Advertisement

Construction of \((n+1)\)-dimensional dual-mode nonlinear equations: multiple shock wave solutions for \((3+1)\)-dimensional dual-mode Gardner-type and KdV-type

  • Ali Jaradat
  • M. M. M. JaradatEmail author
  • Mohd Salmi Md Noorani
  • H. M. Jaradat
  • Marwan Alquran
Open Access
Research
  • 103 Downloads

Abstract

The goal of this study is to offer an exclusive functional conversion to produce \((n+1)\)-dimensional dual-mode nonlinear equations. This transformation has been implemented and new \((3+1)\)-dimensional dual-mode Gradner-type and KdV-type have been established. Finally, the simplified bilinear method is used to tell the necessary conditions on these new models to have multiple singular-solitons.

Keywords

\((3+1)\)-dimensional dual-mode Gradner-type \((3+1)\)-dimensional dual-mode KdV-type Simplified bilinear method Multiple singular-soliton shock wave solutions 

1 Introduction

The main upstream of understanding the physical nature of mathematical models arising in different disciplines of science is to extract their traveling wave solutions. Seeking for possible reliable solutions require suggesting and developing mathematical methods with supportive geometric analysis such as conservation laws and symmetry analysis [1, 2, 3, 4, 5, 6, 7, 8, 9, 10].

Traveling wave solutions have different types which give a complete understanding of the dynamics of a particular physical model. Solitons, kinks, and periodics are the most popular types that propagate as single-moving-waves as in KdV, mKdV, and Burgers’. But, in the case of Boussinesq equation, its traveling wave solutions propagate as dual-waves with interaction phase velocity.

The phenomenon of dual-waves has been adopted by Korsunsky and developed by Wazwaz [11, 12] when they considered the KdV equation of second order in time which reads
$$ \phi _{tt}-s^{2} \phi _{xx}+ \biggl( \frac{\partial }{\partial t}-\alpha s \frac{\partial }{\partial x} \biggr) \phi \phi _{x} + \biggl(\frac{ \partial }{\partial t}-\beta s \frac{\partial }{\partial x} \biggr) \phi _{xxx}=0, $$
(1.1)
where \(\phi =\phi (x,t)\) is a field function, s is the interaction phase velocity, α is the nonlinearity factor, and β is the dispersive factor with \(s\geq 0\), \(|\alpha | \leq 1\), \(|\beta | \leq 1\), and we refer to equation (1.1) as the two-mode KdV equation (TMKdV). The equation given in (1.1) was revisited by Alquran and Jarrah, and new Jacobi elliptic sine-cosine solutions were obtained [13].

Inspired by the form of TMKdV, many new two-mode or dual-mode models have been established. Two-mode Burgers equation (TMBE) and two-mode fifth-order KdV equations (TMFKdV) [14, 15], the two-mode higher-order Boussinesq–Burger system [16], two-mode coupled Burgers equation [17], two-mode coupled modified Korteweg–de Vries [18], two-mode coupled Korteweg–de Vries [19], two-mode Korteweg–de Vries–Burgers equation [20], the weak-dissipative two-mode perturbed Burgers and Ostrovsky models [21], two-mode Kuramoto–Sivashinsky [22], the dual-mode nonlinear Schrodinger’s equation and Kerr-law nonlinearity [23], the two-mode second- and third-order dispersive Fisher [24, 25] and the dual-mode Kadomtsev–Petviashvili model with strong-weak surface tension [26]. Single and multiple soliton/kink solutions have been obtained for the aforementioned models by using a simplified bilinear method, tanh method, sine-cosine method, Kudryashov method, and the \((G'/G)\)-expansion method.

The motivation of this work is to introduce for the first time a formulation of \((n+1)\)-dimensional dual-mode equations and to establish new \((3+1)\)-dimensional dual-mode equations of type Gardner and KdV. Also, we aim to find the necessary constraint conditions that enable such equations possess soliton solutions, singular soliton solutions, multiple soliton solutions, and multiple singular soliton solutions by using the simplified bilinear method.

The forms of single-mode \((3+1)\)-dimensional Gardner and KdV-type equations are, respectively, read as
$$ v_{t}+6lvv_{x}+v_{xxx}-\frac{3}{2}k^{2}v^{2}v_{x}+3h^{2} \partial _{x} ^{-1}v_{yy}-3khv_{x} \partial _{x}^{-1}v_{y}+3h^{2}\partial _{x}^{-1}v _{zz}-3khv_{x}\partial _{x}^{-1}v_{z}=0, $$
(1.2)
and
$$ v_{t}+6v_{x}v_{y}+v_{xxy}+v_{xxxxz}+60v_{x}^{2}v_{z}+10v_{xxx}v_{z}+20v _{x}v_{xxz}=0. $$
(1.3)
The above two equations are widely used in physics and its applications such as quantum field theory, plasma physics, and fluid physics. Also, different types of solutions have been obtained by using many methods such as Hirota’s direct method, the Casorati and Grammian determinant solutions, and the inverse scattering method [27, 28, 29, 30].

2 Formulation of \((n+1)\)-dimensional dual-mode equations

Wazwaz and Korsunsky [11, 12, 14, 15] established the \((1+1)\)-dimensional two-mode equation in a scaled form as
$$ v_{tt}-c^{2}v_{xx}+\biggl(\frac{\partial }{\partial t}-cb \frac{\partial }{ \partial x}\biggr)L(v_{mx})+\biggl(\frac{\partial }{\partial t}-cd \frac{\partial }{\partial x}\biggr)N(v,v_{x},\ldots)=0, $$
(2.1)
where \(m\geq 2\), \(L(v_{kx})\) is a linear term, \(N(v,v_{x},\ldots)\) is a nonlinear term, \(c>0\) is the phase velocities, \(x \in (-\infty , \infty )\), \(t>0\), \(\vert b \vert \leq 1\), and \(\vert d \vert \leq 1\).
In this study we propose a new scale for the \((n+1)\)-dimensional dual-mode equations in the variables \(t,x_{1},x_{2},x_{3},\ldots ,x _{n} \). The new scale is suggested to have the following form:
$$ 0=v_{tt}-\sum_{i=1}^{n}c^{2}v_{x_{i}x_{i}}+ \Biggl(\frac{\partial }{\partial t}-\sum_{i=1}^{n}ca_{i} \frac{\partial }{\partial x_{i}}\Biggr)L+\Biggl(\frac{ \partial }{\partial t}-\sum _{i=1}^{n}cb_{i}\frac{\partial }{\partial x_{i}}\Biggr)N, $$
(2.2)
where L and N are, respectively, linear and nonlinear, \(\vert a_{i} \vert \leq 1\), and \(\vert b_{i} \vert \leq 1\), \(i=1,2,\ldots,n\). Note that when \(c=0\) and integrating once with respect to t, (2.2) is reduced to the standard single-mode \((n+1)\)-dimensional equation.

