Structure of solitary wave solutions of the nonlinear complex fractional generalized Zakharov dynamical system

Lu, Dianchen; Seadawy, Aly R.; Khater, Mostafa M. A.

doi:10.1186/s13662-018-1734-4

Structure of solitary wave solutions of the nonlinear complex fractional generalized Zakharov dynamical system

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Published: 03 August 2018

Volume 2018, article number 266, (2018)
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Structure of solitary wave solutions of the nonlinear complex fractional generalized Zakharov dynamical system

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Abstract

The analytical and solitary traveling solutions of the nonlinear complex fractional generalized Zakharov equations are investigated. The nonlinear complex fractional generalized Zakharov equations describe the interaction between dispersive and non-dispersive waves in one dimension. Analytical and solitary traveling wave solutions were obtained through applying a generalized Kudryashov and a novel $(\frac{G'}{G})$-expansion methods. Novel solutions were the results of our investigated model, which illustrated the effectiveness and the power of the obtained methods in regards to accuracy, power, and effectiveness compared to the previously used methods.

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1 Introduction

The nonlinear complex fractional generalized-Zakharov system characterizes the proliferation of Langmuir waves in the ionized plasma. This model was discovered by Irving Langmuir and Lewi Tonks in the 1920s. Langmuir waves are known as plasma oscillations, they represent the fast oscillation of the electron density in conducting media. This oscillation can describe the instability in the dielectric function of free electron gas. While the ionization is defined as an atom which could be positively or negatively charged through acquiring or losing an electron or more, and that is always synchronized with other chemical shifts.

The nonlinear model of nonlinear complex fractional generalized-Zakharov system is very important and has many applications related to this phenomenon. Long time ago, researchers were trying to discover new physical features of the nonlinear model. Vladimir Zakharov in 1972 was the first who introduced this system. Then, further research was done to continue this trend, and the results obtained were analytical and solitary traveling wave solutions. In 1996, Malfliet et al. used the tanh method [1]; in 2009, Borhanifar et al. used the Exp-function method [2]; in 2003, Li et al. used the improved tanh method and symbolic computation [3]; in 2004, Yong et al. used the Riccati equation method [4]; in 2012, Naher Hasibun et al. used the Exp-function method [5]; in 2007, Abdou used the extended tanh method [6]; in 2015, Tuluce Demiray et al. used the extended trial equation method [7]; in 2007, Zhang et al. used a direct algebraic method [8, 9]; in 2008 Wazwaz and Abdul-Majid used the extended tanh method [10]; in 2009, Zhang et al. used the new generalized algebraic method [11]; in 2005, Wazwaz and Abdul-Majid used the sine-cosine algorithm [12]; in 2011, Kabir et al. used the modified Kudryashov method [13]; in 2009, Layeni used the rational auxiliary equation method [14]; in 2010, Betchewe et al. used [15], and so on [16–20].

From the above discussed studies, lots of methods were derived to solve the nonlinear partial differential equation models as follows: the $( \frac{G'}{G} ) $-expansion method extended the $( \frac{G'}{G} ) $-expansion method, the tanh-function method extended a simple equation method, the modified simple equation method extended the Jacobian method, a novel $( \frac{G'}{G} ) $-expansion method, and so on [21–55].

This paper is regulated as follows: In Sect. 2, we apply a generalized Kudryashov method and a novel $( \frac{G'}{G} ) $-expansion method to get the exact and solitary traveling wave solutions of the nonlinear complex fractional generalized-Zakharov equations. In Sect. 3, conclusion is presented and discussed.

2 Formulation for the nonlinear complex fractional generalized-Zakharov equations

Consider the nonlinear complex fractional generalized-Zakharov equations in the form [21–23]:

$$ \textstyle\begin{cases} i \frac{D^{\alpha } P}{D t^{\alpha }}+ P_{x x}-2 \delta \vert P \vert ^{2} P+2 P Q=0, \\ \frac{D^{2 \alpha } Q}{D t^{2 \alpha }}-Q_{x x}+( \vert P \vert ^{2})_{x x}=0. \end{cases} $$

(1)

Using the following traveling wave transformation $[P(x,t)=Y(x,t) e ^{i \phi } , \phi =k x+\frac{c t^{\alpha }}{\alpha } , Y(x,t)=Y( \xi ) , Q(x,t)=Q(\xi ) , \xi =x-\frac{2 k t^{\alpha }}{\alpha } ]$ on system (1), we obtain

$$ \textstyle\begin{cases} Y''-(c+k^{2}) Y - 2 \delta Y^{3} + 2 Y Q=0, \\ (4 k^{2}-1) Q'' + (Y^{2})''=0. \end{cases} $$

(2)

Integrating twice the second equation in system (2), where the constant of integration equals zero, we get $[ Q(\xi )=\frac{Y ^{2}}{1-4 k^{2}} ] $. Substituting $Q(\xi )$ into the first equation of system (2), we obtain

$$ Y''-\bigl(c+k^{2}\bigr) Y+ \biggl(\frac{2}{1-4 k^{2}}-2 \delta \biggr) Y^{3}=0. $$

(3)

Balancing Eq. (3) by using the relation between the highest order derivative term and the nonlinear term, we get $Y'' \& Y^{3}\Rightarrow N+2=3N\Rightarrow N=1$.

2.1 Exact and solitary wave solution of the nonlinear complex fractional generalized-Zakharov equations by using a generalized Kudryashov method

We use a generalized Kudryashov method on the nonlinear complex fractional generalized-Zakharov equations that suppose the general solution of it in the following form:

$$ Y(\xi )={\frac{a_{{0}}+a_{{1}} E( \xi ) +a_{{2}} E^{2}( \xi ) }{b _{{0}}+b_{{1}} E(\xi ) }}. $$

(4)

Substituting Eq. (4) and its derivatives into Eq. (3) and equating the coefficient of different power of $E^{i}( \xi )$ to zero, we obtain a system of algebraic equations by solving it with any computer program like (Maple, Mathematica and so on). We get the following:

$$\begin{aligned}& k=\sqrt{1-c},\quad\quad a_{0}=0, \quad\quad a_{1}=-b_{1} \sqrt{ \frac{4 c-3}{4 c \delta -3 \delta -1}}, \quad\quad a_{2}=b_{1} \sqrt{\frac{4 c-3}{4 c \delta -3 \delta -1}},\\& b_{0}=-\frac{b_{1}}{2}, \end{aligned}$$

where $[ \frac{4 c-3}{4 c \delta -3 \delta -1}>0, c\neq 1 ] $. Consequently, the solitary wave solution for Eqs. (1) is as follows:

$$\begin{aligned}& P(x,t)=\frac{2 \sqrt{{\frac{4 c-3}{-1+ ( 4 c-3 ) \delta }}} \alpha {\mathrm{e}^{{\frac{x\alpha -2 k{t}^{\alpha }}{ \alpha }}}}}{{\alpha }^{2} {\mathrm{e}^{{\frac{-4 k{t}^{\alpha }+2 x \alpha }{\alpha }}}}-1} e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned}$$

(5)

$$\begin{aligned}& Q(x,t)=\frac{ ( -4 c+3 ) {\alpha }^{2}{\mathrm{e}^{{\frac{-4 k{t}^{\alpha }+2 x\alpha }{\alpha }}}}}{ ( 4 c\delta -3 \delta -1 ) ( {\mathrm{e}^{{\frac{-4 k{t}^{\alpha }+2 x \alpha }{\alpha }}}}{\alpha }^{2}-1 ) ^{2} ( -1+c ) }. \end{aligned}$$

