Invariance Analysis for Determining the Classical Solutions, Optimal System, and Conserved Vectors of Generalized Higher-Dimensional Ablowitz-Kaup- Newell-Segur Equation in Theoretical Physics

1. Introduction

Nonlinear partial differential equations (NLNPDEEs) play a significant role in various fields such as applied mathematics, physics, and engineering. These equations are essential for modeling physical phenomena and have numerous practical applications. However, solving NLNPDEEs can be challenging due to the lack of systematic methods for finding analytic solutions, unlike linear partial differential equations (LNPDEEs). Despite the importance of finding analytic solutions for NLNPDEEs, there remains a need for new techniques to discover these solutions efficiently. The exploration of nonlinear wave phenomena has had a significant impact on various natural sciences such as biology, mathematics, and several branches of physics including condensed matter physics, nonlinear optics, chemical physics, plasma physics, solid-state physics, and fluid dynamics. The analysis of nonlinear partial differential equations is a thriving and dynamic area of study in theoretical physics, applied mathematics, and various engineering disciplines. Researchers are particularly interested in discovering exact solutions for traveling waves in these differential equations that describe important physical and dynamic processes [1–16]. However, in recent times, numerous researchers have focused on creating effective methods to analyze solutions of the Nonlinear Nonlocal Partial Differential Equations. Some of these techniques consist of: Kudryashov’s approach [17], simplest equation technique [18], sine-Gordon equation expansion technique [19], Hirota’s bilinear approach [20], $\left({\displaystyle \frac{G^{\prime }}{G}}\right)$-expansion technique [21], power series solution technique [22], Darboux transformation approach [23], and among others. Additional, techniques such as Painlevé expansion technique [24], bifurcation approach [25], homotopy perturbation [26], extended homoclinic test technique [27], tanh-coth technique [28], Adomian decomposition technique [29], Symmetry group analysis [30,31], F-expansion approach [32], Bäcklund transformation method[33], extended simplest equation technique [17], Cole-Hopf transformation approach [34], rational expansion approach [35], and many more have been developed. Since the establishment of Kadomtsev and Petviashvili’s hierarchy of equations over fifty years ago, numerous research papers have been published, each delving into different aspects of this complex field of equations. For instance, see [36,37,38,39,40,41,42], one mentions but a few here.

Sophus Lie, a prominent Norwegian mathematician, revolutionized the study of Lie symmetries from 1842 to 1899. Lie discovered that the seemingly random methods used to solve ordinary differential equations (ODEEs) could be organized into a structured theory based on the consistency of differential equations within a continuous symmetry group. By applying invariant solutions to a symmetry group derived from a partial differential equation (PDE), one can determine a solution to the differential equation (DE) with fewer independent variables. Among these solutions are similarity solutions, traveling wave solutions, and other closed-form solutions that hold significance in the physical world [30,31].

Exploration of differential equations has revealed numerous valuable applications of conserved quantities. One such application is the ability to determine the integrability of a chosen partial differential equation, providing a deeper insight into its properties. Utilizing conserved quantities can also ensure the existence and uniqueness of solutions to a differential equation. Furthermore, these quantities play a crucial role in validating numerical solution methods. Despite these benefits, a challenge lies in identifying the conserved quantities relevant to a specific differential equation. The general multiplier method offers a systematic approach to discovering conserved quantities for differential equations, whether or not they are based on variational principles [43,44,45,46,47,48,49].

A (2+1)-dimensional AKNS equation given as [50]

$$4u_{tx}+\alpha u_{xx}+8u_x{u}_{xy}+4u_y{u}_{xx}+u_{xxxy}=0,$$

(1)

with constant coefficient $\alpha \ne 0$ which indicated the dissipative characteristic effects of the system was investigated in [50] whereby the authors engaged the idea of Hirota’s bilinear approach to seeking for the closed-form solution of equation (1). This technique made it possible to procure multiple singular closed-form soliton solutions of the equation. The researchers in [51] through the involvement of Bilinear Bäcklund transformation explicitly established some wave solutions founded on a multidimensional Riemann theta function which are periodic. Wazwaz [52] utilized the simplified Hirota’s bilinear approach developed by Hereman [53] to find multiple-soliton solutions of equation (1). In [54], some exact travelling wave solutions of equation (1) were gotten for the (2+1)-dimensional AKNS equation by utilizing the improved tanh method. Authors in [55] applied binary Bell polynomials to secure bilinear Bäcklund transformation, the bilinear representation, conservation laws as well as the Lax pair of the equation (1). When $\alpha =0$, equation (1) converts to the (2+1)-dimensional AKNS equation which is closely interrelated with the Schwarz KdV equation through the use of a Miura transform [54,56]. Mothibi in [57] considered the generalized version of (1) and then generated the conservation laws of the equation via Noether theorem. Hiu et al. [58] employed the truncated Painlevè expansion together with consistent Riccati expansion abbreviated as "CRE" to gain soliton-cnoidal wave interaction solutions of equation (1) explicitly. The divers closed-form solutions of AKNS equations (1) are procured by virtue of extended homoclinic, tanh function method as well as symmetry analysis [54,56,59,60].

The Ablowitz-Kaup-Newell-Segur (AKNS) system [61]

$$\begin{array}{cc}\hfill & w_t-v{w}^2+{\displaystyle \frac{1}{2}}{w}_{xx}-w=0,\hfill \\ \hfill & v_t+w{v}^2-{\displaystyle \frac{1}{2}}{v}_{xx}+v=0,\hfill \end{array}$$

(2)

whereby the subscripts stands for the partial derivatives. We also have functions $w=w(x,t)$ and $v=v(x,t)$ depending on x and t. AKNS system (2) is one of the members of the family of AKNS models. In [62] the researchers dissolved the AKNS system (2) to integrable Hamiltonian systems that are finitely-dimensional via Bargmann adjoint symmetry constraints. Moreover, in [63], one-parameter groups of solutions of the single non-linear Schrödinger equation which is regarded as a special reduction type of AKNS (2) was presented through the engagement of Lie symmetry technique. Li et al. [64] got some multisoliton solutions of system (2) by utilizing Darboux transformation. In the same vein, Wang et al. in [65] procured some new solutions which are multisoliton. Furthermore, the authors in [64,65] transformed AKNS system (2) into the (1+1)-dimensional integrable non-linear dispersive-wave system

$$\left\{\begin{array}{c}{\xi }_t+{\left[(1+\xi )p\right]}_x=-{\displaystyle \frac{1}{4}}{p}_{xxx},\hfill \\ p_t+p{p}_x+{\xi }_x=0,\hfill \end{array}\right.$$

via the engagement of the transformation

$$\begin{array}{cc}\hfill & w={\mathrm{e}}^{\int pdx},v=-\left(1+\xi -{\displaystyle \frac{1}{2}}{p}_x\right){\mathrm{e}}^{-\int pdx},\hfill \\ \hfill & \mathrm{or}p={\displaystyle \frac{w_x}{w}},\xi =-1-wv+{\displaystyle \frac{1}{2}}{p}_x.\hfill \end{array}$$

The fourth-order nonlinear (2+1)-dimensional Ablowitz-Kaup-Newell-Segur water equation (AKNS) [66]

$$4u_{tx}-\theta u_{xx}+8u_x{u}_{xy}+4u_y{u}_{xx}+u_{xxxt}=0,$$

(3)

with a perturbation parameter $\theta$, is an important partial differential equation which has numerous applications in various fields of study including physics together with some other nonlinear sciences. In [66], Asghar et al. adopted simple equation approach as well as its modified form to analytically secure travelling wave solutions to the earlier-mentioned AKNS equation (3). Helal et al. [67], conducted stability analysis on AKNS equations (3) whereby extended auxiliary equation method was involved to gain new soliton-like solutions for the underlying equation. Besides, they got symmetrical solution, non-symmetrical kink solution, bell-shaped solitary wave solution, solitary wave solution as well as travelling wave solution. The algorithmic scheme of $(G^{\prime }/G)$-expansion technique was engaged in the construction of new solutions for AKNS (3) in [68]. Authors in [69] made a description of a unified scheme for all equations of AKNS hierarchy as well as their combinations. In furtherance to that, they presented examples of solutions that fulfilled different equations for various parameters by precisely taking into consideration a rank-2 quasi- rational solution which can as well be employed in examining divers integrable models featuring in nonlinear optics.

In this study, we investigate the (2+1)-dimensional generalized Ablowitz-Kaup-Newell-Segur equation (2D-gnrAKNSe)

$$\rho u_{txxx}+\sigma u_{tx}-\theta u_{xx}+2\alpha u_x{u}_{xy}+\alpha u_{xx}{u}_y=0,$$

(4)

where $\alpha$, $\sigma$ and $\theta$ are real-valued constants. It is noteworthy to state here that, we obtained the generalized AKNS (4) by taking $\alpha =\sigma =4$ and $\rho =1$ in equation (3). Explicit analysis of 2D-gnrAKNSe (4) is effectuated in this article as follows; Introduction is explicated in section 1, whereas the essential method of solution (Lie group theory) is implemented in calculating the point symmetries that constitute the Lie algebra of the model (2D-gnrAKNSe (4)) in Section 2. Additionally, groups of transformations in one parametric format is computed. In tabular form, commutation relations as well as adjoint representation of the computed Lie algebra are expressed. Moreover, the subalgebras found are engaged in discharging symmetry reductions of (4) to various ordinary differential equations (ODEE). Thereafter, wave dynamics of the solutions are put into consideration via numerical simulation of the results, in section 3. Thereafter, conservation laws of the considered model (4) are calculated by utilizing the Ibragimov’s theorem for conserved vectors. These are presented in section 4. Finally, at the end, concluding remarks are given in section 5.

2. Lie Point Symmetry Analysis

This section of the article calculates and presents the Lie point symmetries of 2D-gnrAKNSe (4) and thereafter engages them to find exact solutions.

2.1. Determination of Lie point symmetries of (4)

The Lie group of transformation is defined as

$$\begin{array}{cc}\hfill & \tilde{t}\approx t+ϵ{\xi }^1(t,x,y,u)+O\left({ϵ}^2\right),\hfill \\ \hfill & \tilde{x}\approx x+ϵ{\xi }^2(t,x,y,u)+O\left({ϵ}^2\right),\hfill \\ \hfill & \tilde{y}\approx y+ϵ{\xi }^3(t,x,y,u)+O\left({ϵ}^2\right),\hfill \\ \hfill & \tilde{u}\approx u+ϵ\eta (t,x,y,u)+O\left({ϵ}^2\right).\hfill \end{array}$$

(5)

The group of symmetries of 2D-gnrAKNSe (4) will be calculated by contemplating a vector field given as

$$\mathfrak{S}={\xi }^1{\displaystyle \frac{\partial }{\partial t}}+{\xi }^2{\displaystyle \frac{\partial }{\partial x}}+{\xi }^3{\displaystyle \frac{\partial }{\partial y}}+\eta {\displaystyle \frac{\partial }{\partial u}},$$

with ${\xi }^1,{\xi }^2,{\xi }^3$ and $\eta$ taken as functions depending on t, x, y and u, secures a Lie point symmetry of 2D-gnrAKNSe (4) if the invariant condition

$${Pr}^{\left(4\right)}\mathfrak{S}[\sigma u_{tx}-\theta u_{xx}+2\alpha u_x{u}_{xy}+\alpha u_y{u}_{xx}+\rho u_{xxxt}]=0,$$

(6)

holds whenever $\sigma u_{tx}-\theta u_{xx}+2\alpha u_x{u}_{xy}+\alpha u_y{u}_{xx}+\rho u_{xxxt}=0$. Precisely Pr${\mathrm{Pr}}^{\left(4\right)}\mathfrak{S}$ denotes the third prolongation of $\mathfrak{S}$ which is expressed in the form

$${Pr}^{\left(3\right)}\mathfrak{S}=\mathfrak{S}+{\zeta }_t{\partial }_{u_t}+{\zeta }_x{\partial }_{u_x}+{\zeta }_y{\partial }_{u_y}+{\zeta }_{tx}{\partial }_{u_{tx}}+{\zeta }_{xx}{\partial }_{u_{xx}}+{\zeta }_{xy}{\partial }_{u_{xy}}+{\zeta }_{xxxt}{\partial }_{u_{xxxt}},$$

(7)

with the coefficients ${\zeta }_t$, ${\zeta }_x$, ${\zeta }_y$, ${\zeta }_{xt}$, ${\zeta }_{xx}$, ${\zeta }_{xy}$, ${\zeta }_{yy}$, ${\zeta }_{xxxt}$, given respectively as

$$\begin{array}{cc}\hfill & {\zeta }_t={\mathfrak{D}}_t\left(\eta \right)-u_t{\mathfrak{D}}_t\left({\xi }^1\right)-u_x{\mathfrak{D}}_t\left({\xi }^2\right)-u_y{\mathfrak{D}}_t\left({\xi }^3\right),\hfill \\ \hfill & {\zeta }_x={\mathfrak{D}}_x\left(\eta \right)-u_t{\mathfrak{D}}_x\left({\xi }^1\right)-u_x{\mathfrak{D}}_x\left({\xi }^2\right)-u_y{\mathfrak{D}}_x\left({\xi }^3\right),\hfill \\ \hfill & {\zeta }_y={\mathfrak{D}}_y\left(\eta \right)-u_t{\mathfrak{D}}_y\left({\xi }^1\right)-u_x{\mathfrak{D}}_y\left({\xi }^2\right)-u_y{\mathfrak{D}}_y\left({\xi }^3\right),\hfill \\ \hfill & {\zeta }_{xt}={\mathfrak{D}}_x\left({\zeta }_t\right)-u_{tt}{\mathfrak{D}}_x\left({\xi }^1\right)-u_{xt}{\mathfrak{D}}_x\left({\xi }^2\right)-u_{yt}{\mathfrak{D}}_x\left({\xi }^3\right),\hfill \\ \hfill & {\zeta }_{xx}={\mathfrak{D}}_x\left({\zeta }_x\right)-u_{xt}{\mathfrak{D}}_x\left({\xi }^1\right)-u_{xx}{\mathfrak{D}}_x\left({\xi }^2\right)-u_{xy}{\mathfrak{D}}_x\left({\xi }^3\right),\hfill \\ \hfill & {\zeta }_{xy}={\mathfrak{D}}_x\left({\zeta }_y\right)-u_{yt}{\mathfrak{D}}_x\left({\xi }^1\right)-u_{yx}{\mathfrak{D}}_x\left({\xi }^2\right)-u_{yy}{\mathfrak{D}}_x\left({\xi }^3\right),\hfill \\ \hfill & {\zeta }_{yy}={\mathfrak{D}}_y\left({\zeta }_y\right)-u_{yt}{\mathfrak{D}}_y\left({\xi }^1\right)-u_{xy}{\mathfrak{D}}_y\left({\xi }^2\right)-u_{yy}{\mathfrak{D}}_y\left({\xi }^3\right),\hfill \\ \hfill & {\zeta }_{xxxt}={\mathfrak{D}}_x\left({\zeta }_{xxt}\right)-u_{xxtt}{\mathfrak{D}}_x\left({\xi }^1\right)-u_{xxxt}{\mathfrak{D}}_x\left({\xi }^2\right)-u_{xxyt}{\mathfrak{D}}_x\left({\xi }^3\right),\hfill \end{array}$$

(8)

and the total derivatives are stated as

$$\begin{array}{cc}\hfill & {\mathfrak{D}}_t={\partial }_t+u_t{\partial }_u+u_{tt}{\partial }_{u_t}+u_{xt}{\partial }_{u_x}+\cdots ,\hfill \\ \hfill & {\mathfrak{D}}_x={\partial }_x+u_x{\partial }_u+u_{xx}{\partial }_{u_x}+u_{xt}{\partial }_{u_t}+\cdots \hfill \\ \hfill & {\mathfrak{D}}_y={\partial }_y+u_y{\partial }_u+u_{yy}{\partial }_{u_y}+u_{yt}{\partial }_{u_t}+\cdots .\hfill \end{array}$$

(9)

