Solitary Wave Solutions to the Modified Zakharov–Kuznetsov and the (2 + 1)-Dimensional Calogero–Bogoyavlenskii–Schiff Models in Mathematical Physics

. Te modifed Zakharov–Kuznetsov (mZK) and the (2+1)-dimensional Calogero–Bogoyavlenskii–Schif (CBS) models convey a signifcant role to instruct the internal structure of tangible composite phenomena in the domain of two-dimensional discrete electrical lattice, plasma physics, wave behaviors of deep oceans, nonlinear optics, etc. In this article, the dynamic, companionable, and further broad-spectrum exact solitary solitons are extracted to the formerly stated nonlinear models by the aid of the recently enhanced auxiliary equation method through the traveling wave transformation. Te implication of the soliton solutions attained with arbitrary constants can be substantial to interpret the involuted phenomena. Te established soliton solutions show that the approach is broad-spectrum, efcient, and algebraic computing friendly and it may be used to classify a variety of wave shapes. We analyze the achieved solitons by sketching fgures for distinct values of the associated parameters by the aid of the Wolfram Mathematica program.


Introduction
Nonlinear evolution equations (NLEEs) are the fundamental tools to model most of the phenomena that arise in engineering, natural sciences, and mathematical physics, such as propagation of shallow water waves, optical fbers, signal processing, condensed matter, electromagnetic, plasma physics, and fuid mechanics. Te solitary wave solutions of NLEEs are used to analyze many natural events. Tis is owing to the fact that NLEEs contain indefnite multivariable functions and their derivatives. Terefore, many researchers have shown their interest in studying and researching on this topic. Te traveling waves are the waves which propagate with respect to time and are gained much attention to the recent researchers. Terefore, a large assortment of mathematical methods which are efcient, powerful, and reliable were suggested for fndings the exact traveling wave solutions of the nonlinear diferential equations (NLDEs) as for instance, the method of exp-function [1], the (G ′ /G)-expansion method [2][3][4][5], the fnite diference approach [6], the extended tanhfunction approach [7], the Jacobi elliptic function method [8], the Bernoulli sub-equation function [9], the MSE method [10][11][12], the frst integral scheme [13], the variational iteration scheme [14], the modifed Kudryashov scheme [15], the Hirota's bilinear form [16], the Shehu transform scheme [17], the modifed trial equation approach [18], the fractional subequation approach [19], the Laplace transform scheme [20], the sine-Gordon equation (SGE) [21,22], the modifed extended direct algebraic technique [23], the auxiliary equation method [24][25][26][27], etc.
Te weakly nonlinear ion-acoustic waves in strongly magnetized lossless plasma in two-dimension are described by the Zakharov-Kuznetsov (ZK) equation, was frst introduced by Zakharov and Kuznetsov in 1974. But, when more realistic situations arise, the nonisothermal electrons are governed by the ZK equation, the equation is changed into a modifed form, known as the modifed Zakharov-Kuznetsov (mZK) equation [28][29][30]. Te mZK equation was frst introduced by Munro and Parkes in 1999 [28]. Te mZK equation is also important in the feld of 2D discrete electrical lattice, plasma physics, etc. Bogoyavlenskii and Schif constructed the Calogero-Bogoyavlenskii-Schif (CBS) equation, where Bogoyavlenskii used the modifed Lax formalism [31]. And, Schif derived the CBS equation by reducing the self-dual Yang-Mills equation [32]. Te (2 + 1)dimensional CBS equation [33,34] is useful in research on wave behavior in deep sea, nonlinear optics, and other felds. In addition, the researchers showed that the CBS equation possesses soliton as well as N-soliton solutions which are smooth in one coordinate. In the literature, the modifed mZK equation was investigated through the frst integral approach [35], the extended tanh-function method [36], the Lie symmetry scheme [37], the exp-function technique [38], the unifed method [39], and some other techniques. Te modifed simple equation approach [40], the Hirota's bilinear approach [41], the improved (G′/G)-expansion scheme [42], the exp(− ϕ(ζ))-expansion method [43], the bilinear method [44], and some other techniques were used to examine the CBS equation.
Te auxiliary equation (AE) method is a relatively new technique. It is observed that the AE method is readily applicable to a large variety of NLEEs as well as coupled NLEEs. Moreover, it not only generates regular solutions but also singular ones involving csch and coth functions. Due to this advantage the AE method, it gains considerable interest to the researchers. To our optimum comprehension, the exact solutions of the modifed ZK and CBS equations have yet not been developed using the auxiliary equation approach [24][25][26][27]. Terefore, the aim of this study is to examine the soliton solutions which means the solutions of a widespread class of weakly NLDEs describing a physical system of the modifed Zakharov-Kuznetsov (mZK) equation and the (2 + 1)-dimensional Calogero-Bogoyavlenskii-Schif (CBS) equation via the auxiliary equation method. With the appropriateness and simplicity of this method, we achieve several realistic and further generic solutions to the equations. Inserting specifc values of arbitrary factors various wave solitons are created and these attained solitons are not established in the previous literature. We portray the diagram of attain solutions and illustrate their physical signifcation.
Te organization of this work is as follows: in Section 2, the auxiliary equation method is described. We study the stated nonlinear evolution equations and examine solutions of the equations in Section 3. Explain the physical importance of the obtained solutions in Section 4 and compare the results in Section 5. Finally, the conclusion is presented.

