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\title{ New exact solutions of the combined and double combined sinh--cosh--Gordon equations via modified Kudryashov method}

\author[1,2*]{Atish Kumar Joardar}
\author[3]{Dipankar Kumar}
\author[4]{K. M. Abdul Al Woadud}
\affil[1]{Department of Mathematics, Islamic University, Kushtia--7003, Bangladesh.}
\affil[2]{Institute of Computational Mathematics and Scientific $/$ Engineering Computing, Academy of Mathematics and System Sciences, University of Chinese Academy of Sciences, Beijing, China.}
\affil[3]{Department of Mathematics, Bangabandhu Sheikh Mujibur Rahman Science and Technology University, Gopalganj--8100, Bangladesh.}
\affil[4]{Department of Electrical \& Electronic Engineering, Uttara University, Dhaka, Bangladesh.}
\affil[*]{Corresponding author Email: atishjoardar@yahoo.com}


\spcAbstract{
 The combined and double combined sinh--cosh--Gordon equations are very important to a wide range of various scientific applications that ranges from chemical reactions to water surface gravity waves. In this article, with the assistance of a function transform and Painlev\'{e} property, the nonlinear combined and double combined sinh--cosh--Gordon equations turn into ordinary differential equations. Later on, modified Kudryashov method is adopted for investigating new analytical solution of the studied equations. As a consequence, a series of new analytical solutions are accquired and we demonstrated the actual behavior of the achieved solutions of the mentioned equations with the aid of 3D and 2D MATLAB graphs. Finally, we also validate the effectiveness of the modified Kudryashov method for the problem of extracting new exact solutions of the combined and double combined sinh--cosh--Gordon equations with the aid of Maple package program. It is shown that the implemented method is capable to extract new solutions and it can also used to other nonlinear partial differential equation (NLPDE's) arising in mathematical physics or other applied field.
}

\spcKeywords{ Combined sinh--cosh--Gordon  equation; Double combined sinh--cosh--Gordon equation; Modified Kudryashov method; New exact solutions; Painlev\'{e} property.}