For the main purpose of determining Lie group (5) admitted by 2D-gnrAKNSe (4), we invoke (8) into criteria (7). We make the coefficients of varying derivatives order of u equal to zero. Thus, gaining overdetermined systems of LNPDEEs with respect to functions ${\xi }^1(t,x,y,u)$, ${\xi }^2(t,x,y,u)$, ${\xi }^3(t,x,y,u)$, and $\eta (t,x,y,u)$, viz;

$$\begin{array}{cc}\hfill & {\xi }_{tx}^3=0,{\xi }_{tu}^3=0,{\xi }_t^3=0,{\xi }_{xx}^3=0,{\xi }_{xx}^3=0,{\xi }_{xuu}^3=0,{\xi }_{xu}^3=0,{\xi }_{xu}^3=0,\hfill \\ \hfill & {\xi }_x^3=0,{\xi }_x^3=0,{\xi }_{uuu}^3=0,{\xi }_{uu}^3=0,{\xi }_{uu}^3=0,{\xi }_u^3=0,{\xi }_u^3=0,{\xi }_{tx}^3=0,{\xi }_{xu}^3=0,\hfill \\ \hfill & {\xi }_{tx}^3=0,{\xi }_{tuuu}^2=0,{\xi }_{tuu}^2=0,{\xi }_{tu}^3=0,{\xi }_{tu}^3=0,{\xi }_{tu}^3=0,{\xi }_{tu}^2=0,{\xi }_x^3=0,\hfill \\ \hfill & {\xi }_t^3=0,{\xi }_t^3=0,{\xi }_t^2=0,{\xi }_{xx}^3=0,{\xi }_{xx}^3=0,{\xi }_{xuuu}^1=0,{\xi }_{xuu}^1=0,{\xi }_{xu}^3=0,\hfill \\ \hfill & {\xi }_{xu}^3=0,{\xi }_{xu}^1=0,{\xi }_x^3=0,{\xi }_x^3=0,{\xi }_{uu}^2=0,{\xi }_{uu}^1=0,{\xi }_u^3=0,{\xi }_u^3=0,\hfill \\ \hfill & {\xi }_u^3=0,{\xi }_u^3=0,{\xi }_u^2=0,{\xi }_u^1=0,\sigma {\xi }_{xu}^1+\rho {\xi }_{xxxu}^1=0,\sigma {\xi }_x^3+\rho {\xi }_{xxx}^3=0,\hfill \\ \hfill & \sigma {\xi }_x^1+\rho {\xi }_{xxx}^1=0,\alpha {\xi }_{uu}^1+\rho {\xi }_{xuuu}^3=0,\theta {\xi }_{xx}^3-\sigma {\xi }_{tx}^3=0,3\alpha {\xi }_{uu}^2+\rho {\xi }_{tuuu}^3=0,\hfill \\ \hfill & 2\alpha {\xi }_u^2+\rho {\xi }_{tuu}^3=0,\sigma {\xi }_{uu}^1+3\rho {\xi }_{xxuu}^1=0,\sigma {\xi }_u^1+3\rho {\xi }_{xxu}^1=0,\sigma {\xi }_{xuu}^3-2\alpha {\xi }_{xxu}^1=0,\hfill \\ \hfill & \sigma {\xi }_{xu}^3-\alpha {\xi }_{xx}^1=0,\sigma {\xi }_{uuu}^3-8\alpha {\xi }_{xuu}^1=0,\sigma {\xi }_{uu}^3-4\alpha {\xi }_{xu}^1=0,\rho \sigma {\xi }_u^3-2\alpha {\xi }_x^1=0,\hfill \\ \hfill & 2\theta {\xi }_{xx}^3-3\sigma {\xi }_{tx}^3=0,3\rho {\xi }_{tuu}^3-2\alpha {\xi }_u^2=0,15\rho \sigma {\xi }_t^3-36\theta \rho {\xi }_x^3=0,3\theta {\xi }_x^3-2\sigma {\xi }_t^3=0,\hfill \\ \hfill & 18\theta {\xi }_x^3-5\sigma {\xi }_t^3=0,2\alpha {\xi }_u^1+3\rho {\xi }_{xuu}^3=0,{\eta }_{uuuu}-3{\xi }_{xuuu}^2-{\xi }_{tuuu}^1=0,\hfill \\ \hfill & {\eta }_{uuu}-3{\xi }_{xuu}^2-{\xi }_{tuu}^1=0,\sigma {\xi }_{xu}^3+\alpha {\xi }_{xx}^1+\rho {\xi }_{xxxu}^3=0,\sigma {\eta }_{tx}-\theta {\eta }_{xx}+\rho {\eta }_{txxx}=0,\hfill \\ \hfill & \sigma {\xi }_{uu}^3+4\alpha {\xi }_{xu}^1+3\rho {\xi }_{xxuu}^3=0,\sigma {\xi }_u^3+2\alpha {\xi }_x^1+3\rho {\xi }_{xxu}^3=0,\sigma {\xi }_u^3+2\alpha {\xi }_x^1+3\rho {\xi }_{xxu}^3=0,\hfill \\ \hfill & \sigma {\xi }_u^3+2\alpha {\xi }_x^1+3\rho {\xi }_{xxu}^3=0,2\alpha {\eta }_{tu}-3\theta {\xi }_{tu}^3-6\alpha {\xi }_{tx}^2=0,2\sigma {\xi }_u^3+2\alpha {\xi }_x^1+3\rho {\xi }_{xxu}^3=0,\hfill \\ \hfill & 2\alpha {\xi }_x^1-3\sigma {\xi }_u^3+3\rho {\xi }_{xxu}^3=0,\alpha {\eta }_{xx}+\theta {\xi }_{xx}^3-\sigma {\xi }_{tx}^3-\rho {\xi }_{txxx}^3=0,\hfill \\ \hfill & \sigma {\xi }_t^3-2\alpha {\eta }_x-2\theta {\xi }_x^3+3\rho {\xi }_{txx}^3=0,2\sigma {\xi }_t^3-2\alpha {\eta }_x-5\theta {\xi }_x^3+3\rho {\xi }_{txx}^3=0,\hfill \\ \hfill & 2\alpha {\eta }_x+6\theta {\xi }_x^3-\sigma {\xi }_t^3-3\rho {\xi }_{txx}^3=0,2\alpha {\eta }_x+2\theta {\xi }_x^3-\sigma {\xi }_t^3-3\rho {\xi }_{txx}^3=0,\hfill \\ \hfill & 2\alpha {\xi }_{yu}^2-\theta {\xi }_{uu}^2-\rho {\eta }_{tuuu}+3\rho {\xi }_{txuu}^2=0,2\alpha {\eta }_{uuu}-\theta {\xi }_{uuu}^3-2\alpha \left(3{\xi }_{xuu}^2+{\xi }_{tuu}^1\right)=0,\hfill \\ \hfill & 2\alpha {\eta }_{uu}-\theta {\xi }_{uu}^3-2\alpha \left(3{\xi }_{xu}^2+{\xi }_{tu}^1\right)=0,2\alpha {\eta }_{uu}-3\theta {\xi }_{uu}^3-2\alpha \left(3{\xi }_{xu}^2+{\xi }_{tu}^1\right)=0,\hfill \\ \hfill & 2{\alpha }^2{\xi }_y^2+\rho \left(-2\alpha {\eta }_{tuu}+\theta {\xi }_{tuu}^3+6\alpha {\xi }_{txu}^2\right)=0,\sigma {\eta }_{xu}+\theta {\xi }_{xx}^1+\rho {\eta }_{xxxu}-\sigma {\xi }_{tx}^1-\rho {\xi }_{txxx}^1=0,\hfill \\ \hfill & 3\alpha {\eta }_{uu}+\theta {\xi }_{uu}^3-2\alpha {\xi }_{yu}^3-4\alpha {\xi }_{xu}^2-3\rho {\xi }_{txuu}^3=0,2\alpha {\eta }_{xu}-2\theta {\xi }_{xu}^3-2\alpha {\xi }_{xx}^2+\sigma {\xi }_{tu}^3-2\alpha {\xi }_{tx}^1=0,\hfill \\ \hfill & \sigma {\xi }_{uu}^2-\theta {\xi }_{uu}^1+2\alpha {\xi }_{yu}^1-3\rho {\eta }_{xuuu}+3\rho {\xi }_{xxuu}^2+3\rho {\xi }_{txuu}^1=0,{\xi }_{uu}^3=0,\hfill \\ \hfill & 4\alpha {\eta }_{xu}+2\theta {\xi }_{xu}^3-2\alpha {\xi }_{xy}^3-\alpha {\xi }_{xx}^2-\sigma {\xi }_{tu}^3-3\rho {\xi }_{txxu}^3=0,{\xi }_{uu}^3=0,\hfill \\ \hfill & 2\alpha {\eta }_u+5\theta {\xi }_u^3-2\alpha {\xi }_y^3+2\alpha {\xi }_x^2+2\alpha {\xi }_t^1-6\rho {\xi }_{txu}^3=0,{\xi }_{uu}^3=0,\hfill \\ \hfill & 2\alpha {\eta }_{xy}-2\theta {\eta }_{xu}+\theta {\xi }_{xx}^2+\sigma {\eta }_{tu}-\sigma {\xi }_{tx}^2+3\rho {\eta }_{txxu}-\rho {\xi }_{txxx}^2=0,\hfill \\ \hfill & 2\alpha {\eta }_{yu}-\theta {\eta }_{uu}+2\theta {\xi }_{xu}^2-2\alpha {\xi }_{xy}^2-\sigma {\xi }_{tu}^2+3\rho {\eta }_{txuu}-3\rho {\xi }_{txxu}^2=0,\hfill \\ \hfill & {\alpha }^2{\xi }_y^1+\theta \alpha {\xi }_u^1+\sigma \alpha {\xi }_u^2-3\rho \alpha {\eta }_{xuu}+3\rho \alpha {\xi }_{xxu}^2+3\rho \alpha {\xi }_{txu}^1+3\theta \rho {\xi }_{xuu}^3=0,\hfill \\ \hfill & 4\theta \alpha {\xi }_u^1-4{\alpha }^2{\xi }_y^1+2\sigma \alpha {\xi }_u^2+12\rho \alpha {\eta }_{xuu}-12\rho \alpha {\xi }_{xxu}^2-12\rho \alpha {\xi }_{txu}^1+3\rho \sigma {\xi }_{tuu}^3=0,\hfill \\ \hfill & \sigma {\xi }_{xu}^2-\sigma {\eta }_{uu}-2\theta {\xi }_{xu}^1+2\alpha {\xi }_{xy}^1-3\rho {\eta }_{xxuu}+\rho {\xi }_{xxxu}^2+\sigma {\xi }_{tu}^1+3\rho {\xi }_{txxu}^1=0,\hfill \\ \hfill & {\alpha }^2{\eta }_y+\theta \alpha {\eta }_u-\theta \alpha {\xi }_y^3-\sigma \alpha {\xi }_t^2+3\rho \alpha {\eta }_{txu}-3\rho \alpha {\xi }_{txx}^2+{\theta }^2{\xi }_u^3-3\theta \rho {\xi }_{txu}^3=0,\hfill \\ \hfill & \alpha \sigma {\eta }_u+\theta \sigma {\xi }_u^3-\alpha \sigma {\xi }_y^3-2\alpha \theta {\xi }_x^1-\alpha \sigma {\xi }_x^2-3\alpha \rho {\eta }_{xxu}+\alpha \rho {\xi }_{xxx}^2+\alpha \sigma {\xi }_t^1-3\rho \sigma {\xi }_{txu}^3\hfill \\ \hfill & +3\alpha \rho {\xi }_{txx}^1=0.\hfill \end{array}$$

(10)

The solution to system (10) secured via computer software furnishes us with the values of coefficient functions expressed as

$${\xi }^1={\mathbf{c}}_4+{\mathbf{c}}_{5}t,{\xi }^2={\mathbf{c}}_1,{\xi }^3={\mathbf{c}}_2+{\mathbf{c}}_{3}y,\eta =f\left(t\right)+{\mathbf{c}}_6+\left({\mathbf{c}}_3-{\mathbf{c}}_5\right)u+{\displaystyle \frac{{\mathbf{c}}_5\theta y}{\alpha }}.$$

(11)

Assuming that arbitrary function $f\left(t\right)={\mathbf{c}}_{7}t$, then we achieve Lie point symmetries presented as

$$\begin{array}{cc}\hfill & {\mathfrak{S}}_1={\displaystyle \frac{\partial }{\partial y}},y-\mathrm{space}\mathrm{translation}\hfill \\ \hfill & {\mathfrak{S}}_2={\displaystyle \frac{\partial }{\partial x}},x-\mathrm{space}\mathrm{translation}\hfill \\ \hfill & {\mathfrak{S}}_3={\displaystyle \frac{\partial }{\partial u}},\mathrm{Galilei}\mathrm{translation}\hfill \\ \hfill & {\mathfrak{S}}_4={\displaystyle \frac{\partial }{\partial t}},\mathrm{time}-\mathrm{translation}\hfill \\ \hfill & {\mathfrak{S}}_5=t{\displaystyle \frac{\partial }{\partial u}},\mathrm{Galilei}\mathrm{translation}\hfill \\ \hfill & {\mathfrak{S}}_6=y{\displaystyle \frac{\partial }{\partial y}}+u{\displaystyle \frac{\partial }{\partial u}},\mathrm{Dilatation}\hfill \\ \hfill & {\mathfrak{S}}_7=t{\displaystyle \frac{\partial }{\partial t}}+\left({\displaystyle \frac{\theta }{\alpha }}y-u\right){\displaystyle \frac{\partial }{\partial u}}.\mathrm{Inversion}\hfill \end{array}$$

(12)

Theorem 2.1.

Consequently, 2D-gnrAKNSe (4) admits a seven-dimensional Lie algebra ${\mathcal{L}}^7$ spanned by vectors ${\mathfrak{S}}_1,{\mathfrak{S}}_2,\cdots ,{\mathfrak{S}}_7$ as established in (12).

2.2. One-parameter Lie group transformation using ${\mathcal{L}}^7$

This subsection furnishes the one-parameter transformation group of 2D-gnrAKNSe (4) using ${\mathcal{L}}^7$. In order to gain the transformation group generated by generators (12), its integral curves are needed to be secured. Thus, we come by the theorem:

Theorem 2.2.

If group $Gi,i=1,2,3,\cdots ,7$ constitutes the one parameter group generated by vectors ${\mathfrak{S}}_1,{\mathfrak{S}}_2,{\mathfrak{S}}_3\cdots ,{\mathfrak{S}}_7$ in (12), then, for each of the vectors with any small positive number ϵ, we have accordingly

$$\begin{array}{cc}\hfill G1:& (\tilde{t},\tilde{x},\tilde{y},\tilde{u})⟶(t,x,y+{ϵ}_1,u),\hfill \\ \hfill G2:& (\tilde{t},\tilde{x},\tilde{y},\tilde{u})⟶(t,x+{ϵ}_2,y,u),\hfill \\ \hfill G3:& (\tilde{t},\tilde{x},\tilde{y},\tilde{u})⟶(t,x,y,u+{ϵ}_3),\hfill \\ \hfill G4:& (\tilde{t},\tilde{x},\tilde{y},\tilde{u})⟶(t+{ϵ}_4,x,y,u),\hfill \\ \hfill G5:& (\tilde{t},\tilde{x},\tilde{y},\tilde{u})⟶(t,x,y,u+{ϵ}_{5}t),\hfill \\ \hfill G6:& (\tilde{t},\tilde{x},\tilde{y},\tilde{u})⟶(t,x,y{e}^{{ϵ}_6},u{e}^{{ϵ}_6}),\hfill \\ \hfill G7:& (\tilde{t},\tilde{x},\tilde{y},\tilde{u})⟶\left(t{e}^{\alpha {ϵ}_7},x,y,u{e}^{-\alpha {ϵ}_7}-{\displaystyle \frac{\theta }{\alpha }}t{e}^{-\alpha {ϵ}_7}+{\displaystyle \frac{\theta }{\alpha }}t\right).\hfill \end{array}$$

Theorem 2.3.

In consequence of Theorem 2.2, we can say that if $u(t,x,y)=H(t,x,y)$ satisfies the 2D-gnrAKNSe (4), so also are the functions established as

$$\begin{array}{cc}\hfill G1:u(t,x,y)=& H(t,x,y-{ϵ}_1),\hfill \\ \hfill G2:u(t,x,y)=& H(t,x-{ϵ}_2,y),\hfill \\ \hfill G3:u(t,x,y)=& H(t,x,y)-{ϵ}_3,\hfill \\ \hfill G4:u(t,x,y)=& H(t-{ϵ}_4,x,y),\hfill \\ \hfill G5:u(t,x,y)=& H(t,x,y)-{ϵ}_{5}t,\hfill \\ \hfill G6:u(t,x,y)=& e^{-{ϵ}_6}H\left(t,x,y{e}^{{ϵ}_6}\right),\hfill \\ \hfill G7:u(t,x,y)=& e^{\alpha {ϵ}_7}H\left(t{e}^{-\alpha {ϵ}_7},x,y\right)+{\displaystyle \frac{\theta }{\alpha }}t{e}^{-\alpha {ϵ}_7}-{\displaystyle \frac{\theta }{\alpha }}t,\hfill \end{array}$$

where $ϵ<<1$ is any positive number.

Remark 2.1.

In general, it is noteworthy to declare here that to each one-parameter subgroups of the full symmetry group system, there will be an associated family of solutions referred to as group-invariant solutions.

2.3. Optimal system of sub-algebras

It is well understood that the Lie group theoretic approach performs a crucial function in securing exact solutions of differential equations together with the performance of symmetry reductions. Having identified the fact that any linear combination of infinitesimal generators is also an infinitesimal generator, there are absolutely infinitely many different symmetry subgroups which basically occasions varying types of solutions. This in turn is significant and also necessary for one to have a complete understanding of the involved invariant solutions. Besides, it is sufficient to seek invariant solutions that are not analogous by transformations in the full symmetry group, since any transformation in the full symmetry group maps a solution to another solution. This has consequently led to the concept of an optimal system [70]. The task involved in obtaining an optimal system of subalgebras is as well equivalent to that of achieving an optimal system of subgroups. This classification problem for one-dimensional subalgebras is fundamentally same as the problem involved in the classification of the orbits of adjoint representation. Thus, this problem is thrashed via the engagement of naive approach whereby a general element taken from the Lie algebra is subjected to different adjoint transformations so that it can be simplified as much as possible. It was due to [70,71] that the idea of imploring adjoint representation to classify group-invariant solution was conceived. We define the Lie series as

$$\begin{array}{cc}\hfill Ad\left(exp\left(ϵ{\mathfrak{S}}_m\right)\right)\left({\mathfrak{S}}_n\right)=& e^{-ϵ{\mathfrak{S}}_m}{\mathfrak{S}}_n{e}^{ϵ{\mathfrak{S}}_m}={\mathfrak{S}}_n-ϵ[{\mathfrak{S}}_m,{\mathfrak{S}}_n]+{\displaystyle \frac{1}{2!}}{ϵ}^2[{\mathfrak{S}}_m,[{\mathfrak{S}}_m,{\mathfrak{S}}_n]]-\cdots \hfill \\ \hfill =& ({\alpha }_1{\mathfrak{S}}_1+\cdots +{\alpha }_7{\mathfrak{S}}_7)-ϵ({\Theta }_1{\mathfrak{S}}_1+\cdots +{\Theta }_7{\mathfrak{S}}_7)+O\left({ϵ}^2\right)\hfill \end{array}$$

which include the adjoint representation[31] with $ϵ$ regarded as group parameter together with $A{d}_{{\mathfrak{S}}_m}{\mathfrak{S}}_n=[{\mathfrak{S}}_m,{\mathfrak{S}}_n]$ defined as the Lie algebra commutator (Lie bracket) and $m,n=1,2,3,\cdots ,7$. Thus we construct the commutator table using the relation $[{\mathfrak{S}}_m,{\mathfrak{S}}_n]$ in Table 1.