Description of the Method
Te auxiliary equation (AE) method, which is promising, powerful, and profcient, has been briefy described in this section.
Consider a general higher dimensional NLEE: where u � u(t, x, y, z) be the unknown wave function andH is the polynomial of u(t, x, y, z) and its diverse partial derivatives, where the utmost order derivatives and nonlinear terms are concerned. By executing the consequent steps we can evaluate the solution of (1) by the aid of the AE method: Step 1: Suppose the wave variable in the form: where ω denotes wave propagation velocity. Te transformation (2) assists us to transform (1) into the following equation: where L is a polynomial in V(ϕ) and the derivatives for Step 2: Here, Equation (3) could be integrated one or more times as per possibility.
Step 3: As per AE method the subsequent solution of Equation (3) can be revealed as follows: Here, the constants c i , a have to evaluated, where c N ≠ 0 (N is a positive integer) and f(ϕ) satisfes the subsequent auxiliary equation: Te prime species the derivative with regard to ϕ; l, m, n are parameters.
Here, the transformation ψ � a f(ξ) converts the auxiliary equation Equation (5) to a Riccati equation in ψ as follows: Te Riccati equation (6) gives a set of generic solutions that provide the exact solutions to the nonlinear equation (3). It is worth noting that series extension of (4) is a particular type of rational function and also a polynomial. Terefore, an interrelation was found between the considered approach and the transformed rational function technique. Te AE method is used in the present study since it is straightforward, easy to compute, and adaptable to user-friendly computation software like Maple to examine closed-form soliton solutions.
Step 4: Te value of N seem in (4) can be acquired by considering the homogeneous balance between the utmost order exponent and the derivative occurring in (3).
Step 5: Inserting Equation (4) jointly with (5)  Te AE method is used to examine the exact soliton solutions of NLEEs. Te key idea of this method is to take full advantage of NLEEs which yields useful, fresher, and further general exact wave solutions. Principally, the AE method is a direct algebraic method, efective algorithm, further generalized to establish several exact traveling wave solutions for NLEEs and can be utilized to arrange the wave velocity. Furthermore, by this method, some physically important of NLEEs are investigated with the aid of symbolic computation.

Mathematical Analysis of the Wave Solutions
In this part, we establish scores of advanced, standard, and wide-spectral closed-form traveling wave solutions to the modifed Zakharov-Kuznetsov (mZK) equation and the (2 + 1)-dimensional Calogero-Bogoyavlenskii-Schif (CBS) equation by executing the thriving AE method.

Te mZK Model.
Herein, we examine a variety of newer closed-form soliton solutions to the mZK model by using the AE method. Tis is a noteworthy model which has vast applications in engineering and mathematical physics, namely, the efects of ionic temperature, ion density gradient, presence of third species (dust), composite phenomena in the domain of two-dimensional discrete electrical lattice, oblique propagation, and others. Te mZK model is an integrable model which is in the form [28][29][30]: Using the compound transformation u(ψ) � U(t, x, y), where ψ � x + y − ωt, (7) turns into a nonlinear equation and integrating, it becomes the subsequent form: By means of homogeneous balancing of the uppermost order nonlinear and linear terms appearing in (8), we fnd N � 1. Terefore, the solution structure of (8) is as follows: Inserting (9) jointly with (5) and (8) and gathering all the terms of similar power of a if(ψ) and equating the coefcients to zero yields a class of (algebraic) equations (these are not shown for simplicity) and addressing these via the computation software Maple, gives the solutions: where ω is traveling wave velocity and l, m, n are constants.
Substituting the values assembled in (10) into (9) and applying the conditions of the AE method, we establish the subsequent solutions of (7): While m 2 − 4 ln < 0 and n ≠ 0, the resulting solutions are as follows: When m 2 − 4 ln > 0 and n ≠ 0, the obtain the solutions are For m 2 + 4 l 2 < 0, n ≠ 0 and n � − l, gives the solutions. When While m 2 − 4l 2 < 0 and n � l, we achieve the solution.