\maketitle

\section{Introduction}
Over the last 2 decades, exact solution of nonlinear partial differential equations (NPDEs) is one of the major excited advanced topic in the field of applied physics, mathematical physics and engineering. As we know, most of the real world problems are nonlinear and its real phenomena may depicts in NPDEs. Generally, its exact solutions are signify not only applied Mathematics but also in different research field like fluid dynamics, plasma physics, quantum field theory, nonlinear optics, signal processing and etc. Though, the procedure for finding exact solution of those equations are generally very complex but the solution of these equation are exists, it gives exact information  for describing real phenomena. The concept of travelling wave solution of physical applications such as fluid dynamics, plasma physics, quantum field theory and so on are very popular and significant. Nowadays, the concept of reducing nonlinear partial differential equation into ordinary differential has proved a successful way to generate an analytical solution of nonlinear partial differential equation. According to this concept, many powerful method grown up and used for acquiring exact solutions of NPDEs such as, Homogeneous balance method~\cite{n1}, ${G^{'}}/{G}-$expansion method~\cite{n2}, Improved ${G^{'}}/{G}-$expansion method~\cite{n3,n4}, Improved ${1}/{G^{'}}-$expansion and $({G^{'}}/{G},{1}/{G})$ methods~\cite{n4}, Extended tanh function method~\cite{n5}, Extended F--expansion method~\cite{n6}, Jacobi elliptic function method ~\cite{n7}, Transformed rational function method ~\cite{n8}, Weierstrass elliptic function expansion method ~\cite{n9}, Hirota's bilinear method~\cite{n10}, Generalized Kudryashov method ~\cite{n11}, Trial equation method~\cite{n12}, Solitary ansatz method ~\cite{n13}, Auxiliary equation method~\cite{n14}, Modified Kudryashov method ~\cite{n15,n16,n17}, Sine--Gordon equation expansion method ~\cite{n17,n18}, Hyperbolic function method ~\cite{n19} and so on. Recently, researchers easily design and applied these method for reducing the computational difficulty using computational software package like Maple, Mathematica, MATLAB etc.\\
In recent past, many researchers were tried to solve combined and double combined sinh--cosh--Gordon equations with the help of distinct analytical methods. In this portion, we will review some related works about the combined and double combined sinh--cosh--Gordon equations and several analytical approaches. In this regard,  Salas and Castillo ~\cite{n20} solved the combined sinh--cosh--Gordon equation through the projective Riccati equation based methods such as Conte's projective Riccati equation method, tanh--coth method, He's exp--function method and sn--ns method. At the end of the paper, authors conclude that sn--ns method more efficient than other applied methods. Gomez and Salas ~\cite{n21} also presented the general projective Riccati equations method to derive exact solutions for the combined sinh--cosh--Gordon equation by means of Painlev\'{e} property. Khalique and Magalakwe ~\cite{n22} derived another type of exact solutions of the combined sinh--cosh--Gordon equation by means of the Lie symmetry analysis along with the simplest equation method. Long Wei ~\cite{n23} find out some new type exact solutions for the combined sinh--cosh--Gordon equation based on the a transformed Painlev\'{e} property and the variable separated ODE method. The combined sinh--cosh--Gordon equation is also solved by Jaramillo--Camacho and his co-researchers ~\cite{n24} through the Hamiltonnian systems. Magalakwe et al. ~\cite{n25} studied a generalized double combined sinh--cosh--Gordon equation by employing Lie group method along with the simplest equation method to search new travelling wave solutions. Author also constructed the conservation laws of the above mentioned techniques. Kheiri and Jabbar ~\cite{n26} utilized $G^{'}/G$- expansion method to find out traveling wave solutions of the combined and double combined sinh--cosh--Gordon equations. Irshad and Mohyud--Din ~\cite{n27} studied some nonlinear differential equations including combined and double combined sinh--cosh--Gordon equations for deriving travelling wave solutions using tanh--coth method. Exact solutions of  a generalized double sine--Gordon equation  is also obtained by using different analytical techniques such as the tanh method and the variable separated ODE method through Painlev\'{e} property ~\cite{n28}, Jacobi Amplitude function ~\cite{n29} and exp--function method ~\cite{n30}.\\  
The main purpose of this study is to apply the modified Kudryashov method for solving combined and double combined sinh--cosh--Gordon equations and produce new exact solutions of the above mentioned equations. To this end, the work is organized as follows: In Section 2, we present the algorithm of the modified Kudryashov method. In Section 3, we derived the solutions of combined sinh--cosh--Gordon equation. Next, we extract the exact solution of the double combined sinh--cosh--Gordon equation in Section 4. Finally, Section 5 is devoted to concluding remarks about acquired results and executed method.