(6)

2.2 Exact and solitary wave solution of the nonlinear complex fractional generalized-Zakharov equations by using novel $(\frac{G'}{G})$-expansion method

We use a novel $(\frac{G'}{G})$-expansion method on the nonlinear complex fractional generalized-Zakharov equations that suppose the general solution of it in the following form:

$$ Y(\xi )=\frac{a_{-1}}{(d+\frac{G'}{G})}+a_{0}+a_{1} \biggl(d+\frac{G'}{G}\biggr). $$

(7)

Substituting Eq. (7) and its derivatives into Eq. (3) and equating the coefficient of different power of $(d+\frac{G'}{G})^{i}$ to zero, we obtain a system of algebraic equations by solving it with any computer program like (Maple, Mathematica, Matlap and so on). We get the following:

Family 1

$$\begin{aligned}& \mu =\frac{1}{4} {\frac{2 {k}^{2}+{\lambda }^{2}+2 c}{v-1}}, \quad\quad a _{{-1}}=0, \\& a_{{0}}=\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{ { \frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}, \quad\quad a_{{1}}=- {\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1} { \frac{1}{ \sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}. \end{aligned}$$

Consequently, the solitary wave solution for Eqs. (1) is as follows:

• When $(\Omega =\lambda^{2}-4 \lambda \mu +4 \mu >0)$ and $(\lambda (v-1)\neq 0)$ or $(\mu (v-1)\neq 0)$:

$$\begin{aligned}& \begin{aligned}[b] P_{11}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}- \delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}- \delta +1}{\frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}- \delta +1}}}}}} \\ & \quad {} \times \biggl( d-\frac{1}{2(v-1)} \biggl( \lambda +\sqrt{\Omega } \tanh \biggl( \frac{\sqrt{\Omega }}{2} \biggl( c-\frac{2 k t^{\alpha }}{ \alpha } \biggr) \biggr) \biggr) \biggr) \biggr] e^{i (k x+\frac{c t^{ \alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(8)

$$\begin{aligned}& \begin{aligned}[b] Q_{11}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d-\frac{1}{2(v-1)} \biggl( \lambda +\sqrt{\Omega } \tanh \biggl( \frac{\sqrt{\Omega }}{2} \biggl( c-\frac{2 k t^{\alpha }}{ \alpha } \biggr) \biggr) \biggr) \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(9)

$$\begin{aligned}& \begin{aligned}[b] P_{12}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}- \delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}- \delta +1}{\frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}- \delta +1}}}}}} \\ & \quad {} \times \biggl( d-\frac{1}{2(v-1)} \biggl( \lambda +\sqrt{\Omega } \coth \biggl( \frac{\sqrt{\Omega }}{2} \biggl( c-\frac{2 k t^{\alpha }}{ \alpha } \biggr) \biggr) \biggr) \biggr) \biggr] e^{i (k x+\frac{c t^{ \alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(10)

$$\begin{aligned}& \begin{aligned}[b] Q_{12}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d-\frac{1}{2(v-1)} \biggl( \lambda +\sqrt{\Omega } \coth \biggl( \frac{\sqrt{\Omega }}{2} \biggl( c-\frac{2 k t^{\alpha }}{ \alpha } \biggr) \biggr) \biggr) \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(11)

$$\begin{aligned}& \begin{aligned}[b] P_{13}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}- \delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}- \delta +1}{\frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}- \delta +1}}}}}} \\ & \quad {} \times \biggl( d-\frac{1}{2(v-1)} \biggl( \lambda +\sqrt{\Omega } \biggl( \tanh \biggl( \sqrt{\Omega } \biggl( c-\frac{2 k t^{\alpha }}{ \alpha } \biggr) \biggr) \\ & \quad {} \pm i \operatorname {sech}\biggl( \sqrt{\Omega } \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr) \biggr] e ^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(12)

$$\begin{aligned}& \begin{aligned}[b] Q_{13}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d-\frac{1}{2(v-1)} \biggl( \lambda +\sqrt{\Omega } \biggl( \tanh \biggl( \sqrt{\Omega } \biggl( c-\frac{2 k t^{\alpha }}{ \alpha } \biggr) \biggr) \\ & \quad {}\pm i \operatorname {sech}\biggl( \sqrt{\Omega } \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(13)

$$\begin{aligned}& \begin{aligned}[b] P_{14}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}- \delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}- \delta +1}{\frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}- \delta +1}}}}}} \\ & \quad {} \times \biggl( d-\frac{1}{2(v-1)} \biggl( \lambda +\sqrt{\Omega } \biggl( \coth \biggl( \sqrt{\Omega } \biggl( c-\frac{2 k t^{\alpha }}{ \alpha } \biggr) \biggr) \\ & \quad {}\pm \operatorname {csch}\biggl( \sqrt{\Omega } \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr) \biggr] e ^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(14)

$$\begin{aligned}& \begin{aligned}[b] Q_{14}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d-\frac{1}{2(v-1)} \biggl( \lambda +\sqrt{\Omega } \biggl( \coth \biggl( \sqrt{\Omega } \biggl( c-\frac{2 k t^{\alpha }}{ \alpha } \biggr) \biggr) \\ & \quad {}\pm \operatorname {csch}\biggl( \sqrt{\Omega } \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(15)

$$\begin{aligned}& \begin{aligned}[b] P_{15}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}- \delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}- \delta +1}{\frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}- \delta +1}}}}}} \\ & \quad {} \times \biggl( d-\frac{1}{4(v-1)} \biggl( 2 \lambda +\sqrt{\Omega } \biggl( \tanh \biggl( \frac{\sqrt{\Omega }}{4} \biggl( c-\frac{2 k t ^{\alpha }}{\alpha } \biggr) \biggr) \\ & \quad {}\pm \coth \biggl( \frac{\sqrt{ \Omega }}{4} \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr) \biggr] e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(16)

$$\begin{aligned}& \begin{aligned}[b] Q_{15}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d-\frac{1}{4(v-1)} \biggl( 2 \lambda +\sqrt{\Omega } \biggl( \tanh \biggl( \frac{\sqrt{\Omega }}{4} \biggl( c-\frac{2 k t ^{\alpha }}{\alpha } \biggr) \biggr) \\ & \quad {}\pm \coth \biggl( \frac{\sqrt{ \Omega }}{4} \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(17)

$$\begin{aligned}& \begin{aligned}[b] P_{16}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}- \delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}- \delta +1}{\frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}- \delta +1}}}}}} \\ & \quad {} \times \biggl( d+\frac{1}{2(v-1)} \biggl( -\lambda + \frac{\pm \sqrt{ \Omega ( A^{2}+B^{2} ) }-A \sqrt{\Omega } \cosh ( \sqrt{ \Omega } ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{A \sinh ( \sqrt{\Omega } ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) +B} \biggr) \biggr) \biggr]\hspace{-20pt} \\ & \quad {}\times e^{i (k x+\frac{c t ^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(18)

$$\begin{aligned}& \begin{aligned}[b] Q_{16}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d+\frac{1}{2(v-1)} \biggl( -\lambda \\ & \quad {}+ \frac{\pm \sqrt{ \Omega ( A^{2}+B^{2} ) }-A \sqrt{\Omega } \cosh ( \sqrt{ \Omega } ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{A \sinh ( \sqrt{\Omega } ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) +B} \biggr) \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(19)