Table 1. Commutator table of the Lie algebra of 2D-gnrAKNSe (4)

$[{\mathfrak{S}}_i,{\mathfrak{S}}_j]$	${\mathfrak{S}}_1$	${\mathfrak{S}}_2$	${\mathfrak{S}}_3$	${\mathfrak{S}}_4$	${\mathfrak{S}}_5$	${\mathfrak{S}}_6$	${\mathfrak{S}}_7$
${\mathfrak{S}}_1$	0	0	0	0	0	${\mathfrak{S}}_1$	$\theta {\mathfrak{S}}_3$
${\mathfrak{S}}_2$	0	0	0	0	0	0	0
${\mathfrak{S}}_3$	0	0	0	0	0	${\mathfrak{S}}_3$	$-\alpha {\mathfrak{S}}_3$
${\mathfrak{S}}_4$	0	0	0	0	${\mathfrak{S}}_3$	0	$\alpha {\mathfrak{S}}_4$
${\mathfrak{S}}_5$	0	0	0	$-{\mathfrak{S}}_3$	0	${\mathfrak{S}}_5$	$-2\alpha {\mathfrak{S}}_5$
${\mathfrak{S}}_6$	$-{\mathfrak{S}}_1$	0	$-{\mathfrak{S}}_3$	0	$-{\mathfrak{S}}_5$	0	0
${\mathfrak{S}}_7$	$-\theta {\mathfrak{S}}_3$	0	$\alpha {\mathfrak{S}}_3$	$-\alpha {\mathfrak{S}}_4$	$2\alpha {\mathfrak{S}}_5$	0	0

Utilizing the commutator table according to [72], we secure

$${\Theta }_i=\Theta (a_1,a_2,a_3,\cdots ,a_7,b_1,b_2,b_3,\cdots ,b_7),\mathrm{for}\mathrm{all}i=1,2,3,\cdots ,7,$$

as

$$\begin{array}{cc}\hfill & {\Theta }_1=-a_1{b}_6+a_6{b}_1,{\Theta }_2=0,\hfill \\ \hfill & {\Theta }_3=\alpha a_3{b}_7-\alpha a_7{b}_3-\theta a_1{b}_7+\theta a_7{b}_1-a_3{b}_6-a_4{b}_5+a_5{b}_4+a_6{b}_3,\hfill \\ \hfill & {\Theta }_4=-\alpha a_4{b}_7+\alpha a_7{b}_4,{\Theta }_5=2\alpha a_5{b}_7-2\alpha a_7{b}_5-a_5{b}_6+a_6{b}_5,{\Theta }_6=0,{\Theta }_7=0.\hfill \end{array}$$

(13)

The real function $\Phi =\Phi (a_1,a_2,a_3,\cdots ,a_7)$ is commonly referred to as an invariant which satisfies the relation $\Phi \left(\mathfrak{S}\right)=\Phi \left(A{d}_g\left(\mathfrak{S}\right)\right)$ for all $\mathfrak{S}\in G$ with any subgroup g. Moreover, $\Phi$ is invariant if and only if one can compute the values of $b_j$, for $1\le j\le 7$ and so we have

$$\begin{array}{cc}\hfill & (a_6-\alpha a_7){\displaystyle \frac{\partial \Phi }{\partial a_3}}=0,a_6{\displaystyle \frac{\partial \Phi }{\partial a_1}}+\theta a_7{\displaystyle \frac{\partial \Phi }{\partial a_3}}=0,\hfill \\ \hfill & a_5{\displaystyle \frac{\partial \Phi }{\partial a_3}}+\alpha a_7{\displaystyle \frac{\partial \Phi }{\partial a_4}}=0,-a_4{\displaystyle \frac{\partial \Phi }{\partial a_3}}+(a_6-2\alpha a_7){\displaystyle \frac{\partial \Phi }{\partial a_5}}=0,\hfill \\ \hfill & a_1{\displaystyle \frac{\partial \Phi }{\partial a_1}}+a_3{\displaystyle \frac{\partial \Phi }{\partial a_3}}+a_5{\displaystyle \frac{\partial \Phi }{\partial a_5}}=0,(\alpha a_3-\theta a_1){\displaystyle \frac{\partial \Phi }{\partial a_3}}-\alpha a_4{\displaystyle \frac{\partial \Phi }{\partial a_4}}+2\alpha a_5{\displaystyle \frac{\partial \Phi }{\partial a_5}}=0.\hfill \end{array}$$

(14)

Solving the system of equations secured in (14), we achieve the value of invariant $\Phi$ as

$$\Phi (a_1,a_2,a_3,a_4,a_5,a_6,a_7)=G(a_2,a_6,a_7),$$

(15)

where G represents an arbitrary function dependent on $a_2$, $a_6$ and $a_7$.

Now, we can anticipate the simplification of a given arbitrary element

$$\mathfrak{S}=a_1{\mathfrak{S}}_1+a_2{\mathfrak{S}}_2+a_3{\mathfrak{S}}_3+a_4{\mathfrak{S}}_4+a_5{\mathfrak{S}}_5+a_6{\mathfrak{S}}_6+a_7{\mathfrak{S}}_7$$

(16)

of the 2D-gnrAKNSe (4) Lie algebra represented as $\mathcal{G}$. We notice at this junction that we can represent elements of $\mathcal{G}$ via vector $a=(a_1,a_2,a_3,\cdots ,a_7)\in {\mathbb{R}}^7$ since each of the elements can be expressed in the structure of (16) for some constants $a_1,a_2,a_3,\cdots ,a_7$. As a result, the adjoint action can be taken in this regard as a group of linear transformations of vector $(a_1,a_2,a_3,\cdots ,a_7)$.

Table 2. Adjoint table of Lie algebra for seven parameters.

$[{\mathfrak{S}}_i,{\mathfrak{S}}_j]$	${\mathfrak{S}}_1$	${\mathfrak{S}}_2$	${\mathfrak{S}}_3$	${\mathfrak{S}}_4$	${\mathfrak{S}}_5$	${\mathfrak{S}}_6$	${\mathfrak{S}}_7$
${\mathfrak{S}}_1$	${\mathfrak{S}}_1$	${\mathfrak{S}}_2$	${\mathfrak{S}}_3$	${\mathfrak{S}}_4$	${\mathfrak{S}}_5$	${\mathfrak{S}}_6-{ϵ}_1{\mathfrak{S}}_1$	${\mathfrak{S}}_7-\theta {ϵ}_1{\mathfrak{S}}_3$
${\mathfrak{S}}_2$	${\mathfrak{S}}_1$	${\mathfrak{S}}_2$	${\mathfrak{S}}_3$	${\mathfrak{S}}_4$	${\mathfrak{S}}_5$	${\mathfrak{S}}_6$	${\mathfrak{S}}_7$
${\mathfrak{S}}_3$	${\mathfrak{S}}_1$	${\mathfrak{S}}_2$	${\mathfrak{S}}_3$	${\mathfrak{S}}_4$	${\mathfrak{S}}_5$	${\mathfrak{S}}_6-{ϵ}_3{\mathfrak{S}}_3$	${\mathfrak{S}}_7+\alpha {ϵ}_3{\mathfrak{S}}_3$
${\mathfrak{S}}_4$	${\mathfrak{S}}_1$	${\mathfrak{S}}_2$	${\mathfrak{S}}_3$	${\mathfrak{S}}_4$	${\mathfrak{S}}_5-{ϵ}_4{\mathfrak{S}}_3$	${\mathfrak{S}}_6$	${\mathfrak{S}}_7-\alpha {ϵ}_4{\mathfrak{S}}_4$
${\mathfrak{S}}_5$	${\mathfrak{S}}_1$	${\mathfrak{S}}_2$	${\mathfrak{S}}_3$	${\mathfrak{S}}_4+{ϵ}_5{\mathfrak{S}}_3$	${\mathfrak{S}}_5$	${\mathfrak{S}}_6-{ϵ}_5{\mathfrak{S}}_5$	${\mathfrak{S}}_7+2\alpha {ϵ}_5{\mathfrak{S}}_5$
${\mathfrak{S}}_6$	$e^{{ϵ}_6}{\mathfrak{S}}_1$	${\mathfrak{S}}_2$	$e^{{ϵ}_6}{\mathfrak{S}}_3$	${\mathfrak{S}}_4$	$e^{{ϵ}_6}{\mathfrak{S}}_5$	${\mathfrak{S}}_6$	${\mathfrak{S}}_7$
${\mathfrak{S}}_7$	${\mathfrak{S}}_1+{\Omega }_0{\mathfrak{S}}_3$	${\mathfrak{S}}_2$	$e^{-\alpha {ϵ}_7}{\mathfrak{S}}_3$	$e^{\alpha {ϵ}_7}{\mathfrak{S}}_4$	$e^{-2\alpha {ϵ}_7}{\mathfrak{S}}_5$	${\mathfrak{S}}_6$	${\mathfrak{S}}_7$

where ${\Omega }_0={\displaystyle \frac{\theta -\theta e^{-\alpha {ϵ}_7}}{\alpha }}$.

Thus, an adjoint action is contemplated for Lie algebra $\mathcal{G}$ and so by imploring Table 2, we can conveniently consider the theorem:

Theorem 2.4.

The one-dimensional subalgebras optimal system for 2D-gnrAKNSe (4) Lie algebra$\mathcal{G}$is presented as

(i) $\mathcal{G}$₁ : ${\mathfrak{S}}_1={\displaystyle \frac{\partial }{\partial y}},$

(ii) $\mathcal{G}$₂ : ${\mathfrak{S}}_2+{\mathfrak{S}}_7={\displaystyle \frac{\partial }{\partial x}}+t{\displaystyle \frac{\partial }{\partial t}}+\left({\displaystyle \frac{\theta }{\alpha }}y-u\right){\displaystyle \frac{\partial }{\partial u}},$

(iii) $\mathcal{G}$₃ : ${\mathfrak{S}}_1+c_1{\mathfrak{S}}_4={\displaystyle \frac{\partial }{\partial y}}+c_1{\displaystyle \frac{\partial }{\partial t}},\mathrm{for}\mathrm{all},c_1\in \{-1,0,1\},$

(iv) $\mathcal{G}$₄ : ${\mathfrak{S}}_6+{\mathfrak{S}}_7=y{\displaystyle \frac{\partial }{\partial y}}+u{\displaystyle \frac{\partial }{\partial u}}+t{\displaystyle \frac{\partial }{\partial t}}+\left({\displaystyle \frac{\theta }{\alpha }}y-u\right){\displaystyle \frac{\partial }{\partial u}},$

(v) $\mathcal{G}$₅ : ${\mathfrak{S}}_2+{\mathfrak{S}}_6={\displaystyle \frac{\partial }{\partial x}}+y{\displaystyle \frac{\partial }{\partial y}}+u{\displaystyle \frac{\partial }{\partial u}},$

(vi) $\mathcal{G}$₆ : ${\mathfrak{S}}_1+c_2{\mathfrak{S}}_5={\displaystyle \frac{\partial }{\partial y}}+c_{2}t{\displaystyle \frac{\partial }{\partial u}},\mathrm{for}\mathrm{all},c_2\in \{-1,0,1\},$

(vii) $\mathcal{G}$₇ : ${\mathfrak{S}}_1+{\mathfrak{S}}_4+{\mathfrak{S}}_5={\displaystyle \frac{\partial }{\partial y}}+{\displaystyle \frac{\partial }{\partial t}}+t{\displaystyle \frac{\partial }{\partial u}},$

(viii) $\mathcal{G}$₈ : ${\mathfrak{S}}_2+{\mathfrak{S}}_6+{\mathfrak{S}}_7={\displaystyle \frac{\partial }{\partial x}}+y{\displaystyle \frac{\partial }{\partial y}}+u{\displaystyle \frac{\partial }{\partial u}}+t{\displaystyle \frac{\partial }{\partial t}}+\left({\displaystyle \frac{\theta }{\alpha }}y-u\right){\displaystyle \frac{\partial }{\partial u}}$.

Proof. A function $F_i^{ϵ}$ : $\mathcal{G}$↦$\mathcal{G}$ defined by $\mathfrak{S}⟼\mathrm{Ad}(exp\left({ϵ}_{\mathrm{i}}{\mathfrak{S}}_{\mathrm{i}}\right)·\mathfrak{S})$ is a linear map, for $i=1,2,3,\cdots ,7$. Furthermore, the matrix $Ad(e^{\left({ϵ}_i{\mathfrak{S}}_i\right)}$ of the function $F_i^{ϵ},i=1,2,3,\cdots ,7$, with regards to the basis $\{{\mathfrak{S}}_1,{\mathfrak{S}}_2,{\mathfrak{S}}_3,\cdots ,{\mathfrak{S}}_7\}$ are given as:

$Ad\left(e^{{ϵ}_1{\mathfrak{S}}_1}\right)=\left(\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\\ -{ϵ}_1& 0& 0& 0& 0& 1& 0\\ 0& 0& -\theta {ϵ}_1& 0& 0& 0& 1\end{array}\right),Ad\left(e^{{ϵ}_2{\mathfrak{S}}_2}\right)=\left(\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\end{array}\right),$

$Ad\left(e^{{ϵ}_3{\mathfrak{S}}_3}\right)=\left(\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& -{ϵ}_3& 0& 0& 1& 0\\ 0& 0& \alpha {ϵ}_3& 0& 0& 0& 1\end{array}\right),Ad\left(e^{{ϵ}_4{\mathfrak{S}}_4}\right)=\left(\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\\ 0& 0& -{ϵ}_4& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& -\alpha {ϵ}_4& 0& 0& 1\end{array}\right),$

$Ad\left(e^{{ϵ}_5{\mathfrak{S}}_5}\right)=\left(\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 0& {ϵ}_5& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& -{ϵ}_5& 1& 0\\ 0& 0& 0& 0& 2\alpha {ϵ}_5& 0& 1\end{array}\right),Ad\left(e^{{ϵ}_6{\mathfrak{S}}_6}\right)=\left(\begin{array}{ccccccc}{e}^{{ϵ}_6}& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 0& 0& e^{{ϵ}_6}& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& e^{{ϵ}_6}& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\end{array}\right),$

$Ad\left(e^{{ϵ}_7{\mathfrak{S}}_7}\right)=\left(\begin{array}{ccccccc}1& 0& {\displaystyle \frac{e^{-\alpha {ϵ}_7}\left(-1+e^{\alpha {ϵ}_7}\right)\theta }{\alpha }}& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 0& 0& e^{-\alpha {ϵ}_7}& 0& 0& 0& 0\\ 0& 0& 0& e^{\alpha {ϵ}_7}& 0& 0& 0\\ 0& 0& 0& 0& e^{-2\alpha {ϵ}_7}& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\end{array}\right).$

Now, suppose we let

$$A({ϵ}_1,{ϵ}_2,{ϵ}_3,\cdots ,{ϵ}_7)=Ad\left(e^{{ϵ}_1{\mathfrak{S}}_1}\right)\circ Ad\left(e^{{ϵ}_2{\mathfrak{S}}_2}\right)\circ Ad\left(e^{{ϵ}_3{\mathfrak{S}}_3}\right),\circ \cdots \circ ,Ad\left(e^{{ϵ}_7{\mathfrak{S}}_7}\right),$$

(17)

where matrix A is defined as the universal matrix calculated and presented as

$A=\left(\begin{array}{ccccccc}{e}^{{ϵ}_6}& 0& {\displaystyle \frac{e^{{ϵ}_6-\alpha {ϵ}_7}\left(-1+e^{\alpha {ϵ}_7}\right)\theta }{\alpha }}& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 0& 0& e^{{ϵ}_6-\alpha {ϵ}_7}& 0& 0& 0& 0\\ 0& 0& {ϵ}_5{e}^{{ϵ}_6-\alpha {ϵ}_7}& e^{\alpha {ϵ}_7}& 0& 0& 0\\ 0& 0& -{ϵ}_4{e}^{{ϵ}_6-\alpha {ϵ}_7}& 0& e^{{ϵ}_6-2\alpha {ϵ}_7}& 0& 0\\ -{ϵ}_1{e}^{{ϵ}_6}& 0& -{\displaystyle \frac{e^{{ϵ}_6-\alpha {ϵ}_7}\left(-1+e^{\alpha {ϵ}_7}\right)\theta {ϵ}_1}{\alpha }}-{ϵ}_3{e}^{{ϵ}_6-\alpha {ϵ}_7}& 0& -{ϵ}_5{e}^{{ϵ}_6-2\alpha {ϵ}_7}& 1& 0\\ 0& 0& \left(-\theta {ϵ}_1+\alpha {ϵ}_3-\alpha {ϵ}_4{ϵ}_5\right)e^{{ϵ}_6-\alpha {ϵ}_7}& -\alpha {ϵ}_4{e}^{\alpha {ϵ}_7}& 2\alpha {ϵ}_5{e}^{{ϵ}_6-2\alpha {ϵ}_7}& 0& 1\end{array}\right).$

Let $\mathfrak{S}={\sum }_{i=1}^7{a}_i{\mathfrak{S}}_i$ be an element in the Lie algebra. The application of the most general adjoint action to $\mathfrak{S}$ where $\mathfrak{S}={\sum }_{i=1}^7{a}_i{\mathfrak{S}}_i$ and $Y={\sum }_{i=1}^7{\tilde{a}}_i{\mathfrak{S}}_i$ can be revealed to be equivalent, can be shown with respect to ${ϵ}_1,{ϵ}_2,{ϵ}_3,\cdots ,{ϵ}_7$ with the relation

$$({\tilde{a}}_1,{\tilde{a}}_2,{\tilde{a}}_3,{\tilde{a}}_4,{\tilde{a}}_5,{\tilde{a}}_6,{\tilde{a}}_7)=(a_1,a_2,a_3,a_4,a_5,a_6,a_7)A,$$

(18)

with A standing for the universal adjoint matrix. We now add a remark here:

Remark 2.2.

We notice that once equation (18) has a solution, then $\mathfrak{S}$ and Y as defined are equivalent under the adjoint action.