\section{Basic steps of modified Kudryashov method}
The general nonlinear  partial differential equation of two independent variables $x$ and $t$ can be written as 
\begin{equation} \label{1}
F\left(u,\frac{\partial u}{\partial t},\frac{\partial u}{\partial x},\frac{\partial^2 u}{\partial t^2},\frac{\partial^2 u}{\partial x^2},....\right)=0,
\end{equation}
where $F$ is a polynomial of $u(x,t)$ with various partial derivatives. Now we consider the following transformation \\
$$u(x,t)=f(\xi),\quad \xi=x-ct,$$ where $c$ is a non zero constant. After transformation the Eq. (\ref{1}) can be written as a non linear ordinary differential equation of the following form
\begin{equation} \label{2}
G(f,f^{'},f^{''},...)=0
\end{equation}
where the total derivative $^{'}$ with respect to the new variable $\xi$. Now consider the solution of the Eq. (\ref{2}) as 
\begin{equation} \label{3}
f(\xi)=\sum\limits_{i=0}^{m}a_{i}Q^{i}(\xi)
\end{equation} 
where $a_i,i=0,1,2,...,m$ are constants but $a_m\neq 0$ which will be calculated and $Q=Q(\xi)$ is the solution of equation  
$$\frac{dQ}{d\xi}=Q(\xi)(Q(\xi)-1)ln(a).$$
If we consider $$Q(\xi)=\frac{1}{1+da^{\xi}},$$ it will easily verify above equation. where $d$ is an arbitrary
constant and $a\neq 0,1$. We need to consider the balance 
of higher order derivative and nonlinear terms of Eq. (\ref{2}) for finding the value of $m$. Now using Eq. (\ref{3}) in Eq. (\ref{2}), we yield a new equation
\begin{equation} \label{4}
T(Q(\xi))=0.
\end{equation}
by equating like power of $Q(\xi)$ in Eq. (\ref{4}), we get a system of equation. Now solving the system of equation we get the values of $a_i$ and 
$c$. Inserting this values in Eq. (\ref{3}) which is the solution of Eq. (\ref{1}).
\section{Solution of the combined sinh--cosh--Gordon equation}
Consider the combined sinh--cosh--Gordon equation is as follows ~\cite{n20,n21,n22,n23,n24}: 
\begin{equation} \label{5}
u_{tt}-ku_{xx}+\alpha sinh(u)+\beta cosh(u)=0,
\end{equation}
where $u$ is the real scalar function of two independent variables $x$ and $t$. The subscript represent the partial derivatives, $\alpha$ and $\beta$ are two arbitrary non zero constants which also represent the model parameters. By implementing Painlev\'{e} property $u=ln(V)$ or $V=e^{u}$, then the $sinh$ and $cosh$ term transformed into the following way:
\begin{equation}  \label{6}
{sinh(u)}=\frac{V-V^{-1}}{2},\quad {cosh(u)}=\frac{V+V^{-1}}{2},
\end{equation}
Now using Eq. (\ref{5}) and Eq. (\ref{6}) along with $u=ln(V)$, we obtain
\begin{equation} \label{7}
2V(V_{tt}^{''}-kV_{xx}^{''})+2(k(V_{x}^{'})^{2}-(V_{t}^{'})^{2})+(\alpha +\beta)V^{3}+(\beta-\alpha)V=0
\end{equation}
Now let $V(x,t)=V(\xi)$, where $\xi=x-ct$, putting this in Eq. (\ref{7}) we get
\begin{equation} \label{8}
2V(c^2-k)V^{''}+2(k-c^2)(V^{'})^{2}+(\alpha+\beta)V^{3}+(\beta-\alpha)V=0
\end{equation}
By blanching the derivative $VV^{''}$ and $(V^{'})^2$, we get $m=2$. So, the finite series is of the form 
\begin{equation}  \label{9}
V(\xi)=a_0+a_1Q(\xi)+a_2Q(\xi)^2 
\end{equation}
putting Eq. (\ref{9})  with first and second  derivative in Eq. (\ref{8}) and equating the like powers of $Q(\xi)$, we the following system of equation:\\
$4 ln(a)^2c^2a_{2}^{2}-4 ln(a)^{2}ka_{2}^{2}+\alpha a_{2}^{3}+\beta a_2^3=0,$\\

$8 ln(a)^2c^2a_1a_2-4 ln(a)^2c^2a_2^2-8 ln(a)^2ka_1a_2+4 ln(a)^2ka_2^2\\+3\alpha a_1a_2^2+3\beta a_1a_2^2=0,$\\

$12 ln(a)^2c^2a_0a_2+2 ln(a)^2c^2a_1^2-10 in(a)^2c^2a_1a_2-12 ln(a)^2ka_0a_2\\-2 ln(a)^2ka_1^2+10 ln(a)^2ka_1a_2+3 \alpha a_0a_2^2+3 \alpha a_1^2a_2+3\beta a_0a_2^2+3\beta a_1^2a_2=0,$ \\

$4 ln(a)^2c^2a_0a_1-20 ln(a)^2c^2a_0a_2-2 ln(a)^2c^2a_1^2+2 ln(a)^2c^2a_1a_2\\-4 ln(a)^2ka_0a_1+20 in(a)^2ka_0a_2+2 ln(a)^2ka_1^2-2 ln(a)^2ka_1a_2+6 \alpha a_0a_1a_2+\alpha a_1^3+6 \beta a_0a_1a_2+\beta a_1^3=0,$\\

$-6 ln(a)^2c^2a_0a_1+8 ln(a)^2c^2a_0a_2+6 ln(a)^2ka_0a_1-8 ln(a)^2ka_0a_2 
+3 \alpha a_0^2a_2+3\alpha a_0a_1^2+3 \ beta a_0^2a_2+3 \beta a_0a_1^2-\alpha a_2+\beta a_2=0,$ \\

