Bright-dark wave envelopes of nonlinear regularized-long-wave and Riemann wave models in plasma physics

The nonlinear regularized-long-wave (RLW) and the Riemann wave (RW) models are physically significant in plasma physics and in further study of nonlinear dispersive waves, namely shallow water, ion-acoustic and magneto-sound waves in plasmas, anharmonic lattice and pressure waves in liquid-gas bubble mixtures, tidal and tsunami waves in rivers and oceans, longitudinal electromagnetic wave propagation in plasma, etc. The advanced generalized (G’/G)-expansion scheme is facilitated in this article for retrieving the bright-dark exact solitary wave solutions generated in the form of hyperbolic, trigonometric, and rational structures to the abovestated models. To figure out the internal contrivance of the solutions, 3D and contour plots concerning bright parabolic wave shape, compacton wave shape, bell-shaped soliton, kink wave type soliton, dark anti-parabolic wave shape, anti-compacton wave shape, bright propagation of solitary wave shape solutions are depicted for definite free parametric values. The characteristic feature of the wave envelops fluctuates taking into consideration the changes of free parameters and they are essentially dominated by the linear and nonlinear impact.


Introduction
Exact solitary wave solutions to the nonlinear evolution equations (NLEEs) contribute extensively in the branches of applied sciences and engineering, such as, optical fibers, condensed matter physics, theory of solitons, chaos theory, fluid dynamics, plasma physics etc. [1]. In mathematical physics, namely meteorology, biology, nuclear physics, and other fields, nonlinear waves involving acoustic waves, hydromagnetic waves, and acoustic gravity waves make significant contributions [2,3]. Many mathematical models have been developed for explaining those wave behaviors.
The water wave propagation in any channel governs the gravitational body forces and surface tension at the interface between the atmosphere and the water [4]. This kind of propagation relies on the water depth, the wavelength and amplitude of the waves, the slopes of the water surface in two orthogonal directions, the capillary length scale etc. Surface tension can be ignored when the ratio of gravitational force to surface tension is very high. Soliton solutions of the evolution equations have a great impact in the direction of pulse propagation via optical fibers for trans-continental and trans-oceanic distances [5].
Dimensionless condition of the dependent and independent variables, boundary conditions and governing equation presents the dynamics of water waves is acknowledged by the anisotropy, amplitude, steepness and topography parameters [6]. The first ratio is between water depth and wave amplitude, and the second ratio is between wavelength and water depth. The parameters of the topography concerned to the slopes of the channel's topography in two orthogonal directions, when the anisotropy represents the dependence of the velocity of propagation on direction.
A group of researchers used several approaches to investigate exact wave solutions to the RLW [39][40][41][42][43][44][45] and RW [46,47] equations. Raslan et al. [1] examined exact solutions through of the coupled generalized RLW equations by taking advantage of the sine-cosine function algorithm and the Kudryashov scheme. Roshid et al. [3] determined solitary wave in plasma, ion-acoustic plasma and shallow water to the RLW equation via the modified simple equation technique. By the aid of exponential and compact-operator approaches, Garcia-Lopez, and Ramos [4] extracted solitary waves generated by bell-shaped initial conditions in the inviscid and viscous generalized RLW equations. Further, the optimal perturbation iteration approach has been applied by Bildik and Deniz [41] to find the analytic approximate solutions to the generalized RLW model. In Ref. [9] the author explained the dark and bright soliton solutions and computational modeling of nonlinear 1D and 2D RLW models. Barman et al. [2] investigated the competent closed form soliton solutions to the RW equation through the generalized Kudryashov technique and accomplished bell-shaped, consolidated bell-shaped, compacton, singular kink shape, flat kink and other soliton solutions. Duran [46] computed the breaking theory of solitary waves for the RW equation in fluid dynamics by utilizing the generalized exponential rational function scheme. Burman [47] established the physically important wave solutions to the RW model using the expanded tanh-function approach and achieved dark soliton, bright soliton, peakon, compacton, periodic and other type soliton solutions in ion-acoustic and magneto-sound waves in plasma. However, no one has used the sophisticated generalized (G ' /G)-expansion technique yet to analyze the above reported models. Therefore, motivated by the recent focus of research, the purpose of this study is to explore the exact soliton solutions by adopting the advanced and generalized (G ' /G)-expansion method having substantial applications in different branches of nonlinear science and engineering. Moreover, we investigate the dynamic wave envelops which are very realistic and timely, and also they are widely used in plasma physics.
The coordination of the rest parts are as follows: In part 2, we mention briefly the methodology of the advanced and generalized (G ' /G)-expansion scheme. The applications of the RLW and RW models are described smoothly in part 3. Part 4 contains the graphical structures involving 3D and contour plots and discusses the physical interpretations elaborately. Finally, the concluding remark is drawn in details in part 5.