Therefore, $A({ϵ}_1,{ϵ}_2,{ϵ}_3,\cdots ,{ϵ}_7)$ transforms $\mathfrak{S}$ this way:

$$\begin{array}{cc}\hfill & A({ϵ}_1,{ϵ}_2,{ϵ}_3,\cdots ,{ϵ}_7)·\mathfrak{S}=\left(a_1{e}^{{ϵ}_6}-a_6{ϵ}_1{e}^{{ϵ}_6}\right){\mathfrak{S}}_1+a_2{\mathfrak{S}}_2+\left\{a_1{e}^{{ϵ}_6}\left({\displaystyle \frac{\theta }{\alpha }}-{\displaystyle \frac{\theta }{\alpha }}{e}^{-\alpha {ϵ}_7}\right)\right.\hfill \\ \hfill & \left.+a_3{e}^{{ϵ}_6}{e}^{-\alpha {ϵ}_7}+a_4{ϵ}_5{e}^{{ϵ}_6}{e}^{-\alpha {ϵ}_7}-a_5{ϵ}_4{e}^{{ϵ}_6}{e}^{-\alpha {ϵ}_7}-a_6\left({ϵ}_1{e}^{{ϵ}_6}\left({\displaystyle \frac{\theta }{\alpha }}-{\displaystyle \frac{\theta }{\alpha }}{e}^{-\alpha {ϵ}_7}\right)+{ϵ}_3{e}^{{ϵ}_6}{e}^{-\alpha {ϵ}_7}\right)\right\}{\mathfrak{S}}_3\hfill \\ \hfill & +\left(a_4{e}^{\alpha {ϵ}_7}-a_7\alpha {ϵ}_4{e}^{\alpha {ϵ}_7}\right){\mathfrak{S}}_4+\left(a_5{e}^{{ϵ}_6}{e}^{-2\alpha {ϵ}_7}-a_6{ϵ}_5{e}^{{ϵ}_6}{e}^{-2\alpha {ϵ}_7}+2a_7\alpha {ϵ}_5{e}^{{ϵ}_6}{e}^{-2\alpha {ϵ}_7}\right){\mathfrak{S}}_5+a_6{\mathfrak{S}}_6\hfill \\ \hfill & +a_7{\mathfrak{S}}_7.\hfill \end{array}$$

Now, the construction of one dimensional subalgebra optimal system of 2D-gnrAKNSe (4) can be performed. We contemplate the invariant function G depending on its sign[31,72] and so we do these cases, viz:

Case 1. $a_2=1$, $a_6=1$, $a_7=1$,

Imploring the selection of parameters as ${\tilde{a}}_i=0,i=1,3,4,5$, as well as ${\tilde{a}}_i=1,i=2,6,7$ in adjoint equation (18), we secured a solution

$${ϵ}_1=a_1,{ϵ}_3={\displaystyle \frac{\alpha \theta a_1-\alpha a_3+a_4{a}_5}{\alpha \left(\alpha -1\right)}},{ϵ}_4={\displaystyle \frac{a_4}{\alpha }},{ϵ}_5=-{\displaystyle \frac{a_5}{2\alpha -1}},\alpha \ne 0,1,{\displaystyle \frac{1}{2}}.$$

(19)

Therefore, we gain the optimal representative $\mathfrak{S}={\mathfrak{S}}_2+{\mathfrak{S}}_6+{\mathfrak{S}}_7$.

Case 2. $a_2{a}_6{a}_7=0$,

This case, dependent on the value of the constant parameters occasions four subcases. We highlight these subcases in the subsequent part of the study.

Case 2.1.$a_2=0$, $a_6=1$, $a_7=1$,

We insert the parametric values ${\tilde{a}}_i=0,i=1,2,3,4,5$, alongside ${\tilde{a}}_i=1,i=6,7$ in system (18), then we get

$${ϵ}_1=a_1,{ϵ}_3={\displaystyle \frac{a_4{a}_5+\alpha \theta a_1-\alpha a_3}{\alpha \left(\alpha -1\right)}},{ϵ}_4={\displaystyle \frac{a_4}{\alpha }},{ϵ}_5=-{\displaystyle \frac{a_5}{2\alpha -1}},\alpha \ne 0,1,{\displaystyle \frac{1}{2}}.$$

(20)

So, we select the optimal representative $\mathfrak{S}={\mathfrak{S}}_6+{\mathfrak{S}}_7$.

Case 2.2.$a_2=1$, $a_6=0$, $a_7=1$,

Moreover, we invoke the parametric values of ${\tilde{a}}_i,i=1,2,3,4,5$ as ${\tilde{a}}_i=0,i=1,3,4,5,6$, alongside ${\tilde{a}}_i=1,i=2,7$ in adjoint system (18), we achieve the solution

$${ϵ}_3={\displaystyle \frac{\alpha \theta {ϵ}_1-\alpha a_3+a_4{a}_5}{{\alpha }^2}},{ϵ}_4={\displaystyle \frac{a_4}{\alpha }},{ϵ}_5=-{\displaystyle \frac{a_5}{2\alpha }},\alpha \ne 0.$$

(21)

In consequence, we select the optimal representative $\mathfrak{S}={\mathfrak{S}}_2+{\mathfrak{S}}_7$.

Case 2.3.$a_2=1$, $a_6=1$, $a_7=0$,

We further substitute parameters ${\tilde{a}}_i,i=1,2,3,4,5$ in adjoint system (18) as ${\tilde{a}}_i=0,i=1,3,4,5,7$, alongside ${\tilde{a}}_i=1,i=2,6$ and obtain solutions with regards to ${ϵ}_1$, ${ϵ}_3$, ${ϵ}_4$ and ${ϵ}_5$ as

$${ϵ}_1=a_1,{ϵ}_3=a_3-a_5{ϵ}_4,{ϵ}_5=a_5.$$

(22)

Hence, we select the optimal representative $\mathfrak{S}={\mathfrak{S}}_2+{\mathfrak{S}}_6$.

Case 2.4.$a_2=0$, $a_6=0$, $a_7=0$,

In this subcase, we insert the values $a_2=0$, $a_6=0$ and $a_7=0$ in the system of partial differential equations (14), we gain a new invariant $\Phi (a_1,a_3,a_4,a_5)$ solution as

$$\Phi (a_1,a_3,a_4,a_5)=G\left({\displaystyle \frac{a_5{a}_4^2}{a_1}}\right).$$

(23)

We proceed to consider two major cases of the new invariants as ${\displaystyle \frac{a_5{a}_4^2}{a_1}}=1$ and ${\displaystyle \frac{a_5{a}_4^2}{a_1}}=0$ and explore them.

Case 2.4.1.$a_1=1$, $a_4=1$, $a_5=1$,

We observe that if ${\displaystyle \frac{a_5{a}_4^2}{a_1}}=1$, then we can say $a_1=a_5{a}_4^2$ and so without loss of generality, we can choose parametric values of ${\tilde{a}}_i,i=1,3,4,5$ as ${\tilde{a}}_i=0,i=2,3,6,7$, along with ${\tilde{a}}_i=1,i=1,4,5$ in adjoint system (18). Thus, we secure the solution

$${ϵ}_5={\displaystyle \frac{\alpha a_3+\theta a_4{a}_5-\alpha a_5{ϵ}_4-\theta a_4^2{a}_5}{\alpha a_4}},{ϵ}_6=ln\left({\displaystyle \frac{1}{a_4^2{a}_5}}\right),{ϵ}_7=-{\displaystyle \frac{1}{\alpha }}ln\left(a_4\right).$$

(24)

Therefore, we choose the optimal representative $\mathfrak{S}={\mathfrak{S}}_1+{\mathfrak{S}}_4+{\mathfrak{S}}_5$. Now, for invariant ${\displaystyle \frac{a_5{a}_4^2}{a_1}}=0$, we contemplate three possible options.

Case 2.4.2.$a_1\ne 0$, $a_4\ne 0$, $a_5=0$,

In the first place, we contemplate $a_1\ne 0$, $a_4\ne 0$, $a_5=0$ and by substituting the appropriate values of parameters ${\tilde{a}}_i,i=1,2,3,\cdots ,7$, that is, ${\tilde{a}}_i=0,i=2,3,5,6,7$, along with ${\tilde{a}}_i=1,i=1,4$, in system (18), one obtains

$${ϵ}_5=-{\displaystyle \frac{\theta a_1+\alpha a_3{a}_4-\theta a_1{a}_4}{\alpha a_4^2}},{ϵ}_6=-ln\left(a_1\right),{ϵ}_7=-{\displaystyle \frac{1}{\alpha }}ln\left(a_4\right).$$

(25)

Thus, we select the optimal representative $\mathfrak{S}={\mathfrak{S}}_1+{\mathfrak{S}}_4$. In addition, if we choose ${\tilde{a}}_i=0,i=2,3,5,6,7$, together with ${\tilde{a}}_i=-1,i=1,4$, in system (18), one then gets

$${ϵ}_5=-{\displaystyle \frac{\theta a_1+\alpha a_3{a}_4-\theta a_1{a}_4}{\alpha a_4^2}},{ϵ}_6=ln\left(\left|-{\displaystyle \frac{1}{a_1}}\right|\right),{ϵ}_7={\displaystyle \frac{1}{\alpha }}ln\left(\left|-{\displaystyle \frac{1}{a_4}}\right|\right).$$

(26)

Thus, we select the optimal representative $\mathfrak{S}={\mathfrak{S}}_1-{\mathfrak{S}}_4$.

Case 2.4.3.$a_1\ne 0$, $a_5\ne 0$, $a_4=0$,

Now, we consider $a_4=0$, $a_1\ne 0$, $a_5\ne 0$ and insert the values of parameters ${\tilde{a}}_i=0,i=2,3,4,6,7$, along with ${\tilde{a}}_i=1,i=1,5$, in adjoint system (18), we gain

$${ϵ}_4={\displaystyle \frac{1}{\alpha a_5}}\left(\theta a_5\sqrt{{\displaystyle \frac{a_1}{a_5}}}+\alpha a_3-\theta a_1\right),{ϵ}_6=-ln\left(a_5\right)-2ln\left(\sqrt{{\displaystyle \frac{a_1}{a_5}}}\right),{ϵ}_7=-{\displaystyle \frac{1}{\alpha }}ln\left(\sqrt{{\displaystyle \frac{a_1}{a_5}}}\right).$$

(27)

In consequence, we choose the optimal representative $\mathfrak{S}={\mathfrak{S}}_1+{\mathfrak{S}}_5$. Besides, if we choose ${\tilde{a}}_i=0,i=2,3,4,6,7$, together with ${\tilde{a}}_i=-1,i=1,5$, in system (18), we then achieve the result

$$\begin{array}{cc}\hfill & {ϵ}_4={\displaystyle \frac{1}{\alpha a_5}}\left(\theta a_5\sqrt{{\displaystyle \frac{a_1}{a_5}}}+\alpha a_3-\theta a_1\right),{ϵ}_6=ln\left(\left|-{\displaystyle \frac{1}{a_5}}\right|\right)-2ln\left(\sqrt{{\displaystyle \frac{a_1}{a_5}}}\right),\hfill \\ \hfill & {ϵ}_7=-{\displaystyle \frac{1}{\alpha }}ln\left(\sqrt{{\displaystyle \frac{a_1}{a_5}}}\right).\hfill \end{array}$$

As a result, we select the optimal representative $\mathfrak{S}={\mathfrak{S}}_1-{\mathfrak{S}}_5$. Finally if we contemplate $a_1\ne 0$, $a_5=0$, $a_4=0$, make proper choice of ${\tilde{a}}_i,i=1,2,3,\cdots ,7$ and substitute them in (18) as we have earlier done, then we achieve the solution

$${ϵ}_6=-ln\left(a_1\right),{ϵ}_7=-{\displaystyle \frac{1}{\alpha }}ln\left(\left|-{\displaystyle \frac{\theta a_1}{\alpha a_3-\theta a_1}}\right|\right).$$

(28)

Therefore, we select the optimal representative ${\mathfrak{S}}_1$. Now, based on the outcome of our computations, we achieve eight members of the one-dimensional subalgebras optimal system for 2D-gnrAKNSe (4) as: ${\mathfrak{S}}_1$, ${\mathfrak{S}}_2+{\mathfrak{S}}_7$, ${\mathfrak{S}}_1+c_1{\mathfrak{S}}_4$, ${\mathfrak{S}}_6+{\mathfrak{S}}_7$, ${\mathfrak{S}}_2+{\mathfrak{S}}_6$, ${\mathfrak{S}}_1+c_2{\mathfrak{S}}_5$, ${\mathfrak{S}}_1+{\mathfrak{S}}_4+{\mathfrak{S}}_5$ and ${\mathfrak{S}}_2+{\mathfrak{S}}_6+{\mathfrak{S}}_7$, for all $c_1,c_2\in \{-1,0,1\}$. This ends the proof of the theorem.

2.4. Lie Group-Invariants and Similarity Solutions of (4)

This section furnishes the utilization of the obtained optimal system of subalgebras explicated earlier in reducing 2D-gnrAKNSe (4) to ordinary differential equations and subsequently obtaining possible exact solutions of the model. In achieving this, we take into account the relation purveys as

$${\displaystyle \frac{dx}{{\xi }^1(t,x,y,u)}}={\displaystyle \frac{dy}{{\xi }^2(t,x,y,u)}}={\displaystyle \frac{dt}{{\xi }^3(t,x,y,u)}}={\displaystyle \frac{du}{\eta (t,x,y,u)}},$$

(29)

which is often referred to as the characteristic equations or Lagrangian system.

2.4.1. Invariant Solutions Using Subalgebra Component ${\mathfrak{S}}_1$

In the first place, we contemplate time translation symmetry ${\mathfrak{S}}_1$ which when apply (29) engenders invariants $P=t$ and $Q=x$ with group invariant $W(P,Q)=u(t,x,y)$. Inserting the function in equation (4) gives the NLNPDEE

$$\sigma W_{PQ}-\theta W_{QQ}+\rho W_{PQQQ}=0.$$

(30)

Thus, we have the solution to 2D-gnrAKNSe (4) in this regard as

$$\begin{array}{cc}\hfill u(t,x,y)=& f\left(t\right)+A_1{A}_{2}exp\left(c_{1}t+{\displaystyle \frac{\theta }{2c_1\rho }}x+{\displaystyle \frac{1}{2\rho c_1}}\sqrt{{\theta }^2-4\rho \sigma c_1^2}x\right)\hfill \\ \hfill & \times {\left[{\displaystyle \frac{\theta }{2\rho c_1}}+{\displaystyle \frac{1}{2\rho c_1}}\sqrt{{\theta }^2-4\rho \sigma c_1^2}\right]}^{-1}+A_1{A}_3{\left[{\displaystyle \frac{\theta }{2\rho c_1}}-{\displaystyle \frac{1}{2\rho c_1}}\sqrt{{\theta }^2-4\rho \sigma c_1^2}\right]}^{-1}\hfill \\ \hfill & \times exp\left(c_{1}t+{\displaystyle \frac{\theta }{2c_1\rho }}x-{\displaystyle \frac{1}{2\rho c_1}}\sqrt{{\theta }^2-4\rho \sigma c_1^2}x\right),\hfill \end{array}$$

(31)

where $c_1,A_1,A_2$ and $A_3$ are arbitrary constants. Further exploration of (30) gives

$$\begin{array}{cc}\hfill L_1=& {\displaystyle \frac{\partial }{\partial P}},L_2={\displaystyle \frac{\partial }{\partial Q}},L_3={\displaystyle \frac{\partial }{\partial P}}+{\displaystyle \frac{\partial }{\partial Q}},L_4={\displaystyle \frac{\partial }{\partial P}}+{\displaystyle \frac{\partial }{\partial W}},\hfill \\ \hfill L_5=& {\displaystyle \frac{\partial }{\partial Q}}+{\displaystyle \frac{\partial }{\partial W}},L_6={\displaystyle \frac{\partial }{\partial P}}+{\displaystyle \frac{\partial }{\partial Q}}+W{\displaystyle \frac{\partial }{\partial W}},\hfill \\ \hfill L_7=& {\displaystyle \frac{\partial }{\partial P}}-\left[c_1\sigma -{\displaystyle \frac{1}{2}}\left(\sqrt{{\theta }^2-4\rho \sigma c_1^2}-\theta \right)exp\left(c_{1}t+{\displaystyle \frac{\theta }{2c_1\rho }}x-{\displaystyle \frac{1}{2\rho c_1}}\sqrt{{\theta }^2-4\rho \sigma c_1^2}x\right)\right]\hfill \\ \hfill & -{\displaystyle \frac{\partial }{\partial W}},L_8={\displaystyle \frac{\partial }{\partial P}}+{\displaystyle \frac{\partial }{\partial Q}}\hfill \\ \hfill & +\left[c_1\sigma -{\displaystyle \frac{1}{2}}\left(\sqrt{{\theta }^2-4\rho \sigma c_1^2}-\theta \right)exp\left(c_{1}t+{\displaystyle \frac{\theta }{2c_1\rho }}x-{\displaystyle \frac{1}{2\rho c_1}}\sqrt{{\theta }^2-4\rho \sigma c_1^2}x\right)\right]{\displaystyle \frac{\partial }{\partial W}}.\hfill \end{array}$$

In utilizing the computed symmetries to reduce (4), we first contemplate $L_1$ which gives $W(P,Q)=\Theta \left(w\right)$, where $w=Q$. Thus, we use the function and secure ${\Theta }^{\prime \prime }\left(w\right)=0$ which when solved leads to a solution of 2D-gnrAKNSe (4) as

$$u(t,x,y)=C_{0}x+C_1$$

(32)

with integration constants $C_0$ and $C_1$. Meanwhile, $L_2$ gives a trivial solution of (4). We now explore $L_3$ and achieve $W(P,Q)=\Theta \left(w\right)$, $w=Q-P$ reducing (4) to

$$u(t,x,y)=A_0+A_1\left(x-t\right)+A_{2}sin\left(\sqrt{{\displaystyle \frac{\sigma +\theta }{\rho }}}\left(x-t\right)\right)+A_{3}cos\left(\sqrt{{\displaystyle \frac{\sigma +\theta }{\rho }}}\left(x-t\right)\right),$$

(33)

which is a trigonometric solution of 2D-gnrAKNSe (4), where $A_0,\cdots ,A_3$ are arbitrary constants. Exploring $L_4$ leads to $W(P,Q)=\Theta \left(w\right)+P$, where $w=Q$ leading to linear ordinary differential equation (LNODE) just like the case of $L_1$, and giving

$$u(t,x,y)=t+B_{0}x+B_1$$

(34)

with arbitrary integration constants $B_0$ and $B_1$. Now, we take on $L_6$, since $L_5$ purveys a trivial solution of model (4) and so it gives $W(P,Q)=\Theta \left(w\right)exp\left(P\right)$, where we have $w=Q-P$. Substituting the obtained function in (4) secures

$$\sigma {\Theta }^{\prime }\left(w\right)-\sigma {\Theta }^{\prime \prime }\left(w\right)-\theta {\Theta }^{\prime \prime }\left(w\right)+\rho {\Theta }^{\prime \prime \prime }\left(w\right)-\rho {\Theta }^{\prime \prime \prime \prime }\left(w\right)=0.$$

(35)

