Dynamic study of multi-peak solitons and other wave solutions of new coupled KdV and new coupled Zakharov–Kuznetsov systems with their stability

In this paper, our aim is to further expand the use of the two-variable -expansion approach to a new coupled KdV and Z-K system, which has various significant applications in different fields of applied sciences. The KdV equation, along with shallow-water waves and long internal waves in oceans, basically explains how long, one-dimensional waves propagate in a variety of physical conditions. The study of coastal waves on the basis of the ocean is done using the Zakharov–Kuznetsov (Z-K) equation and this model is utilized to illustrate ion-acoustic wave propagation. By using this method, different forms of analytical solutions of the new coupled KdV (NCKdV) system and the new coupled Z-K (NCZ-K) system, such as solitons, multi-peak solitons, solitary waves, trigonometric, hyperbolic and rational functions and other wave solutions are constructed. The significant features of multi-peak solitons induced by the higher-order effects, including velocity variations, localization or periodicity attenuation and state transitions, are revealed. When the localization disappears then the multi-peak soliton becomes a periodic wave. The constructed solutions are also presented graphically having their applications in engineering, etc. The stability of the solution is examined by utilizing modulation instability. The results obtained show that the proposed technique is universal and efficient. In addition, this technique can also be applied to lots of other new coupled systems arising in other areas of applied sciences.


Introduction
Non-linear phenomena play a vital part in applied sciences.Soliton theory heavily depends on the travelling wave solutions of non-linear models in applied sciences [1][2][3][4].The generalized KdV and Zakharov-Kuznetsov (Z-K) equations are essential models for many physical phenomena in partial differential equations (PDEs), including shallow-water waves, non-linear optics, applications in astrophysical and space environments, the interaction between a water wave and a floating ice cover, gravity-capillary waves, shock waves, ionacoustic waves, hydro-dynamic, stratified waves and many others.The study of explicit analytical solutions to new coupled systems is crucial for illuminating the properties of non-linear phenomena and soliton theory.The travelling wave solutions of these non-linear coupled systems and high-dimensional non-linear equations may enable researchers to thoroughly investigate the aforementioned natural phenomena.Therefore, it is very important to build up non-linear evolution equations and analytical solutions of new coupled systems.
In mathematical physics and the applied sciences, the study of analytical solutions in the explicit form for new coupled systems is crucial for describing the characteristics of non-linear issues and soliton theory.
To gain explicit results in the form of solitons, solitary wave and elliptic function solutions, many precise and influential techniques have been developed such as Inverse scattering transform technique [5], (G /G)expansion method [6], the Classical Darboux transformations [7,8], Bäcklund transformation [9][10][11], expansion scheme, Hirota's bilinear [12,13], Painlevé analysis [14,15], extended tanh scheme [16], auxiliary equation scheme [17], direct algebraic methods [18,19], the mapping and extended mapping methods [20], elliptic function scheme [21], rational expansion method and several others [22,23].A fantastic progress has been made in the analysis of the solutions, structures, interactions and other properties of solitary waves and solitons, also distinct profound results have been effectively obtained [24,25].Explicit and numerical solutions were illustrated of KdV and coupled KdV equations by decomposition approaches [26][27][28].' The concept of soliton was initiated to show that the solitary waves, which keep their characters unaffected, throughout the proliferation process and after interface, are like particles [29][30][31].Because of these characteristics, solitons are believed as standard data bits [32].Owing to the potential applications of optical solitons in telecommunications and ultrafast signal processing systems, it is the main target of theoretical and experimental analysis [32,33].As the exact results of the PDEs, the breather plays a central task in semi-conductor quantum wells [34], arrays of micro-mechanical oscillators [35] and junctions of Josephson [36].Breather specifies that the solutions' behaviour is periodic in space or time having the property of spatial or temporal localization.Two types of known breathers are the Akhmediev breather of space-periodic and the Kuznetsov-Ma soliton of time-periodic [37][38][39][40][41][42].Some other analytical results are the results of rogue waves [?, [42][43][44][45][46].These rogue wave solutions are confined in together space and time, arise from nowhere and vanish starved of a trace [43,44].These results may be acquired through the Taylor expansion of the breather results [47][48][49][50].
In this paper, the two-variable (G /G, 1/G)-expansion approach is used to achieve the analytical solutions of the new coupled KdV and Z-K systems.As a result, distinct types of solitons and other wave solutions are obtained.The stability of the solution is examined by utilizing modulation instability (MI).The significant features of multi-peak solitons induced via the higher-order effects, including velocity variations, localization or periodicity attenuation and state transitions, are revealed.When the localization disappears, then the multi-peak soliton becomes a periodic wave.The main advantage of this method is that different forms of analytical solutions of such as solitons, solitary waves, rational solitons, trigonometric function, hyperbolic function and other wave solutions are constructed using this technique.The proposed technique is also universal and efficient.This technique can also be applied to lots of other new coupled systems arising in other areas of applied sciences.
The remaining article's main structure is as follows: Section 2 describes the two-variable (G /G, 1/G) technique in detail.In the third Section, the given technique is applied on the NCKdV and NCZ-K systems for producing precise wave results.The analysis of modulation instability is discussed in the fourth section to explain the stability of models.The results discussion and its physical justification are provided in the fifth section.Finally, the sixth section summarizes the entire work.