3 Analysis of the method

In this section, we give a brief description of the simplified bilinear method to find N-soliton solutions for nonlinear partial differential equations (NPDEs) as follows:

First, we substitute
$$ v(x_{1},x_{2},\ldots,x_{n},t)=e^{\omega _{i}(x_{1},x_{2},\ldots,x_{n},t)}, $$
where
$$ \omega _{i}(x_{1},x_{2},\ldots,x_{n},t)= \sum_{j=1}^{n}l_{j_{i}}x _{j}-\lambda _{i}t,\quad i=1,2,3,\ldots,N, $$
in the problem under consideration, to find the relation among \(l_{j_{i}}\) and \(\lambda _{i}\). To find the soliton solutions, we use an appropriate transformation formula. We often use one of the following formulas:
$$\begin{aligned}& v(x_{1},x_{2},\ldots,x_{n},t) =R\ln f(x_{1},x_{2},\ldots,x_{n},t) \\& v(x_{1},x_{2},\ldots,x_{n},t) =R\bigl(\ln f(x_{1},x_{2},\ldots,x_{n},t)\bigr)_{x _{i}}, \\& v(x_{1},x_{2},\ldots,x_{n},t) =R\bigl(\ln f(x_{1},x_{2},\ldots,x_{n},t)\bigr)_{x _{i}x_{j}}, \\& \text{where } i,j \in \{ 1,2,\ldots,N \} ,R\in \mathbb{R}. \end{aligned}$$
For one-soliton solutions, we use the auxiliary function
$$ f(x_{1},x_{2},\ldots,x_{n},t)=1+c_{1}e^{\omega _{i}(x_{1},x_{2},\ldots,x_{n},t)}, \quad c _{1}=\pm 1. $$
For two-soliton solutions, we use the auxiliary function
$$ f(x_{1},x_{2},\ldots,x_{n},t)=1+c_{1}e^{\omega _{1}}+c_{2}e^{\omega _{2} }+c _{1}c_{2}\nu _{12}e^{\omega _{1}+\omega _{2}},\quad c_{1}=c_{2}=\pm 1. $$
For three-soliton solutions, we use the auxiliary function
$$\begin{aligned}& \begin{aligned} f(x_{1},x_{2},\ldots,x_{n},t) & =1+c_{1}e^{\omega _{1}}+c_{2}e^{\omega _{2}}+c_{3}e^{\omega _{3} }+c_{1}c_{2} \nu _{12}e^{\omega _{1}+\omega _{2}} \\ &\quad {} +c_{1}c_{3}\nu _{13}e^{\omega _{1}+\omega _{3}}+c_{2}c_{3} \nu _{23}e ^{\omega _{2}+\omega _{3}}+c_{1}c_{2}c_{3} \nu _{123}e^{\omega _{1}+\omega _{2}+ \omega _{3}}, \end{aligned} \\& c_{1} =c_{2}=c_{3}=\pm 1, \end{aligned}$$
provided that three-soliton solutions exist if \(\nu _{123}=\nu _{12}\nu _{13}\nu _{23}\). Moreover, for any nonlinear PDEs that have three-soliton solutions, they also have N-soliton solutions for \(N\geq 4\).

4 Applications

The purposes of this section is to apply the above described method to solve new \((3+1)\)-dimensional dual-mode nonlinear PDEs.