Next, we contemplate $L_7$ and on solving the associated system (29) leads to

$$\begin{array}{cc}\hfill W(P,Q)=& {\displaystyle \frac{1}{2\sigma c_1^2}}\left\{2\sigma c_1^2\Theta \left(w\right)+\theta exp\left[{\displaystyle \frac{1}{2\rho c_1}}\left(2\rho c_1^{2}P+\theta Q+\sqrt{{\theta }^2-4\rho \sigma c_1^2}Q\right)\right]\right.\hfill \\ \hfill & \left.+2\sigma c_1^{2}P-\sqrt{{\theta }^2-4\rho \sigma c_1^2}exp\left[{\displaystyle \frac{1}{2\rho c_1}}\left(2\rho c_1^{2}P+\theta Q+\sqrt{{\theta }^2-4\rho \sigma c_1^2}Q\right)\right]\right\},\hfill \end{array}$$

(36)

where $w=Q$. Inserting the relation in (36) into (4) gives LNODE as we have in the case of $L_1$ which solves to secure a solution of 2D-gnrAKNSe (4) here as

$$\begin{array}{cc}\hfill u(t,x,y)=& {\displaystyle \frac{1}{2\sigma c_1^2}}\left\{2\sigma c_1^2\left(C_{0}x+C_1\right)+\theta exp\left[{\displaystyle \frac{1}{2\rho c_1}}\left(2\rho c_1^{2}t+\theta x+\sqrt{{\theta }^2-4\rho \sigma c_1^2}x\right)\right]\right.\hfill \\ \hfill & \left.+2\sigma c_1^{2}t-\sqrt{{\theta }^2-4\rho \sigma c_1^2}exp\left[{\displaystyle \frac{1}{2\rho c_1}}\left(2\rho c_1^{2}t+\theta x+\sqrt{{\theta }^2-4\rho \sigma c_1^2}x\right)\right]\right\},\hfill \end{array}$$

(37)

where $C_0$ and $C_1$ are arbitrary integration constants. Finally, under ${\mathfrak{S}}_1$, we engage symmetry operator $L_8$ which yields the invariant expressed here as

$$\begin{array}{cc}\hfill W(P,Q)=& {\displaystyle \frac{1}{\sigma \left(\sqrt{{\theta }^2-4\rho \sigma c_1^2}-2\rho c_1^2-\theta \right)}}\{-2\rho \sigma c_1^2\Theta \left(w\right)+\sigma \sqrt{{\theta }^2-4\rho \sigma c_1^2}\Theta \left(w\right)\hfill \\ \hfill & -2\rho \sigma c_1^{2}P-\sigma \theta P-\theta \rho exp\left[-{\displaystyle \frac{1}{2\rho c_1}}\left(-2\rho c_1^{2}P-\theta Q+\sqrt{{\theta }^2-4\rho \sigma c_1^2}Q\right)\right]\hfill \\ \hfill & -\rho \sqrt{{\theta }^2-4\rho \sigma c_1^2}exp\left[-{\displaystyle \frac{1}{2\rho c_1}}\left(-2\rho c_1^{2}P-\theta Q+\sqrt{{\theta }^2-4\rho \sigma c_1^2}Q\right)\right]\hfill \\ \hfill & -\sigma \theta \Theta \left(w\right)+\sigma \sqrt{{\theta }^2-4\rho \sigma c_1^2}P\},\mathrm{where}w=Q-P.\hfill \end{array}$$

(38)

On invoking the expression computed for $W(P,Q)$ in (38) into (4), we have

$$\sigma {\Theta }^{\prime \prime }\left(w\right)+\theta {\Theta }^{\prime \prime }\left(w\right)+\rho {\Theta }^{\prime \prime \prime \prime }\left(w\right)=0.$$

(39)

Solving (39) yields a solution of 2D-gnrAKNSe (4) as

$$\begin{array}{cc}\hfill u(t,x,y)=& {\displaystyle \frac{1}{\sigma \left(\sqrt{{\theta }^2-4\rho \sigma c_1^2}-2\rho c_1^2-\theta \right)}}\{\left(\sigma \sqrt{{\theta }^2-4\rho \sigma c_1^2}-2\rho \sigma c_1^2\right)\hfill \\ \hfill & \times \left[A_0+A_1\left(x-t\right)+A_{2}sin\left(\sqrt{{\displaystyle \frac{\sigma +\theta }{\rho }}}\left(x-t\right)\right)+A_{3}cos\left(\sqrt{{\displaystyle \frac{\sigma +\theta }{\rho }}}\left(x-t\right)\right)\right]\hfill \\ \hfill & -2\rho \sigma c_1^{2}t-\sigma \theta t-\theta \rho exp\left[-{\displaystyle \frac{1}{2\rho c_1}}\left(-2\rho c_1^{2}t-\theta x+\sqrt{{\theta }^2-4\rho \sigma c_1^2}x\right)\right]\hfill \\ \hfill & -\rho \sqrt{{\theta }^2-4\rho \sigma c_1^2}exp\left[-{\displaystyle \frac{1}{2\rho c_1}}\left(-2\rho c_1^{2}t-\theta x+\sqrt{{\theta }^2-4\rho \sigma c_1^2}x\right)\right]\hfill \\ \hfill & -\sigma \theta \left[A_0+A_1\left(x-t\right)+A_{2}sin\left(\sqrt{{\displaystyle \frac{\sigma +\theta }{\rho }}}\left(x-t\right)\right)\right.\hfill \\ \hfill & \left.+A_{3}cos\left(\sqrt{{\displaystyle \frac{\sigma +\theta }{\rho }}}\left(x-t\right)\right)\right]+\sigma \sqrt{{\theta }^2-4\rho \sigma c_1^2}t\},\hfill \end{array}$$

(40)

where $A_0$, $A_1$, $A_2$ and $A_3$ are arbitrary constants.

2.4.2. Invariant Solutions Using Subalgebra Component ${\mathfrak{S}}_2+{\mathfrak{S}}_7$

Now, we take on the second member of the achieved-optimal system which is ${\mathfrak{S}}_2+{\mathfrak{S}}_7$. Solving the related Lagrangian system purveys an invariant explicated as

$$u(t,x,y)={\displaystyle \frac{1}{t}}W\left(P,Q\right)+{\displaystyle \frac{\theta }{\alpha }}Q,\mathrm{where}P=x-ln\left(t\right),Q=y.$$

(41)

Utilizing the gained-function transforms (4) to a NLNPDEE furnished as

$$2\alpha W_P{W}_{PQ}+\alpha W_Q{W}_{PP}-\sigma W_P-\sigma W_{PP}-\rho W_{PPP}-\rho W_{PPPP}=0.$$

(42)

Here, we secure a solution of model (4) in this instance as

$$u(t,x,y)={\displaystyle \frac{1}{t}}\left\{f_1\left[x-ln\left(t\right)\right]f_2\left(y\right)\right\}+{\displaystyle \frac{\theta }{\alpha }}y,$$

(43)

where arbitrary functions $f_1$ and $f_2$ are depending on their respective arguments. Further exploration of the gained NLNPDEE (42) reveals the four symmetries, viz

$$L_1={\displaystyle \frac{\partial }{\partial Q}},L_2={\displaystyle \frac{\partial }{\partial P}},L_3={\displaystyle \frac{\partial }{\partial W}},L_4=Q{\displaystyle \frac{\partial }{\partial Q}}+W{\displaystyle \frac{\partial }{\partial W}}.$$

Using symmetry operator $L_1$ furnishes a trivial solution of (4) and so in the case of $L_2$ one secures $W(P,Q)=\Theta \left(w\right)$, where we have $w=P$ which reduces (4) to

$$\sigma {\Theta }^{\prime }\left(w\right)+\sigma {\Theta }^{\prime \prime }\left(w\right)+\rho {\Theta }^{\prime \prime \prime }\left(w\right)+\rho {\Theta }^{\prime \prime \prime \prime }\left(w\right)=0.$$

(44)

Hence, we gain a trigonometry solution of (4) in this case as

$$\begin{array}{cc}\hfill u(t,x,y)=& {\displaystyle \frac{1}{t}}\{C_0+C_{2}sin\left(\sqrt{{\displaystyle \frac{\sigma }{\rho }}}\left(x-ln\left(t\right)\right)\right)+C_{3}cos\left(\sqrt{{\displaystyle \frac{\sigma }{\rho }}}\left(x-ln\left(t\right)\right)\right)\hfill \\ \hfill & +C_1{e}^{-\left(x-ln\left(t\right)\right)}\}+{\displaystyle \frac{\theta }{\alpha }}y\hfill \end{array}$$

(45)

with arbitrary constants $C_0$, $C_2$ and $C_3$. In the case of symmetry generator $L_3$, one gains $W(P,Q)=\Theta \left(w\right)+P$, where we have $w=Q-P$. Thus, we have (4) become

$$\sigma {\Theta }^{\prime }\left(w\right)-2\alpha {\Theta }^{\prime \prime }\left(w\right)-\sigma {\Theta }^{\prime \prime }\left(w\right)+3\alpha {\Theta }^{\prime }\left(w\right){\Theta }^{\prime \prime }\left(w\right)+\rho {\Theta }^{\prime \prime \prime }\left(w\right)-\rho {\Theta }^{\prime \prime \prime \prime }\left(w\right)-\sigma =0.$$

(46)

In addition, for $L_4$, one achieves $W(P,Q)=Q\Theta \left(w\right)$, where $w=P$, so (4) becomes

$$2\alpha {\Theta }^{\prime }{\left(w\right)}^2-\sigma {\Theta }^{\prime }\left(w\right)-\sigma {\Theta }^{\prime \prime }\left(w\right)+\alpha {\Theta }^{\prime }\left(w\right){\Theta }^{\prime \prime }\left(w\right)-\rho {\Theta }^{\prime \prime \prime }\left(w\right)-\rho {\Theta }^{\prime \prime \prime \prime }\left(w\right)=0.$$

(47)

2.4.3. Invariant Solutions Using Subalgebra Component ${\mathfrak{S}}_1+c_1{\mathfrak{S}}_4$

The associated characteristic equation to ${\mathfrak{S}}_1+c_1{\mathfrak{S}}_4$ solves to give the function $u(t,x,y)=W\left(P,Q\right)$, where $P=x$, $Q=1/c_1\left(c_{1}y-t\right)$. Using these reduce (4) to

$$\alpha c_1{W}_Q{W}_{PP}-\sigma W_{PQ}-\theta c_1{W}_{PP}+2\alpha c_1{W}_P{W}_{PQ}-\rho W_{PPPQ}=0.$$

(48)

In this case, we have a solution of (48) in this regard as the tan-hyperbolic function

$$u(t,x,y)=A_2-{\displaystyle \frac{4\rho A_1}{\alpha c_1}}tanh\left[A_{1}x-{\displaystyle \frac{\theta c_1{A}_1}{4\rho A_1^2+\sigma }}\left({\displaystyle \frac{1}{c_1}}\left(c_{1}y-t\right)\right)+A_0\right]$$

(49)

with arbitrary constants $A_0,A_1$ and $A_2$. Besides, (48) furnishes symmetries

$$\begin{array}{cc}\hfill L_1=& {\displaystyle \frac{\partial }{\partial P}}+F_1\left(Q\right){\displaystyle \frac{\partial }{\partial Q}}+{\displaystyle \frac{\theta }{\alpha }}{F}_1\left(Q\right){\displaystyle \frac{\partial }{\partial W}},L_2=F_2\left(Q\right){\displaystyle \frac{\partial }{\partial Q}}+\left(1+{\displaystyle \frac{\theta }{\alpha }}{F}_2\left(Q\right)\right){\displaystyle \frac{\partial }{\partial W}},\hfill \\ \hfill L_3=& P{\displaystyle \frac{\partial }{\partial P}}+F_3\left(Q\right){\displaystyle \frac{\partial }{\partial Q}}+\left\{{\displaystyle \frac{\sigma }{\alpha c_1}}P+{\displaystyle \frac{\theta }{\alpha }}\left[Q+F_3\left(Q\right)\right]-W\right\}{\displaystyle \frac{\partial }{\partial W}}.\hfill \end{array}$$

In order to attain interesting results, we let $F_i\left(Q\right)=Q,i=1,\cdots ,3$. So, applying Lie theoretic approach to $L_1$, one achieves $W(P,Q)=\Theta \left(w\right)+{\displaystyle \frac{\theta }{\alpha }}Q$, where we have $w=Qexp(-P)$. Thus, substituting the function in (48) further reduces it to

$$\begin{array}{cc}\hfill & \sigma w{\Theta }^{\prime }\left(w\right)+3\alpha c_1{w}^2{\Theta }^{\prime }{\left(w\right)}^2+\rho w{\Theta }^{\prime }\left(w\right)+\sigma w^2{\Theta }^{\prime \prime }\left(w\right)+7\rho w^2{\Theta }^{\prime \prime }\left(w\right)\hfill \\ \hfill & +3\alpha c_1{w}^3{\Theta }^{\prime }\left(w\right){\Theta }^{\prime \prime }\left(w\right)+6\rho w^3{\Theta }^{\prime \prime \prime }\left(w\right)+\rho w^4{\Theta }^{\prime \prime \prime \prime }\left(w\right)=0.\hfill \end{array}$$

(50)

Next, we contemplate $L_2$ and this produces $W(P,Q)=\Theta \left(w\right)+ln\left(Q\right)+{\displaystyle \frac{\theta }{\alpha }}Q$, with $w=P$. This further reduces (4) to an LNODE giving a solution of the model as

$$u(t,x,y)=C_{0}x+ln\left[{\displaystyle \frac{1}{c_1}}\left(c_{1}y-t\right)\right]+{\displaystyle \frac{\theta }{\alpha }}\left({\displaystyle \frac{1}{c_1}}\left(c_{1}y-t\right)\right)+C_1$$

(51)

with integration constants $C_0$ and $C_1$. Now, we consider $L_3$ and by the doing the usual, one attains $W(P,Q)={\displaystyle \frac{1}{P}}\Theta \left(w\right)+{\displaystyle \frac{\theta }{\alpha }}Q+{\displaystyle \frac{\sigma }{2\alpha c_1}}P$, $w=Q/P$, reducing (4) to

$$\begin{array}{cc}\hfill & 2\alpha c_{1}w\Theta \left(w\right){\Theta }^{\prime \prime }\left(w\right)+8\alpha c_{1}w{\Theta }^{\prime }{\left(w\right)}^2+24\rho {\Theta }^{\prime }\left(w\right)+6\alpha c_1\Theta \left(w\right){\Theta }^{\prime }\left(w\right)+36\rho w{\Theta }^{\prime \prime }\left(w\right)\hfill \\ \hfill & +3\alpha c_1{w}^2{\Theta }^{\prime }\left(w\right){\Theta }^{\prime \prime }\left(w\right)+12\rho w^2{\Theta }^{\prime \prime \prime }\left(w\right)+\rho w^3{\Theta }^{\prime \prime \prime \prime }\left(w\right)=0.\hfill \end{array}$$

(52)

2.4.4. Invariant Solutions Using Subalgebra Component ${\mathfrak{S}}_6+{\mathfrak{S}}_7$

In the case of subalgebra ${\mathfrak{S}}_6+{\mathfrak{S}}_7$, we calculate its invariants and so have

$$P=x,Q={\displaystyle \frac{y}{t}},\mathrm{where}\mathrm{we}\mathrm{have}\mathrm{group}\mathrm{invariant},W(P,Q)+{\displaystyle \frac{\theta }{\alpha }}y=u(t,x,y).$$

On applying the calculated-invariants in (4), one attains NLNPDEE

$$\alpha W_Q{W}_{PP}-\sigma Q{W}_{PQ}+2\alpha W_P{W}_{PQ}-\rho Q{W}_{PPPQ}.$$

(53)

Consequently, we secure a solution of model (4) in this regard as

$$u(t,x,y)=f_1\left(x\right)f_2\left({\displaystyle \frac{y}{t}}\right)+{\displaystyle \frac{\theta }{\alpha }}y,$$

(54)

where we have arbitrary functions $f_1$ and $f_2$ depending on their arguments. Here, (53) admits three symmetries $L_1={\partial }_P$, $L_2={\partial }_W$, and $L_3=Q{\partial }_Q+W{\partial }_W$. Having found out that $L_1$ and $L_1+L_2$ give trivial solutions of (4), we then explore $L_3$ and achieve $W(P,Q)=Q\Theta \left(w\right)$, where we have $w=P$. This further transforms (4) to

$$2\alpha {\Theta }^{\prime }{\left(w\right)}^2-\sigma {\Theta }^{\prime }\left(w\right)+\alpha \Theta \left(w\right){\Theta }^{\prime \prime }\left(w\right)-\rho {\Theta }^{\prime \prime \prime }\left(w\right)=0.$$

(55)

2.4.5. Invariant Solutions Using Subalgebra Component ${\mathfrak{S}}_2+{\mathfrak{S}}_6$

Here, we take a look at ${\mathfrak{S}}_2+{\mathfrak{S}}_6$ and by following the usual steps secures; $exp\left(x\right)W(P,Q)=u(t,x,y)$, where $P=t$ and $Q=yexp(-x)$. Invoking these in (4) gives NLNPDEE

$$\begin{array}{cc}\hfill & \sigma W_P+\rho W_P-\theta W-\sigma Q{W}_{PQ}-\rho Q{W}_{PQ}+\theta Q{W}_Q+\alpha W{W}_Q-\alpha Q{W}_Q^2\hfill \\ \hfill & -\theta Q^2{W}_{QQ}-2\alpha QW{W}_{QQ}+3\alpha Q^2{W}_Q{W}_{QQ}-\rho Q^3{W}_{PQQQ}=0.\hfill \end{array}$$

(56)

Exploring (56) further via Lie symmetry approach leads to the attainment of

$$\begin{array}{cc}\hfill L_1=& {\displaystyle \frac{\partial }{\partial P}},L_2=Q{\displaystyle \frac{\partial }{\partial Q}}+W{\displaystyle \frac{\partial }{\partial W}},L_3=P{\displaystyle \frac{\partial }{\partial P}}+\left\{{\displaystyle \frac{\theta }{\alpha }}Q-W\right\}{\displaystyle \frac{\partial }{\partial W}}.\hfill \end{array}$$

In case of $L_1$, we have invariant $W(P,Q)=\Theta \left(w\right)$, where $w=P$, reducing (4) to

$$\begin{array}{cc}\hfill & \theta w{\Theta }^{\prime }\left(w\right)-2\alpha w\Theta \left(w\right){\Theta }^{\prime \prime }\left(w\right)-\alpha w{\Theta }^{\prime }{\left(w\right)}^2+\alpha \Theta \left(w\right){\Theta }^{\prime }\left(w\right)-\theta w^2{\Theta }^{\prime \prime }\left(w\right)\hfill \\ \hfill & +3\alpha w^2{\Theta }^{\prime }\left(w\right){\Theta }^{\prime \prime }\left(w\right)-\theta \Theta \left(w\right)=0.\hfill \end{array}$$

