^2-c_0^2P^2\right)}+\frac{\beta K^2}{\left({\omega }^2-c_0^2P^2\right)}U^2</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (15)
|}

where ''C'' is the integrating constant. By using  [[#e0075|(15)]] into the first equation of [[#e0070|(14)]], we obtain

<span id='e0080'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>{\left({\omega }^2-c_0^2P^2\right)}^2U^{{''}}+\left\{\left({\omega }^2-\right. \right. </math><math>\left. \left. c_0^2P^2\right)\left(-{\omega }^2+c_0^2K^2+\right. \right. </math><math>\left. \left. f_0^2\right)+\delta C\right\}U+\delta \beta K^2U^3=</math><math>0</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (16)
|}

This is the elliptic-like equation with cubic nonlinearity. The pole of this equation is <math display="inline">N=1</math>. Therefore, the <math display="inline">exp(-\Phi (\xi ))</math>-expansion method allows us to use the solution in the following form:

<span id='e0085'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>U(\xi )=A_0+A_1exp(-\Phi (\xi ))\mbox{,}\quad A_1\quad \not =</math><math>\quad 0</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (17)
|}

where <math display="inline">A_0</math> and <math display="inline">A_1</math> are unknown constants. By means of Eq. [[#e0085|(17)]], Eq. [[#e0080|(16)]] turns out into a polynomial of <math display="inline">exp(-\Phi (\xi ))</math> and equating the coefficient of <math display="inline">{\left(exp(-\Phi (\xi ))\right)}^i\mbox{,}\quad (i=</math><math>0\mbox{,}1\mbox{,}2\mbox{,}3)</math> to zero, we obtain a system of algebraic equations as follows:

<span id='e0090'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\delta \beta K^2A_1^3+2A_1c_0^4P^4+2{\omega }^4A_1-</math><math>4{\omega }^2c_0^2P^2A_1=0\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (18)
|}

<span id='e0095'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>3A_1{\omega }^4\lambda -6A_1{\omega }^2c_0^2P^2\lambda +</math><math>3\delta \beta K^2A_0A_1^2+3A_1c_0^4P^4\lambda =0</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (19)
|}

<span id='e0100'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>2A_1{\omega }^4\mu -4A_1{\omega }^2c_0^2P^2\mu +3\delta \beta K^2A_0^2A_1+</math><math>\delta {CA}_1+A_1{\omega }^4{\lambda }^2+{\omega }^2c_0^2P^2A_1-</math><math>c_0^2P^2f_0^2A_1+A_1c_0^4P^4{\lambda }^2+{\omega }^2f_0^2A_1-</math><math>2A_1{\omega }^2c_0^2P^2{\lambda }^2-c_0^4P^2K^2A_1+</math><math>{\omega }^2K^2+c_0^2A_1+2A_1c_0^4P^4\mu -{\omega }^4A_1=</math><math>0</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (20)
|}

<span id='e0105'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>-c_0^4P^2K^2A_0-2A_1{\omega }^2c_0^2P^2\lambda \mu +</math><math>A_1{\omega }^4\lambda \mu +{\omega }^2c_0^2P^2A_0+</math><math>{\omega }^2K^2c_0^2A_0+\delta {CA}_0+\delta \beta K^2A_0^3-</math><math>c_0^2P^2f_0^2A_0+{\omega }^2f_0^2A_0-{\omega }^4A_0+</math><math>A_1c_0^4P^4\lambda \mu =0</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (21)
|}

Solving the system algebraic Eqs.  [[#e0090|(18)]], [[#e0095|(19)]], [[#e0100|(20)]] and [[#e0105|(21)]], we obtain the following set of solutions:

<span id='e0110'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>C=-\frac{1}{2\delta }4{\omega }^4\mu -8{\omega }^2c_0^2P^2\mu -</math><math>{\omega }^4{\lambda }^2\mu +2{\omega }^2c_0^2P^2-2c_0^2P^2f_0^2-</math><math>c_0^4P^4{\lambda }^2+2{\omega }^2f_0^2+2{\omega }^2c_0^2P^2{\lambda }^2-</math><math>2c_0^4P^2K^2+2{\omega }^2K^2c_0^2+4c_0^4P^4\mu -2{\omega }^4A_0=</math><math>\frac{\pm i}{K\sqrt{2\delta \beta }}\left(c_0^2P^2-\right. </math><math>\left. {\omega }^2\right)\lambda \mbox{,}\quad A_1=</math><math>\frac{\pm i}{K\sqrt{2\delta \beta }}\left(c_0^2P^2-\right. </math><math>\left. {\omega }^2\right)\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (22)
|}

where <math display="inline">B_0\mbox{,}\lambda </math> and <math display="inline">\mu </math> are arbitrary constants.

Using [[#e0060|(12)]], [[#e0075|(15)]], [[#e0085|(17)]] and [[#e0110|(22)]], the traveling wave solutions of the coupled Klein–Gordon–Zakharov equation are obtained as follows:

When <math display="inline">\Theta ={\lambda }^2-4\mu >0\mbox{,}\quad \mu \quad \not =</math><math>\quad 0</math>, we have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u_1(\overset{\rightarrow}{r}\mbox{,}t)=e^{i\left(\overset{\rightarrow}{k}\cdot \overset{\rightarrow}{r}-\Omega t\right)}\left(c_0^2P^2-\right. </math><math>\left. {\omega }^2\right)\frac{\pm i\lambda }{K\sqrt{2\delta \beta }}-</math><math>\frac{\pm i}{K\sqrt{2\delta \beta }}\left(\frac{2\mu }{\sqrt{\Theta }tanh\left\{0.5\sqrt{\Theta }(\overset{\rightarrow}{p}\cdot \overset{\rightarrow}{r}-\omega t+E)\right\}+\lambda }\right)</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (23)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>v_1(\overset{\rightarrow}{r}\mbox{,}t)=\frac{C}{\left({\omega }^2-c_0^2P^2\right)}-</math><math>\beta K^2\left({\omega }^2-c_0^2P^2\right)\frac{\pm \lambda }{K\sqrt{2\delta \beta }}\mp \frac{1}{K\sqrt{2\delta \beta }}{\left(\frac{2\mu }{\sqrt{\Theta }tanh\left\{0.5\sqrt{\Theta }(\overset{\rightarrow}{p}\cdot \overset{\rightarrow}{r}-\omega t+E)\right\}+\lambda }\right)}^2</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (24)
|}

When <math display="inline">\Theta ={\lambda }^2-4\mu <0</math>, we have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u_2(\overset{\rightarrow}{r}\mbox{,}t)=e^{i\left(\overset{\rightarrow}{k}\cdot \overset{\rightarrow}{r}-\Omega t\right)}\times \left(c_0^2P^2-\right. </math><math>\left. {\omega }^2\right)\times \left\{\frac{\pm i\lambda }{K\sqrt{2\delta \beta }}+\right. </math><math>\left. \frac{\pm i}{K\sqrt{2\delta \beta }}\left(\frac{2\mu }{\sqrt{-\Theta }tan\left\{0.5\sqrt{-\Theta }(\overset{\rightarrow}{p}\cdot \overset{\rightarrow}{r}-\omega t+E)\right\}-\lambda }\right)\right\}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (25)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>v_2(\overset{\rightarrow}{r}\mbox{,}t)=\frac{C}{\left({\omega }^2-c_0^2P^2\right)}-</math><math>\beta K^2\left({\omega }^2-c_0^2P^2\right){\left\{\frac{\pm \lambda }{K\sqrt{2\delta \beta }}+\frac{\pm 1}{K\sqrt{2\delta \beta }}\left(\frac{2\mu }{\sqrt{-\Theta }tan\left\{0.5\sqrt{-\Theta }(\overset{\rightarrow}{p}\cdot \overset{\rightarrow}{r}-\omega t+E)\right\}-\lambda }\right)\right\}}^2</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (26)
|}