4.1 Soliton solutions for \((3+1)\)-dimensional dual-mode Gardner equation

Applying the suggested prescribed scale (2.2) on (1.2), the \((3+1)\)-dimensional dual-mode Gardner equation will have the following form:
$$\begin{aligned} 0 & =v_{tt}-c^{2}v_{xx}-c^{2}v_{yy}-c^{2}v_{zz} \\ &\quad {}+ \biggl(\frac{\partial }{ \partial t}-ca_{1}\frac{\partial }{\partial x}-ca_{2} \frac{\partial }{ \partial y}-ca_{3}\frac{\partial }{\partial z}\biggr) \bigl\{ v_{xxx}+3h ^{2}\partial _{x}^{-1}v_{yy}+3h^{2} \partial _{x}^{-1}v_{zz} \bigr\} \\ &\quad{} +\biggl(\frac{\partial }{\partial t}-cb_{1}\frac{\partial }{\partial x}-cb _{2} \frac{\partial }{\partial y}-cb_{3}\frac{\partial }{\partial z}\biggr) \\ &\quad {}\times\biggl\{ 6lvv_{x}-\frac{3}{2}k^{2}v^{2}v_{x}-3khv_{x} \partial _{x}^{-1}v _{y}-3khv_{x} \partial _{x}^{-1}v_{z} \biggr\} . \end{aligned}$$
(4.1)
We aim to find the needed necessary conditions in order to obtain multiple soliton and multiple singular-soliton solutions by using a simplified bilinear method [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. To drop the presence of the operator \(\partial _{x}^{-1}\), we use the transformation
$$ v(x,y,z,t)=w_{x}(x,y,z,t). $$
Accordingly, a new equivalent version of \((3+1)\)-TMGE (4.1) is given by
$$\begin{aligned} 0 & =w_{xtt}-c^{2}w_{xxx}-c^{2}w_{xyy}-c^{2}w_{xzz}+w_{xxxxt}-ca_{1}w _{xxxxx}-ca_{2}w_{xxxxy}-ca_{3}w_{xxxxz} \\ &\quad{} +3h^{2}w_{yyt}-3h^{2}ca_{1}w_{yyx}-3h^{2}ca_{2}w_{yyy}-3h^{2}ca _{3}w_{yyz} \\ &\quad{} +3h^{2}w_{zzt}-3h^{2}ca_{1}w_{zzx}-3h^{2}ca_{2}w_{zzy}-3h^{2}ca _{3}w_{zzz} \\ &\quad{} +6l(w_{x}w_{xx})_{t}-6lcb_{1}(w_{x}w_{xx})_{x}-6lcb_{2}(w_{x}w_{xx})_{y}-6lcb _{3}(w_{x}w_{xx})_{z} \\ & \quad{}-\frac{3}{2}k^{2}\bigl(w_{x}^{2}w_{xx} \bigr)_{t}+\frac{3}{2}k^{2}cb_{1}\bigl(w _{x}^{2}w_{xx}\bigr)_{x}+ \frac{3}{2}k^{2}cb_{2}\bigl(w_{x}^{2}w_{xx} \bigr)_{y}+ \frac{3}{2}k^{2}cb_{3} \bigl(w_{x}^{2}w_{xx}\bigr)_{z} \\ & \quad{}-3kh(w_{xx}w_{y})_{t}+3khcb_{1}(w_{xx}w_{y})_{x}+3khcb_{2}(w_{xx}w _{y})_{y}+3khcb_{3}(w_{xx}w_{y})_{z} \\ & \quad{}-3kh(w_{xx}w_{z})_{t}+3khcb_{1}(w_{xx}w_{z})_{x}+3khcb_{2}(w_{xx}w _{z})_{y}+3khcb_{3}(w_{xx}w_{z})_{z}. \end{aligned}$$
(4.2)
Inserting
$$ w(x,y,z,t)=e^{\omega _{i}(x,y,z,t)} $$
with
$$ \omega _{i}(x,y,z,t)=\alpha _{i}x+\beta _{i}y+ \zeta _{i}z-\gamma _{i}t,\quad i=1,2,3,\ldots,N, $$
into the linear terms of (4.2), we get the dispersion relations
$$ \gamma _{i}=\frac{-(\alpha _{i}^{4}+3h^{2}\beta _{i}^{2}+3h^{2}\zeta _{i} ^{2})\pm \sqrt{(\alpha _{i}^{4}+3h^{2}\beta _{i}^{2}+3h^{2}\zeta _{i} ^{2})^{2}+4\alpha _{i}\Delta _{i}}}{2\alpha _{i}}, $$
(4.3)
where
$$\begin{aligned} \Delta _{i} & =\bigl(c^{2}\alpha _{i}^{3}+ca_{1} \alpha _{i}^{5}+ca_{2}\alpha _{i}^{4} \beta _{i}+ca_{3}\alpha ^{4}\zeta _{i}+c^{2}\beta _{i}^{2}\alpha _{i} \\ &\quad {}+3h^{2}ca_{1}\beta _{i}^{2} \alpha _{i}+3h^{2}ca_{2}\beta _{i}^{3}+3h ^{2}ca_{3}\beta _{i}^{2}\zeta _{i} \\ &\quad{} +c^{2}\alpha _{i}\zeta _{i}^{2}+3h^{2}ca_{1} \zeta _{i}^{2}\alpha _{i}+3h ^{2}ca_{2} \zeta _{i}^{2}\beta _{i}+3h^{2}ca_{3} \zeta _{i}^{3}\bigr). \end{aligned}$$
(4.4)
Now, we consider the Cole–Hopf transformation
$$ v(x,y,t)=R\bigl(\ln f(x,y,z,t)\bigr)_{x} $$
(4.5)
which leads to
$$ w(x,y,t)=R\ln f(x,y,z,t) $$
(4.6)
provided that R is a constant and \(f(x,y,z,t)\) is an auxiliary function. For the one-soliton solution, we consider
$$ f(x,y,z,t)=1+c_{1}e^{\alpha _{1}x+\beta _{1}y+\zeta _{1}z-\frac{-(\alpha _{1}^{4}+3k^{2}\beta _{1}^{2}+3h^{2}\zeta _{1}^{2})\pm \sqrt{(\alpha _{1}^{4}+3k^{2}\beta _{1}^{2}+3h^{2}\zeta _{1}^{2})^{2}+4\alpha _{1} \Delta _{1}}}{2\alpha _{1}}t}, $$
(4.7)
where \(c_{1}=\pm 1\). Following (4.3), inserting (4.6) and (4.7) into (4.2) and solving for R, the one soliton solution of (4.2) exists if
$$\begin{aligned}& R =\frac{2}{k}, \\& a_{1} =a_{2}=a_{3}=b_{1}=b_{2}=b_{3}, \\& \beta _{1} =\frac{2\alpha _{1}l-k\alpha _{1}^{2}-hk\zeta _{1}}{hk}. \end{aligned}$$
(4.8)
By the Cole–Hopf transformation (4.6), we conclude the one-soliton solution of (4.2) as
$$ w(x,y,z,t)=\frac{2}{k}\ln \bigl( 1+c_{1}e^{\alpha _{1}x+\frac{2\alpha _{1}l-k \alpha _{1}^{2}-hk\zeta _{1}}{hk}y+\zeta _{1}z-\frac{-(\alpha _{1}^{4}+3k ^{2}\beta _{1}^{2}+3h^{2}\zeta _{1}^{2})\pm \sqrt{(\alpha _{1}^{4}+3k ^{2}\beta _{1}^{2}+3h^{2}\zeta _{1}^{2})^{2}+4\alpha _{1}\Delta _{1}}}{2 \alpha _{1}}t} \bigr), $$
and then,
$$ v(x,y,z,t)=\frac{2}{k}\frac{\alpha _{1}c_{1}e^{{^{\alpha _{1}x+\frac{2 \alpha _{1}l-k\alpha _{1}^{2}-hk\zeta _{1}}{hk}y+\zeta _{1}z-\frac{-( \alpha _{1}^{4}+3k^{2}\beta _{1}^{2}+3h^{2}\zeta _{1}^{2})\pm \sqrt{( \alpha _{1}^{4}+3k^{2}\beta _{1}^{2}+3h^{2}\zeta _{1}^{2})^{2}+4\alpha _{1}\Delta _{1}}}{2\alpha _{1}}t}}}}{1+c_{1}e^{\alpha _{1}x+\frac{2\alpha _{1}l-k\alpha _{1}^{2}-hk\zeta _{1}}{hk}y+\zeta _{1}z-\frac{-(\alpha _{1} ^{4}+3k^{2}\beta _{1}^{2}+3h^{2}\zeta _{1}^{2})\pm \sqrt{(\alpha _{1} ^{4}+3k^{2}\beta _{1}^{2}+3h^{2}\zeta _{1}^{2})^{2}+4\alpha _{1}\Delta _{1}}}{2\alpha _{1}}t}}. $$
In the case \(c_{1}=1\), we get the single-soliton solution as follows:
$$\begin{aligned} v(x,y,z,t) & =\frac{2}{k}\frac{\alpha _{1}e^{\alpha _{1}x+\frac{2\alpha _{1}l-k \alpha _{1}^{2}-hk\zeta _{1}}{hk}y+\zeta _{1}z-\frac{-(\alpha _{1}^{4}+3k ^{2}\beta _{1}^{2}+3h^{2}\zeta _{1}^{2})\pm \sqrt{(\alpha _{1}^{4}+3k ^{2}\beta _{1}^{2}+3h^{2}\zeta _{1}^{2})^{2}+4\alpha _{1}\Delta _{1}}}{2 \alpha _{1}}t}}{1+e^{\alpha _{1}x+\frac{2\alpha _{1}l-k\alpha _{1}^{2}-hk \zeta _{1}}{hk}y+\zeta _{1}z-\frac{-(\alpha _{1}^{4}+3k^{2}\beta _{1}^{2}+3h ^{2}\zeta _{1}^{2})\pm \sqrt{(\alpha _{1}^{4}+3k^{2}\beta _{1}^{2}+3h ^{2}\zeta _{1}^{2})^{2}+4\alpha _{1}\Delta _{1}}}{2\alpha _{1}}t}} \\ & =\frac{\alpha _{1}}{k} \biggl[ 1+\tanh \biggl( \frac{\omega _{1}(x,y,z,t)}{2}\biggr) \biggr]. \end{aligned}$$