Methodology
Let us suppose a general nonlinear partial differential equation (PDE) in the form where F is a polynomial in u, u(x, y, t) = u(ζ) is an undetermined function, and the derivatives involving nonlinear terms and the subscripts refers to the partial derivatives.
Step 1: We compile the real variables x, y and t by the traveling coordinate ζ u(x, y, t) = u(ζ), ζ = gx + hy − μt (2) where g, h denote the wave numbers and μ be the velocity of the solitary wave. The assumption (2) transmutes Eq. (1) into a nonlinear differential equation (NLDE) due for u = u(ζ): where Q be the polynomial of u with its derivatives and the superscripts reveal the ordinary derivatives in terms of ζ. When g = 0 and h = 1, the wave transformation (2) turns into Step 2: As per possibility, we integrate Eq. (3) one or more times, arises integration constant(s) that should be set zero due to avoid complexity.
Step 3: The exact wave solution of Eq. (3) considered as: where at either a m or b m may be zero, but both a m and b m cannot be zero at the same time, a l (l = 0, 1, 2, ⋯, m) and b l (l = 1, 2, ⋯, m) and d are arbitrary cstants to be estimated subsequently and H (ζ) is given by where G = G(ζ) satisfies the nonlinear DE given underneath: The prime stands for the ordinary derivatives in terms of ζ; E, F, P and Q are taken as real parameters.
Step 4: To estimate the balance number m, we apply the balancing technique between the greatest derivatives and the highest order nonlinear terms arise in Eq. (3).

Analysis of solutions
In this subdivision, we discuss the application of the approaches through two important examples, the nonlinear RLW and RW models via the advanced and generalized (G ' /G)-expansion method.

The generalized Regularized-Long-Wave model
Let us consider the generalized RLW model [8] as: where p and b are positive constants and q be a positive integer.
Choosing q = 2 in Eq. (13) it takes a special form named regularized long wave (RLW) equation: herein a = 2p is considered. Peregrine was the first who introduced the RLW model for small-amplitude and long-wave on the surface water passing through a channel [48]. The model works as an alternative model to the Kortweg-de Vries equation in practical fields like longcrested waves in near-shore zones, unidirectional propagating waves in a water channel and many other similar issues. Applying the transformation (4) into Eq. (14), we obtain a nonlinear equation in the following form: Balancing technique between u 2 and u'' in (14) yields the balance number m = 2. Thus, the solution structure of the Eq. (15) takes the form: where a 0 , a 1 , b 1 , a 2 , b 2 , d are constants to be computed. Replacing (16) along with (6 and 7) into Eq. (15), the left side is changed into poly- Taking each coefficient of the attained polynomial and putting them to zero becomes an over-estimated set of algebraic equations for a 0 , a 1 , b 1 , a 2 , b 2 , d and μ. The algebraic equations are ignored herewith for simplicity.
Computing the algebraic equations by means of the Maple computation software, we accomplish the six sets of solutions as appeared underneath: , a 1 = 0, We have considered the set-1 as a solution set from the above stated set of solutions set-1 to set-6 owing to achieve best outcomes.

The Riemann wave equation
The (2 + 1)-dimensional generalized breaking soliton equation [53] is given by where at α, β, γ, δ, ε be the real arbitrary parameters. The (2 + 1)dimensional interaction of the Riemann wave is interpreted by the Eq. (23) in which overlapping solutions have been created for the particular case α = 0. The spectral parameter is entitled by the reputed breaking behaviour which is the principal nature of such types of equations. Otherwise, the spectral value is treated as multivalued function. Thus, the solution types of these equations would be multivalued. Therefore, the Eq. (23) is associated with the (2 +1) dimensional generalized breaking soliton equations [49] as: Xu [49] showed the Eqs. (24a) and (24b) have many special cases investigated by Calogero and Degasperis [50], Bogoyavlenskii [51], Radha and Laksmanan [52], and Zhang et al. [53] for different particular values. Following this technique, we consider the RW model yield from the Eqs. (24a and 24b) is of the structure [47]: where into a, b, and f stand for the real numbers. To extract the traveling wave solutions, placing the assumption (2) into (25a and 25b), and decomposed into the ensuing system hu' = gw'.
Integrating the Eq. (27) gives wherein the integral constant is presumed zero. Removing w and w' from the equation (26) provide the following nonlinear equation The new form of equation (29) is obtained by integrating first time The equation (30) consists of linear term u'' and nonlinear term u 2 . Actually, the solitary wave is generated by the combination of two factors. The terms u'' and u 2 leads to an index number m = 2. Therefore, the solution structure (5) is written as where about a 0 , a 1 , b 1 , a 2 , b 2 , d are constants to be computed. In accordance with, we achieve the new and rich sets of solutions as follows: where at ϕ = E − P, d, E, F, P, Q are free parameters.
The more appropriate output named set-1 has been taken from the above stated solutions set-1 to set-3 so that we could acquire more new and suitable results. Now, transferring Eq. (32) into Eq. (31) having (8) and Family-1, and also simplifying, we accomplish the subsequent traveling wave results (K 11 ∕ = 0, K 12 = 0 and K 12 ∕ = 0, K 11 = 0): Connecting these solutions into equation (25) provides another set of solutions as below: In similar manner, inserting the values of (32) into (31) accompany with (9) and elucidating, we attain the wave solutions with the aid of Family-2 as follows (K 11 ∕ = 0, K 12 = 0 and K 12 ∕ = 0, K 11 = 0): Similarly, by taking help of the above solutions, we achieve another set of solutions from equation (25), Putting the values of Eq. (32) in (31) along with (10) and after simplification the result takes the form utilizing Family-3,              By utilizing above results into equation (25), the solution set yields: Furthermore, substituting Eq. (32) into (31) accompany with (12) and after computation, we find out the subsequent solitary wave solutions by taking help of the Family-5 (K 11 ∕ = 0, K 12 = 0 and K 12 ∕ = 0, The Eq. (25) becomes by getting help of the above solutions as: The established results are illustrative, further generic, and they have not been examined in the previous works. The dynamic wave solutions have been widely used in the communication industry with optical fibers, nuclear physics with nuclear fission and fusion reactions, and oceanography with natural disasters such as cyclones, tornadoes, and tidal bores, etc.