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By means of m 2 − 4l 2 > 0 and n � l, leads the solutions.
While m 2 � 4 ln, the wave velocity becomes zero, therefore, the soliton solution does not exist in this case.
When l � n � 0, the soliton reach into trivial form and it is physical signifcance less. So the solution dropped here.
While l � m � f and n � 0, we achieve a trivial soliton solution and that is negligible here.
For m � n � f and l � 0, the solutions turns into, When m � l + n, we get a trivial solution, which is not represented here.
For m � − (l + n), we attain the trivial solution, therefore it is omitted here because it has no physical signifcance.
When l � 0, we obtain the exponential function solution which is signifcant.
For n � l � 0 and l � m � 0, the obtained solution is in trivial form and therefore omitted here.
While n � l and m � 0, we determine.
When n � 0, we achieve a trivial form soliton solution, which is also skipped here.
In all solutions, ψ � x + y − ωt, where ω is the traveling wave velocity and l, m, n are free parameters.
It is remarkable to observe that the established wave solutions of the considered models which are further general and for the individual values of the arbitrary parameters some exact solutions are reinstated and some other solutions are ascertained which are not found in the earlier study.

Te CBS Model.
In this subsection, we determine novel, useful, and further general closed-form solitary wave solutions to the (2 + 1)-dimensional CBS model by the aid of the auxiliary equation method. Te model has a vital role in mathematical physics and engineering such as, explore several nonlinear dynamics of interaction phenomena in fuids and plasmas felds, nonlinear wave in optics, wave behaviors of deep oceans, and more. Te CBS model [33,34] which is an integrable model is as follows: Te traveling wave variable u(ξ) � V(x, y, t), where ξ � x + y − ωt converts the model (30) into a nonlinear equation and after integration with zero integral constant it turns into the following equation: Now, using the balance number N � 1, the form of the solution of (30) is written as follows: Setting the solution (32), and (5) into (31) and summing up the coefcients of identical powers of a if(ξ) and assigning them to zero, we fnd out a set of algebraic equations (for simplicity which are not assembled here) for c 0 , c 1 , l, m, n.
Unraveling the system of equations (algebraic) by applying computation software (Maple), it provides the following solutions: where ω is wave velocity and l, m, n are free parameters. Now, applying the results (33) and (32) and by means of the constraints on the free parameters, we establish the following solutions of (30): While m 2 − 4 ln < 0 and n ≠ 0, we attain the singular periodic solutions: When m 2 − 4 ln > 0 and n ≠ 0, we attain kink and singular kink shape solutions. 4 Journal of Mathematics For m 2 + 4 l 2 < 0, n ≠ 0 and n � − l, we reach the soliton solutions.

v(ξ)
While m 2 � 4 ln, the wave velocity becomes zero, so the soliton solution does not exist for this case.
When ln < 0, m � 0 and n ≠ 0, we attain the standard kink and singular kink soliton.

v(ξ)
When l � n � 0, or l � m � f and n � 0, introducing a trivial solution thus has no physical signifcance.
For m � n � f and l � 0, we accomplish the following solution: When m � l + n, and/or m � − (l + n), we have the afterwards trivial soliton solution and that is negligible.
While l � 0, the solution turns into, For n � m � l ≠ 0, we accomplish the solution. v For n � m � 0 , the resulting soliton is a trivial solution and has no physically signifcance. So the result dropped here.
When l � m � 0, we get the trivial solution, which is not essential to show here.
When n � 0, the established soliton is not present here because the resulting solution is physically signifcance less.
For all solutions, ξ � x + y − ωt, where ω is traveling wave velocity and l, m, n are indefnite constants.
Te above-established solutions of the considered CBS model are further general and the particular values for the associated constants some exact solutions are available in the literature which are abundant novel and not found in the previous research.

The Graphical Representations and Physical Description
In this part, we will demonstrate the 3-dimensional and contour structure for the obtained solitons of the models using the Wolfram Mathematica program to visualize the internal mechanism of them. For simplicity, some fgures of the gained solitons are depicted and some are skipped here.

Te Graphical Representations.
Here, we have demonstrated the diverse nature of attained solitons graphically of the considered nonlinear evolution equations for dissimilar values of the integrated parameters inside appropriate interval but the values of traveling wave velocity ω varies. Te infuence of wave velocity is exposed in the following shapes. Te profles of the wave solution (11) for diverse traveling wave velocity ω are shown in Figures 1-3.

