The Propagating Exact Solitary Waves Formation of Generalized Calogero–Bogoyavlenskii–Schiff Equation with Robust Computational Approaches

Abstract

The generalized Calogero–Bogoyavlenskii–Schiff equation (GCBSE) is examined and analyzed in this paper. It has several applications in plasma physics and soliton theory, where it forecasts the soliton wave propagation profiles. In order to obtain the analytically exact solitons, the model under consideration is a nonlinear partial differential equation that is turned into an ordinary differential equation by using the next traveling wave transformation. The new extended direct algebraic technique and the modified auxiliary equation method are applied to the generalized Calogero–Bogoyavlenskii–Schiff equation to get new solitary wave profiles. As a result, novel and generalized analytical wave solutions are acquired in which singular solutions, mixed singular solutions, mixed complex solitary shock solutions, mixed shock singular solutions, mixed periodic solutions, mixed trigonometric solutions, mixed hyperbolic solutions, and periodic solutions are included with numerous soliton families. The propagation of the acquired soliton solution is graphically presented in contour, two- and three-dimensional visualization by selecting appropriate parametric values. It is graphically demonstrated how wave number impacts the obtained traveling wave structures.

1. Introduction

Recently, applications of nonlinear evolution equations (NLEEs) have been showing their imperative rules in physical science, applied science, and design, beginning not long ago with a widespread interest among various scientists, specialists, and researchers. Evolution equations can be used to predict the path of a number of physical problems or complex issues. Due to these abilities, optics, dense-matter physical science, astronomy, biomechanics, high-energy material science, electrical design, synthetic kinematics, electrodynamics, plasma physical science, gas elements, quantum and sea design, and other fields have begun to use NLEEs to solve physical problems. Similarly, the study of traveling wave profiles by nonlinear partial differential equations is useful in various fields such as quantum mechanics, fluid mechanics, and many other engineering and science branches [1,2]. Due to this, researchers have increased their interest in using them for the understanding and study of different physical models. For example, Akram et al. [3] discussed the optical soliton solution in fiber optics, a numerical approximation of the time-fractional equation [4], investigated the water waves in oceanography [5], and the mathematical Noyes–Field model of the nonlinear homogeneous oscillatory Belousov–Zhabotinsky reaction [6]. The stochastic form of the Newell–Whitehead–Segel equation [7] and the fractional-order pseudo-parabolic differential equations [8] were studied by Akgül, and he also analyzed the effects of the interfacial nanolayer and Lorentz force on a nanofluid flow [9].

Soliton waves are types of pulses that can maintain their shapes while moving at a constant speed; that is why exchanging information from one electrical source to another is very smooth and easy due to their use. With the development of technology, optical solitons have succeeded in making their mark through innovations in fiber optics. Moreover, with each passing day, soliton innovations are becoming a part of our regular lives in the forms of Facebook correspondence, Internet websites, Twitter, and so on. Due to the widespread potentiality of solitons, the researchers have worked on their applications. Afridi worked on the solution of the generalized Kadomtsev–Petviashvili modified equal-width Burgers equation [10], analyzed the $(2+1)$-dimensional elliptic nonlinear Schrödinger equation [11], investigated the Heisenberg spin chain process [12], and determined the number of solutions for the nonlinear Schrödinger equation’s dissipative form [13]. Jhangeer generated the soliton solutions of the Chen–Lee–Liu governing model [14], double dispersive equation [15], and also found the solitonic structures of the nonlinear electrical transmission lattice [16]. Osman used the sine Gordon expansion approach to obtain the different wave profiles for nonlinear evolution equations [17], studied the nonlinear Schrödinger equation of higher order and obtained analytical solutions [18], and figured out the solution of the generalized shallow water-wave equation [19]. Stochastic Fisher type equations [20] and the (3+1)-dimensional generalized Korteweg–de Vries–Zakharov–Kuznetsov equation [21] were analyzed by Seadwy. Sachin found soliton solutions for higher-order BKP–Boussinesq (gBKP-B) equations [22] and Kadomtsev–Petviashvili (KP) equations [23]. Wazwaz discovered optical, exact, dark, and bright solutions of different nonlinear models [24,25,26,27]. Tariq also proposed optical solutions for multiple models [28,29,30,31]. Fabio conducted an experimental study that showed that the perennial solitary waves in quadratic nonlinear environments could emit resonant dispersive waves in the absence of the larger derivative order [32]. Shen et al. [33] analyzed the nonlocal nonlinear Schrödinger equation and developed the complex-valued astigmatic cosine–Gaussian soliton solution. The superimposed field of Laguerre–Gaussian and Hermite–Gaussian solitons were investigated by Song et al. [34]. Guo et al. [35] applied the variational approach on the nonlocal Schrödinger equation and demonstrated the propagation dynamics of optical breathers in nonlinear media. The complex-valued hyperbolic–sine–Gaussian beams (CVHSGBs) were discussed and it was proved that the evolution of CVHSGBs was variable depending on the parameters of a complex-valued hyperbolic sine function [36,37].

Apart from the benefits of using nonlinear partial differential equations (NLPDE), the difficulty lies in finding their exact analytical solutions. As a result, a number of different systems have been designed to find accurate analytical solutions for NLPDEs. These schemes include the variational iteration method [38], $\left(\frac{G^{\prime }}{G}\right)$-expansion method [39], soliton perturbation theory [40], inverse scattering method [41], integral scheme [42], etc. The present manuscript looks at soliton solutions of the GBSE with the help of the new extended direct algebraic technique [43] and modified auxiliary equation method [44]. These two approaches have their own significance, such as the fact that the modified auxiliary equation method produces exact traveling wave soliton solutions, including trigonometric as well as hyperbolic functions, whereas the new extended direct algebraic technique generates a facile and universal system to cover 37 solitonic wave profiles’ solutions, which include exponential, trigonometric, rational, and hyperbolic functions.

Consider the following form of the generalized Calogero–Bogoyavlenskii–Schiff equation (GCBSE) [45],

$${\pounds }_{xt}+{\pounds }_{xxxy}+3{\pounds }_x{\pounds }_{xy}+3{\pounds }_{xx}{\pounds }_y+{\delta }_1{\pounds }_{xy}+{\delta }_2{\pounds }_{yy}=0,$$

(1)

where $\pounds =\pounds (x,y,t)$ is a continuous function that stands to explain the structure of nonlinear soliton waves, with spatial components x, y, and t as temporal components, and nonzero parameters ${\delta }_1$ and ${\delta }_2$. The GCBSE was first calculated by Schiff and Bogoyavlenskii, but in different ways. A modified Lax formalism was used by Bogoyavlenskii, while Schiff derived the same equation by reducing the self-dual Yang–Mills equation. The GBSE has a number of applications in plasma physics and soliton theory, where it predicts the propagation profiles of soliton waves. Many scientists have worked on the solution of different forms of this model by using several approaches.

Jarad et al. [45] analyzed the generalized Calogero–Bogoyavlenskii–Schiff equation and used the Lie symmetry approach and the new auxiliary equation method to find the analytical solutions. As a result, they obtained few types of soliton solutions such as trigonometric function and hyperbolic function solutions. It meant that a lot of soliton types were not found and there is a gap in the literature. In order to address this gap, the new extended direct algebraic method (NEDAM) and the modified auxiliary equation method is utilized. The new extended algebraic method is more generalized than the new auxiliary equation method (NAEM) because the NEDAM depends upon a second-degree differential equation. However, the NAEM depends on a first-order differential equation. The NEDAM provides thirty-seven types of soliton solutions which cover almost all types of soliton families.

In this manuscript, Section 2 contains a detailed description of the proposed techniques. Section 3 contains all the solutions obtained for the model with the help of different values of parameters obtained by the applied techniques, and graphs of some of them are also included in the same section. Section 4 is dedicated to the discussion of the different wave profiles, and the conclusion of the whole work is explained in detail in Section 5.

2. Description of the Proposed Technique

In this section, we explain the two different analytical approaches that we utilize to develop the solutions of the generalized Calogero–Bogoyavlenskii–Schiff equation.

Suppose a general nonlinear partial differential equation of the type [43]:

$$\begin{array}{c}\hfill Y(\pounds ,{\pounds }_t,{\pounds }_x,{\pounds }_{tt},{\pounds }_{xx},\cdots )=0.\end{array}$$

(2)

Its nonlinear ordinary differential equations (LNODE) are [43]:

$$\begin{array}{c}\hfill \mathbb{Q}(\mathbb{R},{\mathbb{R}}^{\prime },{\mathbb{R}}^{″},\cdots )=0.\end{array}$$

(3)

Consider the next traveling wave transformation [43]:

$$\pounds (x,y,t)=\pounds (\Omega ),$$

(4)

where $\Omega =k_1(x+y)+k_{2}t$. The prime signs in Equation (3) indicate the differentiation’s order for different attributes.