$$\begin{aligned}& \begin{aligned}[b] P_{17}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}- \delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}- \delta +1}{\frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}- \delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d+\frac{1}{2(v-1)} \biggl( -\lambda \\ & \quad {}+ \frac{\pm \sqrt{ \Omega ( A^{2}+B^{2} ) }+A \sqrt{\Omega } \cosh ( \sqrt{ \Omega } ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{A \sinh ( \sqrt{\Omega } ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) +B} \biggr) \biggr) \biggr] e^{i (k x+\frac{c t ^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(20)

$$\begin{aligned}& \begin{aligned}[b] Q_{17}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d+\frac{1}{2(v-1)} \biggl( -\lambda \\ & \quad {}+ \frac{\pm \sqrt{ \Omega ( A^{2}+B^{2} ) }+A \sqrt{\Omega } \cosh ( \sqrt{ \Omega } ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{A \sinh ( \sqrt{\Omega } ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) +B} \biggr) \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(21)

$$\begin{aligned}& \begin{aligned}[b] P_{18}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}- \delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}- \delta +1}{\frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}- \delta +1}}}}}} \\ & \quad {} \times \biggl( d+\frac{2 \mu \cosh ( \frac{\sqrt{\Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{ \Omega } \sinh ( \frac{\sqrt{\Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) -\lambda \cosh ( \frac{\sqrt{ \Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) } \biggr) \biggr] e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(22)

$$\begin{aligned}& \begin{aligned}[b] Q_{18}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d+\frac{2 \mu \cosh ( \frac{\sqrt{\Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{ \Omega } \sinh ( \frac{\sqrt{\Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) -\lambda \cosh ( \frac{\sqrt{ \Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) } \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(23)

$$\begin{aligned}& \begin{aligned}[b] P_{19}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}- \delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}- \delta +1}{\frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}- \delta +1}}}}}} \\ & \quad {} \times \biggl( d+\frac{2 \mu \sinh ( \frac{\sqrt{\Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{ \Omega } \cosh ( \frac{\sqrt{\Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) -\lambda \sinh ( \frac{\sqrt{ \Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) } \biggr) \biggr] e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(24)

$$\begin{aligned}& \begin{aligned}[b] Q_{19}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d+\frac{2 \mu \sinh ( \frac{\sqrt{\Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{ \Omega } \cosh ( \frac{\sqrt{\Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) -\lambda \sinh ( \frac{\sqrt{ \Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) } \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(25)

$$\begin{aligned}& \begin{aligned}[b] P_{110}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}- {\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1} {\frac{1}{ \sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}} \\ & \quad {} \times \biggl( d+\frac{2 \mu \cosh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{\Omega } \sinh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) -\lambda \cosh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) \pm i \sqrt{ \Omega }} \biggr) \biggr] \\ & \quad {} \times e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(26)

$$\begin{aligned}& \begin{aligned}[b] Q_{110}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-30pt} \\ & \quad {} \times \biggl( d+\frac{2 \mu \cosh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{\Omega } \sinh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) -\lambda \cosh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) \pm i \sqrt{ \Omega }} \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(27)

$$\begin{aligned}& \begin{aligned}[b] P_{111}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}- {\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1} {\frac{1}{ \sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}} \\ & \quad {} \times \biggl( d+\frac{2 \mu \sinh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{\Omega } \cosh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) -\lambda \sinh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) \pm i \sqrt{ \Omega }} \biggr) \biggr] \\ & \quad {} \times e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(28)

$$\begin{aligned}& \begin{aligned}[b] Q_{111}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d+\frac{2 \mu \sinh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{\Omega } \cosh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) -\lambda \sinh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) \pm i \sqrt{ \Omega }} \biggr) \biggr]^{2}, \end{aligned} \end{aligned}$$

(29)

where A, B are arbitrary real constants and $A^{2}+B^{2}>0$.

• When $( \Omega =\lambda^{2}-4 \lambda \mu +4 \mu <0 ) $ and $( \lambda (v-1)\neq 0 ) $ or $( \mu (v-1) \neq 0 ) $:

$$\begin{aligned}& \begin{aligned}[b] P_{112}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}- {\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1} {\frac{1}{ \sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}} \\ & \quad {} \times \biggl( d+\frac{1}{2(v-1)} \biggl( -\lambda +\sqrt{-\Omega } \tan \biggl( \frac{\sqrt{-\Omega }}{2} \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr] e^{i (k x+\frac{c t ^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(30)

$$\begin{aligned}& \begin{aligned}[b] Q_{112}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d+\frac{1}{2(v-1)} \biggl( -\lambda +\sqrt{-\Omega } \tan \biggl( \frac{\sqrt{-\Omega }}{2} \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(31)

$$\begin{aligned}& \begin{aligned}[b] P_{113}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}- {\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1} {\frac{1}{ \sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}} \\ & \quad {} \times \biggl( d-\frac{1}{2(v-1)} \biggl( \lambda +\sqrt{-\Omega } \cot \biggl( \frac{\sqrt{-\Omega }}{2} \biggl( c-\frac{2 k t^{\alpha }}{ \alpha } \biggr) \biggr) \biggr) \biggr) \biggr] e^{i (k x+\frac{c t^{ \alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(32)

$$\begin{aligned}& \begin{aligned}[b] Q_{113}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d-\frac{1}{2(v-1)} \biggl( \lambda +\sqrt{-\Omega } \cot \biggl( \frac{\sqrt{-\Omega }}{2} \biggl( c-\frac{2 k t^{\alpha }}{ \alpha } \biggr) \biggr) \biggr) \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(33)

$$\begin{aligned}& \begin{aligned}[b] P_{114}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}- {\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1} {\frac{1}{ \sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}} \\ & \quad {} \times \biggl( d+\frac{1}{2(v-1)} \biggl( -\lambda +\sqrt{-\Omega } \biggl( \tan \biggl( \sqrt{-\Omega } \biggl( c-\frac{2 k t^{\alpha }}{ \alpha } \biggr) \biggr) \\ & \quad {} \pm \sec \biggl( \sqrt{-\Omega } \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr) \biggr] e ^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(34)

$$\begin{aligned}& \begin{aligned}[b] Q_{114}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d+\frac{1}{2(v-1)} \biggl( -\lambda +\sqrt{-\Omega } \biggl( \tan \biggl( \sqrt{-\Omega } \biggl( c-\frac{2 k t^{\alpha }}{ \alpha } \biggr) \biggr) \\ & \quad {} \pm \sec \biggl( \sqrt{-\Omega } \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(35)

$$\begin{aligned}& \begin{aligned}[b] P_{115}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}- {\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1} {\frac{1}{ \sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}} \\ & \quad {} \times \biggl( d-\frac{1}{2(v-1)} \biggl( \lambda +\sqrt{-\Omega } \biggl( \cot \biggl( \sqrt{-\Omega } \biggl( c-\frac{2 k t^{\alpha }}{ \alpha } \biggr) \biggr) \\ & \quad {} \pm \csc \biggl( \sqrt{-\Omega } \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr) \biggr] e ^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(36)

$$\begin{aligned}& \begin{aligned}[b] Q_{115}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d-\frac{1}{2(v-1)} \biggl( \lambda +\sqrt{-\Omega } \biggl( \cot \biggl( \sqrt{-\Omega } \biggl( c-\frac{2 k t^{\alpha }}{ \alpha } \biggr) \biggr) \\ & \quad {} \pm \csc \biggl( \sqrt{-\Omega } \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(37)

