Diverse Multiple Lump Analytical Solutions for Ion Sound and Langmuir Waves

: In this work, we study a time-fractional ion sound and Langmuir waves system (FISLWS) with Atangana–Baleanu derivative (ABD). We use a fractional ABD operator to transform our system into an ODE. We investigate multiwaves, periodic cross-kink, rational, and interaction solutions by the combination of rational, trigonometric, and various bilinear functions. Furthermore, 3D, 2D, and relevant contour plots are presented for the natural evolution of the gained solutions under the selection of proper parameters.

where me −iω p t and n illustrate the normalized electric-field of the Langmuir oscillation and perturbation of density, respectively. Both x and t are normalized variables and AB D α t is the AB fractional operator in t direction.

ABD operator is well defined as
where G α is Mittag-Leffler function, defined as and B(α) is the normalization function that satisfies B(1) = B(0) = 1. Thus, for more properties of this operator. This leads towards the following form, where β and γ are arbitrary constants. This wave alteration converts Equation (1) into the following ODE.
Here, u and v are the functions of ξ. By separating the Img part from the first part of Equation (6), γ + ab = 0 =⇒ γ = −ab. (7) and then by integrating the second part of Equation (6) by two times the w.r.t ξ, we obtain Equations (7) and (8) transform Equation (6) into the following form: The contents of this paper are arranged as follows: In Section 2, we present M-shaped rational solitons. In Section 3, we evaluate M-shaped interaction solutions. In Section 4, we find the multiwaves solution. In Section 5, we study homoclinic breather. In Section 6, we investigate periodic cross-kink solutions. In Section 7, we present results and discussions and Section 8 contains concluding remarks.

M-Shaped Rational Solitons
By using the following log transformation, Equation (10) transforms Equation (9) into the following bilinear form: We choose M-shaped rational solution in bilinear form for Φ, as follows [31]: where b i (1 ≤ i ≤ 5) all are real-valued parameters to be measured. Inserting Φ into Equation (11) and collecting all powers of ξ, we obtain proper results, as follows (See Figures 1 and 2): Using this in Equation (12), and then by using Equations (8) and (10), we obtain To obtain final results, we use Equation (5): where Using this in Equation (12), and then by using Equations (8) and (10) in Equation (5), we obtain where , respectively as three-dimensions in (a); contour in (b) and two-dimensions in (c) Using this in Equation (12), and then by using Equations (8) and (10), we obtain To obtain final results, we use Equation (5):

M-Shaped Rational Soliton Interactions with
In this part, we evaluate M-shaped rational interactions with periodic and kink waves by using exponential and cos function in bilinear combinations.

Two-Kink Soliton
For two-kink interaction, the bilinear solution for Φ is as follows (See Figures 7-9): where b i (1 ≤ i ≤ 9) and all are real-valued parameters to be found. Inserting Φ into Equation (11) and collecting all powers of ξ, and e 3(b 5 ξ+b 6 ) , 6 , we obtain proper results, as follows: Using Equation (32) in Equation (31), and then by using Equations (8) and (10), we obtain Using Equation (5), we obtain the required solution for Equation (1), where Set II.

Periodic Waves
For periodic-wave interaction solutions, the bilinear form for Φ is as follows (See Figures 10 and 11): where b i (1 ≤ i ≤ 7) and all are real-valued parameters to be found. Inserting Φ into Equation (11) and collecting all powers of ξ and cos(b 5 , we obtain proper results as follows: By using these parameters in Equation (38), and then by using Equations (8) and (10), we obtain Now, using Equation (5), we obtain the required solution for Equation (1): By using these parameters in Equation (38), and then by using Equations (8) and (10) in Equation (5), we obtain (a) (b) (c)

Multiwave Solutions
For multiwave solutions, Φ in bilinear form can be assumed as [32] where z i s and b i s all are real-valued parameters to be measured. Inserting Φ into Equation (11) and collecting all coefficients of cosh we obtain proper results, as follows (See Figures 12 and 13): By using these values in Equation (44) and then by using Equations (8) and (10), we obtain Using Equation (5), we obtain the following multiwave solutions for Equation (1): By using these values in Equation (44) and then by using Equations (8) and (10), we obtain Using Equation (5), we obtain the following multiwave solutions for Equation (1):

The Periodic Cross-Kink Wave Solutions
For this, Φ in bilinear form can be assumed as [33] where z i s and b i s all are real-valued parameters to be measured. Inserting Φ into Equation (11) and collecting all coefficients of e b 1 ξ+b 2 , , and e −(b 1 ξ+b 2 )+2b 1 ξ+2b 2 ) sin(b 4 + b 3 ξ) sinh(b 6 + b 5 ξ), after solving them, we attain the following parameters (See Figures 16 and 17): By using these values in Equation (57), and then by using Equations (8) and (10), we obtain Now, using Equation (5), we obtain the following solutions for Equation (1): where Now, by using these values in Equation (57), and then by using Equations (8) and (10) in Equation (5), we obtain the following solutions for Equation (1):

Results and Discussion
The study of new imposed solutions for the ion sound and Langmuir waves (ISLWs) has huge importance among scientists. Much of the work has been carried out on ISLWs, for example, Mohammed et al. constructed new traveling wave solutions for ISLWs by using He's semi-inverse and extended Jacobian elliptic function method [34]. Shakeel et al. studied new wave behaviors for ISLWs with the aid of modified exp-function approach [35]. Seadawy et al. used direct algebraic and auxiliary equation mapping to obtain the families of new exact traveling wave solutions for ISLWs [36]. Tripathy and Sahoo studied a variety of analytical solutions for ISLWs [37]. Seadawy et al. studied a variety of exact solutions with modified Kudraysov and hyperbolic-function scheme for ISLWs [38].
Here, we obtained a variety of analytical solutions with rational and trigonometric forms for ISLWs, in which some of them are represented graphically in 3D, contour, and 2D shapes. In Figures 1 and 2, we present M-shaped solutions for m 23 and m 25 with contour and 2D plots, respectively. In Figures 3-6, we see the interactional phenomena with M-shaped and one-kink for m 31 , n 32 , m 35 , and n 36 at different values of the parameters. In these figures, we see M-shaped waves with multiple bright and dark solutions. In Figure 4, waves strongly increased their amplitude according to time. In Figures 7-9, we see the interactional phenomena with M-shaped and two-kink for n 38 , m 39 , and n 40 . In Figure 7, multiple bright, dark, and M-size solitons appear. In Figures 8 and 9, large-sized dark and bright waves appear. Figures 10 and 11 represent the evolution of M-shaped and periodic waves for m 3 and n 4 . Figures 12 and 13 represent the evolution of multiwaves solution for m 43 and n 44 at different values. In Figures 14 and 15, two solutions, m 51 and n 54 , of homoclinic breather are presented graphically, and we also see the changes in graphs by varying the value of a. In Figures 16 and 17, we present periodic cross-kink solutions m 63 and n 44 graphically, and we also see the change in waves into bright and dark solutions by varying the value of a. As α ∈ (0, 1], in all these solutions, we can see that when α = 1, ∑ ∞ s=0 (− α 1−α ) s does not converge.

Conclusions
In this work, we successfully derived some new analytic solutions for FISLWS with Atangana-Baleanu derivative. These exact solutions are derived in the form of bilinear, trigonometric, and exponential functions. As a result, new traveling wave solutions are gained in the form of rational, periodic, multiwaves, multi-kink, solitary waves, bright and dark solitons that are shown graphically in 3D, 2D, and contour structures. These solutions play an important role in different areas of physics, engineering, and other branches of sciences.