2.1. New Extended Direct Algebraic Method

This subsection is devoted to explain the general approach of the new extended direct algebraic method.

Consider the solution of Equation (3) as follows [43]:

$$\begin{array}{c}\hfill \pounds (\Omega )=\sum _{r=0}^i{a}_r{\left(P(\Omega )\right)}^r.\end{array}$$

(5)

The first derivative of $P(\Omega )$ has the following value:

$$\begin{array}{c}\hfill P^{\prime }(\Omega )=ln\left(\rho \right)\left(\psi +\vartheta P(\Omega )+\wp {\left(P(\Omega )\right)}^2\right),\rho \ne 0,1.\end{array}$$

(6)

Here, $\wp ,\vartheta ,$ and $\psi$ are real constants, and $\omega ={\vartheta }^2-4\psi \wp$. The general forms of the solutions of Equation (6) with respect to the present real number parameters are the following:

• For ${\vartheta }^2-4\psi \wp <0$ and $\wp \ne 0,$

  $$\begin{array}{c}\hfill P_1(\Omega )=-\frac{\vartheta }{2\wp }+\frac{\sqrt{-\omega }}{2\wp }{tan}_{\rho }\left(\frac{\sqrt{-\omega }}{2}\Omega \right),\end{array}$$

  (7)

  $$\begin{array}{c}\hfill P_2(\Omega )=-\frac{\vartheta }{2\wp }-\frac{\sqrt{-\omega }}{2\wp }{cot}_{\rho }\left(\frac{\sqrt{-\omega }}{2}\Omega \right),\end{array}$$

  (8)

  $$\begin{array}{c}\hfill P_3(\Omega )=-\frac{\vartheta }{2\wp }+\frac{\sqrt{-\omega }}{2\wp }\left({tan}_{\rho }\left(\sqrt{-\omega }\Omega \right)\pm \sqrt{mn}{sec}_{\rho }\left(\sqrt{-\omega }\Omega \right)\right),\end{array}$$

  (9)

  $$\begin{array}{c}\hfill P_4(\Omega )=-\frac{\vartheta }{2\wp }+\frac{\sqrt{-\omega }}{2\wp }\left({cot}_{\rho }\left(\sqrt{-\omega }\Omega \right)\pm \sqrt{mn}{csc}_{\rho }\left(\sqrt{-\omega }\Omega \right)\right),\end{array}$$

  (10)

  $$\begin{array}{c}\hfill P_5(\Omega )=-\frac{\vartheta }{2\wp }+\frac{\sqrt{-\omega }}{4\wp }\left({tan}_{\rho }\left(\frac{\sqrt{-\omega }}{4}\Omega \right)-{cot}_{\rho }\left(\frac{\sqrt{-\omega }}{4}\Omega \right)\right).\end{array}$$

  (11)
• For ${\vartheta }^2-4\psi \wp >0$ and $\wp \ne 0,$

  $$\begin{array}{c}\hfill P_6(\Omega )=-\frac{\vartheta }{2\wp }-\frac{\sqrt{\omega }}{2\wp }{tanh}_{\rho }\left(\frac{\sqrt{\omega }}{2}\Omega \right),\end{array}$$

  (12)

  $$\begin{array}{c}\hfill P_7(\Omega )=-\frac{\vartheta }{2\wp }-\frac{\sqrt{\omega }}{2\wp }{coth}_{\rho }\left(\frac{\sqrt{\omega }}{2}\Omega \right),\end{array}$$

  (13)

  $$\begin{array}{c}\hfill P_8(\Omega )=-\frac{\vartheta }{2\wp }+\frac{\sqrt{\omega }}{2\wp }\left({tanh}_{\rho }\left(\sqrt{\omega }\Omega \right)\pm i\sqrt{mn}{\mathrm{Sec}\mathrm{h}}_{\rho }\left(\sqrt{\omega }\Omega \right)\right),\end{array}$$

  (14)

  $$\begin{array}{c}\hfill P_9(\Omega )=-\frac{\vartheta }{2\wp }+\frac{\sqrt{\omega }}{2\wp }\left(-{coth}_{\rho }\left(\sqrt{\omega }\Omega \right)\pm \sqrt{mn}{\csc \mathrm{h}}_{\rho }\left(\sqrt{\omega }\Omega \right)\right),\end{array}$$

  (15)

  $$\begin{array}{c}\hfill P_{10}(\Omega )=-\frac{\vartheta }{2\wp }-\frac{\sqrt{\omega }}{4\wp }\left({tanh}_{\rho }\left(\frac{\sqrt{\omega }}{4}\Omega \right)+{coth}_{\rho }\left(\frac{\sqrt{\omega }}{4}\Omega \right)\right).\end{array}$$

  (16)
• For $\psi \wp >0$ and $\vartheta =0,$

  $$\begin{array}{c}\hfill P_{11}(\Omega )=\sqrt{\frac{\psi }{\wp }}{tan}_{\rho }\left(\sqrt{\psi \wp }\Omega \right),\end{array}$$

  (17)

  $$\begin{array}{c}\hfill P_{12}(\Omega )=-\sqrt{\frac{\psi }{\wp }}{cot}_{\rho }\left(\sqrt{\psi \wp }\Omega \right),\end{array}$$

  (18)

  $$\begin{array}{c}\hfill P_{13}(\Omega )=\sqrt{\frac{\psi }{\wp }}\left({tan}_{\rho }\left(2\sqrt{\psi \wp }\Omega \right)\pm \sqrt{mn}{sec}_{\rho }\left(2\sqrt{\psi \wp }\Omega \right)\right),\end{array}$$

  (19)

  $$\begin{array}{c}\hfill P_{14}(\Omega )=\sqrt{\frac{\psi }{\wp }}\left(-{cot}_{\rho }\left(2\sqrt{\psi \wp }\Omega \right)\pm \sqrt{mn}{csc}_{\rho }\left(2\sqrt{\psi \wp }\Omega \right)\right),\end{array}$$

  (20)

  $$\begin{array}{c}\hfill P_{15}(\Omega )=\frac{1}{2}\sqrt{\frac{\psi }{\wp }}\left({tan}_{\rho }\left(\frac{\sqrt{\psi \wp }}{2}\Omega \right)-{cot}_{\rho }\left(\frac{\sqrt{\psi \wp }}{2}\Omega \right)\right).\end{array}$$

  (21)
• For $\psi \wp <0$ and $\vartheta =0,$

  $$\begin{array}{c}\hfill P_{16}(\Omega )=-\sqrt{-\frac{\psi }{\wp }}{tanh}_{\rho }\left(\sqrt{-\psi \wp }\Omega \right),\end{array}$$

  (22)

  $$\begin{array}{c}\hfill P_{17}(\Omega )=-\sqrt{-\frac{\psi }{\wp }}{coth}_{\rho }\left(\sqrt{-\psi \wp }\Omega \right),\end{array}$$

  (23)

  $$\begin{array}{c}\hfill P_{18}(\Omega )=\sqrt{-\frac{\psi }{\wp }}\left(-{tanh}_{\rho }\left(2\sqrt{-\psi \wp }\Omega \right)\pm i\sqrt{mn}{\sec \mathrm{h}}_{\rho }\left(2\sqrt{-\psi \wp }\Omega \right)\right),\end{array}$$

  (24)

  $$\begin{array}{c}\hfill P_{19}(\Omega )=\sqrt{-\frac{\psi }{\wp }}\left(-{coth}_{\rho }\left(2\sqrt{-\psi \wp }\Omega \right)\pm \sqrt{mn}{\csc \mathrm{h}}_{\rho }\left(2\sqrt{-\psi \wp }\Omega \right)\right),\end{array}$$

  (25)

  $$\begin{array}{c}\hfill P_{20}(\Omega )=-\frac{1}{2}\sqrt{-\frac{\psi }{\wp }}\left({tanh}_{\rho }\left(\frac{\sqrt{-\psi \wp }}{2}\Omega \right)+{coth}_{\rho }\left(\frac{\sqrt{-\psi \wp }}{2}\Omega \right)\right).\end{array}$$

  (26)
• For $\vartheta =0$ and $\psi =\wp ,$

  $$\begin{array}{c}\hfill P_{21}(\Omega )={tan}_{\rho }\left(\psi \Omega \right),\end{array}$$

  (27)

  $$\begin{array}{c}\hfill P_{22}(\Omega )=-{cot}_{\rho }\left(\psi \Omega \right),\end{array}$$

  (28)

  $$\begin{array}{c}\hfill P_{23}(\Omega )={tan}_{\rho }\left(2\psi \Omega \right)\pm \sqrt{mn}{sec}_{\rho }\left(2\psi \Omega \right),\end{array}$$