$$\begin{aligned}& \begin{aligned}[b] P_{116}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}- {\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1} {\frac{1}{ \sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}} \\ & \quad {} \times \biggl( d+\frac{1}{4(v-1)} \biggl( -2 \lambda +\sqrt{- \Omega } \biggl( \tan \biggl( \frac{\sqrt{-\Omega }}{4} \biggl( c-\frac{2 k t ^{\alpha }}{\alpha } \biggr) \biggr) \\ & \quad {}- \cot \biggl( \frac{\sqrt{-\Omega }}{4} \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr) \biggr] e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(38)

$$\begin{aligned}& \begin{aligned}[b] Q_{116}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d+\frac{1}{4(v-1)} \biggl( -2 \lambda +\sqrt{- \Omega } \biggl( \tan \biggl( \frac{\sqrt{-\Omega }}{4} \biggl( c-\frac{2 k t ^{\alpha }}{\alpha } \biggr) \biggr) \\ & \quad {} - \cot \biggl( \frac{\sqrt{-\Omega }}{4} \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(39)

$$\begin{aligned}& \begin{aligned}[b] P_{117}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}- {\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1} {\frac{1}{ \sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}} \\ & \quad {} \times \biggl( d+\frac{1}{2(v-1)} \biggl( -\lambda \\ & \quad {} + \frac{\pm \sqrt{- \Omega ( A^{2}-B^{2} ) }-A \sqrt{-\Omega } \cos ( \sqrt{- \Omega } ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{A \sin ( \sqrt{-\Omega } ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) +B} \biggr) \biggr) \biggr] e^{i (k x+\frac{c t ^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(40)

$$\begin{aligned}& \begin{aligned}[b] Q_{117}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d+\frac{1}{2(v-1)} \biggl( -\lambda \\ & \quad {}+ \frac{\pm \sqrt{- \Omega ( A^{2}-B^{2} ) }-A \sqrt{-\Omega } \cos ( \sqrt{- \Omega } ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{A \sin ( \sqrt{-\Omega } ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) +B} \biggr) \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(41)

$$\begin{aligned}& \begin{aligned}[b] P_{118}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}- {\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1} {\frac{1}{ \sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}} \\ & \quad {} \times \biggl( d+\frac{1}{2(v-1)} \biggl( -\lambda \\ & \quad {}+ \frac{\pm \sqrt{- \Omega ( A^{2}-B^{2} ) }+A \sqrt{-\Omega } \cos ( \sqrt{- \Omega } ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{A \sin ( \sqrt{-\Omega } ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) +B} \biggr) \biggr) \biggr] e^{i (k x+\frac{c t ^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(42)

$$\begin{aligned}& \begin{aligned}[b] Q_{118}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d+\frac{1}{2(v-1)} \biggl( -\lambda \\ & \quad {}+ \frac{\pm \sqrt{- \Omega ( A^{2}-B^{2} ) }+A \sqrt{-\Omega } \cos ( \sqrt{- \Omega } ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{A \sin ( \sqrt{-\Omega } ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) +B} \biggr) \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(43)

$$\begin{aligned}& \begin{aligned}[b] P_{119}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}- {\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1} {\frac{1}{ \sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}} \\ & \quad {} \times \biggl( d-\frac{2 \mu \cos ( \frac{\sqrt{-\Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{- \Omega } \sin ( \frac{\sqrt{-\Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) +\lambda \cos ( \frac{\sqrt{- \Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) } \biggr) \biggr] e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(44)

$$\begin{aligned}& \begin{aligned}[b] Q_{119}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d-\frac{2 \mu \cos ( \frac{\sqrt{-\Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{- \Omega } \sin ( \frac{\sqrt{-\Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) +\lambda \cos ( \frac{\sqrt{- \Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) } \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(45)

$$\begin{aligned}& \begin{aligned}[b] P_{120}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}- {\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1} {\frac{1}{ \sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}} \\ & \quad {} \times \biggl( d+\frac{2 \mu \sin ( \frac{\sqrt{-\Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{- \Omega } \cos ( \frac{\sqrt{-\Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) -\lambda \sin ( \frac{\sqrt{- \Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) } \biggr) \biggr] e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(46)

$$\begin{aligned}& \begin{aligned}[b] Q_{120}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d+\frac{2 \mu \sin ( \frac{\sqrt{-\Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{- \Omega } \cos ( \frac{\sqrt{-\Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) -\lambda \sin ( \frac{\sqrt{- \Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) } \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(47)

$$\begin{aligned}& \begin{aligned}[b] P_{121}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}- {\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1} {\frac{1}{ \sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}} \\ & \quad {} \times \biggl( d-\frac{2 \mu \cos ( {\sqrt{-\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{-\Omega } \sin ( {\sqrt{-\Omega }} ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) +\lambda \cos ( {\sqrt{-\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) \pm \sqrt{- \Omega }} \biggr) \biggr]\hspace{-20pt} \\ & \quad {} \times e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(48)

$$\begin{aligned}& \begin{aligned}[b] Q_{121}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}} \\ & \quad {} \times \biggl( d-\frac{2 \mu \cos ( {\sqrt{-\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{-\Omega } \sin ( {\sqrt{-\Omega }} ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) +\lambda \cos ( {\sqrt{-\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) \pm \sqrt{- \Omega }} \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(49)

$$\begin{aligned}& \begin{aligned}[b] P_{122}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}- {\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1} {\frac{1}{ \sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d+\frac{2 \mu \sin ( {\sqrt{-\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{-\Omega } \cos ( {\sqrt{-\Omega }} ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) -\lambda \sin ( {\sqrt{-\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) \pm \sqrt{- \Omega }} \biggr) \biggr]\hspace{-20pt} \\ & \quad {} \times e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(50)

$$\begin{aligned}& \begin{aligned}[b] Q_{122}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d+\frac{2 \mu \sin ( {\sqrt{-\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{-\Omega } \cos ( {\sqrt{-\Omega }} ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) -\lambda \sin ( {\sqrt{-\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) \pm \sqrt{- \Omega }} \biggr) \biggr]\hspace{-20pt} \\ & \quad {} \times e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(51)

where A, B are arbitrary real constants and $A^{2}-B^{2}>0$.

• When $\mu =0$ and $\lambda (v-1)\neq 0$, we have:

$$\begin{aligned}& \begin{aligned}[b] P_{123}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}- {\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1} {\frac{1}{ \sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}} \\ & \quad {} \times \biggl( d-\frac{\lambda k}{(v-1) ( k+\cosh (\lambda ( c-\frac{2 k t^{\alpha }}{\alpha } ) )-\sinh (\lambda ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) ) } \biggr) \biggr] e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(52)

$$\begin{aligned}& \begin{aligned}[b] Q_{123}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d-\frac{\lambda k}{(v-1) ( k+\cosh (\lambda ( c-\frac{2 k t^{\alpha }}{\alpha } ) )-\sinh (\lambda ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) ) } \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(53)

$$\begin{aligned}& \begin{aligned}[b] P_{124}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}- {\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1} {\frac{1}{ \sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}} \\ & \quad {} \times \biggl( d-\frac{\lambda ( \cosh (\lambda ( c-\frac{2 k t^{\alpha }}{\alpha } ) )+\sinh (\lambda ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) ) }{(v-1) ( k+\cosh (\lambda ( c-\frac{2 k t^{\alpha }}{\alpha } ) )+\sinh (\lambda ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) ) } \biggr) \biggr] e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(54)

$$\begin{aligned}& \begin{aligned}[b] Q_{124}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d-\frac{\lambda ( \cosh (\lambda ( c-\frac{2 k t^{\alpha }}{\alpha } ) )+\sinh (\lambda ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) ) }{(v-1) ( k+\cosh (\lambda ( c-\frac{2 k t^{\alpha }}{\alpha } ) )+\sinh (\lambda ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) ) } \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(55)

$$\begin{aligned}& \begin{aligned}[b] P_{125}(x,t) &=\biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}- {\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1} {\frac{1}{ \sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}} \\ & \quad {} \times \biggl( d-\frac{1}{(v-1) ( c-\frac{2 k t^{\alpha }}{ \alpha } ) + C} \biggr) \biggr] e^{i (k x+\frac{c t^{\alpha }}{ \alpha })}, \end{aligned} \end{aligned}$$