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Te above-given fgures of (11) represent singular bell shape soliton and this shape is depicted for the value of the traveling wave velocity ω � 5.0 in the interval 0 ≤ x, t ≤ 1 and the contour graph is depicted at t � 0 for the same velocity and shown in Figure 1. But for the velocity decreasing, i.e., ω � 3.0, the solution (11) displays the singular periodic soliton in the interval − 2 ≤ x, t ≤ 2. Te contour graph of this solution for the velocity ω � 3.0 is depicted at t � 0 and portrayed in Figure 2. Again when the traveling wave velocity is gradually tends to zero i.e., ω � 0.8, the compacton type soliton of (11) is demonstrated inside the range − 2 ≤ x, t ≤ 2. Te contour graph of this solution for the velocity ω � 0.8 is depicted at t � 0 and sketched in Figure 3.
Te profles of the wave solution (13) for diferent traveling wave velocities ω are depicted in  It is observed from the above-given analysis, the background of the soliton (13) is a spike for the traveling wave velocity ω � 44.53 portrayed surrounded by interval − 1 ≤ x, t ≤ 1. Te contour graph of this soliton for the velocity ω � 44.53 is depicted at t � 0 and displayed in Figure 4. But, when the wave velocity is decreased i.e., ω � 3.43, the solution function (13) represents singular bell shape soliton within − 2 ≤ x, t ≤ 2 and for this velocity the contour shape of the solution is depicted at t � 0 and presented in Figure 5. When the wave velocity is further decreased i.e., ω � 2.25, the solution function (13) represents bell shape soliton inside range − 2 ≤ x, t ≤ 2 and also the contour graph is depicted at t � 0 (for same velocity) in Figure 6. Again, if the wave velocity gradually tends to zero i.e., ω � 0.25, the solution (13) represents the singular Journal of Mathematics periodic soliton in the period − 10 ≤ x, t ≤ 10 and the contour design of this solution for the same velocity is drawn at t � 0 and shown in Figure 7. Te profles of the traveling wave solution (34) for diverse wave velocity ω are given as follows.
Te soliton (34) represents the singular kink soliton for the wave velocity ω � 3.2 portrayed within the interval − 1 ≤ x, t ≤ 1. Te graph of the contour of the soliton for the wave velocity ω � 3.2 is depicted at t � 0 and reported in Figure 8. When the traveling wave velocity is ω � 3, the solution (34) represents a singular periodic surrounded by − 2 ≤ x, t ≤ 2 and its contour graph of the soliton is designated in Figure 9 for the same velocity ω � 3. Also, when the wave velocity is ω � 1, the solution (34) also represents singular periodic in − 8 ≤ x, t ≤ 8 and in this case contour shape is depicted in Figure 10 at t � 0.
Te profles of the wave solution (36) for diferent traveling wave velocity ω are given as follows.
Te solution (36) represents kink type soliton for the traveling wave velocity ω � 12 shown inside the range − 2 ≤ x, t ≤ 2. Te shape (contour) of the soliton is depicted at t � 0 for the velocity ω � 12 and documented in Figure 11. But, for decreasing wave velocity, i.e., ω � 2.25, the solution (36) represents the fat kink shape soliton in the space − 2 ≤ x, t ≤ 2. Te graph of the solution (contour) for the wave velocity ω � 2.25 is depicted at t � 0 and displayed in Figure 12. Again, when the wave velocity is gradually tends to zero i.e., ω � 0.44, the solution (36) represents general soliton within the space − 2 ≤ x, t ≤ 8. Te contour fgure for the traveling wave velocity ω � 0.44 of solution is represented at t � 0 and indicated in Figure 13.
Te profles of the wave solution (44) for various traveling wave velocity ω are given as follows.
When the traveling wave velocity is ω � 0.84, the solution (44) stand for singular periodic soliton contained by the space − 8 ≤ x, t ≤ 8. Te contour graph of this solution   for the wave velocity ω � 0.84 is depicted at t � 0 and portrayed in Figure 14. If the wave velocity gradually tends to zero i.e., ω � 0.004, the solution (44) represents a concave parabolic wave contained by the range − 8 ≤ x, t ≤ 8. Te graph of this soliton (contour) for the wave velocity ω � 0.004 is depicted at t � 0 and documented in Figure 15. Te profles of the wave solution (47) for various traveling wave velocity ω are given as follows.
Te solution (47) signifes singular soliton for the traveling wave velocity ω � 10 described in the range − 2 ≤ x, t ≤ 2. Te fgure of this solution (contour) for the wave velocity ω � 10 is depicted at t � 0 and displayed in Figure 16. When the wave velocity decreases i.e., ω � 4, the solution (47) represents a singular kink type soliton within the period − 2 ≤ x, t ≤ 2. Te contour design of this soliton for the traveling wave velocity ω � 4 is depicted at t � 0 and specifed in Figure 17. Besides, when the wave velocity gradually decreases (i, e ω � 2), the solution (47) represents a singular anti-bell soliton within − 8 ≤ x, t ≤ 8 and contour graph of this solution for the same velocity is depicted at t � 0 and specifed in Figure 18.