  (29)

  $$P_{24}(\Omega )=-{cot}_{\rho }\left(2\psi \Omega \right)\pm \sqrt{mn}{csc}_{\rho }\left(2\psi \Omega \right),$$

  (30)

  $$\begin{array}{c}\hfill P_{25}(\Omega )=\frac{1}{2}\left({tan}_{\rho }\left(\frac{\psi }{2}\Omega \right)-{cot}_{\rho }\left(\frac{\psi }{2}\Omega \right)\right).\end{array}$$

  (31)
• For $\vartheta =0$ and $\wp =-\psi ,$

  $$\begin{array}{c}\hfill P_{26}(\Omega )=-{tanh}_{\rho }\left(\psi \Omega \right),\end{array}$$

  (32)

  $$\begin{array}{c}\hfill P_{27}(\Omega )=-{coth}_{\rho }\left(\psi \Omega \right),\end{array}$$

  (33)

  $$\begin{array}{c}\hfill P_{28}(\Omega )=-{tanh}_{\rho }\left(2\psi \Omega \right)\pm i\sqrt{mn}{\sec \mathrm{h}}_{\rho }\left(2\psi \Omega \right),\end{array}$$

  (34)

  $$\begin{array}{c}\hfill P_{29}(\Omega )=-{coth}_{\rho }\left(2\psi \Omega \right)\pm \sqrt{mn}{\csc \mathrm{h}}_{\rho }\left(2\psi \Omega \right),\end{array}$$

  (35)

  $$\begin{array}{c}\hfill P_{30}(\Omega )=-\frac{1}{2}\left({tanh}_{\rho }\left(\frac{\psi }{2}\Omega \right)+{coth}_{\rho }\left(\frac{\psi }{2}\Omega \right)\right).\end{array}$$

  (36)
• For ${\vartheta }^2=4\psi \wp$,

  $$\begin{array}{c}\hfill P_{31}(\Omega )=\frac{-2\psi (\vartheta \Omega ln(\rho )+2)}{{\vartheta }^2\Omega ln\left(\rho \right)}.\end{array}$$

  (37)
• For $\vartheta =p,\psi =pq,(q\ne 0)$ and $\wp =0,$

  $$\begin{array}{c}\hfill P_{32}(\Omega )={\rho }^{p\Omega }-q.\end{array}$$

  (38)
• For $\vartheta =\wp =0,$

  $$\begin{array}{c}\hfill P_{33}(\Omega )=\psi \Omega ln\left(\rho \right).\end{array}$$

  (39)
• For $\vartheta =\psi =0,$

  $$\begin{array}{c}\hfill P_{34}(\Omega )=\frac{-1}{\wp \Omega ln\left(\rho \right)}.\end{array}$$

  (40)
• For $\psi =0$ and $\vartheta \ne 0,$

  $$\begin{array}{c}\hfill P_{35}(\Omega )=-\frac{m\vartheta }{\wp \left({cosh}_{\rho }\left(\vartheta \Omega \right)-{sinh}_{\rho }\left(\vartheta \Omega \right)+m\right)},\end{array}$$

  (41)

  $$\begin{array}{c}\hfill P_{36}(\Omega )=-\frac{\vartheta \left({sinh}_{\rho }\left(\vartheta \Omega \right)+{cosh}_{\rho }\left(\vartheta \Omega \right)\right)}{\wp \left({sinh}_{\rho }\left(\vartheta \Omega \right)+{cosh}_{\rho }\left(\vartheta \Omega \right)+n\right)}.\end{array}$$

  (42)
• For $\vartheta =p,\wp =pq,(q\ne 0$ and $\psi =0),$

  $$\begin{array}{c}\hfill P_{37}(\Omega )=-\frac{m{\rho }^{p\Omega }}{m-qn{\rho }^{p\Omega }}.\end{array}$$

  (43)

Here,

$$\begin{array}{c}\hfill {sinh}_{\rho }(\Omega )=\frac{m{\rho }^{\Omega }-n{\rho }^{(-\Omega )}}{2},{cosh}_{\rho }(\Omega )=\frac{m{\rho }^{\Omega }+n{\rho }^{(-\Omega )}}{2},{tanh}_{\rho }(\Omega )=\frac{m{\rho }^{\Omega }-n{\rho }^{(-\Omega )}}{m{\rho }^{\Omega }+n{\rho }^{(-\Omega )}},\end{array}$$

$$\begin{array}{c}\hfill {\csc \mathrm{h}}_{\rho }(\Omega )=\frac{2}{m{\rho }^{\Omega }-n{\rho }^{(-\Omega )}},{\sec \mathrm{h}}_{\rho }(\Omega )=\frac{2}{m{\rho }^{\Omega }+n{\rho }^{(-\Omega )}},{coth}_{\rho }(\Omega )=\frac{m{\rho }^{\Omega }+n{\rho }^{-\Omega }}{m{\rho }^{\Omega }-n{\rho }^{-\Omega }},\end{array}$$

$$\begin{array}{c}\hfill {sin}_{\rho }(\Omega )=\frac{m{\rho }^{i\Omega }-n{\rho }^{(-i\Omega )}}{2i},{cos}_{\rho }(\Omega )=\frac{m{\rho }^{i\Omega }+n{\rho }^{(-i\Omega )}}{2},{tan}_{\rho }(\Omega )=-i\frac{m{\rho }^{i\Omega }-n{\rho }^{(-i\Omega )}}{m{\rho }^{i\Omega }+n{\rho }^{(-i\Omega )}},\end{array}$$

$$\begin{array}{c}\hfill {csc}_{\rho }(\Omega )=\frac{2i}{m{\rho }^{\Omega }-n{\rho }^{(-\Omega )}}{sec}_{\rho }(\Omega )=\frac{2}{m{\rho }^{\Omega }+n{\rho }^{(-\Omega )}},{cot}_{\rho }(\Omega )=i\frac{m{\rho }^{i\Omega }+n{\rho }^{(-i\Omega )}}{m{\rho }^{i\Omega }-n{\rho }^{(-i\Omega )}},\end{array}$$

where m and n are arbitrary constants and $m,n>0$. Moreover, they are considered as parameters of deformation.

2.2. Modified Auxiliary Equation Method

This subsection is devoted to explain the general approach of the modified auxiliary equation method.

Suppose the general solution of Equation (3) [44] as:

$$\begin{array}{c}\hfill \pounds (\Omega )={\alpha }_0+\sum _{i=1}^N[{\alpha }_i(z^{h(\Omega )}+{\beta }_i{z}^{-h(\Omega )})],\end{array}$$

(44)

where $\Omega =k_1(x+y)+k_{2}t$ and ${\alpha }_{i}s$, ${\beta }_{i}s$ are arbitrary constants.

Here, h($\Omega$) follows the following auxiliary equation,

$$\begin{array}{c}\hfill h^{\prime }(\Omega )=\frac{\beta +\alpha z^{-h(\Omega )}+\gamma z^{h(\Omega )}}{lnz}.\end{array}$$

(45)

$\alpha$, $\beta$, $\gamma$, and z are constants, with conditions on z as $z>0$ and $z\ne 1$. Further, ${\alpha }_{i}s$ and ${\beta }_{i}s$ can also not be zero simultaneously.

The solutions of Equation (45) are obtained as follows:

1. If ${\beta }^2-4\alpha \gamma <0$ and $\gamma \ne 0$,

$$\begin{array}{c}{z}^{h(\Omega )}=-\frac{\beta +\sqrt{4\alpha \gamma -{\beta }^2}tan\left(\frac{\sqrt{4\alpha \gamma -{\beta }^2}\Omega }{2}\right)}{2\gamma },\\ \hfill \mathrm{or}\\ \hfill z^{h(\Omega )}=-\frac{\beta +\sqrt{4\alpha \gamma -{\beta }^2}cot\left(\frac{\sqrt{4\alpha \gamma -{\beta }^2}\Omega }{2}\right)}{2\gamma }.\end{array}$$

(46)

2. If ${\beta }^2-4\alpha \gamma >0$ and $\gamma \ne 0$,

$$\begin{array}{c}\hfill z^{h(\Omega )}=-\frac{\beta +\sqrt{{\beta }^2-4\alpha \gamma }tanh\left(\frac{\sqrt{{\beta }^2-4\alpha \gamma }\Omega }{2}\right)}{2\gamma },\\ \hfill \mathrm{or}\\ \hfill z^{h(\Omega )}=-\frac{\beta +\sqrt{{\beta }^2-4\alpha \gamma }coth\left(\frac{\sqrt{{\beta }^2-4\alpha \gamma }\Omega }{2}\right)}{2\gamma }\end{array}$$

(47)

3. If ${\beta }^2-4\alpha \gamma =0$ and $\gamma \ne 0$,

$$\begin{array}{c}\hfill z^{h(\Omega )}=-\frac{2+\beta \Omega }{2\gamma \Omega }\end{array}$$

(48)

3. Construction of Soliton Structures for Equation (3)

The above-mentioned techniques, new extended algebraic method, and modified auxiliary equation scheme were applied on the generalized Calogero–Bogoyavlenskii–Schiff equation.