When <math display="inline">{\lambda }^2-4\mu >0\mbox{,}\quad \mu =0</math>, we have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u_3(\overset{\rightarrow}{r}\mbox{,}t)=e^{i\left(\overset{\rightarrow}{k}\cdot \overset{\rightarrow}{r}-\Omega t\right)}\left(c_0^2P^2-\right. </math><math>\left. {\omega }^2\right)\times \left\{\frac{\pm i\lambda }{K\sqrt{2\delta \beta }}+\right. </math><math>\left. \frac{\pm i}{K\sqrt{2\delta \beta }}\left(\frac{\lambda }{exp\left\{\lambda (\overset{\rightarrow}{p}\cdot \overset{\rightarrow}{r}-\omega t+E)\right\}-1}\right)\right\}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (27)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>v_3(\overset{\rightarrow}{r}\mbox{,}t)=\frac{C}{\left({\omega }^2-c_0^2P^2\right)}-</math><math>\beta K^2\left({\omega }^2-c_0^2P^2\right){\left\{\frac{\pm \lambda }{K\sqrt{2\delta \beta }}\pm \frac{1}{K\sqrt{2\delta \beta }}\left(\frac{\lambda }{exp\left\{\lambda (\overset{\rightarrow}{p}\cdot \overset{\rightarrow}{r}-\omega t+E)\right\}-1}\right)\right\}}^2</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (28)
|}

When <math display="inline">\mu \quad \not =\quad 0\mbox{,}\quad \lambda \quad \not =</math><math>\quad 0</math>, and <math display="inline">{\lambda }^2-4\mu =0</math>, we have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u_4(\overset{\rightarrow}{r}\mbox{,}t)=e^{i\left(\overset{\rightarrow}{k}\cdot \overset{\rightarrow}{r}-\Omega t\right)}\times \left(c_0^2P^2-\right. </math><math>\left. {\omega }^2\right)\times \left\{\frac{\pm i\lambda }{K\sqrt{2\delta \beta }}-\right. </math><math>\left. \frac{\pm i}{K\sqrt{2\delta \beta }}\times \left(\frac{{\lambda }^2\left\{\lambda (\overset{\rightarrow}{p}\cdot \overset{\rightarrow}{r}+E)\right\}}{2\left\{\lambda (\overset{\rightarrow}{p}\cdot \overset{\rightarrow}{r}-\omega t+E)\right\}+4}\right)\right\}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (29)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>v_4(\overset{\rightarrow}{r}\mbox{,}t)=\frac{C}{\left({\omega }^2-c_0^2P^2\right)}-</math><math>\beta K^2\left({\omega }^2-c_0^2P^2\right)\left\{\frac{\pm \lambda }{K\sqrt{2\delta \beta }}-\right. </math><math>\left. \frac{\pm 1}{K\sqrt{2\delta \beta }}\times \left(\frac{{\lambda }^2(\overset{\rightarrow}{p}\cdot \overset{\rightarrow}{r}+E)}{2\left\{\lambda (\overset{\rightarrow}{p}\cdot \overset{\rightarrow}{r}-\omega t+E)\right\}+4}\right)\right\}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (30)
|}

When <math display="inline">\mu =0\mbox{,}\quad \lambda =0</math>, and <math display="inline">{\lambda }^2-4\mu =0</math>, we have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u_5(\overset{\rightarrow}{r}\mbox{,}t)=e^{i\left(\overset{\rightarrow}{k}\cdot \overset{\rightarrow}{r}-\Omega t\right)}\times \left(c_0^2P^2-\right. </math><math>\left. {\omega }^2\right)\times \left\{\frac{\pm i\lambda }{K\sqrt{2\delta \beta }}-\right. </math><math>\left. \frac{\pm i}{K\sqrt{2\delta \beta }}\left(\frac{1}{\overset{\rightarrow}{p}\cdot \overset{\rightarrow}{r}-\omega t+E}\right)\right\}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (31)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>v_5(\overset{\rightarrow}{r}\mbox{,}t)=\frac{C}{\left({\omega }^2-c_0^2P^2\right)}-</math><math>\beta K^2\left({\omega }^2-c_0^2P^2\right){\left\{\frac{\pm \lambda }{K\sqrt{2\delta \beta }}-\frac{\pm 1}{K\sqrt{2\delta \beta }}\left(\frac{1}{\overset{\rightarrow}{p}\cdot \overset{\rightarrow}{r}-\omega t+E}\right)\right\}}^2</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (32)
|}

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.

