1 Introduction

It is well known that various physical phenomena are characterized by nonlinear evolution equations. As mathematical models of the phenomena, the investigation of exact explicit solutions of nonlinear equations will help us to understand these phenomena in a better way. Various methods have been formulated and developed for obtaining explicit exact solutions of nonlinear partial differential equations (NLPDEs). These methods include exp-function method [31], Jacobi elliptic function method [7], Hirota’s direct method [41], F-expansion method [35], tanh function method [42], modified extended tanh-function method [3], modified \((G'/G)\) - expansion method [43], first integral method [9], \((G'/G^2)\)-expansion method [10, 46], \((G'/G)\)-expansion method [30], \((G'/G, 1/G)\)-expansion method [2], classical symmetry method [5, 6, 13, 29] and so on. Using Ibragimov’s approach, conservation laws of nonlinear equations were introduced in [37]. Lie symmetry analysis, conservation laws, and various exact invariant solutions of differential equations were also furnished in [14,15,16,17,18,19].

Sophus Lie [23] introduced the concept of Lie symmetry method. It is one of the best methods for constructing the exact solutions of the NLPDEs by reducing the number of independent variables, exploiting continuous symmetries of the nonlinear system. The obtained exact solutions are known as similarity solutions. Various mathematicians employed this method and constructed the exact solutions [8, 11, 24, 25, 27, 39].

The Klein–Gordon–Zakharov equations [4] are given by

$$\begin{aligned} \phi _{tt}-\phi _{xx}+\phi +\lambda \psi \phi =0,\nonumber \\ \psi _{tt}-\psi _{xx}-\sigma (|\phi |^2)_{xx}=0, \end{aligned}$$
(1.1)

where \(\sigma \), \(\lambda \) are nonzero real constants. The complex-valued function \(\phi \) denotes the fast time scale component of electric field raised by electrons, and the real-valued function \(\psi \) represents the deviation of ion density from its equilibrium. This equation describes the interaction between the Langmuir wave and the ion-acoustic wave in high-frequency plasma physics [4]. Recently, numerous computational approaches have been taken into account to investigate the equations (1.1) such as extended hyperbolic function method [34], extended sinh-Gordon equation expansion method [4], exp-\((\phi (\xi ))\)-expansion method [12], solitary wave ansazt [38], bifurcation method [45], He’s variational principle method [1], simplest equation method [28, 22], conservative finite difference scheme [40], sine–cosine and some extended function method [36], and so on.

In recent times, the focus on the examination of variable coefficient nonlinear equations has been increased drastically. This is due to the fact that these equations provide a more realistic description of numerous nonlinear phenomena when compared to constant-coefficient counterparts. Therefore, it is imperative to consider and incorporate these variable coefficients. Many researchers have dedicated their efforts to the study of nonlinear evolution equations that incorporate time-dependent variable coefficients, and they have put forth precise, explicit solutions as a result [20, 21, 26].

Herein, we will consider the variable coefficient Klein–Gordon–Zakharov equations of the following form:

$$\begin{aligned} \phi _{tt}-\phi _{xx}+\phi +\lambda (t)\psi \phi =0,\nonumber \\ \psi _{tt}-\psi _{xx}-\sigma (t)(|\phi |^2)_{xx}=0, \end{aligned}$$
(1.2)

where the complex-valued function \(\phi \) represents the fast time scale component of electric field, and the real-valued function \(\psi \) represents the deviation of ion density from its equilibrium [4].

This study aims to investigate the Lie symmetries of the variable coefficient Klein–Gordon–Zakharov equations (1.2) and present the exact analytical solutions.

This paper’s structure involves an overview of the classical Lie symmetry in Sect. 2. In Sect. 3, we present the symmetries of Eq. (1.2) along with the forms of variable coefficients. In Sect. 4, the similarity variables and similarity reductions have been established. We observed that the similarity reductions are complex in nature and lack Lie symmetry. Therefore, in Sect. 5, we employed the power series method on the systems of reduced ODEs and proposed the exact series solutions. Concluded remarks are given in Sect. 6.

2 Classical Lie group method: an algorithmic overview

The classical Lie Symmetry approach primarily involves achieving a reduction of partial differential equations (PDEs) by identifying symmetries using determining equations derived from the requirement of invariance within the PDE system. More specifically, when a given set of PDEs is rendered invariant under a one-parameter symmetry group of transformations, it leads to an over-defined system of linear homogeneous partial differential equations. This system is employed to determine the infinitesimal elements of the transformation group. These infinitesimals of the transformations are instrumental in achieving a reduction of the PDE system. The classical Lie symmetry method involves the following key steps:

  • Symmetry transformations: The first step is to find the Lie group that generates the symmetries which maintain the invariance of a given differential equation or system of differential equations. A symmetry transformation represents a change of variables that preserves the essential structure of the differential equation.

  • Lie algebra: The next step is to compute the Lie algebra corresponding to the Lie group which characterizes the infinitesimal versions of these transformations, often represented as vector fields.

  • Infinitesimal generators: Determine the infinitesimal generators within the Lie algebra. These generators serve as the basis for generating one-parameter groups of transformations.

  • Invariance condition: Consider whether the differential equation remains invariant under the action of these infinitesimal generators. The invariance condition is met when the Lie derivative of the equation with respect to the infinitesimal generator equals zero. This gives rise to a set of determining equations.

  • Solving the determining equations: solve the determining equations to identify the infinitesimal generators and, consequently, the Lie symmetries inherent to the differential equation.

  • Reduction: Upon discovering the symmetries, the differential equation can be reduced, potentially leading to simpler or even separable equations. This simplification is advantageous for finding exact solutions, offering insights into the behavior of the system governed by these equations.

3 Symmetry analysis

In this section, we work out the Lie point symmetry of the equation (1.2) by adopting the group theoretic technique. Substituting \(\phi = u+iv, \psi =w \) in Eq. (1.2). By separating the corresponding real and imaginary parts in the resulting expression, we establish a set of three equations

$$\begin{aligned} u_{tt}-u_{xx}+u+\lambda (t)uw=0,\nonumber \\ v_{tt}-v_{xx}+v+\lambda (t)vw=0,\nonumber \\ w_{tt}-w_{xx}-\sigma (t)(u^2+v^2)_{xx}=0. \end{aligned}$$
(3.1)

Consider a one-parameter Lie group of infinitesimal transformations acting on the space of independent and dependent variables under which the equations (3.1) remains invariant of the form [13]:

$$\begin{aligned} x_{1}&=x+\epsilon \xi (t,x,u,v,w)+O(\epsilon ^2),\nonumber \\ t_{1}&=t+\epsilon \eta (t,x,u,v,w)+O(\epsilon ^2),\nonumber \\ u_{1}&=u+\epsilon \tau (t,x,u,v,w)+O(\epsilon ^2),\nonumber \\ v_{1}&=v+\epsilon \zeta (t,x,u,v,w)+O(\epsilon ^2),\nonumber \\ w_{1}&=w+\epsilon \Theta (t,x,u,v,w)+O(\epsilon ^2), \end{aligned}$$
(3.2)

where \(\epsilon \) is a group parameter and \(\xi , \eta , \zeta , \tau , \Theta \) are the group infinitesimals. On using the group of point transformations (3.2) in Eq. (3.1), and equating the coefficients of various partial derivatives of u, v and w with respect to independent variables x and t, the following list of determining equations are obtained from the first equation of (3.1),

$$\begin{aligned}&\xi _{u}=\xi _{v}=\xi _{w}=0, \,\,\eta _{u}=\eta _{v}=\eta _{w}=0,\nonumber \\&\tau _{uu}=\tau _{vv}=\tau _{ww}=-2\tau _{xv}=-2\tau _{xw}=0,\nonumber \\&\tau _{vt}=\tau _{wt}=\tau _{uv}=\tau _{uw}=\tau _{vw}=\tau _{w}=0,\nonumber \\&-\xi _{t}+\eta _{x}=0, \,\, \xi _{x}-\eta _{t}=0,\nonumber \\&2\tau _{ut}-\eta _{tt}+\eta _{xx}=0, \,\, -\xi _{tt}-2\tau _{xu}+\xi _{xx}=0,\nonumber \\&\tau _{tt}-\tau _{xx}+\lambda ^{'}(t)uw\eta +\tau +\lambda (t)w\tau +\lambda (t) u\Theta -u\tau _{u}+2u\eta _{t} -\lambda (t)uw\tau _{u}\nonumber \\&+2\lambda (t)uw\eta {t}-v\tau _{v}-\lambda (t)vw\tau _{v}=0. \end{aligned}$$

Similarly, from the second equation of the system (3.1), the list of determining equations are as follows:

$$\begin{aligned}&\zeta _{w}= \zeta _{uv}=\zeta _{uu}=\zeta _{vv}=2\zeta _{xu}=\zeta _{ut}=0,\nonumber \\&\zeta _{ww}=\zeta _{uw}=\zeta _{vw}=\zeta _{xw}=\zeta _{tw}=0,\nonumber \\&\xi _{x}-\eta _{t}=0,\,\, \xi _{xx}-\xi _{tt}-2\zeta _{xv}=0,\nonumber \\&2\zeta _{tv}-\eta _{tt}+\eta _{xx}=0,\nonumber \\&\zeta _{tt}-\zeta _{xx}-u\zeta _{u}-\lambda (t)uw\zeta _{u}-v\zeta _{v}+\zeta +\lambda ^{'}(t)vw\eta +\lambda (t)v\Theta +\lambda (t)w\zeta \nonumber \\&-\lambda (t)vw\zeta _{v}+2v\eta _{t}+2\lambda (t)vw\eta _{t}=0. \end{aligned}$$