(56)

$$\begin{aligned}& \begin{aligned}[b] Q_{125}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[\frac{1}{2} \bigl( ( 2 v-2 ) d-\lambda \bigr) \sqrt{{\frac{4 {k}^{2}-1}{4 \delta {k}^{2}-\delta +1}}}-{\frac{4 {k}^{2}v-4 {k}^{2}-v+1}{4 \delta {k}^{2}-\delta +1}{ \frac{1}{\sqrt{{\frac{4 {k}^{2}+1}{4 \delta {k}^{2}-\delta +1}}}}}}\hspace{-20pt} \\ & \quad {} \times \biggl( d-\frac{1}{(v-1) ( c-\frac{2 k t^{\alpha }}{ \alpha } ) + C} \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(57)

Family 2

$$ k=\sqrt{\frac{\delta -1}{4 \delta }}, \quad\quad v=1, \quad\quad a_{-1}=0, \quad\quad a_{0}=- \frac{4 \delta \lambda a_{1} (d \lambda -\mu )}{4 c \delta +\delta -1}. $$

So, the solitary traveling wave solutions are as follows:

• When $(\Omega =\lambda^{2}-4 \lambda \mu +4 \mu >0)$ and $(\lambda (v-1)\neq 0)$ or $(\mu (v-1)\neq 0)$:

$$\begin{aligned}& \begin{aligned}[b] P_{21}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1} \\ &\quad{} +a_{1} \biggl( d- \frac{1}{2(v-1)} \biggl( \lambda +\sqrt{ \Omega } \tanh \biggl( \frac{\sqrt{\Omega }}{2} \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr] e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(58)

$$\begin{aligned}& \begin{aligned}[b] Q_{21}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[- \frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl( d- \frac{1}{2(v-1)} \biggl( \lambda +\sqrt{\Omega } \tanh \biggl( \frac{\sqrt{ \Omega }}{2} \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(59)

$$\begin{aligned}& \begin{aligned}[b] P_{22}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl( d- \frac{1}{2(v-1)} \biggl( \lambda +\sqrt{ \Omega } \coth \biggl( \frac{\sqrt{\Omega }}{2} \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr] e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(60)

$$\begin{aligned}& \begin{aligned}[b] Q_{22}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl( d- \frac{1}{2(v-1)} \biggl( \lambda +\sqrt{\Omega } \coth \biggl( \frac{\sqrt{ \Omega }}{2} \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(61)

$$\begin{aligned}& \begin{aligned}[b] P_{23}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1}+a_{1} \biggl(d- \frac{1}{2(v-1)}\biggl(\lambda +\sqrt{ \Omega } \biggl( \tanh \biggl(\sqrt{\Omega } \biggl( c-\frac{2 k t^{\alpha }}{ \alpha }\biggr) \biggr)\hspace{-20pt} \\ & \quad {} \pm i \operatorname {sech}\biggl(\sqrt{\Omega } \biggl( c-\frac{2 k t^{\alpha }}{\alpha }\biggr) \biggr)\biggr)\biggr)\biggr)\biggr] e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(62)

$$\begin{aligned}& \begin{aligned}[b] Q_{23}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl(d- \frac{1}{2(v-1)} \biggl(\lambda +\sqrt{\Omega } \biggl( \tanh \biggl( \sqrt{\Omega } \biggl( c-\frac{2 k t^{\alpha }}{\alpha }\biggr) \biggr) \\ & \quad {} \pm i \operatorname {sech}\biggl(\sqrt{\Omega } \biggl( c-\frac{2 k t^{\alpha }}{\alpha }\biggr) \biggr)\biggr)\biggr)\biggr)\biggr]^{2}. \end{aligned} \end{aligned}$$

(63)

$$\begin{aligned}& \begin{aligned}[b] P_{24}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1}+a_{1} \biggl(d- \frac{1}{2(v-1)}\biggl(\lambda +\sqrt{ \Omega } \biggl( \coth \biggl(\sqrt{\Omega } \biggl( c-\frac{2 k t^{\alpha }}{ \alpha }\biggr)\biggr)\hspace{-20pt} \\ & \quad {} \pm \operatorname {csch}\biggl(\sqrt{\Omega } \biggl( c-\frac{2 k t^{\alpha }}{\alpha }\biggr) \biggr)\biggr)\biggr)\biggr)\biggr] e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(64)

$$\begin{aligned}& \begin{aligned}[b] Q_{24}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl(d- \frac{1}{2(v-1)} \biggl(\lambda +\sqrt{\Omega } \biggl( \coth \biggl( \sqrt{\Omega } \biggl( c-\frac{2 k t^{\alpha }}{\alpha }\biggr)\biggr)\hspace{-20pt} \\ & \quad {} \pm \operatorname {csch}\biggl(\sqrt{\Omega } \biggl( c-\frac{2 k t^{\alpha }}{\alpha }\biggr) \biggr)\biggr)\biggr)\biggr)\biggr]^{2}. \end{aligned} \end{aligned}$$

(65)

$$\begin{aligned}& \begin{aligned}[b] P_{25}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1}+a_{1} \biggl(d- \frac{1}{4(v-1)} \biggl(2 \lambda +\sqrt{\Omega } \biggl( \tanh \biggl( \frac{\sqrt{\Omega }}{4} \biggl( c-\frac{2 k t^{\alpha }}{\alpha }\biggr) \biggr)\hspace{-20pt} \\ & \quad {} \pm \coth \biggl(\frac{\sqrt{\Omega }}{4} \biggl( c-\frac{2 k t^{\alpha }}{ \alpha } \biggr) \biggr)\biggr)\biggr)\biggr)\biggr] e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(66)

$$\begin{aligned}& \begin{aligned}[b] Q_{25}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1} \\ & \quad {}+a_{1} \biggl(d- \frac{1}{4(v-1)} \biggl(2 \lambda +\sqrt{\Omega } \biggl( \tanh \biggl( \frac{\sqrt{ \Omega }}{4} \biggl( c-\frac{2 k t^{\alpha }}{\alpha }\biggr) \biggr) \\ & \quad {} \pm \coth \biggl(\frac{\sqrt{\Omega }}{4} \biggl( c-\frac{2 k t^{\alpha }}{ \alpha } \biggr) \biggr)\biggr)\biggr)\biggr)\biggr]^{2}. \end{aligned} \end{aligned}$$