In order to determine the solution, we used the next traveling wave transformation [45],

$$\pounds (x,y,t)=\pounds (\Omega ),\Omega =k(x+y)+ct,$$

(49)

where k is the speed of the soliton and c is the wave number. By plugging the considered transformation of Equation (49) into Equation (1), we obtained the following ODE:

$$k^3{\pounds }^{⁗}+6k^2{\pounds }^{\prime }{\pounds }^{″}+(c+({\delta }_1+\delta 2)k){\pounds }^{″}=0.$$

(50)

After integrating Equation (50) once,

$$k^3{\pounds }^{‴}+3k^2{\left({\pounds }^{\prime }\right)}^2+(c+({\delta }_1+\delta 2)k){\pounds }^{\prime }=0.$$

(51)

Here, by considering $W(\Omega )={\pounds }^{\prime }(\Omega )$ in Equation (51), we obtained

$$k^3{W}^{″}+3k^2{\left(W\right)}^2+(c+({\delta }_1+{\delta }_2)k)W=0.$$

(52)

3.1. Solution with Modified Auxiliary Equation Method

The modified auxiliary equation method was used to form the soliton structures for Equation (1).

By calculating the homogeneous balancing constant of Equation (52), the solution with the MAE can be written as follows:

$$W(\Omega )={\alpha }_0+{\alpha }_1{z}^{h(\Omega )}+{\beta }_1{z}^{-h(\Omega )}+{\alpha }_2{z}^{2h(\Omega )}+{\beta }_2{z}^{-2h(\Omega )}.$$

(53)

The system of equations was obtained by plugging solution Equation (53) into Equation (52) and then calculating the coefficients of different powers of $z^{h(\Omega )}$. This obtained system was an algebraic equation system solved with Mathematica software, yielding four distinct families of values for ${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$, ${\beta }_1$, and ${\beta }_2$. By using these families one by one, we obtained the results whose integration gave the solutions of the considered model as follows:

Family 1:

$$\begin{array}{cc}\hfill {\alpha }_0=& -2\alpha \gamma k,{\alpha }_1=0,{\beta }_1=-2\alpha \beta k,{\alpha }_2=0,{\beta }_2=-2{\alpha }^{2}k,\hfill \\ \hfill & c=-k\left({\delta }_1+{\delta }_2-4\alpha \gamma k^2+{\beta }^2{k}^2\right).\hfill \end{array}$$

(54)

The general solution for family one was

$$W(\Omega )=-2\alpha k{z}^{-2h(\Omega )}\left(\alpha +z^{h(\Omega )}\left(\beta +\gamma z^{h(\Omega )}\right)\right).$$

(55)

Since we know that $\pounds (\Omega )=\int W(\Omega )d\Omega$, the solutions from result Equation (55), after integration, are given below, where $\Delta ={\beta }^2-4\alpha \gamma$.

Case 1: If ${\beta }^2-4\alpha \gamma <0,\gamma \ne 0;$

$$\begin{array}{c}\hfill {\pounds }_{1,1}(x,y,t)=-\frac{k\left(\beta \Delta +2\alpha \gamma \sqrt{-\Delta }sin\left(\Omega \sqrt{-\Delta }\right)\right)}{{\beta }^2-2\alpha \gamma +2\alpha \gamma cos\left(\Omega \sqrt{-\Delta }\right)},\end{array}$$

(56)

or

$$\begin{array}{c}\hfill {\pounds }_{1,2}(x,y,t)=\frac{k\left(\beta \Delta +2\alpha \gamma \sqrt{-\Delta }sin\left(\Omega \sqrt{-\Delta }\right)\right)}{{\beta }^2-2\alpha \gamma -2\alpha \gamma cos\left(\Omega \sqrt{-\Delta }\right)}.\end{array}$$

(57)

Case 2: If ${\beta }^2-4\alpha \gamma >0,\gamma \ne 0;$

$$\begin{array}{c}\hfill {\pounds }_{1,3}(x,y,t)=\frac{k\left(\beta \Delta +2\alpha \gamma \sqrt{\Delta }sinh\left(\Omega \sqrt{\Delta }\right)\right)}{{\beta }^2-2\alpha \gamma +2\alpha \gamma cosh\left(\Omega \sqrt{\Delta }\right)},\end{array}$$

(58)

or

$$\begin{array}{c}\hfill {\pounds }_{1,4}(x,y,t)=\frac{k\left(\beta \Delta -2\alpha \gamma \sqrt{\Delta }sinh\left(\Omega \sqrt{\Delta }\right)\right)}{{\beta }^2-2\alpha \gamma -2\alpha \gamma cosh\left(\Omega \sqrt{\Delta }\right)}.\end{array}$$

(59)

Case 3: If ${\beta }^2-4\alpha \gamma =0,\gamma \ne 0;$

$$\begin{array}{c}\hfill {\pounds }_{1,5}(x,y,t)=\frac{2\alpha \gamma k\left(-4\Delta log(\beta \Omega +2)-4\alpha \beta \gamma \Omega +\frac{16\alpha \gamma }{\beta \Omega +2}+{\beta }^3\Omega \right)}{{\beta }^3}.\end{array}$$

(60)

Family 2:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\alpha }_0=& \frac{1}{3}\left({\beta }^2(-k)-2\alpha \gamma k\right),{\alpha }_1=0,{\beta }_1=-2\alpha \beta k,{\alpha }_2=0,{\beta }_2=-2{\alpha }^{2}k,\hfill \\ & c=-4\alpha \gamma k^3+{\beta }^2{k}^3-{\delta }_{1}k-{\delta }_{2}k.\hfill \end{array}\end{array}$$

(61)

The general solution for family two was

$$W(\Omega )=-2{\alpha }^{2}k{z}^{-2h(\Omega )}-2\alpha \beta k{z}^{-h(\Omega )}+\frac{1}{3}\left({\beta }^2(-k)-2\alpha \gamma k\right).$$

(62)

Since we know that $\pounds (\Omega )=\int W(\Omega )d\Omega$, the solutions from result Equation (62), after integration, are given below:

Case 1: If ${\beta }^2-4\alpha \gamma <0,\gamma \ne 0;$

$$\begin{array}{cc}\hfill {\pounds }_{2,1}(x,y,t)=& \frac{1}{3}k(\frac{2(4\alpha \gamma -3{\beta }^2){tan}^{-1}\left(tan\left(\frac{1}{2}\Omega \sqrt{-\Delta }\right)\right)}{\sqrt{-\Delta }}-3\beta \left(1+\frac{i\beta }{\sqrt{-\Delta }}\right)\hfill \\ \hfill & log\left(-tan\left(\frac{1}{2}\Omega \sqrt{-\Delta }\right)+i\right)-3\beta \left(1-\frac{i\beta }{\sqrt{-\Delta }}\right)log\left(tan\left(\frac{1}{2}\Omega \sqrt{-\Delta }\right)+i\right)\hfill \\ \hfill & +6\beta log\left(\sqrt{-\Delta }tan\left(\frac{1}{2}\Omega \sqrt{-\Delta }\right)+\beta \right)+3\beta log\left({sec}^2\left(\frac{1}{2}\Omega \sqrt{-\Delta }\right)\right)\hfill \\ \hfill & +\frac{12\alpha \gamma {cos}^2\left(\frac{1}{2}\Omega \sqrt{-\Delta }\right)\left(\left(\Delta \right)tan\left(\frac{1}{2}\Omega \sqrt{-\Delta }\right)+\beta \sqrt{-\Delta }\right)}{\sqrt{-\Delta }\left(2\alpha \gamma cos\left(\Omega \sqrt{-\Delta }\right)-2\alpha \gamma +{\beta }^2\right)}\hfill \\ \hfill & -6\beta {tanh}^{-1}\left(\frac{\sqrt{-\Delta }tan\left(\frac{1}{2}\Omega \sqrt{-\Delta }\right)}{\beta }\right)-3\beta log(-((2\alpha \gamma cos\left(\Omega \sqrt{-\Delta }\right)\hfill \\ \hfill & -2\alpha \gamma +{\beta }^2\left){sec}^2\left(\frac{1}{2}\Omega \sqrt{-\Delta }\right)\right))-{\beta }^2\Omega ),\hfill \end{array}$$