Similarly, the third equation of the system (3.1) gives the following additional equations:

$$\begin{aligned}&\eta _{x}=0, \xi _{t}=0, \Theta _{tv}=\Theta _{tu}=0,\nonumber \\&\Theta _{uu}=\Theta _{ww}=\Theta _{uv}=\Theta _{uw}=\Theta _{vw}=\Theta _{vv}=0,\nonumber \\ {}&2\Theta _{tw}-\eta _{tt}=0,\,\,-2\Theta _{xw}-4\sigma (t)u\tau _{xw}+\xi _{xx}-4\sigma (t)v\zeta _{xw}=0,\nonumber \\ {}&-2\Theta _{xu}-4\sigma (t)\tau _{x}-4\sigma (t)u\tau _{xu}+2\sigma (t)u\xi _{xx}-4\sigma (t)v\zeta _{xu}=0,\nonumber \\ {}&-2\Theta _{xv}-4\sigma (t)\zeta _{x}-4\sigma (t)u\tau _{xv}-4\sigma (t)v\zeta _{xv}+2\sigma (t)v\xi _{xx}=0, \nonumber \\ {}&-\Theta _{vv}-2\sigma ^{'}(t)\eta -4\sigma (t)\tau _{u}+4\sigma (t)\xi _{x}+2\sigma (t)\Theta _{w}-4\sigma (t)\eta _{t}=0,\nonumber \\ {}&-\Theta _{vv}-2\sigma ^{'}(t)\eta -4\sigma (t)\zeta _{v}+4\sigma (t)\xi _{x}+2\sigma (t)\Theta _{w}-4\sigma (t)\eta _{t}=0,\nonumber \\ {}&-2\sigma ^{'}(t)v\eta -2\sigma (t)\zeta +4\sigma (t)v\xi _{x}-2\sigma (t)v\zeta _{v} +2v\sigma (t)\Theta _{w}-4v\sigma (t)\eta _{t}-2\sigma (t)u\tau _{v}=0,\nonumber \\ {}&-2\sigma ^{'}(t)u\eta -2\sigma (t)\tau +4\sigma (t)u\xi _{x}-2\sigma (t)u\tau _{u} +2u\sigma (t)\Theta _{w}-4u\sigma (t)\eta _{t}-2\sigma (t)v\zeta _{u}=0,\nonumber \\ {}&2\xi _{x}-2\sigma (t)u\tau _{w}-2\sigma (t)v\zeta _{w}-2\eta _{t}=0,\nonumber \\ {}&4\sigma (t)\tau _{v}+4\sigma (t)\zeta _{u}+\sigma (t)u\tau _{uv}+4\sigma (t)v\zeta _{vu}=0,\nonumber \\&\Theta _{tt}-\Theta _{xx}-2\sigma (t)u\tau _{xx}-2\sigma (t)v\zeta _{xx}-u\Theta _{u} -\lambda (t)uw\Theta _{u}-v\Theta _{v}-\lambda (t)vw\Theta _{v}=0. \end{aligned}$$

On simplifying the above three sets of equations, we get the following list of determining equations as follows:

$$\begin{aligned}&\xi =\xi _{x},\,\, \eta =\eta _{t},\,\,\,\xi _{x}-\eta _{t}=0,\,\,\, \xi _{xx}-2\zeta _{xv}=0,\nonumber \\&\tau _{uu}=\tau _{vv}=\tau _{vt}=\tau _{uv}=\tau _{xv}=\tau _{w}=\zeta _{w}=0,\nonumber \\ {}&\zeta _{w}=\zeta _{uv}=\zeta _{uu}=\zeta _{vv}=2\zeta _{xu}=\zeta _{ut}=0,\nonumber \\ {}&\Theta _{tv}=\Theta _{tu}=\Theta _{uu}=\Theta _{ww}=\Theta _{uv}=\Theta _{uw}=\Theta _{vw}=\Theta _{vv}=0,\nonumber \\ {}&2\tau _{ut}-\eta _{tt}=0,\,\,\,-2\tau _{xu}+\xi _{xx}=0,\nonumber \\ {}&2\zeta _{tv}-\eta _{tt}=0,\,\,\, 2\Theta _{tw}-\eta _{tt}=0,\nonumber \\ {}&-2\Theta _{xw}-4\sigma (t)u\tau _{xw}+\xi _{xx}-4\sigma (t)v\zeta _{xw}=0,\nonumber \\ {}&-2\Theta _{xu}-4\sigma (t)\tau _{x}-4\sigma (t)u\tau _{xu}+2\sigma (t)u\xi _{xx}-4\sigma (t)v\zeta _{xu}=0,\nonumber \\ {}&-2\Theta _{xv}-4\sigma (t)\zeta _{x}-4\sigma (t)u\tau _{xv}-4\sigma (t)v\zeta _{xv}+2\sigma (t)v\xi _{xx}=0, \nonumber \\ {}&-2\sigma ^{'}(t)\eta -4\sigma (t)\tau _{u}+4\sigma (t)\xi _{x}+2\sigma (t)\Theta _{w}-4\sigma (t)\eta _{t}=0,\nonumber \\ {}&-2\sigma ^{'}(t)\eta -4\sigma (t)\zeta _{v}+4\sigma (t)\xi _{x}+2\sigma (t)\Theta _{w}-4\sigma (t)\eta _{t}=0,\nonumber \\ {}&\Theta _{tt}-\Theta _{xx}-2\sigma (t)u\tau _{xx}-2\sigma (t)v\zeta _{xx}-u\Theta _{u}-\lambda (t)uw\Theta _{u} -v\Theta _{v}-\lambda (t)vw\Theta _{v}=0,\nonumber \\ {}&-2\sigma ^{'}(t)v\eta -2\sigma (t)\zeta +4\sigma (t)v\xi _{x}-2\sigma (t)v\zeta _{v}+2v\sigma (t) \Theta _{w}-4v\sigma (t)\eta _{t}-2\sigma (t)u\tau _{v}=0,\nonumber \\ {}&-2\sigma ^{'}(t)u\eta -2\sigma (t)\tau +4\sigma (t)u\xi _{x}-2\sigma (t)u\tau _{u}+2u\sigma (t) \Theta _{w}-4u\sigma (t)\eta _{t}-2\sigma (t)v\zeta _{u}=0,\nonumber \\ {}&\zeta _{tt}-\zeta _{xx}-u\zeta _{u}-\lambda (t)uw\zeta _{u}-v\zeta _{v}+\zeta +\lambda ^{'}(t)vw\eta +\lambda (t)v\Theta +\lambda (t)w\zeta -\lambda (t)vw\zeta _{v}\nonumber \\ {}&+2v\eta _{t}+2\lambda (t)vw\eta _{t}=0,\nonumber \\ {}&\tau _{tt}-\tau _{xx}+\lambda ^{'}(t)uw\eta +\tau +\lambda (t)w\tau +\lambda (t) u\Theta -u\tau _{u}+2u\eta _{t} -\lambda (t)uw\tau _{u}\nonumber \\ {}&+2\lambda (t)uw\eta _{t}-v\tau _{v}-\lambda (t)vw\tau _{v}=0. \end{aligned}$$
(3.3)

Next, on solving the equations (3.3), we get the infinitesimals as follows:

$$\begin{aligned} \xi&=\alpha x+\beta ,\nonumber \\ \eta&=\alpha t+\delta ,\nonumber \\ \tau&=\rho u-\gamma v\nonumber \\ \zeta&=\gamma u+\rho v,\nonumber \\ \Theta&=\mu w-\frac{2\alpha }{\lambda (t)}, \end{aligned}$$
(3.4)

where \(\alpha , \delta , \gamma , \rho , \beta , \mu \) are arbitrary constants and the arbitrary variable coefficients \(\lambda (t)\) and \(\sigma (t)\) are determined by the following equations:

$$\begin{aligned}&2\alpha \Big (\frac{\lambda (t)\lambda ^{''}(t)-2(\lambda ^{'}(t))^2}{\lambda ^3(t)}\Big )=0, \end{aligned}$$
(3.5)
$$\begin{aligned}&-2\alpha t \sigma ^{'}(t)-2\delta \sigma ^{'}(t)-4\rho \sigma (t)+2\mu \sigma (t)=0, \end{aligned}$$
(3.6)
$$\begin{aligned}&\alpha t \lambda ^{'}(t)+\delta \lambda ^{'}(t)+2\alpha \lambda (t)+\mu \lambda (t)=0. \end{aligned}$$
(3.7)

From Eq. (3.5), we consider two cases:

Case 1: For \(\alpha =0\), the infinitesimals reduce to

$$\begin{aligned} \xi&=\beta ,\nonumber \\ \eta&=\delta ,\nonumber \\ \tau&=\rho u-\gamma v\nonumber \\ \zeta&=\gamma u+\rho v,\nonumber \\ \Theta&=\mu w, \end{aligned}$$
(3.8)

and equations (3.6), (3.7) give the forms of variable coefficients as:

$$\begin{aligned} \sigma (t)&=Ae^{\frac{2\mu -4\rho }{2\delta }t},\nonumber \\ \lambda (t)&=De^{-\frac{\mu }{\delta }t}, \delta \ne 0. \end{aligned}$$
(3.9)

Case 2: For \(\alpha \ne 0\) in (3.5), we get

$$\begin{aligned} \lambda (t)=-\frac{1}{\kappa _{1}t+\kappa _{2}}. \end{aligned}$$
(3.10)

On using Eq. (3.10) in Eq. (3.7), we have

$$\begin{aligned} (\alpha +\mu )\kappa _{1}=0,\nonumber \\ \kappa _{1}\delta -(2\alpha +\mu )\kappa _{2}=0. \end{aligned}$$
(3.11)

Now, from the first equation (3.11), we consider the following two subcases:

Subcase 1: For \(\kappa _{1}=0\), we get \(\mu =-2\alpha \), and the infinitesimals reduce to

$$\begin{aligned} \xi&=\alpha x+\beta ,\nonumber \\ \eta&=\alpha t+\delta ,\nonumber \\ \tau&=\rho u-\gamma v\nonumber \\ \zeta&=\gamma u+\rho v,\nonumber \\ \Theta&=\mu w+2\alpha \kappa _{2}. \end{aligned}$$
(3.12)

From Eq. (3.6), we get

$$\begin{aligned} \sigma (t)=(\alpha t+\delta )^{-\frac{2\rho }{\alpha }-2}, \alpha \ne 0, \end{aligned}$$
(3.13)

and

$$\begin{aligned} \delta (t)=-\frac{1}{\kappa _{2}}. \end{aligned}$$
(3.14)

Subcase 2: For \(\mu =-\alpha \), we have \(\kappa _{2}=\frac{\kappa _{1}\delta }{\alpha }\), and the infinitesimals reduce to

$$\begin{aligned} \xi&=\alpha x+\beta ,\nonumber \\ \eta&=\alpha t+\delta ,\nonumber \\ \tau&=\rho u-\gamma v\nonumber \\ \zeta&=\gamma u+\rho v,\nonumber \\ \Theta&=\mu w+2\alpha \Big (\kappa _{1}t+\frac{\kappa _{1}\delta }{\alpha }\Big ). \end{aligned}$$
(3.15)

From Eq. (3.6), we get

$$\begin{aligned} \sigma (t)=(\alpha t+\delta )^{-\frac{2\rho }{\alpha }-1}, \alpha \ne 0, \end{aligned}$$
(3.16)

and

$$\begin{aligned} \delta (t)=-\frac{1}{\kappa _{1}t+\frac{\kappa _{1}\delta }{\alpha }}. \end{aligned}$$
(3.17)

4 Similarity reductions

Here, we will wor kout the similarity variables and the similarity reductions for each case given in section (3). Similarity variables are obtained by solving the following characteristics: equations [29]

$$\begin{aligned} \frac{\textrm{d}x}{\xi }&=\frac{\textrm{d}t}{\eta }=\frac{\textrm{d}u}{\tau }=\frac{\textrm{d}v}{\zeta }=\frac{\textrm{d}w}{\Theta }, \end{aligned}$$
(4.1)

where \(\xi , \eta , \tau , \zeta , \Theta \) are given in section (3).

Case 1: The similarity variables corresponding to the infinitesimals (3.8) can be obtained by solving the characteristics equations (4.1) given by

$$\begin{aligned} \frac{\textrm{d}x}{\beta }&=\frac{\textrm{d}t}{\delta }=\frac{\textrm{d}u}{\rho u-\gamma v}=\frac{\textrm{d}v}{\gamma u+\rho v}=\frac{\textrm{d}w}{\mu w}, \end{aligned}$$
(4.2)

which is further equal to

$$\begin{aligned} \frac{\textrm{d}x}{\beta }&=\frac{\textrm{d}t}{\delta }=\frac{\textrm{d}\phi }{(\rho +i\gamma )\phi }=\frac{d\psi }{\mu \psi }. \end{aligned}$$
(4.3)

On solving it, we get the similarity variables as follows:

$$\begin{aligned} \theta&=\delta x-\beta t,\nonumber \\ \phi&=e^{\frac{\rho +i\gamma }{\delta }t}F(\theta ),\nonumber \\ \psi&=e^{\frac{\mu }{\delta }}G(\theta ). \end{aligned}$$
(4.4)

Using this equation in Eq. (1.2), we get the reduced system of ordinary differential equations (ODEs):

$$\begin{aligned}&\Big ({\frac{c_{3} +i\gamma }{\delta }}\Big )^2F-2\beta \Big ({\frac{\rho +i\gamma }{\delta }t}\Big )F^{'} +\beta ^2F^{''}+F-\delta ^2F^{''}+DFG=0,\nonumber \\&\frac{\mu ^2}{\delta ^2}G-\frac{2\beta \mu }{\delta }G^{'}+(\beta ^2-\delta ^2) G^{''}-A\delta ^2(|F|^{''})^2=0. \end{aligned}$$
(4.5)

Next, on putting \(F=U(\theta )e^{iV(\theta )}\), and on splitting the real and imaginary parts, we obtain the system of three nonlinear ODEs as follows:

$$\begin{aligned}&\Big ({\frac{\rho ^2-\gamma ^2}{\delta ^2}}\Big )U-\frac{2\beta \rho }{\delta }U^{'} +\frac{2\beta \gamma }{\delta }UV^{'}+(\beta ^2-\delta ^2)U^{''}-(\beta ^2-\delta ^2)U(V^{'})^2+U+DUG=0,\nonumber \\&\frac{2\gamma \rho }{\delta ^2}U-\frac{2\beta \gamma }{\delta }U^{'}-\frac{2\beta \rho }{\delta }UV^{'} +2(\beta ^2-\delta ^2)U^{'}V^{'}+(\beta ^2-\delta ^2)UV^{''}=0,\nonumber \\&\frac{\mu ^2}{\delta ^2}G-\frac{2\beta \mu }{\delta }G^{'}+(\beta ^2-\delta ^2) G^{''}-2A\delta ^2UU^{''}-2A\delta ^2(U^{'})^2=0. \end{aligned}$$
(4.6)

Case 2: For \(\alpha \ne 0\), the similarity variables are given by the following subcases:

Subcase 1: The similarity variables corresponding to the infinitesimal symmetries (3.12) are given by

$$\begin{aligned} \theta&=\frac{\alpha x+\beta }{\alpha t+\delta },\nonumber \\ \phi&=(\alpha t+\delta )^{\frac{\rho +i\gamma }{\alpha }}F(\theta ),\nonumber \\ \psi&=\kappa _{2}-(\alpha t+\delta )^{-2}G(\theta ). \end{aligned}$$
(4.7)

Using this equation in Eq. (1.2), we get the reduced system of ODEs

$$\begin{aligned}&\Big ({\frac{c_{3} +i\gamma }{\alpha }}\Big )^2F-\Big ({\frac{\rho +i\gamma }{\alpha }}\Big )F-2(\rho +i\gamma )\alpha \theta F^{'}+\alpha ^2\theta ^2F^{''}+2\alpha ^2\theta F^{'}-\alpha ^2F^{''}+\frac{1}{\kappa _{2}}FG=0,\nonumber \\&6G+4\theta G^{'}+\theta ^2G^{'}+(|F|^{''})^2-(1-\theta ^2) G^{''}=0. \end{aligned}$$
(4.8)

Further, on using \(F=U(\theta )e^{iV(\theta )}\) in the equations (4.8), and on separating the real and imaginary parts, we have

$$\begin{aligned}&\Big ({\frac{\rho ^2-\gamma ^2}{\alpha ^2}}\Big )U\!-\!{\frac{\rho }{\alpha }}U+2\alpha \theta \rho U^{'} \!-\!2\alpha \gamma \theta UV^{'}+2\alpha ^2\theta U^{'}+\alpha ^2(\theta ^2-1) U^{''} \!-\!\alpha ^2(1-\theta ^2)U(V^{'})^2+\frac{1}{\kappa _{2}}UG=0,\nonumber \\&\frac{2\gamma \rho }{\alpha ^2}U-\frac{\gamma }{\alpha }U+2\alpha \rho \theta UV^{'} +2\alpha \gamma \theta U^{'}+2\alpha ^2\theta UV^{'}+2\alpha ^2(\theta ^2-1)U^{'}V^{'} +\alpha ^2(\theta ^2-1)UV^{''}=0,\nonumber \\&6G+4\theta G^{'}+\theta ^2G^{'}+2UU^{''}+2(U^{'})^2-(1-\theta ^2) G^{''}=0. \end{aligned}$$
(4.9)

Subcase 2: Similarity variable corresponding to symmetries (3.15) are given by

$$\begin{aligned} \theta&=\frac{\alpha x+\beta }{\alpha t+\delta },\nonumber \\ \phi&=(\alpha t+\delta )^{\frac{\rho +i\gamma }{\alpha }}F(\theta ),\nonumber \\ \psi&=\frac{\kappa _{1}(\alpha t+\delta )}{\alpha }+(\alpha t+\delta )^{-1}G(\theta ). \end{aligned}$$
(4.10)

Using this equation in Eq. (1.2), we get the reduced system of ODEs:

$$\begin{aligned}&(\rho +i\gamma )^2F-(\rho +i\gamma )\alpha F-2(\rho +i\gamma )\alpha \theta F^{'} +\alpha ^2(\theta ^2-1)F^{''}+\alpha ^2\theta F^{'}-\frac{\alpha }{\kappa _{2}}FG=0,\nonumber \\&2G+4\theta G^{'}-(|F|^{''})^2+(\theta ^2-1) G^{''}=0. \end{aligned}$$
(4.11)

On using \(F=U(\theta )e^{iV(\theta )}\) in equations (4.11), and on separating the real and imaginary parts, we obtain

$$\begin{aligned}&(\rho ^2-\gamma ^2)U-\alpha \rho U-2\alpha \theta \rho U^{'}+2\alpha \gamma \theta UV^{'} +\alpha ^2\theta U^{'}+\alpha ^2(\theta ^2-1) U^{''}-\alpha ^2(\theta ^2-1)U(V^{'})^2+\frac{\alpha }{\kappa _{1}}UG=0,\nonumber \\&2\gamma \rho U-\gamma \alpha U-2\alpha \gamma \theta U^{'}-2\alpha \rho \theta UV^{'} +\alpha ^2\theta UV^{'}+2\alpha ^2(\theta ^2-1)U^{'}V^{'}+\alpha ^2(\theta ^2-1)UV^{''}=0\nonumber \\&2G+4\theta G^{'}-2UU^{''}-2(U^{'})^2+(\theta ^2-1) G^{''}=0. \end{aligned}$$
(4.12)

Subsequently, our objective is to find the exact explicit solutions of the system of nonlinear ODEs (4.6), (4.9), and (4.12) in terms of some known mathematical functions. However, this task is rendered unattainable due to the absence of Lie symmetries within these second-order nonlinear ODEs. As a result, we resort to establish the exact series solutions to these nonlinear ODEs. However, different mathematicians have utilized the power series method to extract the exact series solutions for various nonlinear ODEs [24]–[39]. Thus, in the next section, we put forward the applications of the power series method and subsequently provide the series solutions for the ODE system.