(67)

$$\begin{aligned}& \begin{aligned}[b] P_{26}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1}+a_{1} \biggl( d+ \frac{1}{2(v-1)} \biggl( - \lambda \\ &\quad{} +\frac{\pm \sqrt{\Omega ( A^{2}+B^{2} ) }-A \sqrt{ \Omega } \cosh ( \sqrt{\Omega } ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{A \sinh ( \sqrt{\Omega } ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) +B} \biggr) \biggr) \biggr] e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(68)

$$\begin{aligned}& \begin{aligned}[b] Q_{26}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1}+a_{1} \biggl( d+ \frac{1}{2(v-1)} \biggl( -\lambda \\ &\quad{}+\frac{\pm \sqrt{\Omega ( A ^{2}+B^{2} ) }-A \sqrt{\Omega } \cosh ( \sqrt{\Omega } ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{A \sinh ( \sqrt{\Omega } ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) +B} \biggr) \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(69)

$$\begin{aligned}& \begin{aligned}[b] P_{27}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1}+a_{1} \biggl( d+ \frac{1}{2(v-1)} \biggl( - \lambda \\ &\quad{}+\frac{\pm \sqrt{\Omega ( A^{2}+B^{2} ) }+A \sqrt{ \Omega } \cosh ( \sqrt{\Omega } ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{A \sinh ( \sqrt{\Omega } ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) +B} \biggr) \biggr) \biggr] e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(70)

$$\begin{aligned}& \begin{aligned}[b] Q_{27}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1}+a_{1} \biggl( d+ \frac{1}{2(v-1)} \biggl( -\lambda \\ &\quad{}+\frac{\pm \sqrt{\Omega ( A ^{2}+B^{2} ) }+A \sqrt{\Omega } \cosh ( \sqrt{\Omega } ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{A \sinh ( \sqrt{\Omega } ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) +B} \biggr) \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(71)

$$\begin{aligned}& \begin{aligned}[b] P_{28}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl( d+ \frac{2 \mu \cosh ( \frac{\sqrt{\Omega }}{2} ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) }{\sqrt{\Omega } \sinh ( \frac{\sqrt{ \Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) - \lambda \cosh ( \frac{\sqrt{\Omega }}{2} ( c-\frac{2 k t ^{\alpha }}{\alpha } ) ) } \biggr) \biggr] e^{i (k x+\frac{c t ^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(72)

$$\begin{aligned}& \begin{aligned}[b] Q_{28}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl( d+\frac{2 \mu \cosh ( \frac{\sqrt{\Omega }}{2} ( c-\frac{2 k t ^{\alpha }}{\alpha } ) ) }{\sqrt{\Omega } \sinh ( \frac{\sqrt{ \Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) - \lambda \cosh ( \frac{\sqrt{\Omega }}{2} ( c-\frac{2 k t ^{\alpha }}{\alpha } ) ) } \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(73)

$$\begin{aligned}& \begin{aligned}[b] P_{29}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl( d+ \frac{2 \mu \sinh ( \frac{\sqrt{\Omega }}{2} ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) }{\sqrt{\Omega } \cosh ( \frac{\sqrt{ \Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) - \lambda \sinh ( \frac{\sqrt{\Omega }}{2} ( c-\frac{2 k t ^{\alpha }}{\alpha } ) ) } \biggr) \biggr] e^{i (k x+\frac{c t ^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(74)

$$\begin{aligned}& \begin{aligned}[b] Q_{19}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl( d+\frac{2 \mu \sinh ( \frac{\sqrt{\Omega }}{2} ( c-\frac{2 k t ^{\alpha }}{\alpha } ) ) }{\sqrt{\Omega } \cosh ( \frac{\sqrt{ \Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) - \lambda \sinh ( \frac{\sqrt{\Omega }}{2} ( c-\frac{2 k t ^{\alpha }}{\alpha } ) ) } \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(75)

$$\begin{aligned}& \begin{aligned}[b] P_{210}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl( d+ \frac{2 \mu \cosh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{\Omega } \sinh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) -\lambda \cosh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) \pm i \sqrt{ \Omega }} \biggr) \biggr] \\ &\quad{}\times e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(76)

$$\begin{aligned}& \begin{aligned}[b] Q_{210}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1}+a_{1} \biggl( d \\ &\quad{}+\frac{2 \mu \cosh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{\Omega } \sinh ( {\sqrt{ \Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) - \lambda \cosh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{ \alpha }}{\alpha } ) ) \pm i \sqrt{\Omega }} \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(77)

$$\begin{aligned}& \begin{aligned}[b] P_{211}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl( d+ \frac{2 \mu \sinh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{\Omega } \cosh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) -\lambda \sinh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) \pm i \sqrt{ \Omega }} \biggr) \biggr] \\ &\quad{}\times e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(78)

$$\begin{aligned}& \begin{aligned}[b] Q_{211}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1}+a_{1} \biggl( d \\ &\quad{}+\frac{2 \mu \sinh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{\Omega } \cosh ( {\sqrt{ \Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) - \lambda \sinh ( {\sqrt{\Omega }} ( c-\frac{2 k t^{ \alpha }}{\alpha } ) ) \pm i \sqrt{\Omega }} \biggr) \biggr]^{2}, \end{aligned} \end{aligned}$$

(79)

where A, B are arbitrary real constants and $A^{2}+B^{2}>0$.

• When $( \Omega =\lambda^{2}-4 \lambda \mu +4 \mu <0 ) $ and $( \lambda (v-1)\neq 0 ) $ or $( \mu (v-1) \neq 0 ) $:

$$\begin{aligned}& \begin{aligned}[b] P_{212}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl( d+ \frac{1}{2(v-1)} \biggl( - \lambda +\sqrt{-\Omega } \tan \biggl( \frac{\sqrt{-\Omega }}{2} \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr] \\ &\quad{}\times e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(80)

$$\begin{aligned}& \begin{aligned}[b] Q_{212}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[- \frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl( d+ \frac{1}{2(v-1)} \biggl( -\lambda +\sqrt{-\Omega } \tan \biggl( \frac{\sqrt{- \Omega }}{2} \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(81)

$$\begin{aligned}& \begin{aligned}[b] P_{213}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl( d- \frac{1}{2(v-1)} \biggl( \lambda +\sqrt{- \Omega } \cot \biggl( \frac{\sqrt{-\Omega }}{2} \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr] e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(82)

$$\begin{aligned}& \begin{aligned}[b] Q_{213}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl( d-\frac{1}{2(v-1)} \biggl( \lambda +\sqrt{-\Omega } \cot \biggl( \frac{\sqrt{- \Omega }}{2} \biggl( c-\frac{2 k t^{\alpha }}{\alpha } \biggr) \biggr) \biggr) \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(83)

$$\begin{aligned}& \begin{aligned}[b] P_{214}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1}+a_{1} \biggl(d+ \frac{1}{2(v-1)} \biggl(-\lambda +\sqrt{-\Omega } \biggl( \tan \biggl(\sqrt{- \Omega } \biggl( c-\frac{2 k t^{ \alpha }}{\alpha }\biggr)\biggr)\hspace{-30pt} \\ & \quad {} \pm \sec \biggl(\sqrt{-\Omega } \biggl( c-\frac{2 k t^{\alpha }}{\alpha }\biggr) \biggr)\biggr)\biggr)\biggr) \biggr] e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(84)

$$\begin{aligned}& \begin{aligned}[b] Q_{214}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl(d+ \frac{1}{2(v-1)} \biggl(-\lambda +\sqrt{-\Omega } \biggl( \tan \biggl( \sqrt{- \Omega } \biggl( c-\frac{2 k t^{\alpha }}{\alpha }\biggr)\biggr) \\ & \quad {} \pm \sec \biggl(\sqrt{-\Omega } \biggl( c-\frac{2 k t^{\alpha }}{\alpha }\biggr) \biggr)\biggr)\biggr)\biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(85)