For the case \(c_{1}=-1\), we get the singular single-soliton solution as follows:
$$\begin{aligned} v(x,y,z,t) & =\frac{2}{k}\frac{\alpha _{1}e^{\alpha _{1}x+\frac{2\alpha _{1}l-k \alpha _{1}^{2}-hk\zeta _{1}}{hk}y+\zeta _{1}z-\frac{-(\alpha _{1}^{4}+3k ^{2}\beta _{1}^{2}+3h^{2}\zeta _{1}^{2})\pm \sqrt{(\alpha _{1}^{4}+3k ^{2}\beta _{1}^{2}+3h^{2}\zeta _{1}^{2})^{2}+4\alpha _{1}\Delta _{1}}}{2 \alpha _{1}}t}}{-1+e^{\alpha _{1}x+\frac{2\alpha _{1}l-k\alpha _{1}^{2}-hk \zeta _{1}}{hk}y+\zeta _{1}z-\frac{-(\alpha _{1}^{4}+3k^{2}\beta _{1}^{2}+3h ^{2}\zeta _{1}^{2})\pm \sqrt{(\alpha _{1}^{4}+3k^{2}\beta _{1}^{2}+3h ^{2}\zeta _{1}^{2})^{2}+4\alpha _{1}\Delta _{1}}}{2\alpha _{1}}t}} \\ & =\frac{\alpha _{1}}{k} \biggl[ 1+\coth \biggl( \frac{\omega _{1}(x,y,z,t)}{2}\biggr) \biggr], \end{aligned}$$
where
$$\begin{aligned} \omega _{1}(x,y,z,t) & =\alpha _{1}x+\frac{2\alpha _{1}l-k\alpha _{1}^{2}-hk \zeta _{1}}{hk}y+ \zeta _{1}z \\ & \quad{}-\frac{-(\alpha _{1}^{4}+3h^{2}\beta _{1}^{2}+3h^{2}\zeta _{1}^{2}) \pm \sqrt{(\alpha _{1}^{4}+3h^{2}\beta _{1}^{2}+3h^{2}\zeta _{1}^{2})^{2}+4 \alpha _{1}\Delta _{1}}}{2\alpha _{1}}t. \end{aligned}$$
For the two-soliton solutions, we set the auxiliary function
$$ f(x,y,z,t)=1+c_{1}e^{\omega _{1}(x,y,z,t) }+c_{2}e^{\omega _{2}(x,y,z,t)t }+c_{1}c_{2} \nu _{12}e^{\omega _{1}(x,y,z,t)+\omega _{2}(x,y,z,t)}, $$
(4.9)
where \(c_{i}=\pm 1\) and \(i=1,2\). Inserting (4.6), (4.8), and (4.9) in (4.2), the two-soliton solution of (4.2) exists if
$$\begin{aligned}& a_{1} =a_{2}=a_{3}=b_{2}=b_{2}=b_{3}= \pm 1, \end{aligned}$$
(4.10)
$$\begin{aligned}& \beta _{1} =\frac{2\alpha _{1}l-k\alpha _{1}^{2}-hk\zeta _{1}}{hk}, \end{aligned}$$
(4.11)
$$\begin{aligned}& \beta _{2} =\frac{2\alpha _{2}l-k\alpha _{2}^{2}-hk\zeta _{2}}{hk}, \end{aligned}$$
(4.12)
$$\begin{aligned}& \zeta _{1} =a\alpha _{1},\qquad \zeta _{2}=a\alpha _{2}, \end{aligned}$$
(4.13)
$$\begin{aligned}& \nu _{12} =0, \end{aligned}$$
(4.14)
where a is any real number.
Combining (4.9)–(4.14) and (4.6), the two-soliton solution is
$$\begin{aligned} w(x,y,z,t) & =\frac{2}{k}\ln \bigl( 1+c_{1}e^{\alpha _{1}x+\frac{2 \alpha _{1}l-k\alpha _{1}^{2}-ahk\alpha _{1}}{hk}y+a\alpha _{1}z-\frac{-( \alpha _{1}^{4}+3k^{2}\beta _{1}^{2}+3h^{2}\zeta _{1}^{2})\pm \sqrt{( \alpha _{1}^{4}+3k^{2}\beta _{1}^{2}+3h^{2}\zeta _{1}^{2})^{2}+4\alpha _{1}\Delta _{1}}}{2\alpha _{1}}t } \\ & \quad {} +c_{2}e^{\alpha _{2}x+\frac{2\alpha _{2}l-k\alpha _{2}^{2}-ahk \alpha _{2}}{hk}y+a\alpha _{2}z-\frac{-(\alpha _{2}^{4}+3k^{2}\beta _{2} ^{2}+3h^{2}\zeta _{2}^{2})\pm \sqrt{(\alpha _{2}^{4}+3k^{2}\beta _{2} ^{2}+3h^{2}\zeta _{2}^{2})^{2}+4\alpha _{2}\Delta _{2}}}{2\alpha _{2}}t } \bigr), \end{aligned}$$
and thus,
$$ v(x,y,z,t)=\frac{2}{k}\frac{\alpha _{1}c_{1}e^{\omega _{1}(x,y,z,t) }+ \alpha _{2}c_{2}e^{\omega _{2}(x,y,z,t) }}{1+c_{1}e^{\omega _{1}(x,y,z,t) }+c_{2}e^{\omega _{2}(x,y,z,t) }}. $$
(4.15)
By setting \(c_{1}=c_{2}=1\) in (4.15), the two-soliton solution is
$$ v(x,y,z,t)=\frac{2}{k}\frac{\alpha _{1}e^{\omega _{1}(x,y,z,t) }+\alpha _{2}e^{\omega _{2}(x,y,z,t) }}{1+e^{\omega _{1}(x,y,z,t) }+e^{\omega _{2}(x,y,z,t) }}, $$
(4.16)
and the singular two-soliton solution by substituting \(c_{1}=c_{2}=-1\) is
$$ v(x,y,z,t)=\frac{2}{k}\frac{\alpha _{1}e^{\omega _{1}(x,y,z,t) }+\alpha _{2}e^{\omega _{2}(x,y,z,t) }}{-1+e^{\omega _{1}(x,y,z,t) }+e^{\omega _{2}(x,y,z,t) }}. $$
(4.17)
For the three-soliton solution, we use the auxiliary function
$$ f(x,y,z,t)=1+c_{1}e^{\omega _{1}(x,y,z,t) }+c_{2}e^{\omega _{2}(x,y,z,t)}+c _{3}e^{\omega _{3}(x,y,z,t) }, $$
(4.18)
where \(c_{i}=\pm 1\), \(i=1,2,3\). Accordingly, the three-soliton solution is
$$\begin{aligned} w(x,y,z,t) & =\frac{2}{k}\ln \bigl( 1+c_{1}e^{\alpha _{1}x+\frac{2 \alpha _{1}l-k\alpha _{1}^{2}-ahk\alpha _{1}}{hk}y+a\alpha _{1}z-\frac{-( \alpha _{1}^{4}+3k^{2}\beta _{1}^{2}+3h^{2}\zeta _{1}^{2})\pm \sqrt{( \alpha _{1}^{4}+3k^{2}\beta _{1}^{2}+3h^{2}\zeta _{1}^{2})^{2}+4\alpha _{1}\Delta _{1}}}{2\alpha _{1}}t } \\ & \quad {} +c_{2}e^{\alpha _{2}x+\frac{2\alpha _{2}l-k\alpha _{2}^{2}-ahk \alpha _{2}}{hk}y+a\alpha _{2}z-\frac{-(\alpha _{2}^{4}+3k^{2}\beta _{2} ^{2}+3h^{2}\zeta _{2}^{2})\pm \sqrt{(\alpha _{2}^{4}+3k^{2}\beta _{2} ^{2}+3h^{2}\zeta _{2}^{2})^{2}+4\alpha _{2}\Delta _{2}}}{2\alpha _{2}}t } \\ & \quad {} +c_{3}e^{\alpha _{3}x+\frac{2\alpha _{3}l-k\alpha _{3}^{2}-ahk \alpha _{3}}{hk}y+a\alpha _{3}z-\frac{-(\alpha _{3}^{4}+3k^{2}\beta _{3} ^{2}+3h^{2}\zeta _{3}^{2})\pm \sqrt{(\alpha _{3}^{4}+3k^{2}\beta _{3} ^{2}+3h^{2}\zeta _{3}^{2})^{2}+4\alpha _{3}\Delta _{3}}}{2\alpha _{3}}t } \bigr) \end{aligned}$$
and then,
$$ v(x,y,z,t)=\frac{2}{k}\frac{\alpha _{1}c_{1}e^{\omega _{1}(x,y,z,t) }+ \alpha _{2}c_{2}e^{\omega _{2}(x,y,z,t) }+\alpha _{3}c_{3}e^{\omega _{3}(x,y,z,t)}}{1+c _{1}e^{\omega _{1}(x,y,z,t) }+c_{2}e^{\omega _{2}(x,y,z,t) }+c_{3}e ^{\omega _{3}(x,y,z,t) }}. $$
(4.19)
By setting \(c_{i}=1\) for \(i=1,2,3\), we obtain the three-soliton solution
$$ v(x,y,z,t)=\frac{2}{k}\frac{\alpha _{1}e^{\omega _{1}(x,y,z,t) }+\alpha _{2}e ^{\omega _{2}(x,y,z,t) }+\alpha _{3}e^{\omega _{3}(x,y,z,t) }}{1+e^{ \omega _{1}(x,y,z,t)}+e^{\omega _{2}(x,y,z,t) }+e^{\omega _{3}(x,y,z,t) }}. $$
(4.20)
Setting \(c_{i}=-1\) for \(i=1,2,3\), the singular three-soliton solution is
$$ v(x,y,t)=\frac{2}{k}\frac{\alpha _{1}e^{\omega _{1}(x,y,z,t) }+\alpha _{2}e ^{\omega _{2}(x,y,z,t) }+\alpha _{3}e^{\omega _{3}(x,y,z,t) }}{-1+e ^{\omega _{1}(x,y,z,t)}+e^{\omega _{2}(x,y,z,t) }+e^{\omega _{3}(x,y,z,t)}}. $$
(4.21)