5 Power series solutions

Herein, we proposed the series solutions of the reduced ordinary differential equations by adopting the power series method.

Case 1: Let us suppose the power series solution of the system (4.6) in the form:

$$\begin{aligned} U(\theta )=\sum \limits ^{\infty }_{n=0}p_{n}\theta ^n,\,\,V(\theta )=\sum \limits ^{\infty }_{n=0}q_{n} \theta ^n,\,\,G=\sum \limits ^{\infty }_{n=0}r_{n}\theta ^{n}, \end{aligned}$$
(5.1)

where \(p_{n},\,\, q_{n},\,\,r_{n}\) are real constants.

Substituting equations (5.1) into system (4.6), we get

$$\begin{aligned}&\Big ({\frac{\rho ^2-\gamma ^2}{\delta ^2}}\Big )\sum \limits ^{\infty }_{n=0}p_{n}\theta ^n-\frac{2\beta \rho }{\delta }\sum \limits ^{\infty }_{n=0}(n+1)p_{n+1}\theta ^{n}+\frac{2\beta \gamma }{\delta }\sum \limits ^{\infty }_{n=0}\sum \limits ^{n}_{j=0}(n-j+1))p_{j}q_{n-j+1}\theta ^{n}\nonumber \\&\quad +(\beta ^2-\delta ^2)\sum \limits ^{\infty }_{n=0}(n+1)(n+2)p_{n+2}\theta ^{n}\nonumber \\&\quad -(\beta ^2-\delta ^2) \sum \limits ^{\infty }_{n=0}\sum \limits ^{n}_{j=0}\sum \limits ^{j}_{i=0}(i+1)(j-i+1)q_{i+1}q_{j-i+1}p_{n-j}\theta ^{n}\nonumber \\&\quad +\sum \limits ^{\infty }_{n=0}p_{n}\theta ^n+D\sum \limits ^{\infty }_{n=0}\sum \limits ^{n}_{j=0}p_{j}r_{n-j}\theta ^{n}=0, \end{aligned}$$
(5.2)
$$\begin{aligned}&\frac{2\gamma \rho }{\delta ^2}\sum \limits ^{\infty }_{n=0}p_{n}\theta ^n-\frac{2\beta \gamma }{\delta } \sum \limits ^{\infty }_{n=0}(n+1)p_{n+1}\theta ^{n}+2(\beta ^2-\delta ^2)\sum \limits ^{\infty }_{n=0} \sum \limits ^{n}_{j=0}(j+1)(n-j+1)p_{j+1}q_{n-j+1}\theta ^{n}\nonumber \\&\quad -\frac{2\beta \rho }{\delta }\sum \limits ^{\infty }_{n=0}\sum \limits ^{n}_{j=0}(n-j+1)p_{j}q_{n-j+1}\theta ^{n}\nonumber \\&\quad +(\beta ^2-\delta ^2)\sum \limits ^{\infty }_{n=0}\sum \limits ^{n}_{j=0}(n-j+1)(n-j+2)p_{j}q_{n-j+2}\theta ^{n}=0, \end{aligned}$$
(5.3)
$$\begin{aligned}&\frac{\mu ^2}{\delta ^2}\sum \limits ^{\infty }_{n=0}r_{n}\theta ^{n} -\frac{2\beta \mu }{\delta }\sum \limits ^{\infty }_{n=1}(n+1)r_{n+1}\theta ^{n}-2A\delta ^2\sum \limits ^{\infty }_{n=0}\sum \limits ^{n}_{j=0}(j+1)(n-j+1)p_{j+1}p_{n-j+1}\theta ^{n}\nonumber \\&\quad +(\beta ^2-\delta ^2)\sum \limits ^{\infty }_{n=0}(n+1)(n+2)r_{n+2}\theta ^{n}\nonumber \\&\quad -2A\delta ^2\sum \limits ^{\infty }_{n=0}\sum \limits ^{n}_{j=0}(n-j+1)(n-j+2)p_{j}p_{n-j+2}\theta ^{n}=0. \end{aligned}$$
(5.4)

On comparing the coefficients of \(\theta ^{n}\) from equations (5.2)–(5.4), for \(n=0\), we get

$$\begin{aligned}&p_{2}=\frac{1}{2(\beta ^2-\delta ^2)}\Big (\frac{2\beta \rho }{\delta }p_{1}-p_{0} -\frac{\rho ^2-\gamma ^2}{\delta ^2}p_{0}-\frac{2\beta \gamma }{\delta }p_{0}q_{1}-p_{0}r_{0}+(\beta ^2-\delta ^2)p_{0}q_{1}^2\Big ), \end{aligned}$$
(5.5)
$$\begin{aligned}&q_{2}=\frac{1}{(\beta ^2-\delta ^2)p_{0}}\Big (\frac{\beta \gamma }{\delta }p_{1} -\frac{\rho \gamma }{\delta ^2}p_{0}+\frac{\beta \rho }{\delta }p_{0}q_{1}-(\beta ^2-\delta ^2)p_{1}q_{1}\Big ), \end{aligned}$$
(5.6)
$$\begin{aligned}&r_{2}=\frac{1}{2(\beta ^2-\delta ^2)}\Big (\frac{2\beta \mu }{\delta }r_{1} -\frac{\mu ^2}{\delta ^2}r_{0}+2A\delta ^2p_{1}^2+4A\delta ^2p_{0}p_{2}\Big ), \end{aligned}$$
(5.7)

where \(p_{2}\) is given by Eq. (5.5).

For \(n=1, 2, \ldots \), the general recurrence relations are follows:

$$\begin{aligned} p_{n+2}&=\frac{1}{(\beta ^2-\delta ^2)(n+1)(n+2)}\Bigg [\frac{2\beta \rho }{\delta }(n+1)p_{n+1} -p_{n}-\frac{\rho ^2-\gamma ^2}{\delta ^2}p_{n}-\sum \limits ^{n}_{j=0}p_{j}r_{n-j}\nonumber \\&\quad -\frac{2\beta \gamma }{\delta }\sum \limits ^{n}_{j=0}(n-j+1)p_{j}q_{n-j+1}+(\beta ^2-\delta ^2) \sum \limits ^{n}_{j=0}\sum \limits ^{j}{i=0}(i+1)(j-i+1)q_{i+1}q_{j-i+1}p_{n-j}\Bigg ], \end{aligned}$$
(5.8)
$$\begin{aligned} q_{n+2}&=\frac{1}{(\beta ^2-\delta ^2)(n+1)(n+2)p_{0}}\Bigg [\frac{2\beta \gamma }{\delta }(n+1)p_{n+1} -(\beta ^2-\delta ^2)\sum \limits ^{n}_{j=1}(n-j+1)(n-j+2)p_{j}q_{n-j+2}\nonumber \\&\quad -\frac{2\rho \gamma }{\delta ^2}p_{n}+\frac{2\beta \rho }{\delta }\sum \limits ^{n}_{j=0}(n-j+1)p_{j}q_{n-j+1} -2(\beta ^2-\delta ^2)\sum \limits ^{n}_{j=0}(j+1)(n-j+1)p_{j+1}q_{n-j+1}\Bigg ], \end{aligned}$$
(5.9)
$$\begin{aligned} r_{n+2}&=\frac{1}{(\beta ^2-\delta ^2)(n+1)(n+2)}\Bigg [\frac{2\beta \mu }{\delta }(n+1)r_{n+1} -\frac{\mu ^2}{\delta ^2}r_{n}+2A\delta ^2\sum \limits ^{n}_{j=0}(j+1)(n-j+1)p_{j+1}p_{n-j+1}\nonumber \\&\quad +2A\delta ^2\sum \limits ^{n}_{j=0}(n-j+1)(n-j+2)p_{j}p_{n-j+2}\Bigg ]. \end{aligned}$$
(5.10)

From these recurrence relations, we can find all the terms of sequences \(\{ p_{n} \}^{\infty }_{n=0}, \{q_{n} \}^{\infty }_{n=0}\), and \(\{ r_{n} \}^{\infty }_{n=0} \).

Therefore, it indicates that for system of ODEs (4.6), the power series solution (5.1) is feasible with the coefficients which are given by equations (5.8), (5.9), and (5.10). Next, we showed the convergence of series solution (5.1) to the equation (4.6).