(63)

or

$$\begin{array}{cc}\hfill {\pounds }_{2,2}(x,y,t)=& \frac{1}{3}k(\frac{6\beta \sqrt{-\Delta }{tan}^{-1}\left(\frac{\beta tan\left(\frac{1}{2}\Omega \sqrt{-\Delta }\right)}{\sqrt{\Delta }}\right)}{\sqrt{\Delta }}-3\beta log(-2\alpha \gamma cos\left(\Omega \sqrt{-\Delta }\right)\hfill \\ \hfill & -2\alpha \gamma +{\beta }^2)+\frac{2\left(3{\beta }^2-4\alpha \gamma \right){tan}^{-1}\left(cot\left(\frac{1}{2}\Omega \sqrt{-\Delta }\right)\right)}{\sqrt{-\Delta }}\hfill \\ \hfill & +6\beta {tanh}^{-1}\left(\frac{\sqrt{-\Delta }cot\left(\frac{1}{2}\Omega \sqrt{-\Delta }\right)}{\beta }\right)-3\beta log\left({csc}^2\left(\frac{1}{2}\Omega \sqrt{-\Delta }\right)\right)\hfill \\ \hfill & -\frac{12\alpha \gamma {sin}^2\left(\frac{1}{2}\Omega \sqrt{-\Delta }\right)\left(\left(\Delta )cot(\frac{1}{2}\Omega \sqrt{-\Delta }\right)+\beta \sqrt{-\Delta }\right)}{\sqrt{-\Delta }\left(-2\alpha \gamma cos\left(\Omega \sqrt{-\Delta }\right)-2\alpha \gamma +{\beta }^2\right)}\hfill \\ \hfill & +3\beta log\left(\left(2\alpha \gamma cos\left(\Omega \sqrt{-\Delta }\right)+2\alpha \gamma -{\beta }^2\right){csc}^2\left(\frac{1}{2}\Omega \sqrt{-\Delta }\right)\right)+2{\beta }^2\Omega ).\hfill \end{array}$$

(64)

Case 2: If ${\beta }^2-4\alpha \gamma >0,\gamma \ne 0;$

$$\begin{array}{cc}\hfill {\pounds }_{2,3}(x,y,t)=& \frac{1}{3}k(\frac{2\left(4\alpha \gamma -3{\beta }^2\right){tanh}^{-1}\left(tanh\left(\frac{1}{2}\Omega \sqrt{\Delta }\right)\right)}{\sqrt{\Delta }}-6\beta log\left(\sqrt{\Delta }tanh(\frac{1}{2}\Omega \sqrt{\Delta }\right)\hfill \\ \hfill & +\beta )+\frac{12\alpha \beta \gamma log\left(tanh\left(\frac{1}{2}\Omega \sqrt{\Delta }\right)+1\right)}{\beta \sqrt{\Delta }-\Delta }-\frac{12\alpha \beta \gamma log\left(1-tanh\left(\frac{1}{2}\Omega \sqrt{\Delta }\right)\right)}{\beta \sqrt{\Delta }+\Delta }\hfill \\ \hfill & -3\beta log\left({\sec \mathrm{h}}^2\left(\frac{1}{2}\Omega \sqrt{\Delta }\right)\right)-\frac{6\beta \sqrt{\Delta }{tan}^{-1}\left(\frac{\sqrt{\Delta }tanh\left(\frac{1}{2}\Omega \sqrt{\Delta }\right)}{\beta }\right)}{\sqrt{\Delta }}\hfill \\ \hfill & -\frac{12\alpha \gamma {cosh}^2\left(\frac{1}{2}\Omega \sqrt{\Delta }\right)\left(\beta \sqrt{\Delta }-\left(\Delta \right)tanh\left(\frac{1}{2}\Omega \sqrt{\Delta }\right)\right)}{\sqrt{\Delta }\left(2\alpha \gamma cosh\left(\Omega \sqrt{\Delta }\right)-2\alpha \gamma +{\beta }^2\right)}\hfill \\ \hfill & +3\beta log\left(\left(2\alpha \gamma cosh\left(\Omega \sqrt{\Delta }\right)-2\alpha \gamma +{\beta }^2\right){\sec \mathrm{h}}^2\left(\frac{1}{2}\Omega \sqrt{\Delta }\right)\right)-{\beta }^2\Omega ),\hfill \end{array}$$

(65)

or

$$\begin{array}{cc}\hfill {\pounds }_{2,4}(x,y,t)=& \frac{1}{3}k(-6\beta {tanh}^{-1}\left(\frac{\beta tanh\left(\frac{1}{2}\Omega \sqrt{\Delta }\right)}{\sqrt{\Delta }}\right)-3\beta log(-2\alpha \gamma cosh\left(\Omega \sqrt{\Delta }\right)\hfill \\ \hfill & -2\alpha \gamma +{\beta }^2)-\frac{6\beta \sqrt{-\Delta }{tan}^{-1}\left(\frac{\sqrt{-\Delta }coth\left(\frac{1}{2}\Omega \sqrt{\Delta }\right)}{\beta }\right)}{\sqrt{\Delta }}\hfill \\ \hfill & +\frac{2\left(4\alpha \gamma -3{\beta }^2\right){tanh}^{-1}\left(coth\left(\frac{1}{2}\Omega \sqrt{\Delta }\right)\right)}{\sqrt{\Delta }}-3\beta log\left(-{\csc \mathrm{h}}^2\left(\frac{1}{2}\Omega \sqrt{\Delta }\right)\right)\hfill \\ \hfill & +\frac{12\alpha \gamma {sinh}^2\left(\frac{1}{2}\Omega \sqrt{\Delta }\right)\left(\beta \sqrt{\Delta }-\left(\Delta \right)coth\left(\frac{1}{2}\Omega \sqrt{\Delta }\right)\right)}{\sqrt{\Delta }\left(-2\alpha \gamma cosh\left(\Omega \sqrt{\Delta }\right)-2\alpha \gamma +{\beta }^2\right)}\hfill \\ \hfill & +3\beta log\left(\left(2\alpha \gamma cosh\left(\Omega \sqrt{\Delta }\right)+2\alpha \gamma -{\beta }^2\right){\csc \mathrm{h}}^2\left(\frac{1}{2}\Omega \sqrt{\Delta }\right)\right)+2{\beta }^2\Omega ).\hfill \end{array}$$

(66)

Case 3: If ${\beta }^2-4\alpha \gamma =0,\gamma \ne 0;$

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\pounds }_{2,5}(x,y,t)=\frac{1}{3}k\left(-\frac{24{\alpha }^2{\gamma }^2\left({\beta }^2{\Omega }^2+2\beta \Omega -4\right)}{{\beta }^3(\beta \Omega +2)}+\frac{24\alpha \gamma \left(-\Delta \right)log(\beta \Omega +2)}{{\beta }^3}+10\alpha \gamma \Omega -{\beta }^2\Omega \right).\end{array}\end{array}$$

(67)

Family 3:

$$\begin{array}{c}\hfill \begin{array}{cc}& {\alpha }_0=-2\alpha \gamma k,{\alpha }_1=-2\beta \gamma k,{\beta }_1=0,{\alpha }_2=-2{\gamma }^{2}k,\hfill \\ & {\beta }_2=0,c=-k\left({\delta }_1+{\delta }_2-4\alpha \gamma k^2+{\beta }^2{k}^2\right).\hfill \end{array}\end{array}$$

(68)

The general solution for family three was

$$W(\Omega )=-2\beta \gamma k{z}^{h(\Omega )}-2{\gamma }^{2}k{z}^{2h(\Omega )}-2\alpha \gamma k.$$

(69)

Since we know that $\pounds (\Omega )=\int W(\Omega )d\Omega$, the solutions from result Equation (69), after integration, are given below:

Case 1: If ${\beta }^2-4\alpha \gamma <0,\gamma \ne 0;$

$$\begin{array}{c}\hfill {\pounds }_{3,1}(x,y,t)=-k\sqrt{-\Delta }tan\left(\frac{1}{2}\Omega \sqrt{-\Delta }\right),\end{array}$$

(70)

or

$$\begin{array}{c}\hfill {\pounds }_{3,2}(x,y,t)=k\sqrt{-\Delta }cot\left(\frac{1}{2}\Omega \sqrt{-\Delta }\right).\end{array}$$

(71)

Case 2: If ${\beta }^2-4\alpha \gamma >0,\gamma \ne 0;$

$$\begin{array}{c}\hfill {\pounds }_{3,3}(x,y,t)=k\sqrt{\Delta }tanh\left(\frac{1}{2}\Omega \sqrt{\Delta }\right),\end{array}$$

(72)

or

$$\begin{array}{c}\hfill {\pounds }_{3,4}(x,y,t)=k\sqrt{\Delta }coth\left(\frac{1}{2}\Omega \sqrt{\Delta }\right).\end{array}$$

(73)

Case 3: If ${\beta }^2-4\alpha \gamma =0,\gamma \ne 0;$

$$\begin{array}{c}\hfill {\pounds }_{3,5}(x,y,t)=\frac{1}{2}k\left(\Omega \left(\Delta \right)+\frac{4}{\Omega }\right).\end{array}$$