$2 ln(a)^2c^2a_0a_1-2 ln(a)^2ka_0a_1+3 \alpha a_0^2a_1+3 \beta a_0^2a_1-\alpha a_1+\beta a_1=0,$\\

$\alpha a_0^3+\beta a_0^3-\alpha a_0+\beta a_0=0.$

By solving the above systems, we obtain the following different cases:\\
\textbf{Case--I:} $$\quad a_0=\frac{\sqrt{(\alpha+\beta)(\alpha-\beta)}}{\alpha+\beta},\quad a_1=-\frac{4(\alpha-\beta)}{\sqrt{(\alpha+\beta)(\alpha-\beta)}}$$
$$\quad a_2=\frac{\sqrt{(\alpha+\beta)(\alpha-\beta)}}{\alpha+\beta},\quad c=\frac{\sqrt{ln(a)^2k-\sqrt{(\alpha+\beta)(\alpha-\beta)}}}{ln(a)}.$$
Case--I corresponds to the following exact solution of the combined sinh--cosh--Gordon equation is obtained: 
\begin{equation} \label{10}
\begin{split}
u_{1}(x,t)=\ln\left[\frac{\sqrt{(\alpha+\beta)(\alpha-\beta)}}{\alpha+\beta}-\frac{4(\alpha-\beta)}{\sqrt{(\alpha+\beta)(\alpha-\beta)}}\right.\\
\times \left(\frac{1}{1+da^{x-\frac{\sqrt{ln(a)^2k-\sqrt{(\alpha+\beta)(\alpha-\beta)}}}{ln(a)}t}}\right)
+\frac{\sqrt{(\alpha+\beta)(\alpha-\beta)}}{\alpha+\beta}\\
\times\left.\left(\frac{1}{1+da^{x-\frac{\sqrt{ln(a)^2k-\sqrt{(\alpha+\beta)(\alpha-\beta)}}}{ln(a)}t}}\right)^{2}\right].
\end{split}
\end{equation}

\begin{figure}
	\includegraphics [width=\linewidth]{fig11.eps}
	\caption{(a) 3D snapshot of the Eq. (\ref{10}) for the arbitrary parameters $a=3,\alpha=2,\beta=0.5, d=1$ and $k=-1.5$, and (b) 2D snapshot of (a) at $t=-1$, $t=0$, $t=1$ respectively.}
\end{figure}

\textbf{Case--II:} $$\quad a_0=\frac{\sqrt{(\alpha+\beta)(\alpha-\beta)}}{\alpha+\beta},\quad a_1=-\frac{4(\alpha-\beta)}{\sqrt{(\alpha+\beta)(\alpha-\beta)}}$$
$$\quad a_2=\frac{\sqrt{(\alpha+\beta)(\alpha-\beta)}}{\alpha+\beta},\quad c=-\frac{\sqrt{ln(a)^2k-\sqrt{(\alpha+\beta)(\alpha-\beta)}}}{ln(a)}.$$
Case--II corresponds to the following exact solution of the combined sinh--cosh--Gordon equation is determined: 
\begin{equation} \label{11}
\begin{split}
u_{2}(x,t)=\ln\left[\frac{\sqrt{(\alpha+\beta)(\alpha-\beta)}}{\alpha+\beta}-\frac{4(\alpha-\beta)}{\sqrt{(\alpha+\beta)(\alpha-\beta)}}\right.\\
\times\left(\frac{1}{1+da^{x+\frac{\sqrt{ln(a)^2k-\sqrt{(\alpha+\beta)(\alpha-\beta)}}}{ln(a)}t}}\right)+\frac{\sqrt{(\alpha+\beta)(\alpha-\beta)}}{\alpha+\beta}\\
\times\left.\left(\frac{1}{1+da^{x+\frac{\sqrt{ln(a)^2k-\sqrt{(\alpha+\beta)(\alpha-\beta)}}}{ln(a)}t}}\right)^{2}\right].
\end{split}
\end{equation}

\begin{figure}
	\includegraphics[width=\linewidth]{fig12.eps}
		\caption{(a) 3D snapshot of the Eq. (\ref{11}) for the arbitrary parameters $a=3,\alpha=2,\beta=0.5, d=1$ and $k=-1.5$, and (b) 2D snapshot of (a) at $t=-1$, $t=0$, $t=1$ respectively.}
\end{figure}

