A novel analytical technique for analyzing the (3+1)-dimensional fractional calogero- bogoyavlenskii-schiff equation: investigating solitary/shock waves and many others physical phenomena

This work presents a thorough analysis of soliton wave phenomena in the (3+1)-dimensional Fractional Calogero-Bogoyavlenskii-Schiff equation (FCBSE) with Caputo’s derivatives through the use of a novel analytical technique known as the modified Extended Direct Algebraic Method (mEDAM). By converting nonlinear Fractional Partial Differential equations (FPDE) into integer-order Nonlinear Ordinary Differential equations (NODE), and then using closed-form series solutions to translate the NODE into an algebraic system of equations, this method allows us to derive families of soliton solutions, which include kink waves, lump waves, breather waves, and periodic waves, exposing new insights into the behavior and distinctive features of soliton waves in the FCBSE. By including contour and 3D graphics, the behaviors of a few selected soliton solutions are well depicted, showcasing their amplitude, shape, and propagation characteristics. The results enhance our understanding of the FCBSE and show that the mEDAM is a valuable tool for studying soliton wave phenomena. This work creates new opportunities for studying wave phenomena in more intricately constructed nonlinear FPDEs (NFPDEs).

This study aims to explore soliton solutions for studying traveling wave phenomena in FCBSE using a novel method called mEDAM.mEDAM is a highly efficient technique for dealing with FPDEs.Using appropriate transformation, this method converts FPDE into Nonlinear Ordinary Differential equations (NODE).The transformed NODE is then further translated into a system of algebraic equations by assuming a closed-form series solution.Families of soliton solutions for the original FPDE may be obtained by solving this system, providing essential insights into the behavior and dynamics of the processes the equation attempts to represent.The mEDAM approach is a powerful tool for investigating FPDEs and improving different scientific and engineering fields due to its capacity to produce closed-form and soliton solutions.
While multiple descriptions have been developed for fractional derivatives [34][35][36][37][38][39], this study hires Caputo's derivatives in (1) because of their wide adoption and relevance in modeling real-world phenomena.The benefits of Caputo's derivatives, which are appropriate for accurately representing the dynamics of complicated systems, include compatibility with beginning conditions and an easy-to-understand interpretation within the context of fractional calculus.Its incorporation improves the robustness and efficiency of our proposed method by enabling a thorough comprehension of the FCBSE.This derivative operator is expressed as follows [40]: where N − 1 < ρ N. The following two properties of this operator are used in transformation of (1) into NODE: where τòR and y(η) and x(η) are differentiable.The rest of the article is organized as follows: the methodology of mEDAM is outlined in the next section, section 3. contains soliton solutions for FCBSE, section 4. presents a graphical discussion of some plotted solutions, while the last section concludes our study.

Methodology of mEDAM
The working technique used by EDAM is briefly described in this section.Consider the general FPDE of the form: The following steps are used to solve problem (5): 1. Equation ( 5) is first converted into a NODE with the aid of a variable transformation of the form v(t, z 1 , z 2 , z 3 ,...,z m ) = V(χ), where χ = χ(t, z 1 , z 2 , z 3 ,...,z m ) and χ is capable of being described in a variety of ways.
where V in (6) has derivatives respect to χ.To get the integration's constant, (6) may occasionally be integrated once or more.
2. According to the mEDAM, we suppose the following series form solution for (6): where c j , ( j ä [ − τ, τ]) are unknown constants to be determined later, and R(χ) is the general solution of the following ODE: where b ≠ 0, 1 and A, B and C are unknown constant.
3. The positive integer τ that appears in (7) is called the balance number and is obtained by taking the homogeneous balance between the highest order derivative and the most considerable nonlinear term in equation (6).

4.
After that, we place (7) into (6) or in an equation derived by integrating (6), and then we gather all the terms of R(χ) of the similar order, which yields an expression in R(χ).The system of algebraic equations in c j , ( j ä [ − τ, τ]) and other parameters are then created by equating all the coefficients in the expression to zero using the concept of comparison of coefficients.
5. We use Maple software to solve this system of algebraic equations for some targeted parameters.
6.Then, by computing the unknown coefficients and other parameters and inserting them in (7) together with the R(χ)(general solution of (8), we explore the soliton solutions to (5).The families of soliton solutions shown below may be produced using this generic solution to (8): Family.1: When H < 0 and C ≠ 0, then we arrive at the following family of solitons solutions: Family.2: When H > 0andÃ¢C ≠ 0, then we arrive at the following family of solitons solutions: Family.3: When AC > 0 and B = 0, then we arrive at the following family of solitons solutions: Family.4: When AC > 0 and B = 0, then we arrive at the following family of solitons solutions: Family.5: When C = A and B = 0, then we arrive at the following family of solitons solutions: Family.6: When C = − A and B = 0, then we arrive at the following family of solitons solutions: Family.7: When H = 0 then we obtain the subsequent family of soliton solutions: Family.12: When B = r, A = 0, and C = nr(n ≠ 0), then we arrive at the following family of solitons solutions: When H = B 2 − 4AC, q = p = 1 and are referred to as deformation parameters.The generalized trigonometric and hyperbolic functions are expressed as below:

Results
In this section, we concentrate on solving problem (1) analytically.We start by introducing a variable transformation of the following form: which transforms (1) into a NODE, integrating the NODE with respect to χ and considering constant of integration zero yields: To calculate balance number τ, we consider homogeneous balance between V ¢ and V 2 ( ) ¢ , we have τ + 3 = 2τ + 2, so τ = 1.Putting τ = 1 in (7) suggests the following closed form series solution for (10): 11) in (10) and collecting all terms with the same orders of R(χ) gives an expression in R(χ).We get a system of nonlinear algebraic equations by equating the coefficients of the expression to zero.We employ Maple to solve the resultant system, which gives the following four cases of solutions: Case. 1 Using the values of case. 1, we construct the following families of soliton solutions for (1): Family.1: When H < 0 and C ≠ 0, then equations ( 9) and (11)suggest the below listed corresponding families of soliton solutions: ) ) )) Family.2: When H > 0 and C ≠ 0, then equations ( 9) and (11)suggest the below listed corresponding families of soliton solutions: Phys.Scr.99 (2024) 065257 S Noor et al ) ) ) Family.3: When AC > 0 and B = 0, then equations ( 9) and (11)suggest the below listed corresponding families of soliton solutions: ) ( ) )) ( ) Family.4: When AC < 0 and B = 0, then equations ( 9) and (11)suggest the below listed corresponding families of soliton solutions: Phys.Scr.99 (2024) 065257 S Noor et al    where . Using the values of case.2, we construct the following families of soliton solutions for (1): Family.13: When H < 0 and C ≠ 0, then equations ( 9) and (11)suggest the below listed corresponding families of soliton solutions: )) y y Family.14: When H > 0 and C ≠ 0, then equations ( 9) and (11)suggest the below listed corresponding families of soliton solutions: ) Family.15: When AC > 0 and B = 0, then equations (9) and (11)suggest the below listed corresponding families of soliton solutions: Family.16: When AC < 0 and B = 0, then equations ( 9) and (11)suggest the below listed corresponding families of soliton solutions: Family.17: When C = A and B = 0, then equations ( 9) and (11)suggest the below listed corresponding families of soliton solutions: Family.25: When H < 0 and C ≠ 0, then equations ( 9) and (11)suggest the below listed corresponding families of soliton solutions: . Using the values of case.4, we construct the following families of soliton solutions for (1): Family.34: When H < 0 and C ≠ 0, then equations (9) and (11)suggest the below listed corresponding families of soliton solutions: Family.35: When H > 0 and C ≠ 0, then equations (9) and (11)suggest the below listed corresponding families of soliton solutions:  In this way, the rest of the solutions can be discussed graphically in the same way as analyzing solution (21) because we have obtained many solutions.We only focused on studying some of them to clarify and communicate some concepts to the reader to understand the mechanism of the propagation of solitons and how some relevant variables affect the form and type of the solution, as previously elucidated.

Conclusion
A novel analytical technique called mEDAM was applied to this study to analyze the FCBSE.Under the strategy of mEDAM, a series-like solution was assumed for the NODE resulting from the model, and a system of algebraic equations was created from the NODE.The system was solved using Maple to obtain soliton solutions for the model.It is essential to understand the physical behavior of the model to consider the several traveling wave solutions that the generated soliton solutions revealed.Using contour and 3D graphs, the presence of various waveforms, such as lump waves, kink waves, breather waves, and periodic waves within the soliton solutions, was visually expressed.These waves provide essential information on how waves behave in FCBSE.The article highlights how these results may be used in FCBSE fields and demonstrate how the mEDAM technique can be used to study complicated models and their families of solitons solutions.

Family. 8 :Family. 9 :Family. 10 :Family. 11 :
When B = r, A = nr(n ≠ 0), and C = 0, then we arrive at the following family of solitons solutions: When C = B = 0, then we arrive at the following family of solitons solutions: When A = B = 0, then we arrive at the following family of solitons solutions: When B ≠ 0, A = 0, and C ≠ 0, then we arrive at the following family of solitons solutions:

Family. 5 :
When C = A and B = 0, then equations (9) and (11)suggest the below listed corresponding families of soliton solutions:

Family. 6 :
When C = − A and B = 0, then equations (9) and (11)suggest the below listed corresponding families of soliton solutions:

Family. 18 :
When C = − A and B = 0, then equations (9) and (11)suggest the below listed corresponding families of soliton solutions:

Family. 26 :
When H > 0 and C ≠ 0, then equations (9) and (11)suggest the below listed corresponding families of soliton solutions:

solution ( 21 )
is investigated, as depicted in figure3.It is noted that the increase of the value of the parameter A transforms the soliton solution into an oscillatory shock wave solution as illustrated in figure3.Moreover, the width of the shock wave increases with increasing A. Additionally, as the value of parameter A increases further, a single shock wave solution emerges.Also, the impact of the parameters k 1 and k 3 on the shock wave profile is examined as shown in figures 4 and 5 The obtained results highlight the significance of the calculated solutions in elucidating numerous nonlinear processes described by FCBSE.Also, we numerically analyzed the solution (80) as depicted in figures 6 and 7.The results of the analysis show that this solution represents a shock wave.The impact of the parameter A and the fractional parameter γ on the shock wave profile is examined, as illustrated in figures 6 and 7, respectively.Additionally, the solution (128) is analyzed numerically as illustrated in figure8.It is clear that this solution demonstrates the shock wave.