(57)

Invariant of $L_2$ gives a trivial solution of (4) and so for $L_3$, one gains $W(P,Q)=P^{-1}\Theta \left(w\right)+{\displaystyle \frac{\theta }{\alpha }}Q$, where $w=Q$ which eventually transforms (4) to a NLNODE

$$\begin{array}{cc}\hfill & \rho w{\Theta }^{\prime }\left(w\right)-2\alpha w\Theta \left(w\right){\Theta }^{\prime \prime }\left(w\right)-\alpha w{\Theta }^{\prime }{\left(w\right)}^2+\alpha \Theta \left(w\right){\Theta }^{\prime }\left(w\right)+\sigma w{\Theta }^{\prime }\left(w\right)\hfill \\ \hfill & -\sigma \Theta \left(w\right)-\rho \Theta \left(w\right)+3\alpha w^2{\Theta }^{\prime }\left(w\right){\Theta }^{\prime \prime }\left(w\right)+\rho w^3{\Theta }^{\prime \prime \prime }\left(w\right)=0.\hfill \end{array}$$

(58)

2.4.6. Invariant Solutions Using Subalgebra Component ${\mathfrak{S}}_1+c_2{\mathfrak{S}}_5$

Exploring ${\mathfrak{S}}_1+c_2{\mathfrak{S}}_5$, we have a group invariant relation obtained as $W(P,Q)+c_{2}ty=u(t,x,y)$, where $P=t$ as well as $Q=x$. Inserting the function in (4) produces

$$\sigma W_{PQ}-\theta W_{QQ}+\alpha c_{2}P{W}_{QQ}+\rho W_{PQQQ}=0.$$

(59)

Eventually, the result from here gives a solution of 2D-gnrAKNSe (4) as

$$\begin{array}{cc}\hfill u(t,x,y)=& c_{2}ty+f_1\left(t\right)-{\displaystyle \frac{1}{2}}{C}_3\{\int [\sqrt{-Aexp\left(C_{0}x\right){\left[exp\left(-{\displaystyle \frac{C_1}{\rho }}\sqrt{\rho \left(C_0^2\rho -4\sigma \right)}\right)\right]}^{-1}}\hfill \\ \hfill & \times \sqrt{{\left[exp\left(-{\displaystyle \frac{1}{\rho }}\sqrt{\rho \left(C_0^2\rho -4\sigma \right)}x\right)\right]}^{-1}}exp\left({\displaystyle \frac{1}{2C_0\rho }}[2\theta t-\alpha c_2{t}^2\right.\hfill \\ \hfill & \left.+2C_0\left(C_1+x\right)\sqrt{\rho \left(C_0^2\rho -4\sigma \right)}]\right)+exp\left[-{\displaystyle \frac{1}{2C_0\rho }}\left(\alpha c_{2}t-2\theta \right)t\right]\left]dx\right\}\hfill \end{array}$$

(60)

where $A=exp\left(2C_2\right)$ with the existence of $C_0,C_1,C_2$ and $C_3$ as the arbitrary constants. Further investigation of (59) gives no result of interest. Next, we consider subalgebras with triple vector components.

2.4.7. Invariant Solutions Using Subalgebra Component ${\mathfrak{S}}_1+{\mathfrak{S}}_4+{\mathfrak{S}}_5$

Investigation of ${\mathfrak{S}}_1+{\mathfrak{S}}_4+{\mathfrak{S}}_5$ from the Lie symmetry analysis standpoint purveys

$$W(P,Q)+{\displaystyle \frac{1}{2}}{t}^2=u(t,x,y)\mathrm{where},P=x,\mathrm{and}Q=y-t.$$

Invoking the function in (4) gives a fourth order NLNPDEE explicated as

$$\alpha W_Q{W}_{PP}-\theta W_{PP}-\sigma W_{PQ}+\alpha W_Q{W}_{PP}-\rho W_{PPPQ}=0.$$

(61)

Hence, in this regard, we gain a solution of 2D-gnrAKNSe (4) as

$$u(t,x,y)={\displaystyle \frac{1}{2}}{t}^2-{\displaystyle \frac{4\rho B_1}{\alpha }}tanh\left[B_{1}x-{\displaystyle \frac{\theta B_1}{4\rho B_1^2+\sigma }}\left(y-t\right)+B_0\right]+B_2,$$

(62)

where we have arbitrary constants $B_0,B_1$ and $B_2$. In addition, (61) admits

$$\begin{array}{cc}\hfill L_1=& {\displaystyle \frac{\partial }{\partial P}}+Q{\displaystyle \frac{\partial }{\partial Q}}+{\displaystyle \frac{\theta }{\alpha }}Q{\displaystyle \frac{\partial }{\partial W}},L_2=Q{\displaystyle \frac{\partial }{\partial Q}}+\left(1+{\displaystyle \frac{\theta }{\alpha }}Q\right){\displaystyle \frac{\partial }{\partial W}},\hfill \\ \hfill L_3=& P{\displaystyle \frac{\partial }{\partial P}}+Q{\displaystyle \frac{\partial }{\partial Q}}+\left\{{\displaystyle \frac{\sigma }{\alpha }}P+{\displaystyle \frac{2\theta }{\alpha }}Q-W\right\}{\displaystyle \frac{\partial }{\partial W}}.\hfill \end{array}$$

Considering $L_1$, one obtains $W(P,Q)=\Theta \left(w\right)+{\displaystyle \frac{\theta }{\alpha }}P$, where $w=Q-P$. Introducing the obtained function transforms 2D-gnrAKNSe (4) further into the NLNODE

$$\sigma {\Theta }^{\prime \prime }\left(w\right)-3\theta {\Theta }^{\prime \prime }\left(w\right)+3\alpha {\Theta }^{\prime }\left(w\right){\Theta }^{\prime \prime }\left(w\right)+\rho {\Theta }^{\prime \prime \prime \prime }\left(w\right)=0.$$

(63)

Next, we take a look at $L_2$ and this furnishes $W(P,Q)=\Theta \left(w\right)+Q+{\displaystyle \frac{\theta }{\alpha }}Q$, where $w=P$. This changes (4) to a LNODE that occasions the solution of the model as

$$u(t,x,y)={\displaystyle \frac{1}{2}}{t}^2+K_{0}x+y-t+{\displaystyle \frac{\theta }{\alpha }}\left(y-t\right)+K_2$$

(64)

with arbitrary integration constants $K_1$ and $K_2$. Finally under ${\mathfrak{S}}_1+{\mathfrak{S}}_4+{\mathfrak{S}}_5$ we examine $L_3$ and this produces invariants $W(P,Q)=P^{-1}\Theta \left(w\right)+{\displaystyle \frac{\sigma }{2\alpha }}P+{\displaystyle \frac{\theta }{\alpha }}Q$, where $w=Q-ln\left(P\right)$. Inserting the achieved function in (4) leads to the NLNODE

$$\begin{array}{cc}\hfill & 6\rho {\Theta }^{\prime }\left(w\right)+2\alpha \Theta \left(w\right){\Theta }^{\prime \prime }\left(w\right)+5\alpha {\Theta }^{\prime }{\left(w\right)}^2+4\alpha \Theta \left(w\right){\Theta }^{\prime }\left(w\right)+11\rho {\Theta }^{\prime \prime }\left(w\right)\hfill \\ \hfill & +3\alpha {\Theta }^{\prime }\left(w\right){\Theta }^{\prime \prime }\left(w\right)+6\rho {\Theta }^{\prime \prime \prime }\left(w\right)+\rho {\Theta }^{\prime \prime \prime \prime }\left(w\right)=0.\hfill \end{array}$$

(65)

2.4.8. Invariant Solutions Using Subalgebra Component ${\mathfrak{S}}_2+{\mathfrak{S}}_6+{\mathfrak{S}}_7$

We study the sublagebra component ${\mathfrak{S}}_1+{\mathfrak{S}}_4+{\mathfrak{S}}_5$ which has the invariant

$$W(P,Q)+{\displaystyle \frac{\theta }{\alpha }}y=u(t,x,y)\mathrm{where},P=x-ln\left(t\right),\mathrm{and}Q={\displaystyle \frac{y}{t}}.$$

On inserting the function in (4), one secures the equation

$$\alpha W_Q{W}_{PP}-\sigma Q{W}_{PQ}-\sigma W_{PP}+2\alpha W_P{W}_{PQ}-\rho W_{PPPP}-\rho Q{W}_{PPPQ}=0.$$

(66)

Further analysis carried out on (66) revealed that it admits

$$\begin{array}{cc}\hfill L_1=& {\displaystyle \frac{\partial }{\partial P}},L_2={\displaystyle \frac{\partial }{\partial W}},L_3=Q{\displaystyle \frac{\partial }{\partial Q}}+W{\displaystyle \frac{\partial }{\partial W}}.\hfill \end{array}$$

Regarding $L_1$ and the case of $L_1+L_2$, a trivial solution of (4) is found but for $L_3$, we have invariant $W(P,Q)=Q\Theta \left(w\right)$, where $w=P$. This then reduces (4) to

$$\alpha \Theta \left(w\right){\Theta }^{\prime \prime }\left(w\right)+2\alpha {\Theta }^{\prime }{\left(w\right)}^2-\sigma {\Theta }^{\prime \prime }\left(w\right)-\sigma {\Theta }^{\prime }\left(w\right)-\rho {\Theta }^{\prime \prime \prime }\left(w\right)-\rho {\Theta }^{\prime \prime \prime \prime }\left(w\right)=0.$$

(67)

3. Graphical Appraisal of Solutions and Discussion

The notion of presenting the streaming nature of wave motions that depicts the physcial meaning exhibited by exact solutions to differential equation (DE) is of a great essence. This is due to the fact that the phenomena represented by the model that the DEs portray can only be better understood and then appreciated when their solutions are pictorially enhanced. In view of these, this section exhibits various wave structures represented by some of the obtained exact solutions to 2D-gnrAKNSe (4). The graphs are given in form of three dimensional (3-D), two dimensional (2-D) and density plots through the utilization of suitable choice of parameters. Now, in order to copiously view the interesting wave structures depicted by the exact periodic solution obtained in (33) which contains variables t and x ( $y=0$), pictorial demonstration of the result is given in Figure 1, Figure 2, Figure 3 and Figure 4. Thus, the trigonometry solution is first plotted in Figure 1 by selecting the parameter values as; $A_0=1$, $A_1=1$, $A_2=5$, $A_3=3$, $\sigma =6$, $\theta =0.6$, $\rho =10$, where interval $-5\le t,x\le 5$ exits. In addition, Figure 2 also delineates the wave motion of the solution by doing $A_0=1$, $A_1=1$, $A_2=5$, $A_3=3$, $\sigma =6$, $\theta =3$, $\rho =10$, with variable interval $-10\le t,x\le 10$. In similar way, Figure 3 is plotted with the data set; $A_0=1$, $A_1=1$, $A_2=5$, $A_3=3$, $\sigma =6$, $\theta =3$, $\rho =10$, in the interval $-13\le t,x\le 13$. In the case of Figure 4, one adopts the selection of parameter values $A_0=1$, $A_1=1$, $A_2=5$, $A_3=3$, $\sigma =6$, $\theta =5$, $\rho =10$, where one has $-15\le t,x\le 15$. Another solution to be graphically examined is the logarithmic-trigonometric solution (45) which is presented in Figure 5, Figure 6 and Figure 7. In the first place, the log-trigonometric function (45) is plotted as 3-D, 2-D and density graphs in Figure 5 using the unalike constant values $A_0=1$, $A_1=0$, $A_2=5$, $A_3=3$, $\sigma =6$, $\theta =0.3$, $\alpha =0.6$, $\rho =10$, where variable $t=1$ as well as $-5\le x,y\le 5$. Similarly, one plots graphs in Figure 6 using the values $A_0=1$, $A_1=0$, $A_2=5$, $A_3=3$, $\sigma =7$, $\theta =0.4$, $\alpha =0.7$, $\rho =11$, with $t=1$, where $-10\le x,y\le 10$. In the case of Figure 7 the variant data set utilized are $A_0=1$, $A_1=0$, $A_2=5$, $A_3=3$, $\sigma =6$, $\theta =0.4$, $\alpha =0.5$, $\rho =10.2$, where one takes, $t=1$ alongside $-15\le x,y\le 15$. The next three Figures depict the wave motion of tangent hyperbolic solution (49), Therefore, plotting graphs in Figure 8, the dissimilar parameter values invoked are $A_0=1$, $A_1=0.8$, $A_2=5$, $c_1=3$, $\sigma =1$, $\theta =0.3$, $\alpha =0.6$, $\rho =5$, at $t=1$ as well as $-10\le x,y\le 10$. Likewise, in Figure 9, the kink wave structure is plotted using the data set $A_0=1$, $A_1=0.8$, $A_2=5.1$, $c_1=3.2$, $\sigma =1$, $\theta =0.333$, $\alpha =0.6$, $\rho =5.2$, where variable $t=1$ with $-15\le x,y\le 15$. Regarding Figure 10 in 3-D, 2-D and density plots, still representing (49), one assigns $A_0=1$, $A_1=5$, $A_2=5$, $c_1=4$, $\sigma =2$, $\theta =0.3$, $\alpha =0.6$, $\rho =5$, at $t=1$ where $-15\le x,y\le 15$. The algebraic function with arbitrary functions $f_1\left(x\right)$ and $f_2\left(y/t\right)$. Replacing these arbitrary functions with mathematical functions amd plotting the graph give the interesting wave structures obtainable in the rest of the Figures. In Figure 11 one substitutes $f_1\left(x\right)=sin\left(x\right)$ and $f_2=cos\left(y/t\right)$, where $\alpha =10$, $\theta =0.6$ at $t=1$ as well as $-5\le x,y\le 5$. Similarly, with $\alpha =10$, $\theta =0.6$, where $t=2$ and $-5\le x,y\le 5$, plots of Figure 12 are enhanced. Besides, one obtains the graphical demonstrations in Figure 13 with slight variations, via $\alpha =12$, $\theta =0.7$ at $t=3$ in the interval $-10\le x,y\le 10$. Finally, using functions with some changes, Figure 14 is attainable through the use of the parameter values $\alpha =11$, $\theta =0.7$ at $t=5$ as well as $-7\le x,y\le 7$. These plots represent various wave structures emanating from periodic soliton collisions propagating at varying time t from $t=1$, $t=2$, $t=3$ to $t=5$ with different interval of solution along the horizontal axis.

4. Conservation Laws of 2D-gnrAKNSe (4)

In this part, we will calculate the conserved vectors for equation (4) using Ibragimov’s theorem [73,74] for conserved quantities. To start off, we will provide key features of the method.

4.1. The technique’s overview

Ibragimov introduced a fresh theorem in his work cited as [74] regarding conserved vectors in differential equations (DE). This theorem confirms the validity of any DE system where the number of equations matches the number of dependent variables. Interestingly, the theorem doesn’t rely on the presence of a classical Lagrangian. Ibragimov’s method proposes that each infinitesimal generator is connected to a conserved vector. This concept is built upon the adjoint equation for nonlinear DEs. We give a detailed outline of the theorem as follows;

4.2. Formal Lagrangian and Conserved Currents

Consider a system of sth-order $\alpha$ PDEs

$${\Xi }_{\sigma }(x,\Psi ,{\Psi }_{\left(1\right)},\cdots ,{\Psi }_{\left(s\right)})=0,\sigma =1,\cdots ,\alpha ,$$

(68)

with $\kappa$ independent together with $\alpha$ dependent variables given as $x=(x^1,x^2,\cdots ,x^{\kappa })$ and $\Psi =({\Psi }^1,{\Psi }^2,\cdots ,{\Psi }^{\alpha })$. The system of adjoint equations are given by

$${\Xi }_{\sigma }^{*}(x,\Psi ,\Omega ,\cdots ,{\Psi }_{\left(s\right)},{\Omega }_{\left(s\right)})\equiv {\displaystyle \frac{\delta \left({\Omega }^{\beta }{\Xi }_{\beta }\right)}{\delta {\Psi }^{\sigma }}}=0,\sigma =1,\cdots ,\alpha ,$$

(69)

where $\Omega =({\Omega }^1,\cdots ,{\Omega }^{\alpha })$ are new dependent variables, $\Omega =\Omega \left(x\right)$. The operator $\delta /\delta {\Psi }^{\sigma }$, expressed for each $\sigma$, as

$${\displaystyle \frac{\delta }{\delta {\Psi }^{\sigma }}}={\displaystyle \frac{\partial }{\partial {\Psi }^{\sigma }}}+\sum _{s=1}^{\infty }{(-1)}^s{D}_{i_1}\cdots D_{i_s}{\displaystyle \frac{\delta }{\delta {\Psi }_{i_1,i_2,\cdots ,i_s}^{\sigma }}},i=1,\cdots ,\kappa ,$$

(70)

is the Euler-Lagrange operator and

$$D_i={\displaystyle \frac{\partial }{\partial x^i}}+{\Psi }_i^{\sigma }{\displaystyle \frac{\partial }{\partial {\Psi }^{\sigma }}}+{\Psi }_{ij}^{\sigma }{\displaystyle \frac{\partial }{\partial {\Psi }_j^{\sigma }}}+\cdots ,i=1,\cdots ,\kappa ,j=1,\cdots ,\kappa$$

(71)

is the total differential operator.

An n-tuple $C=(C^1,C^2,\cdots ,C^n)$, $1=1,2,\cdots ,n$, such that

$$D_i{C}^i=0,$$

(72)

holds for all solutions of PDEs (68) is referred to as the conserved current of the equation.

The formal Lagrangian of the system (68) and its adjoint (69) is given as

$$\mathcal{L}={\Omega }^{\sigma }{\Xi }_{\sigma }(x,\Psi ,{\Psi }_{\left(1\right)},\cdots ,{\Psi }_{\left(s\right)}).$$

(73)

Theorem 4.1.