\textbf{Case--III:} $$\quad a_0=-\frac{\sqrt{(\alpha+\beta)(\alpha-\beta)}}{\alpha+\beta},\quad a_1=\frac{4(\alpha-\beta)}{\sqrt{(\alpha+\beta)(\alpha-\beta)}}$$
$$\quad a_2=-\frac{\sqrt{(\alpha+\beta)(\alpha-\beta)}}{\alpha+\beta},\quad c=\frac{\sqrt{ln(a)^2k+\sqrt{(\alpha+\beta)(\alpha-\beta)}}}{ln(a)}.$$
Case--III corresponds to the following exact solution of the combined sinh--cosh--Gordon equation is explored: 
\begin{equation} \label{12}
\begin{split}
u_{3}(x,t)=\ln\left[-\frac{\sqrt{(\alpha+\beta)(\alpha-\beta)}}{\alpha+\beta}+\frac{4(\alpha-\beta)}{\sqrt{(\alpha+\beta)(\alpha-\beta)}}\right.\\
\times\left(\frac{1}{1+da^{x-\frac{\sqrt{ln(a)^2k-\sqrt{(\alpha+\beta)(\alpha-\beta)}}}{ln(a)}t}}\right)-\frac{\sqrt{(\alpha+\beta)(\alpha-\beta)}}{\alpha+\beta}\\
\times\left.\left(\frac{1}{1+da^{x-\frac{\sqrt{ln(a)^2k+\sqrt{(\alpha+\beta)(\alpha-\beta)}}}{ln(a)}t}}\right)^{2}\right].
\end{split}
\end{equation}
 
\begin{figure}
	\includegraphics[width=\linewidth]{fig13.eps}
	\caption{(a) 3D snapshot of the Eq. (\ref{12}) for the arbitrary parameters $a=3,\alpha=2,\beta=0.5, d=1$ and $k=-1.5$, and (b) 2D snapshot of (a) at $t=-1$, $t=0$, $t=1$ respectively.}
\end{figure}

\textbf{Case--IV:} $$\quad a_0=-\frac{\sqrt{(\alpha+\beta)(\alpha-\beta)}}{\alpha+\beta},\quad a_1=\frac{4(\alpha-\beta)}{\sqrt{(\alpha+\beta)(\alpha-\beta)}}$$
$$\quad a_2=-\frac{\sqrt{(\alpha+\beta)(\alpha-\beta)}}{\alpha+\beta},\quad c=-\frac{\sqrt{ln(a)^2k+\sqrt{(\alpha+\beta)(\alpha-\beta)}}}{ln(a)}.$$\\
Case--IV corresponds to the following exact solution of the combined sinh--cosh--Gordon equation is found: 
\begin{equation} \label{13}
	\begin{split}
		u_{4}(x,t)=\ln\left[-\frac{\sqrt{(\alpha+\beta)(\alpha-\beta)}}{\alpha+\beta}+\frac{4(\alpha-\beta)}{\sqrt{(\alpha+\beta)(\alpha-\beta)}}\right.\\
		\times\left(\frac{1}{1+da^{x+\frac{\sqrt{ln(a)^2k-\sqrt{(\alpha+\beta)(\alpha-\beta)}}}{ln(a)}t}}\right)
		-\frac{\sqrt{(\alpha+\beta)(\alpha-\beta)}}{\alpha+\beta}\\
		\times\left.\left(\frac{1}{1+da^{x+\frac{\sqrt{ln(a)^2k+\sqrt{(\alpha+\beta)(\alpha-\beta)}}}{ln(a)}t}}\right)^{2}\right].
	\end{split}
\end{equation}

\begin{figure}
	\includegraphics[width=\linewidth]{fig14.eps}
	\caption{(a) 3D snapshot of the Eq. (\ref{13}) for the arbitrary parameters $a=3,\alpha=2,\beta=0.5, d=1$ and $k=-1.5$, and (b) 2D snapshot of (a) at $t=-1$, $t=0$, $t=1$ respectively.}
\end{figure}

Exact solutions of the combined sinh--cosh--Gordon equation is demonstrated by some MATLAB figures which indicating Figure 1 to Figure 4 by choosing free parameters as $a=3,\alpha=2,\beta=0.5, d=1$ and $k=-1.5$. 