<span id='s0020'></span>
==4. Physical significance==

This section presents the physical explanation of traveling wave solutions to the coupled Klein–Gordon–Zakharov equation in the <math display="inline">x</math>–<math display="inline">y</math> plane for different values of time ''t  ''. The coupled Klein–Gordon–Zakharov equation are given various types of traveling wave propagation in the <math display="inline">x</math>–<math display="inline">y</math> 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–Zakharov equation in the <math display="inline">x</math>–<math display="inline">y</math> 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  [[#f0005|Figure 1]], [[#f0010|Figure 2]], [[#f0015|Figure 3]], [[#f0020|Figure 4]], [[#f0025|Figure 5]], [[#f0030|Figure 6]], [[#f0035|Figure 7]], [[#f0040|Figure 8]], [[#f0045|Figure 9]] and [[#f0050|Figure 10]].

<span id='f0005'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_García_157915352-1-s2.0-S1110016816000636-gr1.jpg|center|px|Shape of |u1(x,y)| in x–y plane at (a) t=0, (b) t=10 and (c) t=30.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 1.

Shape of <math display="inline">\vert u_1(x\mbox{,}y)\vert </math> in <math display="inline">x</math>–<math display="inline">y</math> plane at (a) <math display="inline">t=0</math>, (b) <math display="inline">t=10</math> and (c) <math display="inline">t=30</math>.

</span>
|}

<span id='f0010'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_García_157915352-1-s2.0-S1110016816000636-gr2.jpg|center|px|Shape of ν1 in x–y plane at (a) t=0, (b) t=10 and (c) t=30.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 2.

Shape of <math display="inline">{\nu }_1</math> in <math display="inline">x</math>–<math display="inline">y</math> plane at (a) <math display="inline">t=0</math>, (b) <math display="inline">t=10</math> and (c) <math display="inline">t=30</math>.

</span>
|}

<span id='f0015'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_García_157915352-1-s2.0-S1110016816000636-gr3.jpg|center|px|Shape of |u2| in x–y plane at (a) t=0, (b) t=10 and (c) t=1000.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 3.

Shape of <math display="inline">\vert u_2\vert </math> in <math display="inline">x</math>–<math display="inline">y</math> plane at (a) <math display="inline">t=0</math>, (b) <math display="inline">t=10</math> and (c) <math display="inline">t=1000</math>.

</span>
|}

<span id='f0020'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_García_157915352-1-s2.0-S1110016816000636-gr4.jpg|center|px|Shape of ν2 in x–y plane at (a) t=0, (b) t=10 and (c) t=100.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 4.

Shape of <math display="inline">{\nu }_2</math> in <math display="inline">x</math>–<math display="inline">y</math> plane at (a) <math display="inline">t=0</math>, (b) <math display="inline">t=10</math> and (c) <math display="inline">t=100</math>.

</span>
|}

<span id='f0025'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_García_157915352-1-s2.0-S1110016816000636-gr5.jpg|center|px|Shape of |u3| in x–y plane at (a) t=0, (b) t=10 and (c) t=30.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 5.

Shape of <math display="inline">\vert u_3\vert </math> in <math display="inline">x</math>–<math display="inline">y</math> plane at (a) <math display="inline">t=0</math>, (b) <math display="inline">t=10</math> and (c) <math display="inline">t=30</math>.

</span>
|}

<span id='f0030'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_García_157915352-1-s2.0-S1110016816000636-gr6.jpg|center|px|Shape of v3 in x–y plane at (a) t=0, (b) t=10 and (c) t=30.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 6.