Every nonlocal symmetry, Lie-Bäcklund, as well as Lie point symmetry,

$$\mathcal{R}={\xi }^i{\displaystyle \frac{\partial }{\partial x^i}}+{\phi }^{\sigma }{\displaystyle \frac{\partial }{{\partial }_{{\Psi }^{\sigma }}}},{\xi }^i={\xi }^i(x,\Psi ),{\phi }^{\sigma }={\phi }^{\sigma }(x,\Psi ),$$

(74)

admitted by the system (68) produces a conserved vector for equations (68) and its adjoint (69), with the conserved vectors $T=(T^1,\cdots ,T^{\kappa })$ having components $T^i$ given by

$$\begin{array}{cc}\hfill T^i=& {\xi }^i\mathcal{L}+{\Pi }^{\sigma }\left[{\displaystyle \frac{\partial \mathcal{L}}{\partial {\Psi }_i^{\sigma }}}-D_j{\displaystyle \frac{\partial \mathcal{L}}{\partial {\Psi }_{ij}^{\sigma }}}+D_j{D}_k\left({\displaystyle \frac{\partial \mathcal{L}}{\partial {\Psi }_{ijk}^{\sigma }}}\right)+\cdots \right]+D_j\left({\Pi }^{\sigma }\right)[{\displaystyle \frac{\partial \mathcal{L}}{\partial {\Psi }_{ij}^{\sigma }}}\hfill \\ \hfill & -D_k{\displaystyle \frac{\partial \mathcal{L}}{\partial {\Psi }_{ijk}^{\sigma }}}+\cdots ]+D_j{D}_k\left({\Pi }^{\sigma }\right){\displaystyle \frac{\partial \mathcal{L}}{\partial {\Psi }_{ijk}}}+\cdots ,i,j,k=1,\cdots ,\kappa ,\hfill \end{array}$$

(75)

where ${\Pi }^{\sigma }$ is the Lie characteristic function defined by

$${\Pi }^{\sigma }={\phi }^{\sigma }-{\xi }^j{\Psi }_j^{\sigma },\sigma =1,\cdots ,\alpha ,j=1,\cdots ,\kappa .$$

(76)

Remark 4.1.

It is worth noting that a system described by equation (68) is considered self-adjoint when substituting $v=u$ in the corresponding adjoint equation (69), results in the identical system (69). For a comprehensive explanation of the proof and further insights into the findings discussed here, readers are encouraged to refer to the references [73,74].

The multiplier $\Lambda$ of system (68) has the property that

$$D_i{C}^i={\Lambda }^{\sigma }{\Xi }_{\sigma },\sigma =1,\dots ,\alpha .$$

(77)

The governing equations for all multipliers involved are obtained from

$${\displaystyle \frac{\delta }{\delta {\Psi }^{\sigma }}}\left({\Lambda }^{\sigma }{\Xi }_{\sigma }\right)=0,\sigma =1,\dots ,\alpha .$$

(78)

The moment the multipliers are generated via (78), the conserved currents can be procured using (77) as the determining equation.

4.3. Derivation of Conservation Laws via Ibragimov’s Theorem

In this section, we introduce Ibragimov’s conservation theorem [74] which is utilized to establish the conservation principles of the 2D-gnrAKNSe equation (4). Building upon the previously provided information, we can now present the theorem as follows:

Theorem 4.2.

The adjoint equation of 2D-gnrAKNSe (4) is expressed as

$$G^{*}\equiv \rho v_{txxx}+\sigma v_{tx}+v_{xx}\left(\alpha u_y-\theta \right)+2\alpha v_x{u}_{xy}+2\alpha u_x{v}_{xy}+\alpha u_{xx}{v}_y=0$$

(79)

with the formal Lagrangian given as

$$\mathcal{L}=vG\equiv v\left(\rho u_{txxx}+\sigma u_{tx}-\theta u_{xx}+2\alpha u_x{u}_{xy}+\alpha u_{xx}{u}_y\right)$$

(80)

where G is expressed in this regard as

$$G=\rho u_{txxx}+\sigma u_{tx}-\theta u_{xx}+2\alpha u_x{u}_{xy}+\alpha u_{xx}{u}_y.$$

(81)

The adjoint equation (79) and remark (4.1) clearly indicate that the 2D-gnrAKNSe equation (4) is not self-adjoint. Therefore, based on the information provided earlier, we calculate the conserved quantities of equation (4) for the solutions of the optimal system obtained in Theorem 2.4 and present the results accordingly, as