Proposed scheme
In order to arrive at wave results of the aforementioned equations, the (G /G, 1/G)-expansion technique is described in detail in this section.The following is a full introduction to this strategy from [1,51]: The aforementioned approach begins with the second-order linear ODE, which is known as follows: by taking for the accuracy of the calculations.The variables and have the derivatives as There are three types of cases in the general results of Equation (1) Case-I: σ < 0, the solution is Case-II: σ > 0, the solution is as with Case-III: σ = 0, the rational solution is as In all cases, A 1 and A 2 are constants.
The following procedures must be taken in order to use this method to obtain precise solutions to the nonlinear evolution equation.
Step 1: Wave transformation ξ = x − tc is used to convert Equation (10) to ODE (ordinary differential equation); it includes u(x, t) = u(ξ ) as where u (n) is the nth order derivative with respect to ξ .
Step 2: The result could be in the form of and here G adheres to (1).constant coefficients a j , b j , β, c and α are to be determined.N can be determined by using the balancing principle in (11).
Step 3: By considering case 1 as an example, utilizing (12) into (11) while taking into account (3) and ( 5), a polynomial equation in and is attained and produces a system of algebraic equations.
Step 4: A software package program is used to solve the system.Wave solutions in (11) are constructed as three different types of functions by utilizing resultant values a j , b j , c, β, α, A 2 and A 1 .
Step 5: The solution process is completed by producing results in (10) using ξ = x − ct (wave transformation) conversely.

The new coupled KdV system
Qin [40] used a finite-dimensional integrable system to build a new hierarchy of non-linear evolution equations.A novel coupled KdV system [52] is an interesting equation in this hierarchy is as where λ, α, β and γ are arbitrary constants.The author in [17] more extended the NCKdV system into a new coupled KP system and deliberated both systems using Hirota's bilinear approach.Consider the travelling wave solution as where, ξ = δx + ρt, (14) where δ and ρ are wave number and frequency of results, respectively.Using ( 14) in ( 13) and the ODEs obtained as Using the balancing principle on (15) and considering the solutions as Utilizing Equations ( 16) with ( 3) into (15), we obtained system of equations in ρ, μ, 1 , 2 , 3 , a 0 , d 2 , e 2 and f 2 by equating the coefficients of i j to zero.After solving the system, the following families of results are produced: Family 1: Family 2: Family 3: Family 4: Case-I: σ < 0, from the first family, the following soliton solution set is obtained in the hyperbolic function form as , from the second family, the following soliton solution set is also obtained in the hyperbolic function form as Case-II: σ > 0, from the first and second families, the following soliton solution sets are obtained in the trigonometric function form as , . ( 21) Case-III: σ = 0, from the first and second families, the following soliton solution sets are obtained in the rational function form as 2  , The more generalized resemble solitons of Equation ( 13) other families can also be constructed.

The new coupled Z-K system
The coupled KdV system (13) can extend to the NCZ-K system [52] in the following form where λ, α, β and γ are real numbers.The authors in [52] studied this system using the modified extended direct algebraic scheme.By considering the wave transformation as where δ, ρ and ν are the wave numbers and frequency, respectively.Now Equation ( 25) is transformed into ODE by using Equation (25) as On Equation ( 27) using the balancing principle, and presuming solutions as Utilizing Equations ( 28) with ( 3) into ( 27), and the system of algebraic equations obtained in ρ, μ, 1 , 2 , d 2 , e 2 and f 2 by equating the coefficients of i j to zero, the following families of solutions by solving these equations are obtained as follows: Family 1: Family 2: Family 3: Family 4: Case-1: σ < 0, from the first family, the following solution set is obtained in the hyperbolic function form as , From the second family, the following solution set is obtained in the hyperbolic function form as Case-2: σ > 0, from the first and second families, the following solution sets are obtained in the trigonometric function form as Case-III: σ = 0, from the first and second families, the following solution sets are obtained in the rational function form as The more generalized resemble solitons of Equation ( 25) other families can also be constructed.

Modulation stability analysis
Various non-linear PDEs exhibit scattering and nonlinear consequences as a result of uncertainty in the steady-state modulation.Utilizing linear stability analysis, the modulation instability of NCKdV and NCZ-K systems is investigated [36,38].

