Novel Soliton Solutions of the Fractional Riemann Wave Equation via a Mathematical Method

: The Riemann wave equation is an intriguing nonlinear equation in the areas of tsunamis and tidal waves in oceans, electromagnetic waves in transmission lines, magnetic and ionic sound radiations in plasmas, static and uniform media, etc. In this innovative research, the analytical solutions of the fractional Riemann wave equation with a conformable derivative were retrieved as a special case, and broad-spectrum solutions with unknown parameters were established with the improved (G’/G) -expansion method. For the various values of these unknown parameters, the re-nowned periodic, singular, and anti-singular kink-shaped solitons were retrieved. Using the Maple software, we investigated the solutions by drawing the 3D, 2D, and contour plots created to analyze the dynamic behavior of the waves. The discovered solutions might be crucial in the disciplines of science and ocean engineering.


Introduction
Over the last few decades, numerical and exact solutions to nonlinear partial differential equations (NLPDEs) [1,2] have appeared to be of great interest to many scholars due to the amazing popularity of nonlinear sciences and engineering. Recently, the stability analysis of nonlinear fractional partial differential equations (NLFPDEs) has played an important role in the field of solitary wave theory. Analytical solutions of NLFPDEs are crucial in applied nonlinear research. Moreover, the structures of natural phenomena are better described by fractional-order differential equations than by integer-order differential equations. The use of analytical techniques to find traveling wave solutions to NLF-PDEs [3][4][5] can explain the physical behavior of related real-world problems more effectively. Researchers have increasingly concentrated on analytical and numerical solutions to NLFPDEs [6][7][8][9][10][11] with the help of computer science and symbol-based software. Numerous robust computational techniques have been developed in the literature to explore the various types of exact solutions of NLFPDEs, such as the extended rational sinh-cosh method [12], extended tanh-coth method [13], exp-function method [14,15], fractional Riccati expansion method [16], generalized Kudryashov approach [17], enhanced and generalized (G'/G)-expansion scheme [18,19], and Painleve Property [20].
Few years ago, Wang et al. [21] developed the (G'/G)-expansion method, a popular, direct, and brief technique for finding exact traveling wave solutions. Different modifications of the (G'/G)-expansion method have been developed by different researchers, but the improved (G'/G)-expansion method is also one of the most reliable, effective, and consistent methods used by different scholars to extract traveling wave solutions of NLPDEs.
For instance, Islam et al. [19] presented the nonlinear dynamics of magnetic soliton solutions, several soliton solutions of a nonlinear model were constructed by Yokus et al. [22] by means of the (G'/G 1/G)-expansion method, some researchers obtained the dynamical and physical nature of a few novel and accurate trigonometric, hyperbolic, and rational solitary wave solutions with the help of Atangana's conformable differential operator by using an efficient (G'/G)-expansion method [23][24][25], and Younis et al. [26] constructed solitary wave solutions to the Schrödinger-Poisson system with the help of the (G'/G)-expansion method.
The characteristics of the improved (G'/G)-expansion method boosted our interest in a notable and suitable model for the Riemann wave equation (RWE), which is associated with superconductivity, plasma electrostatic waves, and ion-cyclotron wave electrostatic potential in a centrifugally inhomogeneous plasma. Consider a type of generalized (2 + 1) −dimensional breaking soliton equation (BSE) [27]: where , , , , and are real parameters and −1 [28] is the inverse operator of . By using the Painleve property, there are different choices for the values of real parameters. The spectral parameters used in the Lax representations demonstrate breaking behavior, which is a distinctive characteristic of this family of equations. As a consequence of the spectral values the solutions evolve into multivalued functions. As a result, the (2 + 1) −dimensional breaking soliton equations (BSEs) [27] with different overlapping solutions were developed, and they propagate along the − axis while interacting with a long wave traveling toward the − . The Riemann wave equation [29] is a type of generalized (2 + 1) −dimensional BSE. Moreover, this (2 + 1)-dimensional BSE is associated with the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy. In addition to this, these equations have a wide range of applications for the propagation of tidal and tsunami waves in the ocean. These equations also represent the turbulent state by using a mixture of Whistler wave packets and random phases with a finite amplitude. A magnetic sound wave is dampened because the Whistler turbulence's interaction with it dampens a plasma's electrostatic wave [27]. Scholars and researchers have recently been very concerned with finding RWE solutions by using various methods. The Wronskian method was used to extract rational and periodic solutions of a (2 + 1) −dimensional BSE by Hong-hai et al. [30], soliton solutions to the RWE were derived by Barman et al. by using the extended tanh-function technique [27], Barman et al. [29] used the generalized Kudryashov method to establish traveling wave solutions, Roy et al. [31] evaluated exact bright-dark solitary wave solutions with the aid of the generalized (G'/G)-expansion method, and the generalized exponential rational function approach was utilized by the authors of [32] to evaluate RWE solutions that simulated the construction, breaking, and interaction of waves that resulted from any peripheral influence on the ocean envelope.
Taking the prior analysis into account, consider a Riemann wave equation of fractional order: where represents the -order partial derivative of " " with respect to " ". The closedform solutions of Equation (4) were constructed by using the improved (G'/G)-expansion method, which includes the periodic, kink, and singular kink wave solutions. To demonstrate the stability and accuracy of the method, the established results were compared with the existing results. The method was applied for the first time to the given model, and the obtained solutions were more comprehensible, which shows the novelty of the work. Graphical interpretations of the solutions produced for several free parameters are presented, which will prove to be useful in the future. Comparison of the obtained solutions with the existing solutions in the literature is given in the form of Table 1.
The scheme of this article is as follows: The definition and properties of the conformable derivative are presented in Section 2. The methodology is discussed in the Section 3. The nonlinear time-fractional Riemann wave equation is examined via the improved (G'/G)-expansion method in the Section 4. In the Section 5, graphs are presented and the physical interpretations of the outcomes are demonstrated. Finally, we come to a conclusion.  If we put = 1, = 1, ℎ = 1, = 4, = 4, = 1 into Equation (32), we get:

Analysis of the Method
The method's essential points are outlined in this section. Consider the following NLFPDEs: where = ( , ) is a wave variable that needs to be identified, Υ is a polynomial expression in = ( , ), and its partial derivatives include nonlinear terms and the highest-order derivatives. The subscript indicates partial derivatives. To analyze the solution of Equation (3), the following crucial actions must be taken by using the improved (G'/G)expansion method: Step 1: Take the traveling wave transformation where indicates the wave velocity, represents fractional-order derivatives, and 0 < ≤ 1. Equation (7) transforms Equation (6) into the following form: where primes are the derivatives with respect to ζ.
Step 2: Integrate Equation (8) and the integration constants, which are considered to be zero for simplicity.
Step 4: In Equation (9), M is a positive integer that is evaluated with the help of the balancing principle in Equation (8).
Step 5: By putting Equation (9) into Equation (8) together with Equation (10) and by considering every coefficient of ( ′ / ) to zero, the random variables and can be calculated by utilizing a computer algebra system (CAS). By inserting these values into Equation (9), the new traveling wave solutions of Equation (6) are developed.

Application of the Method
In this part, the conformable time-fractional derivatives of the fractional-order RWE that are given in Equation (4) are used to produce reliable and more accurate solutions that include random parameters with the help of the improved (G'/G)-expansion method. By using the transformation into Equation (4) given in Equation (7) and after integration, we get where = 1 . We obtain = 2 with the aid of the homogeneous balance between the terms 2 and ′′ in Equation (17). As a result, the solution of Equation (17) can be expressed as: where 0 , 1 , and 2 are unknown parameters to be calculated thereafter. Inserting Equation (18) with Equation (10) into Equation (17) and considering every coefficient of ( ′ / ) as equal to zero, there exist a set or sets of equations (these are not included here for the sake of ease) for 1 , 2 , 3 , and μ with the assistance of CAS, and we obtain: .

Physical Implications and Graphical Analysis
In this paper, the exact analytic results of a fractional-order RWE were extracted by introducing fractional transformation. With the help of numerical simulations, several analytic solutions of RWE were discovered. The outcomes are depicted in the forms of 3D, 2D, and contour profiles. Consequently, the periodic, singular, and anti-singular kink wave solutions were captured for four forms of wave findings that were developed by using this method-these were exponential, rational hyperbolic, and trigonometric functions. The physical phenomena of the results are presented with the help of CAS in Figures 1-6, which help us to understand the nature, behavior, properties, and characteristics of the RWE. The behaviors of solitary waves are controlled by assigning different values to the free parameters. Hence, by altering the values of the parameters, the nature of the graph changes. On the other hand, the solution profiles depend on the fractional-order α. For = 1, the conformable fractional derivative becomes the classical derivative. Here, we explain how the fractional order affects the graph of the optical solitons found for the fractional nonlinear optics model (RWE) for different sets of variables. For clarification, the figures are shown with respect to (0 < ≤ 1), which varies with small differences. We discovered two sets of solutions. Set-1 comprises the Equations (21)- (33). Meanwhile, the values of the unknown coefficients produced in set-2 provide different, fresh, and constructive Equations (35)-(46).     The obtained solutions consist of the parameters , , , 1 , 2 , , , , , , and . However, for the values = 1, = 4, = 2, 1 = 1, 2 = 1, = 1, = −1, = 1, = 1, = 1, and = 2.5 for the variables in Equation (32), we establish a singular soliton solution. In the case of singular solutions, a travelling wave has endless wings on both sides and a nominal space in between them, which causes the wave to be singular. The profile of this figure is widely escalating on one side. The 3D and contour graphs are depicted within the limits 0 ≤ ≤ 10 and 0 ≤ ≤ 5 for different values of . The 2D profile is constructed for 0 ≤ ≤ 10 and = 1 ( Figure 5).
Again, when = 1, = 4, = 2, 1 = 1, 2 = 1, = 1, = −1, = 1, = 1, = 1, and = 2.5 for the parameters of Equation (32), we establish the singular and anti-singular kink soliton solutions. The 3D and contour profiles are illustrated within the limits −10 ≤ ≤ 10 and 0 ≤ ≤ 5. The 2D profile is constructed for −10 ≤ ≤ 10 and = 1 ( Figure 6). In this work, we described different states with several values of free parameters by using fractional derivatives. Fractional derivatives play a crucial role in the understanding of the nature of the proposed problem. In addition to this, a consistent behavior of the traveling wave was demonstrated throughout the article. When it comes to analyzing different kinds of nonlinear fractional evolution equations, these findings illustrate that this is a more reliable, proficient, and dominant technique. Furthermore, different techniques were used on the given model in the literature, but the graphical representations of the acquired results show that the obtained solutions are novel, more contemporary, and more universal than the results that were previously attained.

Conclusions
In this article, we used the improved (G'/G)-expansion method to study the fractional-order RWE problem. In comparison with earlier research, the findings of this investigation are being published for the first time. A time-fractional derivative operator was used to analyze the model given above. The main benefit of fractional-order models is that changes gradually develop, and these can be used to graphically illustrate the behavior of the solitary waves in relation to time and space. The graphs of the obtained solutions prove the accuracy and consistency of the above-mentioned method. The accuracy of the results was tested using Maple by plugging the acquired results into the established model. Hence, this method could be used to investigate various nonlinear models that arise frequently in a variety of real-world problems.