(74)

Family 4:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\alpha }_0=& -2\alpha \gamma k,{\alpha }_1=0,{\beta }_1=-2\alpha \beta k,{\alpha }_2=0,{\beta }_2=-2{\alpha }^{2}k,\hfill \\ & c=-k\left({\delta }_1+{\delta }_2-4\alpha \gamma k^2+{\beta }^2{k}^2\right).\hfill \end{array}\end{array}$$

(75)

The general solution for family four was

$$W(\Omega )=-2\beta \gamma k{z}^{h(\Omega )}-2{\gamma }^{2}k{z}^{2h(\Omega )}+\frac{1}{3}\left({\beta }^2(-k)-2\alpha \gamma k\right).$$

(76)

Since we know that $\pounds (\Omega )=\int W(\Omega )d\Omega$, the solutions from result Equation (76), after integration, are given below:

Case 1: If ${\beta }^2-4\alpha \gamma <0,\gamma \ne 0;$

$$\begin{array}{c}\hfill {\pounds }_{4,1}(x,y,t)=-\frac{1}{3}k\left(\Omega \left(\Delta \right)+3\sqrt{-\Delta }tan\left(\frac{1}{2}\Omega \sqrt{-\Delta }\right)\right),\end{array}$$

(77)

or

$$\begin{array}{c}\hfill {\pounds }_{4,2}(x,y,t)=\frac{1}{3}k\left(3\sqrt{-\Delta }cot\left(\frac{1}{2}\Omega \sqrt{-\Delta }\right)-\Delta \Omega \right).\end{array}$$

(78)

Case 2: If ${\beta }^2-4\alpha \gamma >0,\gamma \ne 0;$

$$\begin{array}{c}\hfill {\pounds }_{4,3}(x,y,t)=\frac{1}{3}k\left(3\sqrt{\Delta }tanh\left(\frac{1}{2}\Omega \sqrt{\Delta }\right)-\Delta \Omega \right),\end{array}$$

(79)

or

$$\begin{array}{c}\hfill {\pounds }_{4,4}(x,y,t)=\frac{1}{3}k\left(3\sqrt{\Delta }coth\left(\frac{1}{2}\Omega \sqrt{\Delta }\right)-\Delta \Omega \right).\end{array}$$

(80)

Case 3: If ${\beta }^2-4\alpha \gamma =0,\gamma \ne 0;$

$$\begin{array}{c}\hfill {\pounds }_{4,5}(x,y,t)=\frac{1}{6}k\Omega \Delta +\frac{2k}{\Omega }.\end{array}$$

(81)

3.2. Solution with New Extended Direct Algebraic Method

The new extended direct algebraic method was used to form the soliton structures for Equation (1).

According to the new extended direct algebraic method, the solutions of Equation (52) are given as follows,

$$W(\Omega )=a_0+a_{1}P(\Omega )+a_{2}P{(\Omega )}^2,$$

(82)

along with,

$$P^{\prime }{(\Omega )=ln\left(\rho \right)(\psi +\vartheta P+\wp \left(P(\Omega )\right)}^2.$$

(83)

By plugging solution Equation (82) into Equation (52) and then calculating the coefficients of P($\Omega$) for different powers, we obtained an algebraic system of equations. This algebraic system was solved by Mathematica software, and it provided the following two families of values for the constant.

Here are the results:

$$\begin{array}{cc}\hfill a_0=& -2k\psi \wp {log}^2\left(\rho \right),a_1=-2k\vartheta \wp {log}^2\left(\rho \right),a_2=-2k{\wp }^2{log}^2\left(\rho \right),\hfill \\ \hfill & c=-k\left({\delta }_1+{\delta }_2+k^2{\vartheta }^2{log}^2\left(\rho \right)-4k^2\psi \wp {log}^2\left(\rho \right)\right),\hfill \end{array}$$

(84)

and

$$\begin{array}{cc}\hfill a_0=& -\frac{1}{3}k{log}^2\left(\rho \right)\left(2\wp \zeta +{\nu }^2\right),a_1=-2\zeta k\nu {log}^2\left(\rho \right),a_2=-2{\zeta }^{2}k{log}^2\left(\rho \right),\hfill \\ \hfill & c=-4\wp \zeta k^3{log}^2\left(\rho \right)+k^3{\nu }^2{log}^2\left(\rho \right)-{\delta }_{1}k-{\delta }_{2}k.\hfill \end{array}$$

(85)

The general solution of Equation (52) was calculated by plugging Equation (84) into Equation (82):

$$W(\Omega )=-2k\wp {log}^2\left(\rho \right)\left(\alpha +\vartheta P(\Omega )+\wp P{(\Omega )}^2\right).$$

(86)

It should be noted that we could obtain many solutions for $W(\Omega )$ by using different $P_r$ from Equations (7)–(43), given that $\pounds (\Omega )=\int W(\Omega )d\Omega$. Thus, by integrating each solution, we could determine the solutions for $\pounds (\Omega )$.

(1)

For ${\vartheta }^2$ − 4$\psi$℘< 0, ℘≠ 0, the mixed trigonometric solutions were determined as follows:

$${\pounds }_1(x,y,t)=-k{log}^2\left(\rho \right)\sqrt{-\omega }tan\left(\frac{1}{2}\Omega \sqrt{-\omega }\right),$$

(87)

$${\pounds }_2(x,y,t)=k{log}^2\left(\rho \right)\sqrt{-\omega }cot\left(\frac{1}{2}\Omega \sqrt{-\omega }\right),$$

(88)

$${\pounds }_3(x,y,t)=-\frac{1}{2}k{log}^2\left(\rho \right)\sqrt{-\omega }\left((mn+1)tan\left(\Omega \sqrt{-\omega }\right)\pm 2\sqrt{mn}sec\left(\Omega \sqrt{-\omega }\right)\right),$$

(89)

$${\pounds }_{4a}(x,y,t)=\frac{1}{2}k{log}^2\left(\rho \right)\sqrt{-\omega }csc\left(\Omega \sqrt{-\omega }\right)\left((mn+1)cos\left(\Omega \sqrt{-}\right)+2\sqrt{mn}\right),$$

(90)

$${\pounds }_{4b}(x,y,t)=\frac{1}{2}k{log}^2\left(\rho \right)\sqrt{-\omega }\left((mn+1)cot\left(\Omega \sqrt{-\omega }\right)-2\sqrt{mn}csc\left(\Omega \sqrt{-\omega }\right)\right),$$

(91)

$${\pounds }_5(x,y,t)=k{log}^2\left(\rho \right)\sqrt{-\omega }cot\left(\frac{1}{2}\Omega \sqrt{-\omega }\right).$$

(92)

(2)

For ${\vartheta }^2$ − 4$\psi$℘> 0, ℘≠ 0, the shock solution was determined as follows:

$${\pounds }_6(x,y,t)=k{log}^2\left(\rho \right)\sqrt{\omega }tanh\left(\frac{1}{2}\Omega \sqrt{\omega }\right).$$

(93)

The singular solution was determined as follows:

$${\pounds }_7(x,y,t)=k{log}^2\left(\rho \right)\sqrt{\omega }coth\left(\frac{1}{2}\Omega \sqrt{\omega }\right).$$

(94)

The mixed complex solitary-shock solution was determined as follows:

$${\pounds }_8(x,y,t)=\frac{1}{2}k{log}^2\left(\rho \right)\sqrt{\omega }\left((mn+1)tanh\left(\Omega \sqrt{\omega }\right)\mp 2i\sqrt{mn}\sec \mathrm{h}\left(\Omega \sqrt{\omega }\right)\right).$$

(95)

The mixed singular solutions were determined as follows:

$${\pounds }_{9a}(x,y,t)=\frac{1}{2}k{log}^2\left(\rho \right)\sqrt{\omega }\left((mn+1)coth\left(\Omega \sqrt{\omega }\right)-2\sqrt{mn}\csc \mathrm{h}\left(\Omega \sqrt{\omega }\right)\right),$$

(96)

$${\pounds }_{9b}(x,y,t)=\frac{1}{2}k{log}^2\left(\rho \right)\sqrt{\omega }\csc \mathrm{h}\left(\Omega \sqrt{\omega }\right)\left((mn+1)cosh\left(\Omega \sqrt{\omega }\right)+2\sqrt{mn}\right).$$

(97)

The mixed shock singular solution was determined as follows:

$${\pounds }_{10}(x,y,t)=k{log}^2\left(\rho \right)\sqrt{\omega }\left(\Omega \sqrt{\omega }-coth\left(\frac{1}{2}\Omega \sqrt{\omega }\right)\right).$$

(98)

(3)

