Dynamical behavior of solitons of the (2+1)-dimensional Konopelchenko Dubrovsky system

Utilizing nonlinear evolution equations (NEEs) is common practice to establish the fundamental assumptions underlying natural phenomena. This paper examines the weakly dispersed non-linear waves in mathematical physics represented by the Konopelchenko-Dubrovsky (KD) equations. The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(G^\prime /G^2)$$\end{document}(G′/G2)-expansion method is used to analyze the model under consideration. Using symbolic computations, the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(G^\prime /G^2)$$\end{document}(G′/G2)-expansion method is used to produce solitary waves and soliton solutions to the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2+1)$$\end{document}(2+1)-dimensional KD model in terms of trigonometric, hyperbolic, and rational functions. Mathematica simulations are displayed using two, three, and density plots to demonstrate the obtained solitary wave solutions’ behavior. These proposed solutions have not been documented in the existing literature.


Introduction to the (G ′ /G 2 )-expansion method
In this section, we give a general overview of ansatz that is utilized to compute the traveling wave structures of some nonlinear equations.Here are the main steps for the ansatz of (G ′ /G 2 )-expansion method 22,31 .
Step 1.Consider a general system of NPDEs (with two dependent variables and three independent variables) as where u, v are unknown functions of independent variables x, y, t.
Step 2. We apply the following transformation to convert (2) into a system of nonlinear ordinary differential equations (NLODES) After some mathematical calculations system (4) is converted into a single NLODE where ( ′ ) indicates the differentiation w.r.t.ξ.

Case-iii: Rational type solutions
When we have η = 0 , ψ = 0 then from (7) rational function solution can be written as where in all above cases E 1 and E 2 are constants.In the next section we apply the introduced method.
The first equation is integrated, and the result is By incorporating it into the second equation of the NLODEs system and ignoring the integration constant, the following NLODE is produced as By applying the balancing principle to the Eq. ( 9), we get at M + 2 = 3M , which results in M = 1 .Using (6)  for M = 1 formula, we arrive at the trial solution presented as where d 0 , d 1 and d −1 are constants.On inserting (10) and its derivatives along with (7) into (9), we get a system of equations.We obtain the set of algebraic equations by setting the coefficients of powers of (G ′ /G 2 ) to zero We obtain the following set of parameters by solving the aforementioned system using the computer program like Mathematica, Maple, or MATLAB. Set-1 Vol:.( 1234567890)

Set-7
The three possibilities for the promising solution of (G ′ /G 2 ) from Set-1 are listed below; Case-i: Trigonometric type solutions If we have ηψ > 0 , then (7) gives

Case-iii: Rational type solutions
If we have η = 0 , ψ = 0 then from (7) rational solution can be written as where E 1 and E 2 are real parameters.By back substituting the values in (10), we get the solutions of the system (1) in the form; Family-1: Trigonometric type solutions If we have ηψ > 0 , then ( (33) .

Graphical interpretation of some solutions
We can demonstrate the graphical representation of the wave solution profile of various solution surfaces in this section.Using symbolic computations, the (G ′ /G 2 )-expansion is used to produce waveform soliton solutions to the (2 + 1)-dimensional KD model in terms of trigonometric, hyperbolic, and rational function solutions.

Discussion and conclusions
It is common practice to utilize NEEs to establish the fundamental assumptions underlying natural phenomena.In this paper, the weakly dispersed non-linear waves in mathematical physics were represented by the KD equations.The (G ′ /G 2 )-expansion method was used to analyze the model under consideration.Several researchers have obtained the analytical solutions of the KD system using hyperbolic, trigonometric, and rational functions.In summarizing previous work, Wazwaz 24 reported the solutions, introducing kink, soliton, and periodic wave solutions.Subsequently, Feng et al. 25 explored solitary wave, periodic wave, and variable-separation solutions.Concurrently, Kumar et al. 26 derived periodic waves, singular solutions, cnoidal, and snoidal waves.Additionally, Alfalqi et al. 28 outlined solutions, including shock waves, singular solutions, solitary waves, periodic singular waves, and plane waves.Khater et al. 29 established periodic waves, kinks, and solitary waves.In an approach, Kumar et al. 30 obtained soliton solutions such as kink waves, periodic waves, and oscillating waves.Although various approaches in the discussion produced trigonometric, hyperbolic, and rational function solutions with distinct structures, our approach also explored the same class of rational, hyperbolic, and trigonometric functionbased solutions.Comparing results, we conclude that the literature has not featured any of our produced solutions.The obtained solitary wave families validate the method, showcasing applications in solitary wave theory, mathematical sciences, and nonlinear sciences.For future studies, we intend to investigate the KD system using the Jacobi elliptic function method to derive elliptic function-based solutions.

Family- 2 :Family- 3 :Family- 5 :Family- 6 :
https://doi.org/10.1038/s41598-023-46593-zwww.nature.com/scientificreports/Hyperbolic type solutions If we have ηψ < 0 , then Rational type solutions If we have η = 0 , ψ = 0 then rational function solution is given as By substituting the values of Set − 2 in (10), we get the solutions of the system (1) in the form; Family-4: Trigonometric type solutions If we have ηψ > 0 , then Hyperbolic type solutions If we have ηψ < 0 , then Rational type solutions If we have η = 0 , ψ = 0 then rational function solution is given as By substituting the values of Set − 3 in (10), we get the solutions of the system (1) in the form; Family-7: Trigonometric type solutions If we have ηψ > 0 , then

.Family- 8 :Family- 9 :Family- 11 :Family- 12 :
https://doi.org/10.1038/s41598-023-46593-zwww.nature.com/scientificreports/Hyperbolic type solutions If we have ηψ < 0 , then Rational type solutions If we have η = 0 , ψ = 0 then rational function solution is given as By substituting the values of Set − 4 in (10), we get the solutions of the system (1) in the form; Family-10: Trigonometric type solutions If we have ηψ > 0 , then Hyperbolic type solutions When we have ηψ < 0 , then Rational type solutions If we have η = 0 , ψ = 0 then rational function solution is given as By substituting the values of Set − 5 in (10), we get the solutions of the system (1) in the form; Family-13: Trigonometric type solutions When we have ηψ > 0 , then
Figures 1, 2, 3 and 4 show the periodic waves as well as solitary wave solutions for the KD system (1).An essential set of parameters that are mentioned with each case are included in each plot.

Figure 1 .
Figure 1.The nature of solitary wave solution (22) obtained by fixing all parameters to 1.

Figure 2 .Figure 3 .
Figure 2. The nature of periodic waves (25) obtained by fixing all parameters to 1.

Figure 4 .
Figure 4.The wave nature of rational polynomial solution (33) obtained by fixing all parameters to 1.