Shape of <math display="inline">v_3</math> in <math display="inline">x</math>–<math display="inline">y</math> plane at (a) <math display="inline">t=0</math>, (b) <math display="inline">t=10</math> and (c) <math display="inline">t=30</math>.

</span>
|}

<span id='f0035'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_García_157915352-1-s2.0-S1110016816000636-gr7.jpg|center|px|Shape of |u4| in x–y plane at (a) t=0, (b) t=10 and (c) t=30.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 7.

Shape of <math display="inline">\vert u_4\vert </math> in <math display="inline">x</math>–<math display="inline">y</math> plane at (a) <math display="inline">t=0</math>, (b) <math display="inline">t=10</math> and (c) <math display="inline">t=30</math>.

</span>
|}

<span id='f0040'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_García_157915352-1-s2.0-S1110016816000636-gr8.jpg|center|px|Shape of v4 in x–y plane at (a) t=0, (b) t=10 and (c) t=30.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 8.

Shape of <math display="inline">v_4</math> in <math display="inline">x</math>–<math display="inline">y</math> plane at (a) <math display="inline">t=0</math>, (b) <math display="inline">t=10</math> and (c) <math display="inline">t=30</math>.

</span>
|}

<span id='f0045'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_García_157915352-1-s2.0-S1110016816000636-gr9.jpg|center|px|Shape of |u5| in x–y plane at (a) t=0, (b) t=10 and (c) t=30.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 9.

Shape of <math display="inline">\vert u_5\vert </math> in <math display="inline">x</math>–<math display="inline">y</math> plane at (a) <math display="inline">t=0</math>, (b) <math display="inline">t=10</math> and (c) <math display="inline">t=30</math>.

</span>
|}

<span id='f0050'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_García_157915352-1-s2.0-S1110016816000636-gr10.jpg|center|px|Shape of v5 in x–y plane at (a) t=0, (b) t=10 and (c) t=30.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 10.

Shape of <math display="inline">v_5</math> in <math display="inline">x</math>–<math display="inline">y</math> plane at (a) <math display="inline">t=0</math>, (b) <math display="inline">t=10</math> and (c) <math display="inline">t=30</math>.

</span>
|}

The shape of the solutions <math display="inline">u_1</math> and <math display="inline">v_1</math> of the coupled Klein–Gordon–Zakharov equation is presented in [[#f0005|Figure 1]] and [[#f0010|Figure 2]] respectively with some fixed parametric values <math display="inline">k_1=2</math>, <math display="inline">k_2=1</math>, <math display="inline">k_3=2</math>, <math display="inline">p=1</math>, <math display="inline">q=-1</math>, <math display="inline">r=1</math>, <math display="inline">c_0=0.1</math>, <math display="inline">E=1</math>, <math display="inline">\lambda =8</math>, <math display="inline">\mu =2</math>, <math display="inline">f_0=1</math>, <math display="inline">\Omega =5</math>, <math display="inline">\delta =5</math>, and <math display="inline">\beta =3</math> at times <math display="inline">t=0</math>, 10, 25. From [[#f0005|Figure 1]] and [[#f0010|Figure 2]], it is observed that the traveling wave solutions <math display="inline">u_1</math> and <math display="inline">v_1</math> initially form kink type soliton in the <math display="inline">x</math>–<math display="inline">y</math> 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.

[[#f0015|Figure 3]] and [[#f0020|Figure 4]] show the wave profiles corresponding to the solutions <math display="inline">u_2</math> and <math display="inline">v_2</math> considering the fixed values of the parameters <math display="inline">k_1=2</math>, <math display="inline">k_2=-1</math>, <math display="inline">k_3=2</math>, <math display="inline">p=1</math>, <math display="inline">q=1</math>, <math display="inline">r=1</math>, <math display="inline">c_0=0.1</math>, <math display="inline">E=1</math>, <math display="inline">\lambda =4</math>, <math display="inline">\mu =6</math>, <math display="inline">f_0=1</math>, <math display="inline">\Omega =0.5</math>, <math display="inline">\delta =5</math>, and <math display="inline">\beta =3</math> for different values of <math display="inline">t=0</math>, 10, 1000. The exact solutions <math display="inline">u_2</math> and <math display="inline">v_2</math> of Eq. [[#e0055|(11)]] are represented periodic wave solutions in the <math display="inline">x</math>–<math display="inline">y</math> 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.