$$\begin{aligned}& \begin{aligned}[b] P_{215}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1}+a_{1} \biggl(d- \frac{1}{2(v-1)} \biggl(\lambda +\sqrt{-\Omega } \biggl( \cot \biggl(\sqrt{-\Omega } \biggl( c-\frac{2 k t^{ \alpha }}{\alpha }\biggr) \biggr)\hspace{-20pt} \\ & \quad {} \pm \csc \biggl(\sqrt{-\Omega } \biggl( c-\frac{2 k t^{\alpha }}{\alpha }\biggr) \biggr)\biggr)\biggr)\biggr)\biggr] e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(86)

$$\begin{aligned}& \begin{aligned}[b] Q_{215}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl(d- \frac{1}{2(v-1)} \biggl(\lambda +\sqrt{-\Omega } \biggl( \cot \biggl( \sqrt{- \Omega } \biggl( c-\frac{2 k t^{\alpha }}{\alpha }\biggr)\biggr) \\ & \quad {} \pm \csc \biggl(\sqrt{-\Omega } \biggl( c-\frac{2 k t^{\alpha }}{\alpha }\biggr) \biggr)\biggr)\biggr)\biggr)\biggr]^{2}. \end{aligned} \end{aligned}$$

(87)

$$\begin{aligned}& \begin{aligned}[b] P_{216}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl(d+ \frac{1}{4(v-1)} \biggl(-2 \lambda +\sqrt{-\Omega } \biggl( \tan \biggl( \frac{\sqrt{-\Omega }}{4} \biggl( c-\frac{2 k t^{\alpha }}{\alpha }\biggr)\biggr) \\ & \quad {} - \cot \biggl(\frac{\sqrt{-\Omega }}{4} \biggl( c-\frac{2 k t^{\alpha }}{ \alpha }\biggr) \biggr)\biggr)\biggr)\biggr)\biggr] e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(88)

$$\begin{aligned}& \begin{aligned}[b] Q_{216}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl(d+ \frac{1}{4(v-1)} \biggl(-2 \lambda +\sqrt{-\Omega } \biggl( \tan \biggl(\frac{\sqrt{- \Omega }}{4} \biggl( c-\frac{2 k t^{\alpha }}{\alpha }\biggr) \biggr) \\ & \quad {} - \cot \biggl(\frac{\sqrt{-\Omega }}{4} \biggl( c-\frac{2 k t^{\alpha }}{ \alpha }\biggr) \biggr)\biggr)\biggr)\biggr)\biggr]^{2}. \end{aligned} \end{aligned}$$

(89)

$$\begin{aligned}& \begin{aligned}[b] P_{217}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1}+a_{1} \biggl(d+ \frac{1}{2(v-1)} \biggl(-\lambda \\ &\quad{} +\frac{\pm \sqrt{-\Omega (A^{2}-B^{2})}-A \sqrt{-\Omega } \cos (\sqrt{- \Omega } ( c-\frac{2 k t^{\alpha }}{\alpha }) )}{A \sin (\sqrt{- \Omega } ( c-\frac{2 k t^{\alpha }}{\alpha }) )+B}\biggr)\biggr) \biggr] \\ & \quad {} \times e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(90)

$$\begin{aligned}& \begin{aligned}[b] Q_{217}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1}+a_{1} \biggl(d+ \frac{1}{2(v-1)} \biggl(-\lambda \\ &\quad{}+\frac{\pm \sqrt{-\Omega (A^{2}-B^{2})}-A \sqrt{-\Omega } \cos (\sqrt{- \Omega } ( c-\frac{2 k t^{\alpha }}{\alpha }) )}{A \sin (\sqrt{- \Omega } ( c-\frac{2 k t^{\alpha }}{\alpha }) )+B}\biggr)\biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(91)

$$\begin{aligned}& \begin{aligned}[b] P_{218}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1}+a_{1} \biggl(d+ \frac{1}{2(v-1)} \biggl(-\lambda \\ &\quad{}+\frac{\pm \sqrt{-\Omega (A^{2}-B^{2})}+A \sqrt{-\Omega } \cos (\sqrt{- \Omega } ( c-\frac{2 k t^{\alpha }}{\alpha }) )}{A \sin (\sqrt{- \Omega } ( c-\frac{2 k t^{\alpha }}{\alpha }) )+B}\biggr)\biggr) \biggr] e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(92)

$$\begin{aligned}& \begin{aligned}[b] Q_{218}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1}+a_{1} \biggl(d+ \frac{1}{2(v-1)}\biggl(-\lambda \\ &\quad{}+\frac{\pm \sqrt{-\Omega (A^{2}-B^{2})}+A \sqrt{-\Omega } \cos (\sqrt{-\Omega } ( c-\frac{2 k t^{ \alpha }}{\alpha }))}{A \sin (\sqrt{-\Omega } ( c-\frac{2 k t ^{\alpha }}{\alpha }) )+B}\biggr)\biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(93)

$$\begin{aligned}& \begin{aligned}[b] P_{219}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl( d- \frac{2 \mu \cos ( \frac{\sqrt{-\Omega }}{2} ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) }{\sqrt{-\Omega } \sin ( \frac{\sqrt{- \Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) + \lambda \cos ( \frac{\sqrt{-\Omega }}{2} ( c-\frac{2 k t ^{\alpha }}{\alpha } ) ) } \biggr) \biggr] \\ &\quad{}\times e^{i (k x+\frac{c t ^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(94)

$$\begin{aligned}& \begin{aligned}[b] Q_{219}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl( d-\frac{2 \mu \cos ( \frac{\sqrt{-\Omega }}{2} ( c-\frac{2 k t ^{\alpha }}{\alpha } ) ) }{\sqrt{-\Omega } \sin ( \frac{\sqrt{- \Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) + \lambda \cos ( \frac{\sqrt{-\Omega }}{2} ( c-\frac{2 k t ^{\alpha }}{\alpha } ) ) } \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(95)

$$\begin{aligned}& \begin{aligned}[b] P_{220}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl( d+ \frac{2 \mu \sin ( \frac{\sqrt{-\Omega }}{2} ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) }{\sqrt{-\Omega } \cos ( \frac{\sqrt{- \Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) - \lambda \sin ( \frac{\sqrt{-\Omega }}{2} ( c-\frac{2 k t ^{\alpha }}{\alpha } ) ) } \biggr) \biggr] \\ &\quad{}\times e^{i (k x+\frac{c t ^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(96)

$$\begin{aligned}& \begin{aligned}[b] Q_{220}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl( d+\frac{2 \mu \sin ( \frac{\sqrt{-\Omega }}{2} ( c-\frac{2 k t ^{\alpha }}{\alpha } ) ) }{\sqrt{-\Omega } \cos ( \frac{\sqrt{- \Omega }}{2} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) - \lambda \sin ( \frac{\sqrt{-\Omega }}{2} ( c-\frac{2 k t ^{\alpha }}{\alpha } ) ) } \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(97)

$$\begin{aligned}& \begin{aligned}[b] P_{221}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl( d- \frac{2 \mu \cos ( {\sqrt{-\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{-\Omega } \sin ( {\sqrt{-\Omega }} ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) +\lambda \cos ( {\sqrt{-\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) \pm \sqrt{- \Omega }} \biggr) \biggr]\hspace{-20pt} \\ &\quad{}\times e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(98)