For $\psi \wp >0$ and $\vartheta =0$, the trigonometric solutions were determined as follows:

$${\pounds }_{11}(x,y,t)=-2k\sqrt{\psi \wp }{log}^2\left(\rho \right)tan\left(\Omega \sqrt{\psi \wp }\right),$$

(99)

$${\pounds }_{12}(x,y,t)=2k\sqrt{\psi \wp }{log}^2\left(\rho \right)cot\left(\Omega \sqrt{\psi \wp }\right).$$

(100)

The mixed trigonometric solutions were determined as follows:

$$\begin{array}{cc}\hfill {\pounds }_{13}(x,y,t)=& -\frac{k\wp {log}^2\left(\rho \right)}{2\sqrt{\psi \wp }}(-(\vartheta \sqrt{\frac{\psi }{\wp }}log(1-sin(2\Omega \sqrt{\psi \wp }\left)\right)\hfill \\ \hfill & \mp \vartheta \sqrt{mn}\sqrt{\frac{\psi }{\wp }}log(1-sin(2\Omega \sqrt{\psi \wp }\left)\right))-((\vartheta \sqrt{\frac{\psi }{\wp }}log(sin(2\Omega \sqrt{\psi \wp })+1)\hfill \\ \hfill & \pm \vartheta \sqrt{mn}\sqrt{\frac{\psi }{\wp }}log\left(sin(2\Omega \sqrt{\psi \wp })+1\right))\pm 4\psi \sqrt{mn}sec(2\Omega \sqrt{\psi \wp })))+\hfill \\ \hfill & 2mn\psi tan(2\Omega \sqrt{\psi \wp })+4\psi \Omega \sqrt{\psi \wp }-2\psi {tan}^{-1}\left(tan(2\Omega \sqrt{\psi \wp })\right)\hfill \\ \hfill & +2\psi tan(2\Omega \sqrt{\psi \wp })\left)\right),\hfill \end{array}$$

(101)

$$\begin{array}{cc}\hfill {\pounds }_{14}(x,y,t)=& \frac{(k\wp {log}^2\left(\rho \right)csc(\Omega \sqrt{\psi \wp }))}{4\sqrt{\psi \wp }}\left(\right(sec(\Omega \sqrt{\psi \wp })\left(\right(2\psi (mn+1)cos(2\sqrt{\psi \wp }\Omega )\hfill \\ \hfill & \pm 2\vartheta \sqrt{mn}\sqrt{\frac{\psi }{\wp }}sin(2\Omega \sqrt{\psi \wp }){tanh}^{-1}\left(cos(2\Omega \sqrt{\psi \wp })\right))\hfill \\ \hfill & \mp 4\psi \sqrt{mn}+\vartheta \sqrt{\frac{\psi }{\wp }}sin(2\Omega \sqrt{\psi \wp })log({sin}^2(2\Omega \sqrt{\psi \wp })\left)\right)\left)\right),\hfill \end{array}$$

(102)

$${\pounds }_{15}(x,t)=2k\sqrt{\psi \wp }{log}^2\left(\rho \right)cot\left(\Omega \sqrt{\psi \wp }\right).$$

(103)

(4)

For $\psi \wp <0$ and $\vartheta =0$, the shock-wave solution was determined as follows:

$${\pounds }_{16}(x,y,t)=2k{log}^2\left(\rho \right)\sqrt{-\psi \wp }tanh\left(\Omega \sqrt{-\psi \wp }\right).$$

(104)

The singular solution was determined as follows:

$${\pounds }_{17}(x,y,t)=2k{log}^2\left(\rho \right)\sqrt{-\psi \wp }coth\left(\Omega \sqrt{-\psi \wp }\right).$$

(105)

Some distinct complex combo-type solutions were determined as follows:

$${\pounds }_{18}(x,y,t)=k{log}^2\left(\rho \right)\sqrt{-\psi \wp }\left((mn+1)tanh\left(2\Omega \sqrt{-\psi \wp }\right)\mp 2i\sqrt{mn}\sec \mathrm{h}\left(2\Omega \sqrt{-\psi \wp }\right)\right),$$

(106)

$$\begin{array}{cc}\hfill {\pounds }_{19}(x,y,t)=& \frac{1}{2}k{log}^2\left(\rho \right)\sqrt{-\psi \wp }\csc \mathrm{h}\left(\Omega \sqrt{-\psi \wp }\right)\hfill \\ & \sec \mathrm{h}\left(\Omega \sqrt{-\psi \wp }\right)\left((mn+1)cosh\left(2\Omega \sqrt{-\psi \wp }\right)\mp 2\sqrt{mn}\right),\hfill \end{array}$$

(107)

$${\pounds }_{20}(x,y,t)=2k{log}^2\left(\rho \right)\sqrt{-\psi \wp }coth\left(\Omega \sqrt{-\psi \wp }\right).$$

(108)

(5)

For $\vartheta =0$ and $\wp =\psi$, the periodic and mixed-periodic wave solutions were determined as follows:

$${\pounds }_{21}(x,y,t)=-2k\psi {log}^2\left(\rho \right)tan(\psi \Omega ),$$

(109)

$${\pounds }_{22}(x,y,t)=2k\psi {log}^2\left(\rho \right)cot(\psi \Omega ),$$

(110)

$$\begin{array}{cc}\hfill {\pounds }_{23}(x,y,t)=& -\frac{k\psi {log}^2\left(\rho \right)}{{\left(\sqrt{mn}\pm sin(2\psi \Omega )\right)}^2}\left(\right(\left(m^2{n}^{2}tan(2\psi \Omega )\pm 4\sqrt{mn}sin(2\psi \Omega )tan(2\psi \Omega )\right)\hfill \\ \hfill & \pm 2{\left(mn\right)}^{3/2}sin(2\psi \Omega )tan(2\psi \Omega ))+5mntan(2\psi \Omega )+mn{sin}^2(2\psi \Omega )tan(2\psi \Omega )\hfill \\ \hfill & -2\psi \Omega \left(\left(mn\pm 2\sqrt{mn}sin(2\psi \Omega )\right)+{sin}^2(2\psi \Omega )\right)+\hfill \\ \hfill & \left(\left(2mn\psi \Omega \pm 2{\left(mn\right)}^{3/2}sec(2\psi \Omega )\right)\pm 4\psi \Omega \sqrt{mn}sin(2\psi \Omega )\right)+\hfill \\ \hfill & 2\psi \Omega {sin}^2(2\psi \Omega )+{sin}^2(2\psi \Omega )tan(2\psi \Omega )),\hfill \end{array}$$

(111)

$$\begin{array}{cc}\hfill {\pounds }_{24a}(x,y,t)=& \frac{k\psi {log}^2\left(\rho \right)csc(\psi \Omega )sec(\psi \Omega )}{8{\left(\sqrt{mn}-cos(2\psi \Omega )\right)}^2}\left(\right(\left(4m^2{n}^2+23mn+3\right)cos(2\psi \Omega )-\hfill \\ \hfill & 4\sqrt{mn}(mn+2)cos(4\psi \Omega )+mncos(6\psi \Omega )-12{\left(mn\right)}^{3/2}-8\sqrt{mn}+cos(6\psi \Omega )\left)\right),\hfill \end{array}$$

(112)

$${\pounds }_{24b}(x,y,t)=\frac{1}{2}k\psi {log}^2\left(\rho \right)csc(\psi \Omega )sec(\psi \Omega )\left((mn+1)cos(2\psi \Omega )+2\sqrt{mn}\right),$$

(113)

$${\pounds }_{25}(x,y,t)=2k\psi {log}^2\left(\rho \right)cot(\psi \Omega ).$$

(114)

(6)

For $\vartheta =0$ and $\wp =-\psi$, some mixed-periodic and single wave solutions were determined as follows:

$${\pounds }_{26}(x,t)=2k\psi {log}^2\left(\rho \right)tanh(\psi \Omega ),$$

(115)

$${\pounds }_{27}(x,t)=2k\psi {log}^2\left(\rho \right)coth(\psi \Omega ),$$

(116)

$$\begin{array}{cc}\hfill {\pounds }_{28a}(x,y,t)=& -\frac{k\psi {log}^2\left(\rho \right)}{{\left(\sqrt{mn}-sinh(2\psi \Omega )\right)}^2}(m^2{n}^{2}tanh(2\psi \Omega )-2mn\psi \Omega +\hfill \\ \hfill & 4\psi \Omega \sqrt{mn}sinh(2\psi \Omega )-5mntanh(2\psi \Omega )+2{\left(mn\right)}^{3/2}\sec \mathrm{h}(2\psi \Omega )+\hfill \\ \hfill & 2\psi \Omega \left(-2\sqrt{mn}sinh(2\psi \Omega )+mn+{sinh}^2(2\psi \Omega )\right)+mn{sinh}^2(2\psi \Omega )\hfill \\ \hfill & tanh(2\psi \Omega )-2{\left(mn\right)}^{3/2}sinh(2\psi \Omega )tanh(2\psi \Omega )\hfill \\ \hfill & +4\sqrt{mn}sinh(2\psi \Omega )tanh(2\psi \Omega )-2\psi \Omega {sinh}^2(2\psi \Omega )-\hfill \\ \hfill & {sinh}^2(2\psi \Omega )tanh(2\psi \Omega )),\hfill \end{array}$$