Graphical representations of the results
, , nonlinear term). That is for the definite values of the free parameter, we accomplish several knowledgeable exact solitary wave shapes. First of all, we have portrayed several types of 3D and contour surfaces for each of the solutions by means of the mathematical software like Wolfram Mathematica and then explained the behavior of those results. The 3D plot shows a relationship between designated dependent and independent variables. Otherwise, the contour graphics demonstrates the 3D wave envelopes in the format of 2D shape.

Graphical outlines of the results to the generalized RLW equation
The generalized RLW equation delivers new forms of exact solutions with the combination of trigonometric and hyperbolic structures. Some distinct wave profiles, such as, bell-shaped, anti-bell shaped, kink shaped, parabolic shaped and other soliton profiles are developed from these solutions by choosing the appropriate values of the unfamiliar parameters. These types of wave forms cover different nonlinear phenomena. For example, kink wave interprets the fiber-opticcommunication process. It is a technique of transmitting information from one state to another state by sending the pulses of infrared light through the optical fiber. In addition, bell shape profile explains the Rogue wave propagations. Rogue waves are such types of waves whose height are more than twice the height of the significant waves. Rogue waves appear to be caused by a combination of physical aspects, such as tremendous storms and robust currents, which bring about these waves to combine into an extremely unique long wave. The behavior of the solutions u 03 (y, t), u 06 (y, t) and u 09 (y, t) have been focused on Figs Fig. 6 in the region 0 ≤ y ≤ 2, 0 < t ≤ 2 and the contour plot is delineated for t = 0. For other values of the wave speed μ = 2 and anonymous parameters a = b = F = ϕ = − 2,E = 2.72,ε = − 1.64, this solution yields a periodic wave structure whose behaviors are outlined in Fig. 7 in the region − 8 ≤ y ≤ 8, 0 < t ≤ 5.

Graphical outlines of the results to the RW equation
This paragraph covers the graphical depictions of certain derived results to the RW equation. As each result contains some unfamiliar parameters, some well-known profiles, such as, compacton, parabolic soliton, bell type soliton, kink soliton and other solitons are established by receiving the arbitrary values of parameters associated with the linear and nonlinear factors. Such types of waves conduct with like many nonlinear wave phenomena, as for instance, ion-acoustic and magneto-sound waves in plasma, cluster of hydrodynamic model, fission, and fusion interaction etc. In the following, we have discussed about the behavior of solutionu 05 (x, y, t), w 05 (x, y, t), u 06 (x, y, t), w 06 (x, y, t), u 08 (x, y, t), w 08 (x, y, t), u 09 (x, y, t) and w 09 (x, y, t) whose propagations are delineated in Figs

Conclusion
Bright-dark, parabolic, anti-parabolic, compacton, anti-compacton, bell-shaped, anti-bell-shaped, kink, flat kink, smooth kink-shaped solutions have been determined in this article which are originated in plasma, rivers and oceans, transmission lines, optical fibers, and other sources. The fiber-optical communication process, energy dissipation, superconductivity, rogue wave propagations, super deformed nuclei, phenomena of nuclear fission and internal fusion etc. can effectively be explained by means of the 3D and contour portrayals. The advanced and generalized (G'/G)-expansion scheme is capable to yield such types of solutions. It is established that the changes in the nature of the wave are explicitly related to the free parameters involving the linear and nonlinear effects. The computed results comprise the trigonometric, hyperbolic, and rational structures. The results are extremely effective in several disciplines of plasma physics where solitary wave theories are studied. The method is further reliable and applicable due to the small size of computational domain.