\section{Solutions of combined double sinh--cosh--Gordon equation}
 Consider the double combined sinh--cosh--Gordon equation is as follows~\cite{n25,n26,n27}: 
\begin{equation} \label{14}
u_{tt}-ku_{xx}+\alpha sinh(u)+\alpha cosh(u)+\beta sinh(2u)+\beta cosh(2u)=0,
\end{equation}
where $u$ is the real scalar function of two independent variables $x$ and $t$. The subscript represent the partial derivatives, $\alpha$ and $\beta$ are two arbitrary non zero constants which also represent the model parameters. By implementing Painlev\'{e} property $u=ln(V)$ or $V=e^{u}$, then the $sinh$, $sin(2h)$, $cosh$ and $cos(2h)$ term transformed to the following way:
\begin{equation}  \label{15}
\left.
\begin{array}{c}
{sinh(u)}=\frac{V-V^{-1}}{2},\quad {cosh(u)}=\frac{V+V^{-1}}{2},\\ \\
{sinh(2u)}=\frac{V^2-V^{-2}}{2},\quad {cosh(2u)}=\frac{V^2+V^{-2}}{2} \\ 
\end{array}
\right\}
\end{equation}
Now using Eq. (\ref{14}) and Eq. (\ref{15}) along with $u=ln(V)$, we obtain
\begin{equation} \label{16}
V(V_{tt}^{''}-kV_{xx}^{''})+2(k(V_{x}^{'})^{2}-(V_{t}^{'})^{2})+(\alpha +\beta)V^{3}+\beta V^4=0
\end{equation}
Now let $V(x,t)=V(\xi)$, where $\xi=x-ct$, putting this in Eq. (\ref{16}) we get
\begin{equation} \label{17}
V(c^2-k)V^{''}+(k-c^2)(V^{'})^{2}+\alpha V^{3}+\beta V^4=0
\end{equation}
by blanching the derivative $VV^{''}$ and $V^4$, we get $m=1$. So, the finite series is the form 
\begin{equation}  \label{18}
V(\xi)=a_0+a_1Q(\xi) 
\end{equation}
putting Eq. (\ref{18})  with first and second  derivative in Eq. (\ref{17}) and equating the like powers of $Q(\xi)$, we get the following system of equation
\begin{eqnarray*}
2 ln(a)^2c^2a_1^2-2 ln(a)^2ka_1^2+2\beta a_1^4=0,\\
4 ln(a)^2c^2a_0a_1-2 ln(a)^2c^2a_1^2-4 ln(a)^2ka_0a_1+2 ln(a)^2ka_1^2\\+8\beta a_0a_1^3+2\alpha a_1^3=0,\\
-6 ln(a)^2c^2a_0a_1+6 ln(a)^2ka_0a_1+12\beta a_0^2a_1^2+6\alpha a_0a_1^2=0,\\
2 ln(a)^2c^2a_0a_1-2 ln(a)^2ka_0a_1+8\beta a_0^3a_1+6\alpha a_0^2a_1=0,\\
2\beta a_0^4+2\alpha a_0^3=0.
\end{eqnarray*}
\\
By solving the above systems, we received the following different solution sets:\\
\textbf{Case--I:} $$\quad a_0=0,\quad a_1=-\frac{\alpha}{\beta},\quad c=\frac{\sqrt{ ln(a)^2k\beta-\alpha^2}}{\sqrt\beta\ ln(a)}$$
Case--I corresponds to the following exact solution of the double combined sinh--cosh--Gordon equation is extracted:
\begin{equation} \label{19}
u_{1}(x,t)=\ln\left[-\frac{\alpha}{\beta}\left(\frac {1}{1+da^{x-\frac{\sqrt{ ln(a)^2k\beta-\alpha^2}}{\sqrt\beta\ ln(a)}t}}\right)\right]
\end{equation}

\begin{figure}
\includegraphics[width=\linewidth]{fig21.eps}
\caption{(a) 3D snapshot of the Eq. (\ref{19}) for the arbitrary parameters $a=3,\alpha=2,\beta=0.5, d=1$ and $k=-1.5$, and (b) 2D snapshot of (a) at $t=-1$, $t=0$, $t=1$ respectively.}
\end{figure}