(117)

$${\pounds }_{28b}(x,y,t)=k\psi {log}^2\left(\rho \right)\sec \mathrm{h}(2\psi \Omega )\left((1-mn)sinh(2\psi \Omega )+2\sqrt{mn}\right),$$

(118)

$${\pounds }_{29}(x,y,t)=\frac{1}{2}k\psi {log}^2\left(\rho \right)\csc \mathrm{h}(\psi \Omega )\sec \mathrm{h}(\psi \Omega )\left((mn+1)cosh(2\psi \Omega )\mp 2\sqrt{mn}\right),$$

(119)

$${\pounds }_{30}(x,y,t)=2k\psi {log}^2\left(\rho \right)coth(\psi \Omega ).$$

(120)

(7)

For ${\vartheta }^2=4\psi \wp$, we deduced only one solution as follows:

$${\pounds }_{31}(x,t)=\frac{2k}{\wp }.$$

(121)

(8)

For $\vartheta =p$, $\psi =pq$, and $\wp =0$,

$${\pounds }_{32}(x,t)=0.$$

(122)

(9)

For $\vartheta =\wp =0$,

$${\pounds }_{33}(x,t)=0.$$

(123)

(10)

For $\vartheta =\psi =0$, we deduced a single solution as follows:

$${\pounds }_{34}(x,t)=\frac{2k}{\wp }.$$

(124)

(11)

For $\psi =0$ and $\vartheta$≠ 0, some mixed hyperbolic solutions were determined as follows:

$${\pounds }_{35}(x,t)=\frac{2km\vartheta {log}^2\left(\rho \right)}{m+cosh(\vartheta \Omega )-sinh(\vartheta \Omega )},$$

(125)

$${\pounds }_{36}(x,t)=-\frac{2kn\vartheta {log}^2\left(\rho \right)}{n+cosh(\vartheta \Omega )+sinh(\vartheta \Omega )}.$$

(126)

(12)

For $\vartheta =p$, $\wp =pq$, where q≠ 0 and $\wp =0$, the single solution of the plane form was determined as follows:

$${\pounds }_{37}(x,t)=-\frac{2kmplog\left(\rho \right)}{n^2}\left(\frac{m^2}{m-nq{\rho }^{\Omega p}}+(m+n)log\left(m-nq{\rho }^{p\Omega }\right)\right).$$

(127)

4. Graphical Discussion

This section is devoted to display and illustrating the graphic discussion and physical aspects of the obtained results.

Figure 1 displays the propagating behavior of solution ${\pounds }_{3,2}$ with the parametric values $\alpha =0.1,\gamma =0.5,\beta =0.1,{\delta }_1=0.2,$ and ${\delta }_2=5$. This solution predicts the antikink periodic behavior of a traveling soliton with an amplitude-increasing fashion and periodicity as the wave number increases.

Figure 2 displays the propagating behavior of solution ${\pounds }_{4,3}$ with the parametric values $\alpha =0.1,\gamma =0.4,\beta =0.6,{\delta }_1=0.5,$ and ${\delta }_2=0.2.$ This solution predicts the kink periodic behavior of a traveling soliton with an amplitude-increasing fashion as the wave number increases.

Figure 3 displays the propagating behavior of the real part of solution ${\pounds }_3$ with the parametric values $\psi =0.5,\rho =5,m=5,n=2;\wp =4,\vartheta =2,{\delta }_1=0.5,$ and ${\delta }_2=0.5$. This solution predicts the periodic with antipeaked crests and antitroughs behavior of a traveling soliton with an amplitude-increasing fashion and periodicity as wave number increases.

Figure 4 displays the propagating behavior of the imaginary part of solution ${\pounds }_3$ with the parametric values $\psi =0.5,\rho =5,m=5,n=2,\wp =4,\vartheta =2,{\delta }_1=0.5,$ and ${\delta }_2=0.5.$ This solution predicts the periodic with peaked crests and troughs behavior of a traveling soliton with an amplitude-increasing fashion and periodicity as the wave number increases.

Figure 5 displays the propagating behavior of the real part of solution ${\pounds }_{18}$ with the parametric values $\psi =-0.02,\wp =0.3,\rho =5,m=5,n=6,{\delta }_1=0.5,$ and ${\delta }_2=0.5.$ This solution predicts the bright compacton with an amplitude-increasing fashion and the compacton achieves singularity as the wave number increases.

Figure 6 displays the propagating behavior of the imaginary part of solution ${\pounds }_{18}$ with the parametric values $\psi =-0.02,\wp =0.3,\rho =5,m=5,n=6,{\delta }_1=0.5,$ and ${\delta }_2=0.5.$ This solution predicts the dark compacton with an amplitude-increasing fashion and the compacton achieves singularity as the wave number increases.

5. Results and Novelty

This section presents the comparison and novelty of this study.

In the recent literature [45], authors developed the trigonometric, singular, periodic, hyperbolic, and kink solitons while in this study, singular solutions, mixed complex solitary shock solutions, mixed singular solutions, mixed shock singular solutions, mixed trigonometric solutions, mixed periodic solutions, mixed hyperbolic, antikink periodic, kink periodic, periodic with antipeaked crests and antitroughs, periodic with peaked crests and troughs, bright compacton, and dark compacton were found which were more generalized solutions. To our best knowledge, these types of soliton had not been studied before this study.

The result in this study was

$$\begin{array}{cc}\hfill {\pounds }_{28a}(x,y,t)=& -\frac{k\psi {log}^2\left(\rho \right)}{{\left(\sqrt{mn}-sinh(2\psi \Omega )\right)}^2}(m^2{n}^{2}tanh(2\psi \Omega )-2mn\psi \Omega +\hfill \\ \hfill & 4\psi \Omega \sqrt{mn}sinh(2\psi \Omega )-5mntanh(2\psi \Omega )+2{\left(mn\right)}^{3/2}\sec \mathrm{h}(2\psi \Omega )+\hfill \\ \hfill & 2\psi \Omega \left(-2\sqrt{mn}sinh(2\psi \Omega )+mn+{sinh}^2(2\psi \Omega )\right)+mn{sinh}^2(2\psi \Omega )\hfill \\ \hfill & tanh(2\psi \Omega )-2{\left(mn\right)}^{3/2}sinh(2\psi \Omega )tanh(2\psi \Omega )\hfill \\ \hfill & +4\sqrt{mn}sinh(2\psi \Omega )tanh(2\psi \Omega )-2\psi \Omega {sinh}^2(2\psi \Omega )-\hfill \\ \hfill & {sinh}^2(2\psi \Omega )tanh(2\psi \Omega )),\hfill \end{array}$$

(128)

while the result from the literature (see Equation (29) in [45]) was

$$g^{f\left(\rho \right)}=-\frac{\mu }{\gamma }+\frac{\sqrt{({\mu }^2-\nu \gamma )}}{\gamma }tanh\left(\frac{\sqrt{({\mu }^2-\nu \gamma )}}{2}\right)(k(x+y)+ct).$$

(129)

In the above solutions, it is clear that this study presented the more generalized and novel solutions. One can compare more results adopting the same process.

6. Conclusions

The generalized Calogero–Bogoyavlenskii–Schiff equation was investigated and analyzed by analytical techniques to obtain and visualize some deep insights. A combination of the new extended direct algebraic method and modified auxiliary equation method was applied and thus:

• Numerous types of solitons were obtained which covered almost all kinds of solitary waves, such as singular solutions, mixed complex solitary shock solutions, mixed singular solutions, mixed shock singular solutions, mixed trigonometric solutions, mixed periodic solutions, and mixed hyperbolic solutions.
• The real and imaginary wave propagation of the complex solutions was graphically displayed. The antikink periodic, kink periodic, periodic with antipeaked crests and antitroughs, periodic with peaked crests and troughs, bright compacton, and dark compacton behavior were graphically visualized.
• Two-dimensional, 3D, and contour visualization were presented and we observed the influence of the parameters on the traveling behavior of the obtained solutions.
• The wave number of the traveling wave profile was responsible for the control of the amplitude and the traveling behavior of the solitary wave. The singularity of the soliton wave could be controlled by the wave number parameter.

It is hoped that this study will be useful to researchers and analysts for improving experimental work; it can be extended to include multiple solitons, lump interaction, and rogue wave breathers.