[[#f0025|Figure 5]] and [[#f0030|Figure 6]] present the wave profiles corresponding to <math display="inline">u_3</math> and <math display="inline">v_3</math> with fixed parametric values <math display="inline">k_1=2</math>, <math display="inline">k_2=-1</math>, <math display="inline">k_3=2</math>, <math display="inline">p=1</math>, <math display="inline">q=1</math>, <math display="inline">r=1</math>, <math display="inline">c_0=0.1</math>, <math display="inline">E=1</math>, <math display="inline">\lambda =0.1</math>, <math display="inline">\mu =0</math>, <math display="inline">f_0=1</math>, <math display="inline">\Omega =5</math>, <math display="inline">\delta =5</math>, and <math display="inline">\beta =3</math> for <math display="inline">t=0</math>, 10, 30. The exact solutions <math display="inline">u_3</math> and <math display="inline">v_3</math> of Eq. [[#e0055|(11)]] initially formed bell type solitary wave in the <math display="inline">x</math>–<math display="inline">y</math> 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.

[[#f0035|Figure 7]] and [[#f0040|Figure 8]] exhibit the wave profiles of <math display="inline">u_4</math> and <math display="inline">v_4</math> with fixed parametric values <math display="inline">k_1=2</math>, <math display="inline">k_2=-1</math>, <math display="inline">k_3=2</math>, <math display="inline">p=1</math>, <math display="inline">q=1</math>, <math display="inline">r=1</math>, <math display="inline">c_0=0.1</math>, <math display="inline">E=1</math>, <math display="inline">\lambda =1</math>, <math display="inline">\mu =2</math>, <math display="inline">f_0=1</math>, <math display="inline">\Omega =5</math>, <math display="inline">\delta =5\mbox{,}</math> and <math display="inline">\beta =3</math> at times <math display="inline">t=0</math>, 10, 30. The exact solutions <math display="inline">u_4</math> and <math display="inline">v_4</math> of Eq. [[#e0055|(11)]] initially represented singular kink traveling wave solutions in the <math display="inline">x</math>–<math display="inline">y</math> plane.

[[#f0045|Figure 9]] and [[#f0050|Figure 10]] present the wave profiles of the solutions <math display="inline">u_5</math> and <math display="inline">v_5</math> taking the fixed values of the parameters <math display="inline">k_1=2</math>, <math display="inline">k_2=-1</math>, <math display="inline">k_3=2</math>, <math display="inline">p=1</math>, <math display="inline">q=-1</math>, <math display="inline">r=1</math>, <math display="inline">c_0=0.1</math>, <math display="inline">E=1</math>, <math display="inline">\lambda =0</math>, <math display="inline">\mu =0</math>, <math display="inline">f_0=1</math>, <math display="inline">\Omega =5</math>, <math display="inline">\delta =5</math>, and <math display="inline">\beta =3</math> for <math display="inline">t=0</math>, 10, 1000. The exact solutions <math display="inline">u_5</math> and <math display="inline">v_5</math> of Eq. [[#e0055|(11)]] are initially represented cupson and soliton wave propagation respectively in the <math display="inline">x</math>–<math display="inline">y</math> plane. Cupson is a special type of soliton wave propagation.