\textbf{Case--II:} $$\quad a_0=0,\quad a_1=-\frac{\alpha}{\beta},\quad c=-\frac{\sqrt{ ln(a)^2k\beta-\alpha^2}}{\sqrt\beta\ ln(a)}$$
Case--II corresponds to the following exact solution of the double combined sinh--cosh--Gordon equation is generated:
\begin{equation} \label{20}
u_{2}(x,t)=\ln\left[-\frac{\alpha}{\beta}\left(\frac {1}{1+da^{x+\frac{\sqrt{ ln(a)^2k\beta-\alpha^2}}{\sqrt\beta\ ln(a)}t}}\right)\right]
\end{equation}

\begin{figure}
\includegraphics[width=\linewidth]{fig22.eps}
\caption{(a) 3D snapshot of the Eq. (\ref{20}) for the arbitrary parameters $a=3,\alpha=2,\beta=0.5, d=1$ and $k=-1.5$, and (b) 2D snapshot of (a) at $t=-1$, $t=0$, $t=1$ respectively.}
\end{figure}

\textbf{Case--III:} $$\quad a_0=-\frac{\alpha}{\beta},\quad a_1=\frac{\alpha}{\beta},\quad c=\frac{\sqrt{ ln(a)^2k\beta-\alpha^2}}{\sqrt\beta\ ln(a)}$$
Case--III corresponds to the following exact solution of the double combined sinh--cosh--Gordon equation is generated:
\begin{equation} \label{21}
u_{3}(x,t)=\ln\left[-\frac{\alpha}{\beta}+\frac{\alpha}{\beta}\left(\frac {1}{1+da^{x-\frac{\sqrt{ ln(a)^2k\beta-\alpha^2}}{\sqrt\beta\ ln(a)}t}}\right)\right]
\end{equation}

\begin{figure}
\includegraphics[width=\linewidth]{fig23.eps}
\caption{(a) 3D snapshot of the Eq. (\ref{21}) for the arbitrary parameters $a=3,\alpha=2,\beta=0.5, d=1$ and $k=-1.5$, and (b) 2D snapshot of (a) at $t=-1$, $t=0$, $t=1$ respectively.}
\end{figure}

\textbf{Case--IV:} $$\quad a_0=-\frac{\alpha}{\beta},\quad a_1=\frac{\alpha}{\beta},\quad c=-\frac{\sqrt{ ln(a)^2k\beta-\alpha^2}}{\sqrt\beta\ ln(a)}$$
Case--IV corresponds to the following exact solution of the double combined sinh--cosh--Gordon equation is produced:
\begin{equation} \label{22}
u_{4}(x,t)=\ln\left[-\frac{\alpha}{\beta}+\frac{\alpha}{\beta}\left(\frac {1}{1+da^{x+\frac{\sqrt{ ln(a)^2k\beta-\alpha^2}}{\sqrt\beta\ ln(a)}t}}\right)\right]
\end{equation}

\begin{figure}
\includegraphics[width=\linewidth]{fig24.eps}
\caption{(a) 3D snapshot of the Eq. (\ref{22}) for the arbitrary parameters $a=3,\alpha=2,\beta=0.5, d=1$ and $k=-1.5$, and (b) 2D snapshot of (a) at $t=-1$, $t=0$, $t=1$ respectively.}
\end{figure}
Exact solutions of the double combined sinh--cosh--Gordon equation is demonstrated by some MATLAB figures which indicating Figure 5 to Figure 8 by choosing free parameters as $a=3,\alpha=2,\beta=0.5, d=1$ and $k=-1.5$. 

\section{Conclusion} This paper produces exact solution of the combined sinh--cosh--Gordon and double combined sinh--cosh--Gordon equations via modified Kudryashov method. The obtained exact solution is also verified the original equation with the aid of maple package program. We claim that, the derived results are totally new and executed method gives very good consistency as compare to other method. Therefore, the method is robust and concise, and it can also be used to other kinds of nonlinear equations in mathematical physics and others applied field. 

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