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

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.

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 [1620].

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 [2155].

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.

Formulation for the nonlinear complex fractional generalized-Zakharov equations

Consider the nonlinear complex fractional generalized-Zakharov equations in the form [2123]:

$$ \textstyle\begin{cases} i \frac{D^{\alpha } P}{D t^{\alpha }}+ P_{x x}-2 \delta \vert P \vert ^{2} P+2 P Q=0, \\ \frac{D^{2 \alpha } Q}{D t^{2 \alpha }}-Q_{x x}+( \vert P \vert ^{2})_{x x}=0. \end{cases} $$
(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 \(Y'' \& Y^{3}\Rightarrow N+2=3N\Rightarrow N=1\).

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)

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.

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 [120]. 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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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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Keywords

  • Nonlinear complex fractional generalized-Zakharov system
  • Generalized Kudryashov methods
  • Novel \((\frac{G'}{G})\)-expansion method
  • Solitary traveling wave solutions
\