Remark

For a finite N, where \(N\geq 4\) and under the conditions \(a_{1}=a_{2}=a_{3}=b_{1}=b_{2}=b_{3}=\pm 1\) and \(\beta _{i}=\frac{2\alpha _{i}l-k\alpha _{i}^{2}-hk\zeta _{i}}{hk}\), \(\zeta _{i}=a\alpha _{i}\), \(i=1,2,\ldots,N\), (4.2) has N-soliton solutions and singular N-soliton solutions given by [31, 32]
$$ v(x,y,z,t)=\frac{2}{k}\frac{{ \sum_{i=1}^{N}} \alpha _{i}e^{\alpha _{i}x+\frac{2\alpha _{i}l-k \alpha _{i}^{2}-ahk\alpha _{i}}{hk}y+a\alpha _{i}z-\frac{-(\alpha _{i} ^{4}+3h^{2}\beta _{i}^{2}+3h^{2}\zeta _{i}^{2})\pm \sqrt{(\alpha _{i} ^{4}+3h^{2}\beta _{i}^{2}+3h^{2}\zeta _{i}^{2})^{2}+4\alpha _{i}\Delta _{i}}}{2\alpha _{i}}t }}{1+{ \sum_{i=1}^{N}} e^{\alpha _{i}x+\frac{2\alpha _{i}l-k\alpha _{i} ^{2}-ahk\alpha _{i}}{hk}y+a\alpha _{i}z-\frac{-(\alpha _{i}^{4}+3h^{2} \beta _{i}^{2}+3h^{2}\zeta _{i}^{2})\pm \sqrt{(\alpha _{i}^{4}+3h^{2} \beta _{i}^{2}+3h^{2}\zeta _{i}^{2})^{2}+4\alpha _{i}\Delta _{i}}}{2\alpha _{i}}t }}, $$
and
$$ v(x,y,z,t)=\frac{2}{k}\frac{{ \sum_{i=1}^{N}} \alpha _{i}e^{\alpha _{i}x+\frac{2\alpha _{i}l-k \alpha _{i}^{2}-ahk\alpha _{i}}{hk}y+a\alpha _{i}z-\frac{-(\alpha _{i} ^{4}+3h^{2}\beta _{i}^{2}+3h^{2}\zeta _{i}^{2})\pm \sqrt{(\alpha _{i} ^{4}+3h^{2}\beta _{i}^{2}+3h^{2}\zeta _{i}^{2})^{2}+4\alpha _{i}\Delta _{i}}}{2\alpha _{i}}t }}{-1+{ \sum_{i=1}^{N}} e^{\alpha _{i}x+\frac{2\alpha _{i}l-k\alpha _{i} ^{2}-ahk\alpha _{i}}{hk}y+a\alpha _{i}z-\frac{-(\alpha _{i}^{4}+3h^{2} \beta _{i}^{2}+3h^{2}\zeta _{i}^{2})\pm \sqrt{(\alpha _{i}^{4}+3h^{2} \beta _{i}^{2}+3h^{2}\zeta _{i}^{2})^{2}+4\alpha _{i}\Delta _{i}}}{2\alpha _{i}}t }}. $$