The new coupled KdV system
The following is the form of steady-state solution (SSS) of the NCKdV system Here, normalized optical power P. Evolution of the perturbation (x, t) is verified by using the linear stability (LS) analysis.We can linearize by putting Equations ( 35) into (13), we have It is supposed that the solution of Equation ( 36) has as where k and ω are the wave number and frequency of perturbation, respectively.Utilizing Equations ( 37) into (36), the dispersion relation (DR) is acquired as The dispersion relations in (38) indicate that steadystate stability varies upon the wave number, selfphase modulation and stimulating Raman scattering.For wave numbers k, the velocity dispersion ω is real and the steady state is stable along the small perturbation if k 2 P(α 2 + 2αλ + 8γ λ + λ 2 ) > 0. when k 2 P(α 2 + 2αλ + 8γ λ + λ 2 ) < 0, it becomes unstable, i.e. ω is imaginary as the perturbation builds exponentially.It can be easily observed MI(modulation instability) when k 2 P(α 2 + 2αλ + 8γ λ + λ 2 ) < 0. These conditions indicate the increased rate of MI achieved spectrum g(k) can be stated as

The new coupled Z-K system
The following form is the SSS of the NCZ-K system √ P + 1 (x, y, t)e Pδ t = u(x, y, t), √ P + 2 (x, y, t)e Pδ t = v(x, y, t), √ P + 3 (x, y, t)e Pδ t = w(x, y, t). ( Here, the normalized optical power is P. Evolution of the perturbation (x, y, t) is verified by using the LS analysis.We are using Equations ( 25) and ( 40) and linearizing It is supposed that the solution of Equation ( 41) has as where ν and σ 1 , σ 2 are the frequency of perturbation and wave numbers, respectively.By using Equations ( 42) and ( 41), the following dispersion relation is obtained The dispersion relations in (43)) indicate that the steady-state stability varies upon the wave number, self-phase modulation and stimulating Raman scattering.For all wave numbers σ 1 , σ 2 , the velocity dispersion ν is real and the steady state is stable along the small perturbation if α 2 Pσ 2 1 + 2αλPσ

Discussion and physical interpretation of results
The acquired results in this article are dissimilar of the gained results of several academics because Equation ( 1) is dissimilar from the existing methods.By assigning precise values of parameters, different families of solutions have been obtained for ODE (1).The new coupled KdV system was resolved by Hirota's bilinear approach [17].In [52], modified extended direct algebraic approach has been utilized to get precise wave results of ( 13) and ( 25) in travelling wave solutions in trignometric and elliptic functions forms.In [53], the researchers found Soliton solutions for a KdV equation and a generalized Hirota-Satsuma CKdV equation.By a natural decomposition method, a new solution of coupled KdV equation has been obtained in [54].Sine-Gordon and Modified Kudryashov methods were used to evaluate dual-mode Hirota-Satsuma coupled KdV equations in [55].The main advantage of this method is that different forms of analytical solutions of such as solitons, solitary waves, rational solitons, trigonometric function, hyperbolic function and other wave solutions are constructed using this technique.If A 1 = 0 or A 2 = 0 in hyperbolic function solutions, then we can obtain an analytical one-soliton solution; otherwise, we can get two soliton solutions.So, in this work, several innovative results have been accomplished, which never have been stated earlier.
In Figure 1, by specifying suitable values of parameters, the wave solution (19) in disparate constitution is represented as follows: Figure 1(A,B,E) describes the bright soliton and their 2D contour plot figures (Figure 1(B,D ,F)) correspondingly.In Figure 2, by specifying     proper values of parameters, the wave solution ( 20) is shown in disparate forms as follows: Figure 2(A,C,E) are solions in multi-peak forms of dissimilar amplitude and their 2D contour plots in figures (Figure 2(B,D ,F)) correspondingly.In Figure 3, by specifying suitable values of parameters, the solution ( 21) is shown in different forms as follows: Figure 3(A,C,E) are solitons in periodic form of dissimilar amplitude and their 2D contourplots in Figure 3(B,D ,F), correspondingly.
In Figure 4, by specifying suitable values of parameters, the wave solution (30) is illustrated in disparate shapes as follows: Figure 4(A,C,E) are solitons in multipeak forms and their 2D contourplot in Figure 4(B,D ,F) correspondingly.In Figure 5, by specifying suitable parameter values, the solution (32) is illustrated in various shapes as follows: Periodic solitons with varying amplitudes are shown in Figure 5(A,C, E), and their 2D contourplots are shown in Figure 5(B,D, F), respectively.Figure 6(A) depicts the DR between the ω and k of (38) and the DR between ν and wave numbers (ω 1 , ω 2 ) of ( 43) is demonstrated in Figure 6(B), respectively.

Conclusion
We effectively applied the proposed technique to the NCKdV and the NCZ-K systems in this work.The KdV equation, along with shallow-water waves and long internal waves in oceans, basically explains how long, one-dimensional waves propagate in a variety of physical conditions.The study of coastal waves on the basis of the ocean is done using the Zakharov-Kuznetsov (Z-K) equation and this model is utilized to illustrate ion-acoustic wave propagation.As a result, different forms of analytical solutions of the new coupled KdV (NCKdV) system and new coupled Z-K (NCZ-K) system, such as solitons, multi-peak solitons, solitary waves, trigonometric, hyperbolic and rational function solutions and other wave solutions, are constructed in the explicit form using the proposed scheme.By giving the parameters, the appropriate values, novel structures of constructed solutions of these models are represented.Graphical representations of the physical structures of a few obtained results are efficient for conveying the intricate physical existences of both models.The stability of the solution is examined by utilizing modulation instability.The results obtained show that the proposed technique is universal and efficient, and can be applied on lots of other new coupled systems.

Disclosure statement
No potential conflict of interest was reported by the author(s).