<span id='s0025'></span>
==5. Conclusion==

The modified <math display="inline">exp(-\phi (\xi ))</math>-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 <math display="inline">x</math>–<math display="inline">y</math> 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==

<ol style='list-style-type: none;margin-left: 0px;'><li><span id='b0005'></span>
[[#b0005|[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), p. 27</li>
<li><span id='b0010'></span>
[[#b0010|[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), p. 012902</li>
<li><span id='b0015'></span>
[[#b0015|[3]]] M.G. Hafez, M.R. Talukder, M.H. Ali; New analytical solutions for propagation of small but finite amplitude ion-acoustic waves in a dense plasma; Waves Random Complex Media, 26 (2016), pp. 68–79</li>
<li><span id='b0020'></span>
[[#b0020|[4]]] M.J. Ablowitz, P.A. Clarkson; Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform; Cambridge University Press, Cambridge (1991)</li>
<li><span id='b0025'></span>
[[#b0025|[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), pp. 31–34</li>
<li><span id='b0030'></span>
[[#b0030|[6]]]  Wang, H.Q. Zhang; Further improved F-expansion method and new exact solutions of Konopelchenko–Dubrovsky equation; Chaos, Solitons Fract., 25 (2005), pp. 601–610</li>
<li><span id='b0035'></span>
[[#b0035|[7]]] Z. Yan; Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres; Chaos, Solitons Fract., 16 (2003), pp. 759–766</li>
<li><span id='b0040'></span>
[[#b0040|[8]]] N.A. Kudryashov; Exact solutions of the generalized Kuramoto–Sivashinsky equation; Phys. Lett. A, 147 (1990), pp. 287–291</li>
<li><span id='b0045'></span>
[[#b0045|[9]]] W. Malfliet; The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations; J. Comput. Appl. Math., 164 (2004), pp. 529–541</li>
<li><span id='b0050'></span>
[[#b0050|[10]]] M. Wazwaz; The extended tanh method for new compact and noncompact solutions for the KP-BBM and the ZK-BBM equations; Chaos, Solitons Fract., 38 (5) (2008), pp. 1505–1516</li>
<li><span id='b0055'></span>
[[#b0055|[11]]] M.A. Abdou; The extended tanh-method and its applications for solving nonlinear physical models; Appl. Math. Comput., 190 (2007), pp. 988–996</li>
<li><span id='b0060'></span>
[[#b0060|[12]]] M.A. Abdou, A.A. Soliman; Modified extended tanh-function method and its application on nonlinear physical equations; Phys. Lett. A, 353 (2006), pp. 487–492</li>
<li><span id='b0065'></span>
[[#b0065|[13]]] S.A. El-Wakil, M.A. Abdou; New exact travelling wave solutions using modified extended tanh-function method; Chaos, Solitons Fract., 31 (4) (2007), pp. 840–852</li>
<li><span id='b0070'></span>
[[#b0070|[14]]] S. Liu, Z. Fu, S.D. Liu, Q. Zhao; Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations; Phys. Lett. A, 289 (2001), pp. 69–74</li>
<li><span id='b0075'></span>
[[#b0075|[15]]] Y. Chen, Q. Wang; Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic functions solutions to (1+1)-dimensional dispersive long wave equation; Chaos, Solitons Fract., 24 (2005), pp. 745–757</li>
<li><span id='b0080'></span>
[[#b0080|[16]]] J.H. He, X.H. Wu; Exp-function method for nonlinear wave equations; Chaos, Solitons Fract., 30 (3) (2006), pp. 700–708</li>
<li><span id='b0085'></span>
[[#b0085|[17]]] S.T. Mohyud-Din, M.N. Noor, A. Waheed; Exp-function method for generalized travelling solutions of Calogero–Degasperis–Fokas equation; Z. Naturforsch. A: J. Phys. Sci., 65a (2010), pp. 78–84</li>
<li><span id='b0090'></span>