4.2 Soliton solutions for \((3+1)\)-dimensional dual-mode KdV-type

Applying the new formulation (2.2), the new \((3+1)\)-dimensional dual-mode KdV equation (\((3+1)\)-TMKdV) has the following form:
$$\begin{aligned} 0 & =v_{tt}-c^{2}v_{xx}-c^{2}v_{yy}-c^{2}v_{zz}+v_{xxyt}-ca_{1}v_{xxyx}-ca _{2}v_{xxyy}-ca_{3}v_{xxyz} \\ &\quad{} +v_{xxxxzt}-ca_{1}v_{xxxxzx}-ca_{2}v_{xxxxzy}-ca_{3}v_{xxxxzz} \\ &\quad{} +6(v_{x}v_{y})_{t}-6cb_{1}(v_{x}v_{y})_{x}-6cb_{2}(v_{x}v_{y})_{y}-6cb _{3}(v_{x}v_{y})_{z} \\ &\quad{} +60\bigl(v_{x}^{2}v_{z}\bigr)_{t}-60cb_{1} \bigl(v_{x}^{2}v_{z}\bigr)_{x}-60cb_{2} \bigl(v _{x}^{2}v_{z}\bigr)_{y}-60cb_{3} \bigl(v_{x}^{2}v_{z}\bigr)_{z} \\ &\quad{} +10(v_{xxx}v_{z})_{t}-10cb_{1}(v_{xxx}v_{z})_{x}-10cb_{2}(v_{xxx}v _{z})_{y}-10cb_{3}(v_{xxx}v_{z})_{z} \\ &\quad{} +20(v_{x}v_{xxz})_{t}-20cb_{1}(v_{x}v_{xxz})_{x}-20cb_{2}(v_{x}v _{xxz})_{y}-20cb_{3}(v_{x}v_{xxz})_{z}. \end{aligned}$$
(4.22)
Inserting
$$ w(x,y,z,t)=e^{\omega _{i}(x,y,z,t)} $$
with
$$ \omega _{i}(x,y,z,t)=\alpha _{i}x+\beta _{i}y+ \zeta _{i}z-\gamma _{i}t,\quad i=1,2,3,\ldots,N, $$
into the linear terms of (4.22), we get the dispersion relations
$$ \gamma _{i}=\frac{(\alpha _{i}^{2}\beta _{i}+\alpha _{i}^{4}\zeta _{i}) \pm \sqrt{(\alpha _{i}^{2}\beta _{i}+\alpha _{i}^{4}\zeta _{i})^{2}+4 \Delta _{i}}}{2}, $$
(4.23)
where
$$ \begin{aligned} \Delta _{i}&=\bigl(c^{2}\alpha _{i}^{2}+c^{2} \beta _{i}^{2}+c^{2}\zeta _{i}^{2}+ca _{1}\alpha _{i}^{3}\beta _{i}+ca_{2} \alpha _{i}^{2}\beta _{i}^{2}\\ &\quad {}+ca_{3} \alpha _{i}^{2}\beta _{i}\zeta _{i}+ca_{1}\alpha _{i}^{5}\zeta _{i}+ca_{2} \alpha _{i}^{4}\beta _{i}\zeta _{i}+ca_{3}\alpha _{i}^{4} \zeta _{i}^{2}\bigr). \end{aligned} $$
Now, we consider the Cole–Hopf transformation
$$ v(x,y,t)=R\bigl(\ln f(x,y,z,t)\bigr)_{x} $$
(4.24)
and the auxiliary function
$$ f(x,y,z,t)=1+c_{1}e^{\alpha _{1}x+\beta _{1}y+\zeta _{1}z-\frac{(\alpha _{i}^{2}\beta _{i}+\alpha _{i}^{4}\zeta _{i})\pm \sqrt{(\alpha _{i}^{2} \beta _{i}+\alpha _{i}^{4}\zeta _{i})^{2}+4\Delta _{i}}}{2}t}. $$
(4.25)
Using (4.23), inserting (4.24) and (4.25) into the \((3+1)\)-TMGE (4.22) and solving for R, the one-soliton solution of (4.22) exists if
$$\begin{aligned}& R =1, \\& a_{1} =a_{2}=a_{3}=b_{2}=b_{2}=b_{3}. \end{aligned}$$
Therefore, the one-soliton solution of the \((3+1)\)-TMGE is
$$ v(x,y,z,t)=\frac{\alpha _{1}c_{1}e^{\omega _{1}/2}}{1+c_{1}e^{\omega _{1}/2}}. $$
In case \(c_{1}=1\), we get the single-soliton solution
$$\begin{aligned} v(x,y,z,t) & =\frac{\alpha e^{\omega _{1}/2}}{1+e^{\omega _{1}/2}} \\ & =\alpha _{1} \biggl[ 1+\tanh \biggl(\frac{\omega _{1}(x,y,z,t)}{4}\biggr) \biggr], \end{aligned}$$
and for the case \(c_{1}=-1\), we get the singular single-soliton solution
$$\begin{aligned} v(x,y,z,t) & =\frac{\alpha _{1}e^{\omega _{1}/2}}{-1+e^{\omega _{1}/2}} \\ & =\alpha _{1} \biggl[ 1+\coth \biggl(\frac{\omega _{1}(x,y,z,t)}{4}\biggr) \biggr]. \end{aligned}$$
For the two-soliton solutions, we use the auxiliary function
$$ f(x,y,z,t)=1+c_{1}e^{\omega _{1}(x,y,z,t) }+c_{2}e^{\omega _{2}(x,y,z,t)t }+c_{1}c_{2} \nu _{12}e^{\omega _{1}(x,y,z,t)+\omega _{2}(x,y,z,t)}, $$
(4.26)
where \(c_{i}=\pm 1\) and \(i=1,2\). Inserting (4.23), (4.24), and (4.26) in (4.22), the two-soliton solution of (4.22) exists if
$$\begin{aligned}& R =1, \\& a_{1} =a_{2}=a_{3}=b_{2}=b_{2}=b_{3}= \pm 1, \\& v_{12} =\frac{(\alpha _{1}-\alpha _{2}) ( ( \alpha _{1} ^{2}\beta _{1}+2\alpha _{1}\alpha _{2}\beta _{1}-2\alpha _{1}\alpha _{2} \beta _{2}-\alpha _{2}^{2}\beta _{1} ) +\theta _{1} ) }{\theta _{2}}\quad \text{where,} \\& \theta _{1} = \bigl[ \alpha _{1}^{4}\zeta _{2}+ ( 2\alpha _{1} \alpha _{2}\zeta _{1}-2\alpha _{1}\alpha _{2}\zeta _{2} ) \bigl( 2 \alpha _{1}^{2}-3\alpha _{1}\alpha _{2}+2\alpha _{2}^{2} \bigr) -\alpha _{2}^{4}\zeta _{1} \bigr], \\& \theta _{2} =(\alpha _{1}+\alpha _{2}) \bigl[ \bigl( \alpha _{1}^{2} \beta _{2}+2\alpha _{1}\alpha _{2}\beta _{1}+2\alpha _{1} \alpha _{2}\beta _{2}+\alpha _{2}^{2}\beta _{1} \bigr) +\theta _{3} \bigr], \\& \theta _{3} = \bigl[ \alpha _{1}^{4}\zeta _{1}+ ( 2\alpha _{1} \alpha _{2}\zeta _{1}+2\alpha _{1}\alpha _{2}\zeta _{2} ) \bigl( 2 \alpha _{1}^{2}+3\alpha _{1}\alpha _{2}+2\alpha _{2}^{2} \bigr) +\alpha _{1}^{4}\zeta _{1} \bigr]. \end{aligned}$$
Therefore, the two-soliton solution of the \((3+1)\)-TMKDVE is given by
$$ v(x,y,z,t)=\frac{\alpha _{1}c_{1}e^{\omega _{1}/2}+\alpha _{2}c_{1}e^{ \omega _{1}/2}+\frac{c_{1}c_{2}v_{12}e^{(\omega _{1}+\omega _{2})/2}}{( \alpha _{1}+\alpha _{2})}}{1+\alpha _{1}c_{1}e^{\omega _{1}/2}+\alpha _{2}c _{1}e^{\omega _{1}/2}+c_{1}c_{2}v_{12}e^{\omega _{1}}}. $$
In case \(c_{1}=1\), we get the two-soliton solution
$$ v(x,y,z,t)=\frac{\alpha _{1}e^{\omega _{1}/2}+\alpha _{2}e^{\omega _{1}/2}+\frac{v _{12}e^{(\omega _{1}+\omega _{2})/2}}{(\alpha _{1}+\alpha _{2})}}{1+e^{ \omega _{1}/2}+e^{\omega _{2}/2}+v_{12}e^{(\omega _{1}+\omega _{2})/2}}, $$
and for the case \(c_{1}=-1\), we get the singular two-soliton solution
$$ v(x,y,z,t)=\frac{-\alpha _{1}e^{\omega _{1}}-\alpha _{2}e^{\omega _{1}}+\frac{v _{12}e^{(\omega _{1}+\omega _{2})/2}}{(\alpha _{1}+\alpha _{2})}}{1-\alpha _{1}e^{\omega _{1}/2}-\alpha _{2}e^{\omega _{2}/2}+v_{12}e^{(\omega _{1}+ \omega _{2})/2}}. $$
We should remark here that we cannot find three or more soliton solutions for (4.22) because this type of KdV equation is not integrable.