$$\begin{array}{cc}\hfill C_1^t=& {\displaystyle \frac{1}{4}}\rho v_x{u}_{xxy}-{\displaystyle \frac{1}{4}}\rho u_{xxxy}v-{\displaystyle \frac{1}{2}}\sigma u_{xy}v-{\displaystyle \frac{1}{4}}\rho v_{xx}{u}_{xy}+{\displaystyle \frac{1}{4}}\rho u_y{v}_{xxx}+{\displaystyle \frac{1}{2}}\sigma u_y{v}_x,\hfill \\ \hfill C_1^x=& {\displaystyle \frac{3}{4}}\rho u_y{v}_{txx}-\alpha u_y{u}_{xy}v-\alpha u_x{u}_{yy}v+\theta u_{xy}v-{\displaystyle \frac{3}{4}}\rho v{u}_{txxy}-{\displaystyle \frac{1}{2}}\sigma u_{ty}v\hfill \\ \hfill & +{\displaystyle \frac{1}{4}}\rho v_t{u}_{xxy}-{\displaystyle \frac{1}{4}}\rho v_{xx}{u}_{ty}-{\displaystyle \frac{1}{2}}\rho v_{tx}{u}_{xy}+{\displaystyle \frac{1}{2}}\rho v_x{u}_{txy}+{\displaystyle \frac{1}{2}}\sigma v_t{u}_y+\alpha u_y^2{v}_x\hfill \\ \hfill & +\alpha u_x{u}_y{v}_y-\theta u_y{v}_x,\hfill \\ \hfill C_1^y=& \alpha u_x{u}_{xy}v+\alpha u_{xx}{u}_{y}v-\theta u_{xx}v+\rho u_{txxx}v+\sigma u_{tx}v+\alpha u_x{u}_y{v}_x;\hfill \\ \\ \hfill C_2^t=& {\displaystyle \frac{1}{2}}\sigma v_x{u}_x-{\displaystyle \frac{1}{2}}\sigma v{u}_x+2t\alpha v{u}_{xy}{u}_x-{\displaystyle \frac{1}{4}}\rho v_{xx}{u}_x+{\displaystyle \frac{1}{4}}\rho v_{xxx}{u}_x-{\displaystyle \frac{\theta \sigma }{2\alpha }}y{v}_x\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\sigma u{v}_x-t\theta v{u}_{xx}-{\displaystyle \frac{1}{2}}\sigma v{u}_{xx}+t\alpha v{u}_y{u}_{xx}+{\displaystyle \frac{1}{4}}\rho v_x{u}_{xx}-{\displaystyle \frac{1}{4}}\rho u_{xx}{v}_{xx}\hfill \\ \hfill & -{\displaystyle \frac{1}{4}}\rho v{u}_{xxx}+{\displaystyle \frac{1}{4}}\rho v_x{u}_{xxx}-{\displaystyle \frac{\theta \rho }{4\alpha }}y{v}_{xxx}+{\displaystyle \frac{1}{4}}\rho u{v}_{xxx}-{\displaystyle \frac{1}{4}}\rho v{u}_{xxxx}+{\displaystyle \frac{1}{2}}t\sigma v_x{u}_t\hfill \\ \hfill & +{\displaystyle \frac{1}{4}}t\rho v_{xxx}{u}_t+{\displaystyle \frac{1}{2}}t\sigma v{u}_{tx}-{\displaystyle \frac{1}{4}}t\rho v_{xx}{u}_{tx}+{\displaystyle \frac{1}{4}}t\rho v_x{u}_{txx}+{\displaystyle \frac{3}{4}}t\rho v{u}_{txxx},\hfill \\ \hfill C_2^x=& {\displaystyle \frac{{\theta }^2}{\alpha }}y{v}_x+2v{u}_x\theta -y{v}_y{u}_x\theta -u{v}_x\theta -y{u}_y{v}_x\theta -u_x{v}_x\theta -t{v}_x{u}_t\theta -{\displaystyle \frac{\sigma \theta }{2\alpha }}{v}_t\hfill \\ \hfill & +tv{u}_{tx}\theta -{\displaystyle \frac{3\rho \theta }{4\alpha }}y{v}_{txx}+\alpha v_y{u}_x^2-2\alpha v{u}_y{u}_x+\alpha u{v}_y{u}_x+\alpha u{u}_y{v}_x\hfill \\ \hfill & +\alpha u_y{u}_x{v}_x+\alpha v{u}_x{u}_{xy}-\sigma v{u}_t+t\alpha v_y{u}_x{u}_t+t\alpha u_y{v}_x{u}_t-{\displaystyle \frac{1}{2}}\rho v_{xx}{u}_t\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\sigma u{v}_t+{\displaystyle \frac{1}{2}}\sigma u_x{v}_t+{\displaystyle \frac{1}{4}}\rho u_{xx}{v}_t+{\displaystyle \frac{1}{4}}\rho u_{xxx}{v}_t+{\displaystyle \frac{1}{2}}t\sigma u_t{v}_t-t\alpha v{u}_x{u}_{ty}\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\sigma v{u}_{tx}-t\alpha v{u}_y{u}_{tx}+\rho v_x{u}_{tx}-{\displaystyle \frac{1}{4}}\rho v_{xx}{u}_{tx}-{\displaystyle \frac{1}{2}}\rho u_x{v}_{tx}-{\displaystyle \frac{1}{2}}\rho u_{xx}{v}_{tx}\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}t\rho u_{tx}{v}_{tx}-{\displaystyle \frac{3}{2}}\rho v{u}_{txx}+{\displaystyle \frac{1}{2}}\rho v_x{u}_{txx}+{\displaystyle \frac{1}{4}}t\rho v_t{u}_{txx}+{\displaystyle \frac{3}{4}}\rho u{v}_{txx}+{\displaystyle \frac{3}{4}}\rho u_x{v}_{txx}\hfill \\ \hfill & +{\displaystyle \frac{3}{4}}t\rho u_t{v}_{txx}+{\displaystyle \frac{1}{4}}\rho v{u}_{txxx}-{\displaystyle \frac{1}{2}}t\sigma v{u}_{tt}-{\displaystyle \frac{1}{4}}t\rho v_{xx}{u}_{tt}+{\displaystyle \frac{1}{2}}t\rho v_x{u}_{ttx}-{\displaystyle \frac{3}{4}}t\rho v{u}_{ttxx},\hfill \\ \hfill C_2^y=& \alpha v_x{u}_x^2-\alpha v{u}_x^2-y\theta v_x{u}_x+\alpha u{v}_x{u}_x-\alpha v{u}_{xx}{u}_x+t\alpha v_x{u}_t{u}_x-t\alpha v{u}_{tx}{u}_x;\hfill \\ \\ \hfill C_3^t=& {\displaystyle \frac{1}{2}}\sigma u_y{v}_x+{\displaystyle \frac{1}{4}}\rho u_{xxy}{v}_x+{\displaystyle \frac{1}{2}}\sigma c_1{u}_t{v}_x+{\displaystyle \frac{1}{4}}\rho c_1{u}_{txx}{v}_x-{\displaystyle \frac{1}{2}}\sigma v{u}_{xy}+2\alpha c_{1}v{u}_x{u}_{xy}\hfill \\ \hfill & -\theta c_{1}v{u}_{xx}+\alpha c_{1}v{u}_y{u}_{xx}-{\displaystyle \frac{1}{4}}\rho u_{xy}{v}_{xx}+{\displaystyle \frac{1}{4}}\rho u_y{v}_{xxx}-{\displaystyle \frac{1}{4}}\rho v{u}_{xxxy}\hfill \\ \hfill & +{\displaystyle \frac{1}{4}}\rho c_1{v}_{xxx}{u}_t+{\displaystyle \frac{1}{2}}\sigma c_{1}v{u}_{tx}-{\displaystyle \frac{1}{4}}\rho c_1{v}_{xx}{u}_{tx}+{\displaystyle \frac{3}{4}}\rho c_{1}v{u}_{txxx},\hfill \\ \hfill C_3^x=& \alpha v_x{u}_y^2+\alpha v_y{u}_x{u}_y-\theta v_x{u}_y-\alpha v{u}_{xy}{u}_y+\alpha c_1{v}_x{u}_t{u}_y+{\displaystyle \frac{1}{2}}\sigma v_t{u}_y\hfill \\ \hfill & -\alpha c_{1}v{u}_{tx}{u}_y+{\displaystyle \frac{3}{4}}\rho v_{txx}{u}_y-\alpha v{u}_{yy}{u}_x+\theta v{u}_{xy}+\alpha c_1{v}_y{u}_x{u}_t-\theta c_1{v}_x{u}_t\hfill \\ \hfill & +{\displaystyle \frac{1}{4}}\rho u_{xxy}{v}_t+{\displaystyle \frac{1}{2}}\sigma c_1{u}_t{v}_t-{\displaystyle \frac{1}{2}}\sigma v{u}_{ty}-\alpha c_{1}v{u}_x{u}_{ty}-{\displaystyle \frac{1}{4}}\rho v_{xx}{u}_{ty}+\theta c_{1}v{u}_{tx}\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\rho u_{xy}{v}_{tx}-{\displaystyle \frac{1}{2}}\rho c_1{u}_{tx}{v}_{tx}+{\displaystyle \frac{1}{2}}\rho v_x{u}_{txy}+{\displaystyle \frac{1}{4}}\rho c_1{v}_t{u}_{txx}+{\displaystyle \frac{3}{4}}\rho c_1{u}_t{v}_{txx}\hfill \\ \hfill & -{\displaystyle \frac{3}{4}}\rho v{u}_{txxy}-{\displaystyle \frac{1}{2}}\sigma c_{1}v{u}_{tt}-{\displaystyle \frac{1}{4}}\rho c_1{v}_{xx}{u}_{tt}+{\displaystyle \frac{1}{2}}\rho c_1{v}_x{u}_{ttx}-{\displaystyle \frac{3}{4}}\rho c_{1}v{u}_{ttxx},\hfill \\ \hfill C_3^y=& \alpha u_y{u}_x{v}_x+\alpha c_1{u}_x{u}_t{v}_x+\alpha v{u}_x{u}_{xy}-\theta v{u}_{xx}+\alpha v{u}_y{u}_{xx}+\sigma v{u}_{tx}\hfill \\ \hfill & -\alpha c_{1}v{u}_x{u}_{tx}+\rho v{u}_{txxx};\hfill \\ \\ \hfill C_4^t=& {\displaystyle \frac{1}{2}}y\sigma u_y{v}_x-{\displaystyle \frac{\theta \sigma }{2\alpha }}y{v}_x+{\displaystyle \frac{1}{4}}y\rho u_{xxy}{v}_x+{\displaystyle \frac{1}{2}}t\sigma u_t{v}_x+{\displaystyle \frac{1}{4}}t\rho u_{txx}{v}_x-t\theta v{u}_{xx}\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}y\sigma v{u}_{xy}+2t\alpha v{u}_x{u}_{xy}+t\alpha v{u}_y{u}_{xx}-{\displaystyle \frac{1}{4}}y\rho u_{xy}{v}_{xx}-{\displaystyle \frac{\theta \rho }{4\alpha }}y{v}_{xxx}\hfill \\ \hfill & +{\displaystyle \frac{1}{4}}y\rho u_y{v}_{xxx}-{\displaystyle \frac{1}{4}}y\rho v{u}_{xxxy}+{\displaystyle \frac{1}{4}}t\rho v_{xxx}{u}_t+{\displaystyle \frac{1}{2}}t\sigma v{u}_{tx}-{\displaystyle \frac{1}{4}}t\rho v_{xx}{u}_{tx}\hfill \\ \hfill & +{\displaystyle \frac{3}{4}}t\rho v{u}_{txxx},\hfill \\ \hfill C_4^x=& {\displaystyle \frac{{\theta }^2}{\alpha }}y{v}_x+v{u}_x\theta -y{v}_y{u}_x\theta -2y{u}_y{v}_x\theta +yv{u}_{xy}\theta -t{v}_x{u}_t\theta -{\displaystyle \frac{\sigma \theta }{2\alpha }}y{v}_t+tv{u}_{tx}\theta \hfill \\ \hfill & -{\displaystyle \frac{3\rho \theta }{4\alpha }}y{v}_{txx}-\alpha v{u}_y{u}_x+y\alpha u_y{v}_y{u}_x-y\alpha v{u}_{yy}{u}_x+y\alpha u_y^2{v}_x-y\alpha v{u}_y{u}_{xy}\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\sigma v{u}_t+t\alpha v_y{u}_x{u}_t+t\alpha u_y{v}_x{u}_t-{\displaystyle \frac{1}{4}}\rho v_{xx}{u}_t+{\displaystyle \frac{1}{2}}y\sigma u_y{v}_t+{\displaystyle \frac{1}{4}}y\rho u_{xxy}{v}_t\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}t\sigma u_t{v}_t-{\displaystyle \frac{1}{2}}y\sigma v{u}_{ty}-t\alpha v{u}_x{u}_{ty}-{\displaystyle \frac{1}{4}}y\rho v_{xx}{u}_{ty}-t\alpha v{u}_y{u}_{tx}+{\displaystyle \frac{1}{2}}\rho v_x{u}_{tx}\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}y\rho u_{xy}{v}_{tx}-{\displaystyle \frac{1}{2}}t\rho u_{tx}{v}_{tx}+{\displaystyle \frac{1}{2}}y\rho v_x{u}_{txy}-{\displaystyle \frac{3}{4}}\rho v{u}_{txx}+{\displaystyle \frac{1}{4}}t\rho v_t{u}_{txx}\hfill \\ \hfill & +{\displaystyle \frac{3}{4}}y\rho u_y{v}_{txx}+{\displaystyle \frac{3}{4}}t\rho u_t{v}_{txx}-{\displaystyle \frac{3}{4}}y\rho v{u}_{txxy}-{\displaystyle \frac{1}{2}}t\sigma v{u}_{tt}-{\displaystyle \frac{1}{4}}t\rho v_{xx}{u}_{tt}\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}t\rho v_x{u}_{ttx}-{\displaystyle \frac{3}{4}}t\rho v{u}_{ttxx},\hfill \\ \hfill C_4^y=& y\alpha u_y{u}_x{v}_x-y\theta u_x{v}_x+t\alpha u_x{u}_t{v}_x+y\alpha v{u}_x{u}_{xy}-y\theta v{u}_{xx}+y\alpha v{u}_y{u}_{xx}\hfill \\ \hfill & +y\sigma v{u}_{tx}-t\alpha v{u}_x{u}_{tx}+y\rho v{u}_{txxx};\hfill \\ \\ \hfill C_5^t=& {\displaystyle \frac{1}{2}}\sigma v{u}_x+{\displaystyle \frac{1}{2}}\sigma v_x{u}_x+{\displaystyle \frac{1}{4}}\rho v_{xx}{u}_x+{\displaystyle \frac{1}{4}}\rho v_{xxx}{u}_x-{\displaystyle \frac{1}{2}}\sigma u{v}_x+{\displaystyle \frac{1}{2}}y\sigma u_y{v}_x\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}y\sigma v{u}_{xy}-{\displaystyle \frac{1}{2}}\sigma v{u}_{xx}-{\displaystyle \frac{1}{4}}\rho v_x{u}_{xx}-{\displaystyle \frac{1}{4}}y\rho u_{xy}{v}_{xx}-{\displaystyle \frac{1}{4}}\rho u_{xx}{v}_{xx}\hfill \\ \hfill & +{\displaystyle \frac{1}{4}}y\rho v_x{u}_{xxy}+{\displaystyle \frac{1}{4}}\rho v{u}_{xxx}+{\displaystyle \frac{1}{4}}\rho v_x{u}_{xxx}-{\displaystyle \frac{1}{4}}\rho u{v}_{xxx}+{\displaystyle \frac{1}{4}}y\rho u_y{v}_{xxx}\hfill \\ \hfill & -{\displaystyle \frac{1}{4}}y\rho v{u}_{xxxy}-{\displaystyle \frac{1}{4}}\rho v{u}_{xxxx},\hfill \\ \hfill C_5^x=& y\alpha v_x{u}_y^2+\alpha v{u}_x{u}_y+y\alpha v_y{u}_x{u}_y-y\theta v_x{u}_y-\alpha u{v}_x{u}_y+\alpha u_x{v}_x{u}_y\hfill \\ \hfill & -y\alpha v{u}_{xy}{u}_y+{\displaystyle \frac{1}{2}}y\sigma v_t{u}_y+{\displaystyle \frac{3}{4}}y\rho v_{txx}{u}_y+\alpha v_y{u}_x^2-\theta v{u}_x-\alpha u{v}_y{u}_x\hfill \\ \hfill & -y\alpha v{u}_{yy}{u}_x+\theta u{v}_x-\theta u_x{v}_x+y\theta v{u}_{xy}+\alpha v{u}_x{u}_{xy}+{\displaystyle \frac{1}{2}}\sigma v{u}_t\hfill \\ \hfill & +{\displaystyle \frac{1}{4}}\rho v_{xx}{u}_t-{\displaystyle \frac{1}{2}}\sigma u{v}_t+{\displaystyle \frac{1}{2}}\sigma u_x{v}_t-{\displaystyle \frac{1}{4}}\rho u_{xx}{v}_t+{\displaystyle \frac{1}{4}}y\rho u_{xxy}{v}_t+{\displaystyle \frac{1}{4}}\rho u_{xxx}{v}_t\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}y\sigma v{u}_{ty}-{\displaystyle \frac{1}{4}}y\rho v_{xx}{u}_{ty}+{\displaystyle \frac{1}{2}}\sigma v{u}_{tx}-{\displaystyle \frac{1}{2}}\rho v_x{u}_{tx}-{\displaystyle \frac{1}{4}}\rho v_{xx}{u}_{tx}+{\displaystyle \frac{1}{2}}\rho u_x{v}_{tx}\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}y\rho u_{xy}{v}_{tx}-{\displaystyle \frac{1}{2}}\rho u_{xx}{v}_{tx}+{\displaystyle \frac{1}{2}}y\rho v_x{u}_{txy}+{\displaystyle \frac{3}{4}}\rho v{u}_{txx}+{\displaystyle \frac{1}{2}}\rho v_x{u}_{txx}-{\displaystyle \frac{3}{4}}\rho u{v}_{txx}\hfill \\ \hfill & +{\displaystyle \frac{3}{4}}\rho u_x{v}_{txx}-{\displaystyle \frac{3}{4}}y\rho v{u}_{txxy}+{\displaystyle \frac{1}{4}}\rho v{u}_{txxx},\hfill \\ \hfill C_5^y=& \alpha v{u}_x^2+\alpha v_x{u}_x^2-\alpha u{v}_x{u}_x+y\alpha u_y{v}_x{u}_x+y\alpha v{u}_{xy}{u}_x-\alpha v{u}_{xx}{u}_x\hfill \\ \hfill & -y\theta v{u}_{xx}+y\alpha v{u}_y{u}_{xx}+y\sigma v{u}_{tx}+y\rho v{u}_{txxx};\hfill \\ \\ \hfill C_6^t=& {\displaystyle \frac{1}{4}}\rho v_x{u}_{xxy}-{\displaystyle \frac{1}{4}}\rho u_{xxxy}v-{\displaystyle \frac{1}{2}}\sigma u_{xy}v-{\displaystyle \frac{1}{4}}{c}_2\rho t{v}_{xxx}-{\displaystyle \frac{1}{2}}{c}_2\sigma t{v}_x\hfill \\ \hfill & -{\displaystyle \frac{1}{4}}\rho v_{xx}{u}_{xy}+{\displaystyle \frac{1}{4}}\rho u_y{v}_{xxx}+{\displaystyle \frac{1}{2}}\sigma u_y{v}_x,\hfill \\ \hfill C_6^x=& \theta u_{xy}v-\alpha u_y{u}_{xy}v-\alpha u_x{u}_{yy}v-{\displaystyle \frac{3}{4}}\rho v{u}_{txxy}-{\displaystyle \frac{1}{2}}\sigma u_{ty}v+{\displaystyle \frac{1}{2}}{c}_2\sigma v\hfill \\ \hfill & -\alpha c_{2}t{u}_y{v}_x-\alpha c_{2}t{u}_x{v}_y-{\displaystyle \frac{1}{2}}{c}_2\sigma t{v}_t+c_2\theta t{v}_x-{\displaystyle \frac{3}{4}}{c}_2\rho t{v}_{txx}+{\displaystyle \frac{1}{4}}{c}_2\rho v_{xx}\hfill \\ \hfill & +{\displaystyle \frac{3}{4}}\rho u_y{v}_{txx}+{\displaystyle \frac{1}{4}}\rho v_t{u}_{xxy}-{\displaystyle \frac{1}{4}}\rho v_{xx}{u}_{ty}-{\displaystyle \frac{1}{2}}\rho v_{tx}{u}_{xy}+{\displaystyle \frac{1}{2}}\rho v_x{u}_{txy}\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\sigma v_t{u}_y+\alpha u_y^2{v}_x+\alpha u_x{u}_y{v}_y-\theta u_y{v}_x,\hfill \\ \hfill C_6^y=& \alpha u_x{u}_{xy}v+\alpha u_{xx}{u}_{y}v-\theta u_{xx}v+\rho u_{txxx}v+\sigma u_{tx}v-\alpha c_{2}t{u}_x{v}_x\hfill \\ \hfill & +\alpha u_x{u}_y{v}_x;\hfill \\ \\ \hfill C_7^t=& {\displaystyle \frac{1}{2}}\sigma u_y{v}_x-{\displaystyle \frac{1}{2}}t\sigma v_x+{\displaystyle \frac{1}{4}}\rho u_{xxy}{v}_x+{\displaystyle \frac{1}{2}}\sigma u_t{v}_x+{\displaystyle \frac{1}{4}}\rho u_{txx}{v}_x-{\displaystyle \frac{1}{2}}\sigma v{u}_{xy}\hfill \\ \hfill & +2\alpha v{u}_x{u}_{xy}-\theta v{u}_{xx}+\alpha v{u}_y{u}_{xx}-{\displaystyle \frac{1}{4}}\rho u_{xy}{v}_{xx}-{\displaystyle \frac{1}{4}}t\rho v_{xxx}+{\displaystyle \frac{1}{4}}\rho u_y{v}_{xxx}\hfill \\ \hfill & -{\displaystyle \frac{1}{4}}\rho v{u}_{xxxy}+{\displaystyle \frac{1}{4}}\rho v_{xxx}{u}_t+{\displaystyle \frac{1}{2}}\sigma v{u}_{tx}-{\displaystyle \frac{1}{4}}\rho v_{xx}{u}_{tx}+{\displaystyle \frac{3}{4}}\rho v{u}_{txxx},\hfill \\ \hfill C_7^x=& \alpha v_x{u}_y^2+\alpha v_y{u}_x{u}_y-t\alpha v_x{u}_y-\theta v_x{u}_y-\alpha v{u}_{xy}{u}_y+\alpha v_x{u}_t{u}_y+\theta t{v}_x\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\sigma v_t{u}_y-\alpha v{u}_{tx}{u}_y+{\displaystyle \frac{3}{4}}\rho v_{txx}{u}_y+{\displaystyle \frac{1}{2}}\sigma v-t\alpha v_y{u}_x-\alpha v{u}_{yy}{u}_x+\theta v{u}_{xy}\hfill \\ \hfill & +{\displaystyle \frac{1}{4}}\rho v_{xx}+\alpha v_y{u}_x{u}_t-\theta v_x{u}_t-{\displaystyle \frac{1}{2}}t\sigma v_t+{\displaystyle \frac{1}{4}}\rho u_{xxy}{v}_t+{\displaystyle \frac{1}{2}}\sigma u_t{v}_t-{\displaystyle \frac{1}{2}}\sigma v{u}_{ty}\hfill \\ \hfill & -\alpha v{u}_x{u}_{ty}-{\displaystyle \frac{1}{4}}\rho v_{xx}{u}_{ty}+\theta v{u}_{tx}-{\displaystyle \frac{1}{2}}\rho u_{xy}{v}_{tx}-{\displaystyle \frac{1}{2}}\rho u_{tx}{v}_{tx}+{\displaystyle \frac{1}{2}}\rho v_x{u}_{txy}\hfill \\ \hfill & +{\displaystyle \frac{1}{4}}\rho v_t{u}_{txx}-{\displaystyle \frac{3}{4}}t\rho v_{txx}+{\displaystyle \frac{3}{4}}\rho u_t{v}_{txx}-{\displaystyle \frac{3}{4}}\rho v{u}_{txxy}-{\displaystyle \frac{1}{2}}\sigma v{u}_{tt}-{\displaystyle \frac{1}{4}}\rho v_{xx}{u}_{tt}\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\rho v_x{u}_{ttx}-{\displaystyle \frac{3}{4}}\rho v{u}_{ttxx},\hfill \\ \hfill C_7^y=& \alpha u_y{u}_x{v}_x-t\alpha u_x{v}_x+\alpha u_x{u}_t{v}_x+\alpha v{u}_x{u}_{xy}-\theta v{u}_{xx}+\alpha v{u}_y{u}_{xx}\hfill \\ \hfill & +\sigma v{u}_{tx}-\alpha v{u}_x{u}_{tx}+\rho v{u}_{txxx};\hfill \\ \\ \hfill C_8^t=& {\displaystyle \frac{1}{2}}y\sigma u_y{v}_x-{\displaystyle \frac{\theta \sigma }{2\alpha }}y{v}_x+{\displaystyle \frac{1}{2}}\sigma u_x{v}_x+{\displaystyle \frac{1}{4}}y\rho u_{xxy}{v}_x+{\displaystyle \frac{1}{4}}\rho u_{xxx}{v}_x+{\displaystyle \frac{1}{2}}t\sigma u_t{v}_x\hfill \\ \hfill & +{\displaystyle \frac{1}{4}}t\rho u_{txx}{v}_x-{\displaystyle \frac{1}{2}}y\sigma v{u}_{xy}+2t\alpha v{u}_x{u}_{xy}-t\theta v{u}_{xx}-{\displaystyle \frac{1}{2}}\sigma v{u}_{xx}+t\alpha v{u}_y{u}_{xx}\hfill \\ \hfill & -{\displaystyle \frac{1}{4}}y\rho u_{xy}{v}_{xx}-{\displaystyle \frac{1}{4}}\rho u_{xx}{v}_{xx}-{\displaystyle \frac{\theta \rho }{4\alpha }}y{v}_{xxx}+{\displaystyle \frac{1}{4}}y\rho u_y{v}_{xxx}+{\displaystyle \frac{1}{4}}\rho u_x{v}_{xxx}\hfill \\ \hfill & -{\displaystyle \frac{1}{4}}y\rho v{u}_{xxxy}-{\displaystyle \frac{1}{4}}\rho v{u}_{xxxx}+{\displaystyle \frac{1}{4}}t\rho v_{xxx}{u}_t+{\displaystyle \frac{1}{2}}t\sigma v{u}_{tx}-{\displaystyle \frac{1}{4}}t\rho v_{xx}{u}_{tx}+{\displaystyle \frac{3}{4}}t\rho v{u}_{txxx},\hfill \\ \hfill C_8^x=& {\displaystyle \frac{{\theta }^2}{\alpha }}y{v}_x+v{u}_x\theta -y{v}_y{u}_x\theta -2y{u}_y{v}_x\theta -u_x{v}_x\theta +yv{u}_{xy}\theta -t{v}_x{u}_t\theta -{\displaystyle \frac{\sigma \theta }{2\alpha }}y{v}_t\hfill \\ \hfill & +tv{u}_{tx}\theta -{\displaystyle \frac{3\rho \theta }{4\alpha }}y{v}_{txx}+\alpha v_y{u}_x^2-\alpha v{u}_y{u}_x+y\alpha u_y{v}_y{u}_x-y\alpha v{u}_{yy}{u}_x\hfill \\ \hfill & +y\alpha u_y^2{v}_x+\alpha u_y{u}_x{v}_x-y\alpha v{u}_y{u}_{xy}+\alpha v{u}_x{u}_{xy}-{\displaystyle \frac{1}{2}}\sigma v{u}_t+t\alpha v_y{u}_x{u}_t\hfill \\ \hfill & +t\alpha u_y{v}_x{u}_t-{\displaystyle \frac{1}{4}}\rho v_{xx}{u}_t+{\displaystyle \frac{1}{2}}y\sigma u_y{v}_t+{\displaystyle \frac{1}{2}}\sigma u_x{v}_t+{\displaystyle \frac{1}{4}}y\rho u_{xxy}{v}_t+{\displaystyle \frac{1}{4}}\rho u_{xxx}{v}_t\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\sigma t{u}_t{v}_t-{\displaystyle \frac{1}{2}}y\sigma v{u}_{ty}-t\alpha v{u}_x{u}_{ty}-{\displaystyle \frac{1}{4}}y\rho v_{xx}{u}_{ty}+{\displaystyle \frac{1}{2}}\sigma v{u}_{tx}-t\alpha v{u}_y{u}_{tx}\hfill \\ \hfill & -{\displaystyle \frac{1}{4}}\rho v_{xx}{u}_{tx}-{\displaystyle \frac{1}{2}}y\rho u_{xy}{v}_{tx}-{\displaystyle \frac{1}{2}}\rho u_{xx}{v}_{tx}-{\displaystyle \frac{1}{2}}t\rho u_{tx}{v}_{tx}+{\displaystyle \frac{1}{2}}y\rho v_x{u}_{txy}\hfill \\ \hfill & -{\displaystyle \frac{3}{4}}\rho v{u}_{txx}+{\displaystyle \frac{1}{2}}\rho v_x{u}_{txx}+{\displaystyle \frac{1}{4}}t\rho v_t{u}_{txx}+{\displaystyle \frac{3}{4}}y\rho u_y{v}_{txx}+{\displaystyle \frac{3}{4}}\rho u_x{v}_{txx}+{\displaystyle \frac{1}{2}}\rho v_x{u}_{tx}\hfill \\ \hfill & +{\displaystyle \frac{3}{4}}t\rho u_t{v}_{txx}-{\displaystyle \frac{3}{4}}y\rho v{u}_{txxy}+{\displaystyle \frac{1}{4}}\rho v{u}_{txxx}-{\displaystyle \frac{1}{2}}t\sigma v{u}_{tt}-{\displaystyle \frac{1}{4}}t\rho v_{xx}{u}_{tt}+{\displaystyle \frac{1}{2}}t\rho v_x{u}_{ttx}\hfill \\ \hfill & -{\displaystyle \frac{3}{4}}t\rho v{u}_{ttxx},\hfill \\ \hfill C_8^y=& \alpha v_x{u}_x^2-y\theta v_x{u}_x+y\alpha u_y{v}_x{u}_x+y\alpha v{u}_{xy}{u}_x-\alpha v{u}_{xx}{u}_x+t\alpha v_x{u}_t{u}_x\hfill \\ \hfill & -t\alpha v{u}_{tx}{u}_x-y\theta v{u}_{xx}+y\alpha v{u}_y{u}_{xx}+y\sigma v{u}_{tx}+y\rho v{u}_{txxx}.\hfill \end{array}$$

Remark 4.2.

We notice that 2D-gnrAKNSe (4) possesses conserved quantities that represent linear momentum, dilation energy as well as conservation of energy.

5. Concluding Remarks

This article presented a synoptic investigations implemented on the nonlinear generalized Ablowitz-Kaup-Newell-Segur water wave equation (4). Fundamentally, Lie group theory was utilized with a view to securing various outcomes of interest for the model. In the first place, a seven-dimensional Lie algebra associated to (4) was calculated in a step-by-step procedural format. Sequel to that, groups of transformation in one parametric format. Besides, the commutation relations as well as adjoint representations of the achieved Lie algebra are explicated in tabular structures. Furthermore, optimal system of one dimensional Lie subalgebras were computed in a comprehensive format. Thereafter, diverse achieved sub-algebras were explored in effectuating symmetry reductions of the nonlinear generalized Ablowitz-Kaup-Newell-Segur water wave equation (4) to ordinary differential equations, with a view to attaining diverse pertinent closed form solutions. In executing this, the exact solution entrenched while conducting the research include the likes of trigonometric, exponential, logarithmic as well as hyperbolic functions. Moreover, algebraic solutions among which one had arbitrary functions exist. At least a solution in quadrature structure as also achieved. In a bid to entrenched a sound knowledge of these results numerical simulation of some the gained solutions were implemented. In consequence, wave structures of diverse significance were achieved by making adequate choice of parameter values. These include, periodic wave, and kink shape waves. In addition, an interesting part of the work in this regard involved the simulations of algebraic function solutions where known mathematical functions were assigned. This furnished pertinent soliton interactions. Furthermore, conserved vectors of model equation (4) were derived through Ibragimov’s conservation law theorem via its formal Lagrangian. These conserved vectors are associated to the members of the optimal system.