From equations (5.8), we obtained

$$\begin{aligned} |p_{n+2}| \le M\Big [|p_{n}|\!+\!|p_{n+1}|\!+\!|\sum \limits ^{n}_{j=0}|p_{j}||q_{n-j\!+\!1}| \!+\!\sum \limits ^{n}_{j=0}|p_{j}||r_{n-j}|\!+\!\sum \limits ^{n}_{j=0}\sum \limits ^{j}_{i=0}|p_{n-j}||q_{i+1}||q_{j-i+1}|\Big ], n=1,2\ldots \end{aligned}$$
(5.11)

where \( M = \max \Big \{\frac{|2\beta \rho |}{|\delta ||\beta ^2-\delta ^2|}, \frac{|\delta ^2|+|\rho ^2-\gamma ^2|}{|\delta ^2||\beta ^2-\delta ^2|}, \frac{2|\beta \gamma |}{|\delta ||\beta ^2-\delta ^2|}\Big \}\). From Eq. (5.9), we get

$$\begin{aligned} |q_{n+2}| \le N\Big [|p_{n}|+|p_{n+1}|+\sum \limits ^{n}_{j=0}|p_{j}||q_{n-j+1}| +\sum \limits ^{k}_{j=0}|p_{j+1}||q_{n-j+1}|+\sum \limits ^{k}_{j=1}|p_{j}||q_{n-j+2}|\Big ], n=1,2\ldots , \end{aligned}$$
(5.12)

where \( N = \max \Big \{{\frac{|2\beta \gamma |}{|p_{0}||\delta ||\beta ^2-\delta ^2|}, \frac{|2\gamma \rho |}{|p_{0}||\beta ^2-\delta ^2||\delta ^2|}},\,\frac{|2\beta \rho |}{|p_{0}||\beta ^2 -\delta ^2||\delta |}, \frac{2}{|p_{0}|}, \frac{1}{|p_{0}|}\Big \}\).

Similarly, from Eq. (5.10), we obtained

$$\begin{aligned} |r_{n+2}| \le L\Big [|r_{n}|+|r_{n+1}|+\sum \limits ^{n}_{j=0}|p_{j+1}||p_{n-j+1}| +\sum \limits ^{n}_{j=0}|p_{j}||p_{n-j+2}|\Big ], n=1,2\ldots , \end{aligned}$$
(5.13)

where \( L = \max \Big \{{\frac{|2\beta \mu |}{|\delta ||\beta ^2-\delta ^2|}, \frac{|\mu ^2|}{|\beta ^2-\delta ^2||\delta ^2|}}, \,\frac{|2A\delta ^2|}{|\beta ^2-\delta ^2|}\Big \}\).

We now consider the three power series as follows:

$$\begin{aligned} X=X(\theta )=\sum \limits ^{\infty }_{n=0}x_{n}\theta ^{n},\,\,Y=Y(\theta )= \sum \limits ^{\infty }_{n=0}y_{n}\theta ^{n},\,\,Z=Z(\theta )=\sum \limits ^{\infty }_{n=0}z_{n}\theta ^{n}, \end{aligned}$$
(5.14)

with \(x_{i}=|p_{i}|\),   \(y_{i}=|q_{i}|\), and \(z_{i}=|r_{i}|\) for \(n=1,2,\ldots \). Therefore,

$$\begin{aligned}&x_{n+2}=M\Bigg [x_{n+1}+x_{n}+\sum \limits ^{n}_{j=0}x_{j}y_{n-j+1}+\sum \limits ^{n}_{j=0}x_{j}z_{n-j} +\sum \limits ^{n}_{j=0}\sum \limits ^{j}_{i=0}y_{i+1}y_{j-i+1}x_{n-j} \Bigg ],\nonumber \\&y_{n+2}=N\Bigg [x_{n+1}+x_{n}+\sum \limits ^{n}_{j=0}x_{j}y_{n-j+1}+\sum \limits ^{n}_{j=0}x_{j+1}y_{n-j+1} +\sum \limits ^{n}_{j=1}x_{j}y_{n-j+2}\Bigg ],\nonumber \\&z_{n+2}=L\Bigg [z_{n+1}+z_{n}+\sum \limits ^{n}_{j=0}x_{j+1}x_{n-j+1}+\sum \limits ^{n}_{j=0}x_{j}x_{n-j+2}\Bigg ]. \end{aligned}$$
(5.15)

We can see that \(|p_{n}|\le x_{n}\),   \(|q_{n}|\le y_{n}\), and \(|r_{n}|\le z_{n}\) where \(n=1,2,\ldots \).

This indicates that the series (5.14) are the majorant series of the series (5.1). Now, we have to show that the series (5.14) have positive radius of convergence.

Hence, by some mathematical calculations, we have

$$\begin{aligned} X(\theta )&=x_{0}+x_{1}\theta +x_{2}\theta ^2+x_{3}\theta ^3+\sum \limits ^{\infty }_{n=2}x_{n+2}\theta ^{n+2}\nonumber \\&= x_{0}+x_{1}\theta +x_{2}\theta ^2+x_{3}\theta ^3 + M\Bigg [\sum \limits ^{\infty }_{n=2}x_{n+1}\theta ^{n+2} +\sum \limits ^{\infty }_{n=2}x_{n}\theta ^{n+2}+\sum \limits ^{\infty }_{n=2}\sum \limits ^{n}_{j=0}x_{j}y_{n-j+1}\theta ^{n+2}\nonumber \\&\quad +\sum \limits ^{\infty }_{n=2}\sum \limits ^{n}_{j=0}x_{j}z_{n-j}\theta ^{n+2}+\sum \limits ^{\infty }_{n=2} \sum \limits ^{n}_{j=0}\sum \limits ^{j}_{i=0}y_{i+1}y_{j-i+1}x_{n-j}\theta ^{n+2} \Bigg ], \end{aligned}$$
(5.16)

which can be further expressed as:

$$\begin{aligned} X(\theta )&=x_{0}+x_{1}\theta +x_{2}\theta ^2+x_{3}\theta ^3+ M \Big [\theta X-x_{0}\theta -x_{1}\theta ^2-x_{2}\theta ^3+\theta ^2X-x_{0}\theta ^2-x_{1}\theta ^3+\theta ^2XZ\nonumber \\&\quad -x_{0}z_{0}\theta ^2-x_{0}z_{1}\theta ^3-x_{1}z_{0}\theta ^3+Y^2X-2y_{0}XY+2y_{0}^2X -y_{0}^2X-x_{0}y_{1}^2\theta ^2-2y_{1}y_{2}x_{0}\theta ^3\nonumber \\ {}&\quad -x_{1}y_{1}^2\theta ^3+\theta XY-\theta Xy_{0}-x_{0}y_{1}\theta ^2-x_{0}y_{2}\theta ^3-x_{1}y_{1}\theta ^3\Big ]. \end{aligned}$$
(5.17)
$$\begin{aligned} Y(\theta )&=y_{0}+y_{1}\theta +y_{2}\theta ^2+y_{3}\theta ^3+\sum \limits ^{\infty }_{n=2}x_{n+2}\theta ^{n+2}\nonumber \\&=y_{0}+y_{1}\theta +y_{2}\theta ^2+y_{3}\theta ^3+N\Bigg [\sum \limits ^{\infty }_{n=2}x_{n+1}\theta ^{n+2} +\sum \limits ^{\infty }_{n=2}x_{n}\theta ^{n+2}+\sum \limits ^{\infty }_{n=2}\sum \limits ^{n}_{j=0}x_{j}y_{n-j+1}\theta ^{n+2}\nonumber \\&\quad +\sum \limits ^{\infty }_{n=2}\sum \limits ^{n}_{j=0}x_{j+1}y_{n-j+1}\theta ^{n+2} +\sum \limits ^{\infty }_{n=2}\sum \limits ^{n}_{j=1}x_{j}y_{n-j+2}\theta ^{n+2}\Bigg ], \end{aligned}$$
(5.18)

which can be further expressed as:

$$\begin{aligned} Y(\theta )&=y_{0}+y_{1}\theta +y_{2}\theta ^2+y_{3}\theta ^3+N\Big [\theta X-x_{0}\theta -x_{1} \theta ^2-x_{2}\theta ^3+\theta ^2X-\theta ^2 x_{0}-\theta ^3 x_{1}+XY \theta \nonumber \\&\quad -y_{0}X\theta -x_{0}y_{1}\theta ^2-x_{0}y_{2}\theta ^3-y_{1}x_{1}\theta ^3 +2XY+x_{0}y_{1}-x_{1}y_{1}\theta ^2-x_{2}y_{1}\theta ^3-2y_{0}X-2x_{0}Y\nonumber \\&\quad +x_{0}y_{0}+y_{1}x_{0}\theta -y_{1}\theta X\Big ]; \end{aligned}$$
(5.19)

similarly,

$$\begin{aligned} Z(\theta )&=z_{0}+z_{1}\theta +z_{2}\theta ^2+z_{3}\theta ^3+\sum \limits ^{\infty }_{n=2}z_{n+2}\theta ^{n+2}\nonumber \\&=z_{0}+z_{1}\theta +z_{2}\theta ^2+z_{3}\theta ^3+L\Bigg [\sum \limits ^{\infty }_{n=2}z_{n+1}\theta ^{n+2} +\sum \limits ^{\infty }_{n=2}z_{n}\theta ^{n+2} +\sum \limits ^{\infty }_{n=2}\sum \limits ^{n}_{j=0}x_{j+1}x_{n-j+1}\theta ^{n+2}\nonumber \\&\quad +\sum \limits ^{\infty }_{n=2}\sum \limits ^{n}_{j=0}x_{j}x_{n-j+2}\theta ^{n+2}\Bigg ], \end{aligned}$$
(5.20)

which can be further expressed as:

$$\begin{aligned} Z(\theta )&=z_{0}+z_{1}\theta +z_{2}\theta ^2+z_{3}\theta ^3+L \Big [\theta Z-z_{0}\theta -z_{1} \theta ^2-z_{2}\theta ^3+\theta ^2 Z-\theta ^2 z_{0}-z_{1}\theta ^3+X^{2}-x_{0}X-x_{1}X\theta \nonumber \\&\quad -x_{0}x_{2}\theta ^2-x_{0}x_{3}\theta ^3+X^2-2x_{0}X+x_{0}^2-x_{1}^2 \theta ^2-x_{1}x_{2}\theta ^3 \Big ]. \end{aligned}$$
(5.21)