5 Conclusions

In this work, we proposed a functional conversion that produces \((n+1)\)-dimensional dual-mode equations of the form
$$ 0=v_{tt}-\sum_{i=1}^{n}c^{2}v_{x_{i}x_{i}}+ \Biggl(\frac{\partial }{\partial t}-\sum_{i=1}^{n}cb_{i} \frac{\partial }{\partial x_{i}}\Biggr)L+\Biggl(\frac{ \partial }{\partial t}-\sum _{i=1}^{n}cd_{i}\frac{\partial }{\partial x_{i}}\Biggr)N. $$
These types of equations describe the spreading of dual-waves moving simultaneously with interaction phase velocity. The simplified bilinear method with the aid of some Cole–Hopf transformations is used to study \((3+1)\)-dimensional dual-mode Gardner-type and KdV-type. We concluded the following results:
  • Kink solutions and singular kink solutions for \((3+1)\)-TMGE exist only if \(a_{1}=a_{2}=a_{3}=b_{1}=b_{2} =b_{3}\) and \(\beta _{1}=\frac{2 \alpha _{1}l-k\alpha _{1}^{2}-hk\zeta _{1}}{hk}\), while the N-soliton and singular N-soliton solutions exist only if \(a_{1}=a_{2}=a_{3}=b_{1}=b _{2} =b_{3}=\pm 1\), \(\zeta _{i}=a\alpha _{i} \) and \(\beta _{i}=\frac{2 \alpha _{i}l-k\alpha _{i}^{2}-hk\zeta _{i}}{hk}\), \(i=1,2,\ldots, N\).

  • One-soliton solutions for \((3+1)\)-TMKdV exist only if \(a_{1}=a_{2}=a _{3}=b_{2}=b_{2}=b_{3}\) and two-soliton solutions exist if \(a_{1}=a _{2}=a_{3}=b_{2}=b_{2}=b_{3}=\pm 1\). This equation is non-integrable, it possesses no k-soliton solutions for \(k=3, 4, \ldots \) .

As future work, we may consider a fractional version of dual-mode equations and conduct the same analysis as that used in [42, 43, 44, 45, 46, 47].

Notes

Acknowledgements

Authors would like to thank the editor and the anonymous referees for their in-depth reading and insightful comments on an earlier version of this paper.

Availability of data and materials

Not applicable.

Authors’ contributions

All authors contributed equally and read and approved the final version of the manuscript.

Funding

This work is financially supported by UKM Grant: DIP-2017-011 and Ministry of Education Malaysia Grant FRGS/1/2017/STG06/UKM/01/1.

Ethics approval and consent to participate

Not applicable.

Competing interests

The authors declare that there is no conflict of interests regarding the publication of this manuscript. The authors declare that they have no competing interests.

Consent for publication

Not applicable.