Te Physical Signifcance of the Established Wave
Solutions. Te Presented contour and 3D graphical depiction of the established solutions of contemplated models might be practical to explicate the internal contrivances of the phenomena related with the considered models. For particular values of the traveling wave velocity, we obtain the diverse shape of solitons and outlined in the graphical representation section. Furthermore, all of the attained solutions exit numerous traveling wave solutions which are of key signifcance in elucidating several physical circumstances which are useful in the feld of wave         mechanics such as wave behaviors of deep oceans, in the two-dimensional discrete electrical lattice, plasma physics, nonlinear optics, and more. We mention that all presented solutions insert valuable insights into associated studies on soliton solutions and are supplementary general in wave nature. We also study that, the solutions hold diferent natures of recognized profles of solitary wave solutions as for instance, kink, singular kink, periodic and singular periodic, spike, compacton, bell shaped, and singular bell shaped.
In the above-given graphical representation, the attained solutions of the models disclose their diferent dynamical characteristics for diferent wave patterns in their corresponding media, show their diferent behavior such as the bell shaped wave indicates that the considered models are typically continuous or smooth, asymptotically approach to zero and commonly symmetric. Te compacton wave specifes that the solutions of the models are stable and carry fnite wavelength. And, the kink wave rises or falls from one asymptotic state to another and it approaches a constant at infnity. Furthermore, the periodic wave indicates that the wave family involves stable parts and the singular periodic wave shows that the wave family involves unstable parts.

Comparison of the Attained Solutions
In this part, we compare the exact traveling wave solutions of the modifed Zakharov-Kuznetsov (mZK) model by executing the AE method with those solutions attained by the (G ′ /G)-expansion method. Also, we compare the solutions of the (2 + 1)-dimensional Calogero-Bogoyavlenskii-Schif (CBS) model with those solutions obtained by the modifed tanh-function method. It is important to observe that the obtained solutions are compatible, straightforward, and further general. Te attained solutions might be helpful to examine the physical signifcance for the considered models.

Te Modifed Zakharov-Kuznetsov (mZK) Model.
Bekir [45] investigate the mZK model and established only two solutions (see Appendix A) by executing the (G ′ /G) -expansion method. We identify that some of our obtained solutions are identical to Bekir's solutions and some are diferent. In Table 1, we compare the obtained solutions by the two methods [45].
Te AE method delivered nineteen diferent solutions. From Table 1, we notice that the solutions u 1,2 and u 3,4 obtained by the (G ′ /G)-expansion method and the solutions (23) and (29) obtained by the AE method are identical respectively. Te solution u 5,6 (see Appendix A) obtained by the (G ′ /G)-expansion method is a trivial solution and does not carry any physical signifcance. Terefore, the solution u 5,6 is not considered. In addition, we found some other solutions, namely, (11)- (23), and (24)-(28) which are not found in [45]. Terefore, by comparing the solutions obtained by the two methods, it might be concluded, that we attain further solutions which are general and compatible.

Te Calogero-Bogoyavlenskii-Schif (CBS) Model.
By executing the modifed tanh-function method, Taha et al. [46] obtained only three solutions (see Appendix B) for the (2 + 1)-dimensional CBS model. We identify that some of the attained solutions are identical to the Taha et al. solutions and some are diferent. In Table 2, we compare the solutions examined by the two methods [46].
By applying the AE method, we determine nineteen fresh solutions. From Table 2, we see that the solutions u 1 , u 2 , u 3 obtained by the modifed tanh-function method and the solutions (36), (40), and (44) obtained by the AE method are identical, respectively. In addition, we attain other sixteen solutions (34), (35), (37)-(39), (41)-(43), (45)-(52)which are not found by Taha et al. [46]. Terefore, from the comparison of the solutions obtained by the two methods, it might be concluded that we have attained further general, functional solutions than Taha et al. [46] solutions.