Let us take the implicit functional system w. r. t. the independent variable \(\theta \):

$$\begin{aligned} A(\theta , X, Y, Z)&=X-x_{0}-x_{1}\theta -x_{2}\theta ^2-x_{3}\theta ^3 -M \Big [\theta X-x_{0}\theta -x_{1}\theta ^2-x_{2}\theta ^3+\theta ^2X-x_{0}\theta ^2-x_{1}\theta ^3\nonumber \\&\quad +\theta ^2XZ-x_{0}z_{0}\theta ^2-x_{0}z_{1}\theta ^3-x_{1}z_{0}\theta ^3+Y^2X-2y_{0}XY+2y_{0}^2X -y_{0}^2X\nonumber \\&\quad -x_{0}y_{1}^2\theta ^2-2y_{1}y_{2}x_{0}\theta ^3-x_{1}y_{1}^2\theta ^3+\theta XY-\theta Xy_{0}-x_{0}y_{1}\theta ^2-x_{0}y_{2}\theta ^3-x_{1}y_{1}\theta ^3\Big ]=0,\nonumber \\ B(\theta , X, Y, Z)&=Y-y_{0}-y_{1}\theta -y_{2}\theta ^2-y_{3}\theta ^3-N\Big [\theta X-x_{0}\theta -x_{1} \theta ^2-x_{2}\theta ^3+\theta ^2X-\theta ^2 x_{0}-\theta ^3 x_{1} \nonumber \\&\quad +XY\theta -y_{0}X\theta -x_{0}y_{1}\theta ^2-x_{0}y_{2}\theta ^3-y_{1}x_{1}\theta ^3+2XY+x_{0}y_{1}-x_{1}y_{1} \theta ^2\nonumber \\&\quad -x_{2}y_{1}\theta ^3-2y_{0}X-2x_{0}Y+x_{0}y_{0}+y_{1}x_{0}\theta -y_{1}\theta X\Big ]=0,\nonumber \\ C(\theta , X, Y, Z)&=Z-z_{0}-z_{1}\theta -z_{2}\theta ^2-z_{3}\theta ^3-L \Big [\theta Z-z_{0} \theta -z_{1}\theta ^2-z_{2}\theta ^3+\theta ^2 Z-\theta ^2 z_{0}-z_{1}\theta ^3+X^{2}\nonumber \\&-x_{0}X-x_{1}X\theta -x_{0}x_{2}\theta ^2-x_{0}x_{3}\theta ^3+X^2-2x_{0}X+x_{0}^2-x_{1}^2 \theta ^2-x_{1}x_{2}\theta ^3 \Big ]=0. \end{aligned}$$
(5.22)

Since A, B, C are analytic in the neighborhood of \((0,x_{0}, y_{0}, z_{0})\) and \(A(0,x_{0}, y_{0}, z_{0})=0\), \(B(0,x_{0}, y_{0}, z_{0})=0,\,\, C(0,x_{0}, y_{0}, z_{0})=0\). Therefore, the Jacobian is given by

$$\begin{aligned} J=\frac{\partial (A, B, C)}{\partial (X, Y, Z)}=1\ne 0. \end{aligned}$$
(5.23)

If we assume the parameters \(x_{0}=|p_{0}|\), \(y_{0}=|q_{0}|\) and \(z_{0}=|r_{0}|\), then by using the Implicit function theorem [33], we find that \(X=X(\theta )\), \(Y=Y(\theta )\) and \(Z=Z(\theta )\) are analytic in the neighborhood of the point \((0,x_{0}, y_{0}, z_{0})\), and have positive radius of convergence. This indicates that the series solution (5.1) converges in a neighborhood of \((0,x_{0}, y_{0}, z_{0})\), and hence proved. Thus, the series solution of the system of equations (4.6) is written as:

$$\begin{aligned} U(\theta )&=p_{0}+p_{1}\theta +\frac{1}{2(\beta ^2-\delta ^2)}\Bigg (\frac{2\beta c_{3} }{\delta }p_{1}-p_{0}-\frac{\rho ^2-\gamma ^2}{\delta ^2}p_{0}-\frac{2\beta \gamma }{\delta }p_{0}q_{1}-p_{0}r_{0},\nonumber \\&\quad +(\beta ^2-\delta ^2)p_{0}q_{1}^2\Bigg )\theta ^2+\sum \limits ^{\infty }_{k=1}p_{k+2}\theta ^{k+2},\nonumber \\ V(\theta )&=q_{0}+q_{1}\theta +\frac{1}{(\beta ^2-\delta ^2)p_{0}}\Big (\frac{\beta \gamma }{\delta }p_{1} -\frac{\rho \gamma }{\delta ^2}p_{0}+\frac{\beta \rho }{\delta }p_{0}q_{1}-(\beta ^2-\delta ^2)p_{1}q_{1}\Big ) \theta ^2+\sum \limits ^{\infty }_{k=1}q_{k+2}\theta ^{k+2},\nonumber \\ G(\theta )&=r_{0}+r_{1}\theta +\frac{1}{2(\beta ^2-\delta ^2)}\Big (\frac{2\beta \mu }{\delta }r_{1} -\frac{\mu ^2}{\delta ^2}r_{0}+2A\delta ^2p_{1}^2+4A\delta ^2p_{0}p_{2}\Big )\theta ^2+\sum \limits ^{\infty }_{k=1}r_{k+2}\theta ^{k+2}. \end{aligned}$$
(5.24)

Series solutions for Case 2:

Subcase 1: Substituting equations (5.1) into system of equations (4.9), we get

$$\begin{aligned}&\Big ({\frac{\rho ^2-\gamma ^2-\alpha \rho }{\alpha ^2}}\Big )(p_{0}+p_{1}\theta )+\Big ({\frac{\rho ^2-\gamma ^2-\alpha \rho }{\alpha ^2}}\Big )\sum \limits ^{\infty }_{n=2}p_{n}\theta ^n+2\rho \alpha \theta p_{1}+2\rho \alpha \sum \limits ^{\infty }_{n=2}np_{n}\theta ^n+2\alpha ^2\theta p_{1}\nonumber \\&\quad +2\alpha ^2\sum \limits ^{\infty }_{n=2}np_{n}\theta ^{n}-2\alpha \gamma \sum \limits ^{\infty }_{n=2}\sum \limits ^{n-1}_{j=0}(n-j)p_{j}q_{n-j}\theta ^{n} -2\alpha \gamma \theta p_{0}q_{1}+\alpha ^2\sum \limits ^{\infty }_{n=2}n(n-1)p_{n}\theta ^{n}-2\alpha ^2p_{2}\nonumber \\&\quad -6\alpha ^2\theta p_{3}-\alpha ^2\sum \limits ^{\infty }_{n=2}(n+1)(n+2)p_{n+2}\theta ^n+\alpha ^2p_{0}q_{1}^2 +\alpha ^2(p_{1}q_{1}^2+4q_{1}q_{2}p_{0})\theta +\frac{1}{\kappa _{2}}p_{0}r_{0}\nonumber \\&\quad +\alpha ^2\sum \limits ^{\infty }_{n=2}\sum \limits ^{n}_{j=0} \sum \limits ^{j}_{i=0}(i+1)(j-i+1)q_{i+1}q_{j-i+1}p_{n-j}\theta ^{n} +\frac{1}{\kappa _{2}}(p_{1}r_{0}+p_{0}r_{1})\theta +\frac{1}{\kappa _{2}} \sum \limits ^{\infty }_{n=2}\sum \limits ^{n}_{j=0}p_{j}r_{n-j}\theta ^n\nonumber \\&\quad -\alpha ^2\sum \limits ^{\infty }_{n=2}\sum \limits ^{n-2}_{j=0}\sum \limits ^{j}_{i=0}(i+1)(j-i+1)q_{i+1}q_{j-i+1}p_{n-j-2}\theta ^{n} =0, \end{aligned}$$
(5.25)
$$\begin{aligned}&\quad \Big (\frac{2\gamma \rho -\alpha \gamma }{\alpha ^2}\Big )(p_{0}+p_{1}\theta ) +\Big (\frac{2\gamma \rho -\alpha \gamma }{\alpha ^2}\Big )\sum \limits ^{\infty }_{n=2}p_{n}\theta ^n+2\alpha \gamma p_{1}\theta +2\alpha \gamma \sum \limits ^{\infty }_{n=2}np_{n}\theta ^{n}+2\alpha \rho p_{0}p_{1}\theta \nonumber \\&\quad +2\alpha \rho \sum \limits ^{\infty }_{n=2}\sum \limits ^{n-1}_{j=0}(n-j)p_{j}p_{n-j} \theta ^{n}+2\alpha ^2p_{0}q_{1}\theta +2\alpha ^2\sum \limits ^{\infty }_{n=2}\sum \limits ^{n-1}_{j=0} (n-j)p_{j}q_{n-j}\theta ^{n}-2\alpha ^2p_{1}q_{1}\nonumber \\&\quad -4\alpha ^2(p_{1}q_{2}+q_{1}p_{2})\theta +2\alpha ^2\sum \limits ^{\infty }_{n=2}\sum \limits ^{n-2}_{j=0}(j+1)(n-j-1)p_{j+1}q_{n-j-1} \theta ^{n}\nonumber \\&\quad -2\alpha ^2\sum \limits ^{\infty }_{n=2}\sum \limits ^{n}_{j=0}(j+1)(n-j+1)p_{j+1}q_{n-j+1}\theta ^{n}\nonumber \\&-2p_{0}q_{2}\alpha ^2-\alpha ^2(6p_{0}q_{3}+2p_{1}q_{2})\theta +\alpha ^2\sum \limits ^{\infty }_{n=2} \sum \limits ^{n-2}_{j=0}(n-j)(n-j-1)p_{j}q_{n-j}\theta ^{n}\nonumber \\&\quad -\alpha ^2\sum \limits ^{\infty }_{n=2}\sum \limits ^{n}_{j=0}(n-j+1)(n-j+2)p_{j}q_{n-j+2}\theta ^{n}=0, \end{aligned}$$
(5.26)
$$\begin{aligned}&\quad 6r_{0}+6r_{1}\theta +6\sum \limits ^{\infty }_{n=2}r_{n}\theta ^{n}+4r_{1}\theta +4 \sum \limits ^{\infty }_{n=2}nr_{n}\theta ^{n}+\sum \limits ^{\infty }_{n=2}n(n-1)r_{n}\theta ^{n} +\sum \limits ^{\infty }_{n=2}(n-1)r_{n-1}\theta ^{n}\nonumber \\&\quad -2r_{2}-6r_{3}\theta -\sum \limits ^{\infty }_{n=2}(n+1)(n+2)r_{n+2}\theta ^n+2p_{1}^2+8p_{1}p_{2} \theta +4p_{0}p_{2}+12p_{0}p_{3}\theta \nonumber \\&\quad +2\sum \limits ^{\infty }_{n=2}\sum \limits ^{n}_{j=0}(j+1)(n-j+1)p_{j+1}p_{n-j+1}\theta ^{n} +2\sum \limits ^{\infty }_{n=0}\sum \limits ^{n}_{j=0}(n-j+1)(n-j+2)p_{j}p_{n-j+2}\theta ^{n}=0. \end{aligned}$$
(5.27)