References

  1. 1.
    Abdel-Gawad, H.I., Tantawy, M., Inc, M., Yusuf, A.: On multi-fusion solitons induced by inelastic collision for quasi-periodic propagation with nonlinear refractive index and stability analysis. Mod. Phys. Lett. B 32(29), 1850353 (2018) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Yusuf, A., Inc, M., Aliyu, A.I., Baleanu, D.: Conservation laws, soliton-like and stability analysis for the time fractional dispersive long-wave equation. Adv. Differ. Equ. 2018, 319 (2018) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Baleanu, D., Inc, M., Yusuf, A., Aliyu, A.I.: Lie symmetry analysis and conservation laws for the time fractional simplified modified Kawahara equation. Open Phys. 16, 302–310 (2018) zbMATHCrossRefGoogle Scholar
  4. 4.
    Baleanu, D., Inc, M., Yusuf, A., Aliyu, A.I.: Optimal system, nonlinear self-adjointness and conservation laws for generalized shallow water wave equation. Open Phys. 16, 364–370 (2018) zbMATHCrossRefGoogle Scholar
  5. 5.
    Inc, M., Yusuf, A., Aliyu, A.I., Baleanu, D.: Lie symmetry analysis and explicit solutions for the time fractional generalized Burgers–Huxley equation. Opt. Quantum Electron. 50(2), 94 (2018) zbMATHCrossRefGoogle Scholar
  6. 6.
    Inc, M., Aliyu, A.I., Yusuf, A., Baleanu, D.: Optical solitons for complex Ginzburg–Landau model in nonlinear optics. Optik 158, 368–375 (2018) CrossRefGoogle Scholar
  7. 7.
    Baleanu, D., Inc, M., Yusuf, A., Aliyu, A.I.: Traveling wave solutions and conservation laws for nonlinear evolution equation. J. Math. Phys. 59(2), 023506 (2018) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Inc, M., Aliyu, A.I., Yusuf, A., Baleanu, D.: Optical solitons for Biswas–Milovic model in nonlinear optics by Sine–Gordon equation method. Optik 157, 267–274 (2018) CrossRefGoogle Scholar
  9. 9.
    Inc, M., Yusuf, A., Aliyu, A.I., Hashemi, M.S.: Soliton solutions, stability analysis and conservation laws for the Brusselator reaction diffusion model with time- and constant-dependent coefficients. Eur. Phys. J. Plus 133(5), 168 (2018) CrossRefGoogle Scholar
  10. 10.
    Inc, M., Yusuf, A., Aliyu, A.I., Baleanu, D.: Soliton solutions and stability analysis for some conformable nonlinear partial differential equations in mathematical physics. Opt. Quantum Electron. 50(4), 190 (2018) CrossRefGoogle Scholar
  11. 11.
    Korsunsky, S.V.: Soliton solutions for a second-order KdV equation. Phys. Lett. A 185, 174–176 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Wazwaz, A.M.: Multiple soliton solutions and other exact solutions for a two-mode KdV equation. Math. Methods Appl. Sci. 40(6), 1277–1283 (2017) MathSciNetGoogle Scholar
  13. 13.
    Alquran, M., Jarrah, A.: Jacobi elliptic function solutions for a two-mode KdV equation. J. King Saud Univ., Sci. (2017).  https://doi.org/10.1016/j.jksus.2017.06.010 CrossRefGoogle Scholar
  14. 14.
    Wazwaz, A.M.: A two-mode Burgers equation of weak shock waves in a fluid: multiple kink solutions and other exact solutions. Int. J. Appl. Comput. Math. 3(4), 3977–3985 (2017) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Wazwaz, A.M.: Two-mode fifth-order KdV equations: necessary conditions for multiple-soliton solutions to exist. Nonlinear Dyn. 87(3), 1685–1891 (2017) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Jaradat, A., Noorani, M.S.M., Alquran, A., Jaradat, H.M.: Construction and solitary wave solutions of two-mode higher-order Boussinesq–Burger system. Adv. Differ. Equ. 2017, 376 (2017) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Jaradat, H.M.: Two-mode coupled Burgers equation: multiple-kink solutions and other exact solutions. Alex. Eng. J. 57(3), 2151–2155 (2018) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Syam, M., Jaradat, H.M., Alquran, M.: A study on the two-mode coupled modified Korteweg–de Vries using the simplified bilinear and the trigonometric-function methods. Nonlinear Dyn. 90(2), 1363–1371 (2017) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Jaradat, H.M., Syam, M., Alquran, M.: A two-mode coupled Korteweg–de Vries: multiple-soliton solutions and other exact solutions. Nonlinear Dyn. 90(1), 371–377 (2017) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Alquran, M., Jaradat, H.M., Syam, M.: A modified approach for a reliable study of new nonlinear equation: two-mode Korteweg–de Vries–Burgers equation. Nonlinear Dyn. 91(3), 1619–1626 (2018) CrossRefGoogle Scholar
  21. 21.
    Jaradat, I., Alquran, M., Ali, M.: A numerical study on weak-dissipative two-mode perturbed Burgers’ and Ostrovsky models: right-left moving waves. Eur. Phys. J. Plus 133, 164 (2018) CrossRefGoogle Scholar
  22. 22.
    Jaradat, H.M., Alquran, M., Syam, M.: A reliable study of new nonlinear equation: two-mode Kuramoto–Sivashinsky. Int. J. Appl. Comput. Math. 4, 64 (2018) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Jaradat, I., Alquran, M., Momani, S., Biswas, A.: Dark and singular optical solutions with dual-mode nonlinear Schrodinger’s equation and Kerr-law nonlinearity. Optik 172, 822–825 (2018) CrossRefGoogle Scholar
  24. 24.
    Alquran, M., Yassin, O.: Dynamism of two-mode’s parameters on the field function for third-order dispersive Fisher: application for fibre optics. Opt. Quantum Electron. 50(9), 354 (2018) CrossRefGoogle Scholar
  25. 25.
    Yassin, O., Alquran, M.: Constructing new solutions for some types of two-mode nonlinear equations. Appl. Math. Inf. Sci. 12(2), 361–367 (2018) MathSciNetCrossRefGoogle Scholar
  26. 26.
    Abu Irwaq, I., Alquran, M., Jaradat, I., Baleanu, D.: New dual-mode Kadomtsev–Petviashvili model with strong-weak surface tension: analysis and application. Adv. Differ. Equ. 2018, 433 (2018) MathSciNetCrossRefGoogle Scholar
  27. 27.
    Konopelchenko, B.G.: Inverse spectral transform for the \((2+1)\)-dimensional Gardner equation. Inverse Probl. 7, 739–753 (1991) MathSciNetzbMATHGoogle Scholar
  28. 28.
    Yu, G.F., Tam, H.W.: On the \((2+1)\)-dimensional Gardner equation: determinant solutions and pfaffianization. J. Math. Anal. Appl. 330, 989–1001 (2007) MathSciNetzbMATHGoogle Scholar
  29. 29.
    Wazwaz, A.M.: Multiple kink solutions for the \((2+1)\)-dimensional integrable Gardner equation. Proc. Rom. Acad., Ser. A: Math. Phys. Tech. Sci. Inf. Sci. 15, 241–246 (2014) MathSciNetGoogle Scholar
  30. 30.
    Alomari, A.K., Awawdeh, F., Tahat, N., Bani Ahmad, F., Shatanaw, W.: Multiple solutions for fractional differential equations: analytic approach. Appl. Math. Comput. 219(17), 8893–8903 (2013) MathSciNetzbMATHGoogle Scholar
  31. 31.
    Hirota, R.: Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971) zbMATHCrossRefGoogle Scholar
  32. 32.
    Hirota, R.: Exact solution of the modified Korteweg–de Vries equation for multiple collisions of solitons. J. Phys. Soc. Jpn. 33, 1456–1458 (1972) CrossRefGoogle Scholar
  33. 33.
    Wazwaz, A.M.: Multiple kink solutions and multiple singular kink solutions for two systems of coupled Burgers’ type equations. Commun. Nonlinear Sci. Numer. Simul. 14, 2962–2970 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Wazwaz, A.M.: Kinks and travelling wave solutions for Burgers-like equations. Appl. Math. Lett. 38, 174–179 (2014) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Wazwaz, A.M.: Gaussian solitary wave solutions for nonlinear evolution equations with logarithmic nonlinearities. Nonlinear Dyn. 83, 591–596 (2016) MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Hirota, R.: Exact N-soliton solutions of a nonlinear wave equation. J. Math. Phys. 14, 805–809 (1973) MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Jaradat, H.M., Awawdeh, F., Al-Shara’, S., Alquran, M., Momani, S.: Controllable dynamical behaviors and the analysis of fractal Burgers hierarchy with the full effects of inhomogeneities of media. Rom. J. Phys. 60(3–4), 324–343 (2015) Google Scholar
  38. 38.
    Awawdeh, F., Al-Shara’, S., Jaradat, H.M., Alomari, A.K., Alshorman, R.: Symbolic computation on soliton solutions for variable coefficient quantum Zakharov–Kuznetsov equation in magnetized dense plasmas. Int. J. Nonlinear Sci. Numer. Simul. 15(1), 35–45 (2014) MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Jaradat, H.M.: New solitary wave and multiple soliton solutions for the time-space fractional Boussinesq equation. Ital. J. Pure Appl. Math. 36, 367–376 (2016) MathSciNetzbMATHGoogle Scholar
  40. 40.
    Jaradat, H.M.: Dynamic behavior of traveling wave solutions for a class for the time-space coupled fractional kdV system with time-dependent coefficients. Ital. J. Pure Appl. Math. 36, 945–958 (2016) MathSciNetzbMATHGoogle Scholar
  41. 41.
    Alquran, M., Jaradat, H.M., Al-Shara’, S., Awawdeh, F.: A new simplified bilinear method for the N-soliton solutions for a generalized FmKdV equation with time-dependent variable coefficients. Int. J. Nonlinear Sci. Numer. Simul. 16, 259–269 (2015) MathSciNetzbMATHGoogle Scholar
  42. 42.
    Ullah, A., Shah, K.: Numerical analysis of Lane Emden–Fowler equations. J. Taibah Univ. Sci. 12(2), 180–185 (2018) CrossRefGoogle Scholar
  43. 43.
    Khan, T., Shah, K., Khan, A., Khan, R.A.: Solution of fractional order heat equation via triple Laplace transform in two dimensions. Math. Methods Appl. Sci. 41(2), 818–825 (2018) MathSciNetzbMATHGoogle Scholar
  44. 44.
    Khan, H., Khan, A., Chen, W., Shah, K.: Stability analysis and a numerical scheme for fractional Klein Gordon equations. Math. Methods Appl. Sci. 42(2), 723–732 (2019) CrossRefGoogle Scholar
  45. 45.
    Ali, S., Bushnaq, S., Shah, K., Arif, M.: Numerical treatment of fractional order Cauchy reaction diffusion equations. Chaos Solitons Fractals 103, 578–587 (2017) MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Alquran, M., Jaradat, I.: A novel scheme for solving Caputo time-fractional nonlinear equations: theory and application. Nonlinear Dyn. 91(4), 2389–2395 (2018) zbMATHCrossRefGoogle Scholar
  47. 47.
    Jaradat, I., Al-Dolat, M., Al-Zoubi, K., Alquran, M.: Theory and applications of a more general form for fractional power series expansion. Chaos Solitons Fractals 108, 107–110 (2018) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Ali Jaradat
    • 1
  • M. M. M. Jaradat
    • 2
    Email author
  • Mohd Salmi Md Noorani
    • 1
  • H. M. Jaradat
    • 3
  • Marwan Alquran
    • 4
  1. 1.School of Mathematical SciencesUniversity Kebangsaan MalaysiaBangiMalaysia
  2. 2.Department of Mathematics, Statistics and PhysicsQatar UniversityDohaQatar
  3. 3.Department of MathematicsAl al-Bayt UniversityMafraqJordan
  4. 4.Department of Mathematics and StatisticsJordan University of Science and TechnologyIrbidJordan

Personalised recommendations