$$\begin{aligned}& \begin{aligned}[b] Q_{221}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1}+a_{1} \biggl( d \\ &\quad{}-\frac{2 \mu \cos ( {\sqrt{-\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) }{\sqrt{-\Omega } \sin ( {\sqrt{- \Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) + \lambda \cos ( {\sqrt{-\Omega }} ( c-\frac{2 k t^{ \alpha }}{\alpha } ) ) \pm \sqrt{-\Omega }} \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(99)

$$\begin{aligned}& \begin{aligned}[b] P_{222}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl(d+ \frac{2 \mu \sin ({\sqrt{- \Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha }) )}{\sqrt{-\Omega } \cos ({\sqrt{-\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha }) )- \lambda \sin ({\sqrt{-\Omega }} ( c-\frac{2 k t^{\alpha }}{ \alpha }) )\pm \sqrt{-\Omega }}\biggr)\biggr]\hspace{-20pt} \\ &\quad{}\times e^{i (k x+\frac{c t^{ \alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(100)

$$\begin{aligned}& \begin{aligned}[b] Q_{222}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl(d+\frac{2 \mu \sin ({\sqrt{-\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha }) )}{\sqrt{-\Omega } \cos ({\sqrt{-\Omega }} ( c-\frac{2 k t ^{\alpha }}{\alpha }) )-\lambda \sin ({\sqrt{-\Omega }} ( c-\frac{2 k t^{\alpha }}{\alpha }) )\pm \sqrt{-\Omega }}\biggr)\biggr]\hspace{-20pt} \\ &\quad{}\times e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(101)

where A, B are arbitrary real constants and $A^{2}-B^{2}>0$.

• When $\mu =0$ and $\lambda (v-1)\neq 0$, we have:

$$\begin{aligned}& \begin{aligned}[b] P_{223}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1} \\ &\quad{} +a_{1} \biggl( d- \frac{\lambda k}{(v-1) ( k+\cosh (\lambda ( c-\frac{2 k t^{\alpha }}{\alpha } ) )-\sinh (\lambda ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) ) } \biggr) \biggr] \\ &\quad{}\times e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(102)

$$\begin{aligned}& \begin{aligned}[b] Q_{223}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl( d-\frac{ \lambda k}{(v-1) ( k+\cosh (\lambda ( c-\frac{2 k t^{ \alpha }}{\alpha } ) )-\sinh (\lambda ( c-\frac{2 k t^{ \alpha }}{\alpha } ) ) ) } \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(103)

$$\begin{aligned}& \begin{aligned}[b] P_{224}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl( d- \frac{\lambda ( \cosh ( \lambda ( c-\frac{2 k t^{\alpha }}{\alpha } ) )+\sinh ( \lambda ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) ) }{(v-1) ( k+\cosh (\lambda ( c-\frac{2 k t^{\alpha }}{\alpha } ) )+\sinh (\lambda ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) ) } \biggr) \biggr] \\ &\quad{}\times e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(104)

$$\begin{aligned}& \begin{aligned}[b] Q_{224}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1} \\ &\quad{}+a_{1} \biggl( d-\frac{ \lambda ( \cosh (\lambda ( c-\frac{2 k t^{\alpha }}{ \alpha } ) )+\sinh (\lambda ( c-\frac{2 k t^{\alpha }}{ \alpha } ) ) ) }{(v-1) ( k+\cosh (\lambda ( c-\frac{2 k t^{\alpha }}{\alpha } ) )+\sinh (\lambda ( c-\frac{2 k t^{\alpha }}{\alpha } ) ) ) } \biggr) \biggr]^{2}. \end{aligned} \end{aligned}$$

(105)

$$\begin{aligned}& \begin{aligned}[b] P_{225}(x,t) &=\biggl[-\frac{4 \delta \lambda a_{1} (d \lambda - \mu )}{4 c \delta +\delta -1}+a_{1} \biggl( d- \frac{1}{(v-1) ( c-\frac{2 k t^{\alpha }}{\alpha } ) + C} \biggr) \biggr] e^{i (k x+\frac{c t^{\alpha }}{\alpha })}, \end{aligned} \end{aligned}$$

(106)

$$\begin{aligned}& \begin{aligned}[b] Q_{225}(x,t) &=\frac{1}{1-4 k^{2}} \biggl[-\frac{4 \delta \lambda a _{1} (d \lambda -\mu )}{4 c \delta +\delta -1}+a_{1} \biggl( d-\frac{1}{(v-1) ( c-\frac{2 k t^{\alpha }}{\alpha } ) + C} \biggr) \biggr]^{2}, \end{aligned} \end{aligned}$$

(107)

where C, k are arbitrary constants.

3 Conclusion

We implement two recent methods called a generalized Kudryashov method and a novel $( \frac{G'}{G} ) $-expansion method. We obtain many forms of solutions that cover all previous solutions obtained by implementation of different methods [1–20]. We get new forms of analytical wave solutions for the nonlinear complex fractional generalized-Zakharov equations. Studying the physical properties of these kinds of models is very motivating and interesting. As we see, both of these methods are very direct, effective, and powerful, and we also showed the ability of these methods to be applied to different kinds of nonlinear partial differential equations whether they are of the fractional order or of the integer order.

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Department of Mathematics, Faculty of Science, Jiangsu University, Zhenjiang, China
Dianchen Lu & Mostafa M. A. Khater
Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia
Aly R. Seadawy
Mathematics Department, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt
Aly R. Seadawy

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Lu, D., Seadawy, A.R. & Khater, M.M.A. Structure of solitary wave solutions of the nonlinear complex fractional generalized Zakharov dynamical system. Adv Differ Equ 2018, 266 (2018). https://doi.org/10.1186/s13662-018-1734-4

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Received: 11 April 2018
Accepted: 27 July 2018
Published: 03 August 2018
DOI: https://doi.org/10.1186/s13662-018-1734-4

Structure of solitary wave solutions of the nonlinear complex fractional generalized Zakharov dynamical system

Abstract

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A novel approach for solitary wave solutions of the generalized fractional Zakharov-Kuznetsov equation

New Solitary Wave Solutions and Dynamical Behaviors of the Nonlinear Fractional Zakharov System

A novel analytical technique to obtain the solitary solutions for nonlinear evolution equation of fractional order

1 Introduction

2 Formulation for the nonlinear complex fractional generalized-Zakharov equations

2.1 Exact and solitary wave solution of the nonlinear complex fractional generalized-Zakharov equations by using a generalized Kudryashov method

2.2 Exact and solitary wave solution of the nonlinear complex fractional generalized-Zakharov equations by using novel \((\frac{G'}{G})\)-expansion method

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3 Conclusion

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Structure of solitary wave solutions of the nonlinear complex fractional generalized Zakharov dynamical system

Abstract

Similar content being viewed by others

A novel approach for solitary wave solutions of the generalized fractional Zakharov-Kuznetsov equation

New Solitary Wave Solutions and Dynamical Behaviors of the Nonlinear Fractional Zakharov System

A novel analytical technique to obtain the solitary solutions for nonlinear evolution equation of fractional order

1 Introduction

2 Formulation for the nonlinear complex fractional generalized-Zakharov equations

2.1 Exact and solitary wave solution of the nonlinear complex fractional generalized-Zakharov equations by using a generalized Kudryashov method

2.2 Exact and solitary wave solution of the nonlinear complex fractional generalized-Zakharov equations by using novel \((\frac{G'}{G})\)-expansion method

Family 1

Family 2

3 Conclusion

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