On comparing the coefficients of \(\theta ^{n}\) from equations (5.25)–(5.27), for \(n=0\), we get

$$\begin{aligned} p_{2}&=\frac{1}{2\alpha ^2}\Big [\Big ({\frac{\rho ^2-\gamma ^2-\alpha \rho }{\alpha ^2}}\Big )p_{0}-\alpha ^2p_{0}q_{1}^2+\frac{p_{0}r_{0}}{\kappa _{2}}\Big ], \end{aligned}$$
(5.28)
$$\begin{aligned} q_{2}&=\frac{1}{2\alpha ^2p_{0}}\Big [\Big (\frac{2\gamma \rho -\alpha \gamma }{\alpha ^2}\Big )p_{0}-2\alpha ^2p_{1}q_{1}^2\Big ], \end{aligned}$$
(5.29)
$$\begin{aligned} r_{2}&=\frac{1}{2}\Big [6r_{0}+4p_{0}p_{2}+2p_{1}^2\Big ], \end{aligned}$$
(5.30)

where \(p_{2}\) is given by Eq. (5.28).

Next, for \(n\ge 1\), the general recurrence relations are as follows:

$$\begin{aligned} p_{n+2}&=\frac{1}{\alpha ^2(n+1)(n+2)}\Bigg [\Big ({\frac{\rho ^2-\gamma ^2-\alpha \rho }{\alpha ^2}}\Big ) p_{n}+2\alpha \rho np_{n}+2\alpha ^2np_{n}-2\alpha \gamma \sum \limits ^{n-1}_{j=0}(n-j)p_{j}p_{n-j}\nonumber \\&\quad +\alpha ^2n(n-1)p_{n}+ \alpha ^2\sum \limits ^{n}_{j=0}\sum \limits ^{j}_{i=0}(i+1)(j-i+1)q_{i+1}q_{j-i+1}p_{n-j} +\frac{1}{\kappa _{2}}\sum \limits ^{n}_{j=0}p_{j}r_{n-j}\nonumber \\&-\alpha ^2\sum \limits ^{n-2}_{j=0}\sum \limits ^{j}_{i=0}(i+1)(j-i+1)q_{i+1}q_{j-i+1}p_{n-2-j}\Bigg ], \end{aligned}$$
(5.31)
$$\begin{aligned} q_{n+2}&=\frac{1}{\alpha ^2(n+1)(n+2)p_{0}}\Bigg [\Big (\frac{2\gamma \rho -\alpha \gamma }{\alpha ^2}\Big )p_{n}+2\alpha \gamma np_{n}+2\alpha \rho \sum \limits ^{n-1}_{j=0}(n-j)p_{j}p_{n-j}\nonumber \\&\quad +2\alpha ^2\sum \limits ^{n-1}_{j=0}(n-j)p_{j}q_{n-j}+2\alpha ^2\sum \limits ^{n-2}_{j=0}(j+1) (n-j-1)p_{j+1}q_{n-j-1}\nonumber \\ {}&-2\alpha ^2\sum \limits ^{n}_{j=0}(j+1)(n-j+1)p_{j+1}q_{n-j+1}\nonumber \\&\quad +\alpha ^2\sum \limits ^{n-2}_{j=0}(n-j)(n-j-1)p_{j}q_{n-j}-\alpha ^2\sum \limits ^{n}_{j=1}(n-j+1)(n-j+2)p_{j}q_{n-j+2}\Bigg ], \end{aligned}$$
(5.32)
$$\begin{aligned} r_{n+2}&=\frac{1}{(n+1)(n+2)}\Bigg [6r_{n}\!+\!4nr_{n}\!+\!n(n\!-\!1)r_{n}+(n-1)r_{n-1} +2\sum \limits ^{n}_{j=0}(j+1)(n-j+1)p_{j+1}p_{n-j+1}\nonumber \\&\quad +2\sum \limits ^{n}_{j=0}(n-j+1)(n-j+2)p_{j}p_{n-j+2}\Bigg ]. \end{aligned}$$
(5.33)

Using recurrence relations, we can calculate the other terms of the sequences \(\{ p_{n} \}^{\infty }_{n=0},\,\, \{q_{n} \}^{\infty }_{n=0}\) and \(\{ r_{n} \}^{\infty }_{n=0} \). Hence, this indicates that for equations (4.9), a series solutions (5.1) is feasible with the coefficients which are given by equations (5.31), (5.32), and (5.33). Thus, the power series solution of equations (4.9) is given by

$$\begin{aligned} U(\theta )&=p_{0}+p_{1}\theta +\frac{1}{2\alpha ^2}\Big [\Big ({\frac{\rho ^2-\gamma ^2-\alpha \rho }{\alpha ^2}}\Big )p_{0}-\alpha ^2p_{0}q_{1}^2+\frac{p_{0}r_{0}}{\kappa _{2}}\Big ]\theta ^2 +\sum \limits ^{\infty }_{k=1}p_{k+2}\theta ^{k+2},\nonumber \\ V(\theta )&=q_{0}+q_{1}\theta +\frac{1}{2\alpha ^2p_{0}}\Big [\Big (\frac{2\gamma \rho -\alpha \gamma }{\alpha ^2}\Big )p_{0}-2\alpha ^2p_{1}q_{1}^2\Big ]\theta ^2+\sum \limits ^{\infty }_{k=1}q_{k+2}\theta ^{k+2},\nonumber \\ G(\theta )&=r_{0}+r_{1}\theta +\frac{1}{2}\Big [6r_{0}+4p_{0}p_{2}+2p_{1}^2\Big ]\theta ^2+\sum \limits ^{\infty }_{k=1}r_{k+2}\theta ^{k+2}. \end{aligned}$$
(5.34)