[[#b0090|[18]]] H. Naher, A.F. Abdullah, M.A. Akbar; New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the Exp-function method; J. Appl. Math., 2012 (2012) [http://dx.doi.org/10.1155/2012/575387 http://dx.doi.org/10.1155/2012/575387]</li>
<li><span id='b0095'></span>
[[#b0095|[19]]] N.A. Kudryashov; On types of nonlinear non-integrable equations with exact solutions; Phys. Lett. A, 155 (1991), pp. 269–275</li>
<li><span id='b0100'></span>
[[#b0100|[20]]] X. Zhao, D. Tang; A new note on a homogeneous balance method; Phys. Lett. A, 297 (1–2) (2002), pp. 59–67</li>
<li><span id='b0105'></span>
[[#b0105|[21]]] X. Zhao, L. Wang, W. Sun; The repeated homogeneous balance method and its applications to nonlinear partial differential equations; Chaos, Solitons Fract., 28 (2) (2006), pp. 448–453</li>
<li><span id='b0110'></span>
[[#b0110|[22]]] F. Zhaosheng; Comment on “On the extended applications of homogeneous balance method”; Appl. Math. Comput., 158 (2) (2004), pp. 593–596</li>
<li><span id='b0115'></span>
[[#b0115|[23]]] M.N. Alam, M.G. Hafez, M.A. Akbar, H.O. Roshid; Exact traveling wave solutions to the (3+1)-dimensional mKdV-ZK and the (2+1)-dimensional Burgers equations via the <math display="inline">exp(-\Phi (\xi ))</math>-expansion method; Alexandria Eng. J., 54 (3) (2015), pp. 635–644</li>
<li><span id='b0120'></span>
[[#b0120|[24]]] M.G. Hafez, R. Sakthivel, M.R. Talukder; Some new electrostatic potential functions used to analyze the ion-acoustic waves in a thomas fermi plasma with degenerate electrons; Chin. J. Phys., 53 (2015), p. 120901</li>
<li><span id='b0125'></span>
[[#b0125|[25]]] M.G. Hafez, M.A. Kusar, M.T. Akter; Some new exact traveling wave solutions for the Zhiber–Shabat equation; Brit. J. Math. Comput. Sci., 4 (18) (2014), pp. 2582–2593</li>
<li><span id='b0130'></span>
[[#b0130|[26]]] M.G. Hafez, M.N. Alam, M.A. Akbar; Traveling wave solutions for some important coupled nonlinear physical models via the coupled Higgs equation and the Maccari system; J. King Saud Univ.-Sci., 27 (2015), pp. 105–112</li>
<li><span id='b0135'></span>
[[#b0135|[27]]] M.G. Hafez, M.A. Akbar; New exact traveling wave solutions to the (1+1)-dimensional Klein–Gordon–Zakharov equation for wave propagation in plasma using the exp(-<math display="inline">\Phi (\xi ))</math>-expansion method; Propul. Power Res., 4 (2015), pp. 31–39</li>
<li><span id='b0140'></span>
[[#b0140|[28]]] J. Manafian, I. Zamanpour; Analytical treatment of the coupled Higgs equation and Maccari system via Exp-Function Method; Acta Universitatis Apulensis, 33 (2013), pp. 203–216</li>
<li><span id='b0145'></span>
[[#b0145|[29]]] J. Lee, R. Sakthivel; Exact travelling wave solutions for some important nonlinear physical models; Pramana – J. Phys., 80 (2013), pp. 757–769</li>
<li><span id='b0150'></span>
[[#b0150|[30]]] A. Bekri, O. Unsal; Exact solutions for a class of nonlinear wave equations by using first integral method; Int. J. Nonlinear Sci., 15 (2) (2013), pp. 99–110</li>
<li><span id='b0155'></span>
[[#b0155|[31]]] T. Wang, J. Chen, L. Zhang; Conservative difference methods for the Klein–Gordon–Zakharov equations; J. Comput. Appl. Math., 205 (1) (2007), pp. 430–452</li>
<li><span id='b0160'></span>
[[#b0160|[32]]] Z.Y. Zhang, J. Zhang, S.S. Dou, J. Liu, D. Peng, T. Gao; Rom. Rep. Phys., 65 (2013), pp. 1155–1169</li>
<li><span id='b0165'></span>
[[#b0165|[33]]] E. Yomba, The New Soltitary Wave Solution for Nonlinear Wave, CKGZ, GDS, DS and GZ Equations, IMA Preprint Series#2041, 2005.</li>
</ol>