Subcase 2: Substituting equations (5.1) into system of equations (4.12), we get

$$\begin{aligned}&(\rho ^2-\gamma ^2)(p_{0}+p_{1}\theta )+(\rho ^2-\gamma ^2)\sum \limits ^{\infty }_{n=2}p_{n}\theta ^n-\rho \alpha p_{0}-\rho \alpha \theta p_{1}-\rho \alpha \sum \limits ^{\infty }_{n=2}p_{n}\theta ^n-2\alpha \rho \theta p_{1}\nonumber \\ {}&\quad -2\alpha \rho \sum \limits ^{\infty }_{n=2}np_{n}\theta ^{n}+2\alpha \gamma p_{0}q_{1}\theta +\alpha ^2p_{1}\theta +2\alpha \gamma \sum \limits ^{\infty }_{n=2}\sum \limits ^{n-1}_{j=0}(n-j)p_{j}q_{n-j}\theta ^{n} +\alpha ^2\sum \limits ^{\infty }_{n=2}np_{n}\theta ^{n}-2\alpha ^2p_{2}\nonumber \\ {}&\quad -6\alpha ^2\theta p_{3}+\alpha ^2\sum \limits ^{\infty }_{n=2}n(n-1)p_{n}\theta ^{n}-\alpha ^2\sum \limits ^{\infty }_{n=2}(n+1)(n+2)p_{n+2}\theta ^n-\frac{\alpha }{\kappa _{1}}p_{0}r_{0}\nonumber \\ {}&\quad -\frac{\alpha }{\kappa _{1}}p_{0}r_{1}\theta -\frac{\alpha }{\kappa _{1}}p_{1}r_{0}\theta +\alpha ^2p_{0}q_{1}^2+\alpha ^2(p_{1}q_{1}^2+4q_{1}q_{2}p_{0})\theta +\frac{1}{\kappa _{2}}p_{0}r_{0}\nonumber \\ {}&\quad +\alpha ^2\sum \limits ^{\infty }_{n=2}\sum \limits ^{n}_{j=0}\sum \limits ^{j}_{i=0}(i+1)(j-i+1)q_{i+1}q_{j-i+1}p_{n-j}\theta ^{n} -\frac{\alpha }{\kappa _{1}}\sum \limits ^{\infty }_{n=2}\sum \limits ^{n}_{j=0}p_{j}r_{n-j}\theta ^n\nonumber \\ {}&\quad -\alpha ^2\sum \limits ^{\infty }_{n=2}\sum \limits ^{n-2}_{j=0}\sum \limits ^{j}_{i=0}(i+1)(j-i+1)q_{i+1}q_{j-i+1}p_{n-j-2}\theta ^{n} =0, \end{aligned}$$
(5.35)
$$\begin{aligned}&\quad 2\gamma \rho (p_{0}+p_{1}\theta )+2\gamma \rho \sum \limits ^{\infty }_{n=2}p_{n}\theta ^n-\gamma \alpha (p_{0}+p_{1}\theta )-\gamma \alpha \sum \limits ^{\infty }_{n=2}p_{n}\theta ^n-2\alpha \gamma p_{1}\theta -2\alpha \gamma \sum \limits ^{\infty }_{n=2}np_{n}\theta ^{n}-2\alpha \rho p_{0}q_{1}\theta \nonumber \\&\quad -2\alpha \rho \sum \limits ^{\infty }_{n=2}\sum \limits ^{n-1}_{j=0}(n-j)p_{j}q_{n-j}\theta ^{n} +\alpha ^2p_{0}q_{1}\theta +\alpha ^2\sum \limits ^{\infty }_{n=2}\sum \limits ^{n-1}_{j=0}(n-j)p_{j}q_{n-j} \theta ^{n}-2\alpha ^2p_{1}q_{1}-4\alpha ^2(p_{1}q_{2}+q_{1}p_{2})\theta \nonumber \\&\quad +2\alpha ^2\sum \limits ^{\infty }_{n=2}\sum \limits ^{n-2}_{j=0}(j+1)(n-j-1)p_{j+1}q_{n-j-1} \theta ^{n}-2\alpha ^2\sum \limits ^{\infty }_{n=2}\sum \limits ^{n}_{j=0}(j+1)(n-j+1)p_{j+1}q_{n-j+1}\theta ^{n}\nonumber \\&\quad -2p_{0}q_{2}\alpha ^2-\alpha ^2(6p_{0}q_{3}+2p_{1}q_{2})\theta +\alpha ^2\sum \limits ^{\infty }_{n=2} \sum \limits ^{n-2}_{j=0}(n-j)(n-j-1)p_{j}q_{n-j}\theta ^{n}\nonumber \\&\quad -\alpha ^2\sum \limits ^{\infty }_{n=2}\sum \limits ^{n}_{j=0}(n-j+1)(n-j+2)p_{j}q_{n-j+2}\theta ^{n}=0, \end{aligned}$$
(5.36)
$$\begin{aligned}&2r_{0}+2r_{1}\theta +2\sum \limits ^{\infty }_{n=2}r_{n}\theta ^{n}+4r_{1}\theta +4\sum \limits ^{\infty }_{n=2}nr_{n}\theta ^{n}+\sum \limits ^{\infty }_{n=2}n(n-1)r_{n}\theta ^{n}-2r_{2}-6r_{3}\theta \nonumber \\&\quad -\sum \limits ^{\infty }_{n=2}(n+1)(n+2)r_{n+2}\theta ^n-2p_{1}^2-8p_{1}p_{2}\theta -4p_{0}p_{2}-12p_{0}p_{3}\theta \nonumber \\&\quad -\sum \limits ^{\infty }_{n=2}\sum \limits ^{n}_{j=0}(j+1)(n-j+1)p_{j+1}p_{n-j+1}\theta ^{n} -2\sum \limits ^{\infty }_{n=2}\sum \limits ^{n}_{j=0}(n-j+1)(n-j+2)p_{j}p_{n-j+2}\theta ^{n}=0. \end{aligned}$$
(5.37)

On comparing the coefficients of \(\theta ^{n}\) from Eqs. (5.35)–(5.37), for \(n=0\), we obtain

$$\begin{aligned} p_{2}&=\frac{1}{2\alpha ^2}\Big [(\rho ^2-\gamma ^2)p_{0}p_{0}-\rho \alpha p_{0}-\frac{\alpha }{\kappa _{1}}p_{0}r_{0}+\alpha ^2p_{0}q_{1}^2\Big ], \end{aligned}$$
(5.38)
$$\begin{aligned} q_{2}&=\frac{1}{2\alpha ^2p_{0}}\Big [2\gamma \rho p_{0}-\gamma \alpha p_{0}-2\alpha ^2p_{1}q_{1}\Big ], \end{aligned}$$
(5.39)
$$\begin{aligned} r_{2}&=\frac{1}{2}\Big [2r_{0}-4p_{0}p_{2}-2p_{1}^2\Big ], \end{aligned}$$
(5.40)

where \(p_{2}\) is given by Eq. (5.38).

Next, for \(n=1, 2, \ldots \), we presented the general recurrence relations as follows:

$$\begin{aligned} p_{n+2}&=\frac{1}{\alpha ^2(n+1)(n+2)}\Bigg [(\rho ^2-\gamma ^2)p_{n}-2\alpha \rho np_{n}+\alpha ^2p_{n}-\rho \alpha p_{n}+2\alpha \gamma \sum \limits ^{n-1}_{j=0}(n-j)p_{j}p_{n-j}\nonumber \\&\quad +\alpha ^2n(n-1)p_{n}+ \alpha ^2\sum \limits ^{n}_{j=0}\sum \limits ^{j}_{i=0}(i+1)(j-i+1)q_{i+1}q_{j-i+1}p_{n-j} -\frac{\alpha }{\kappa _{1}}\sum \limits ^{n}_{j=0}p_{j}r_{n-j}\nonumber \\&\quad -\alpha ^2\sum \limits ^{n-2}_{j=0}\sum \limits ^{j}_{i=0}(i+1)(j-i+1)q_{i+1}q_{j-i+1}p_{n-2-j}\Bigg ], \end{aligned}$$
(5.41)
$$\begin{aligned} q_{n+2}&=\frac{1}{\alpha ^2(n+1)(n+2)p_{0}}\Bigg [2\gamma \rho p_{n}-2\alpha \gamma np_{n} -\alpha \gamma p_{n}-2\alpha \rho \sum \limits ^{n-1}_{j=0}(n-j)p_{j}q_{n-j}\nonumber \\&\quad +\alpha ^2\sum \limits ^{n-1}_{j=0}(n-j)p_{j}q_{n-j}+2\alpha ^2 \sum \limits ^{n-2}_{j=0}(j+1)(n-j-1)p_{j+1}q_{n-j-1}\nonumber \\&\quad -2\alpha ^2\sum \limits ^{n}_{j=0}(j+1)(n-j+1)p_{j+1}q_{n-j+1}\nonumber \\&\quad +\alpha ^2\sum \limits ^{n-2}_{j=0}(n-j)(n-j-1)p_{j}q_{n-j}-\alpha ^2\sum \limits ^{n}_{j=1}(n-j+1)(n-j+2)p_{j}q_{n-j+2}\Bigg ], \end{aligned}$$
(5.42)
$$\begin{aligned} r_{n+2}&=\frac{1}{(n+1)(n+2)}\Bigg [2r_{n}+4nr_{n}+n(n-1)r_{n}-2 \sum \limits ^{n}_{j=0}(j+1)(n-j+1)p_{j+1}p_{n-j+1}\nonumber \\&\quad -2\sum \limits ^{n}_{j=0}(n-j+1)(n-j+2)p_{j}p_{n-j+2}\Bigg ]. \end{aligned}$$
(5.43)

Using recurrence relations, we can find the other terms of the sequences \(\{ p_{n} \}^{\infty }_{n=0},\,\, \{q_{n} \}^{\infty }_{n=0}\), and \(\{ r_{n} \}^{\infty }_{n=0} \). This indicates that for system (4.12), a series solutions (5.1) is feasible with the coefficients given by Eqs. (5.41), (5.42), and (5.43).

Therefore, the power series solution of equations (4.12) is given by

$$\begin{aligned} U(\theta )&=p_{0}+p_{1}\theta +\frac{1}{2\alpha ^2}\Big [(\rho ^2-\gamma ^2)p_{0}p_{0} -\rho \alpha p_{0}-\frac{\alpha }{\kappa _{1}}p_{0}r_{0}+\alpha ^2p_{0}q_{1}^2\Big ] \theta ^2+\sum \limits ^{\infty }_{k=1}p_{k+2}\theta ^{k+2},\nonumber \\ V(\theta )&=q_{0}+q_{1}\theta +\frac{1}{2\alpha ^2p_{0}}\Big [2\gamma \rho p_{0}-\gamma \alpha p_{0}-2\alpha ^2p_{1}q_{1}\Big ]\theta ^2+\sum \limits ^{\infty }_{k=1}q_{k+2}\theta ^{k+2},\nonumber \\ G(\theta )&=r_{0}+r_{1}\theta +\frac{1}{2}\Big [2r_{0}-4p_{0}p_{2}-2p_{1}^2\Big ] \theta ^2+\sum \limits ^{\infty }_{k=1}r_{k+2}\theta ^{k+2}, \end{aligned}$$
(5.44)

where \(p_{i},\) and \(q_{i} (i=0, 1)\) are arbitrary constants & \(p_{k+2}, q_{k+2}, r_{k+2} (k\ge 1)\) are given by Eqs. (5.41), (5.42) and (5.43). \(\square \)

Remark: Convergence of the power series solution in Case 2 can be proved similar to Case 1.

6 Conclusions

In this work, the Klein–Gordon–Zakharov equations with time-dependent variable coefficients have been examined using the Lie group method of infinitesimals. This approach is efficient and feasible for the analysis of nonlinear evolution equations. We furnished the Lie symmetries of the Eq. (1.2) alongwith the forms of variable coefficients. The obtained symmetries reduce the Eq. (1.2) to the system of nonlinear ordinary differential equations. Due to the lack of Lie point symmetries of the obtained system of second-order nonlinear ODEs, we employed the power series method on the nonlinear ODEs (4.6), (4.9), and (4.12) which allowed us to present the exact series solutions. We also demonstrated the convergence of the series solutions.