Construction of diverse water wave structures for coupled nonlinear fractional Drinfel’d-Sokolov-Wilson model with Beta derivative and its modulus instability

This paper aims to analyze the coupled nonlinear fractional Drinfel’d-Sokolov-Wilson (FDSW) model with beta derivative. The nonlinear FDSW equation plays an important role in describing dispersive water wave structures in mathematical physics and engineering, which is used to describe nonlinear surface gravity waves propagating over horizontal sea bed. We have applied the travelling wave transformation that converts the FDSW model to nonlinear ordinary differential equations. After that, we applied the generalized rational exponential function method (GERFM). Diverse types of soliton solution structures in the form of singular bright, periodic, dark, bell-shaped and trigonometric functions are attained via the proposed method. By selecting a suitable parametric value, the 3D, 2D and contour plots for some solutions are also displayed to visualize their nature in a better way. The modulation instability for the model is also discussed. The results show that the presented method is simple and powerful to get a novel soliton solution for nonlinear PDEs.

has been attained in 47 .The series solution of the DSW model was attained by using the Adomian decomposition method 48 .
The coupled (1+1)-dimensional DSW model 49 which read as, We can write the above system in the form of fractional derivative with respect to time is given by, Here, a, γ 1 , 1 and η 1 are the constant and the α represents the order of fractional derivative with 0 < α ≤ 1 .When α = 1 Eq.( 2) is converted to classical DSW equation, which was first introduced by Drinfel' d and Sokolov 50,51 and studied by Wilson 52 .In this article, we will construct an exact solution for the Drinfel' d-Sokolov-Wilson model using the generalized rational exponential function method approach with the help of well-known Beta derivative.The solutions are attained in the form of singular bright, dark, periodic, bell and lump-type water wave structures.The achieved solutions might be useful to comprehend nonlinear phenomena.It is worth noting that the implemented method for solving NPDEs is efficient, and simple to find further and new-fangled solutions in the area of mathematical physics and coastal engineering.Diverse types of fractional derivatives have been used in the past, such as Caputo fractional 53 , Beta derivative 54 , Conformable fractional 55 , Reimann-Liouville 56 and truncated M-fractional derivative 57 etc. have importance in fractional calculus.The remaining article is distributed into various sections.Section (2) contain definition from fractional calculus relevant to our study.In Sect.(3) we have discussed the main step of the method.In Sect.(4) solitary wave solutions have been described.Numerical simulations of some attained solutions are given in (5).In Sects.( 6) and ( 7) modulus instability, a conclusion is presented.

Beta derivative
Definition Let �(t) be a function defined for all non-negative t.The function �(t) 58 is, Theorem Let and g be any two function, = 0 , and α ∈ (0, 1] then

3:
For c any constant, the following relation can be easily satisfied D α t c = 0,

Methodology
The GERF method is a quite novel technique for nonlinear partial differential equations (NLPDE) 49 .The main steps are given as: Step:1 Consider the NLPDE as, Suppose the travelling wave transformation, Substituting (5) into (4) then we get ODE given as, Step:2 (1) (2) (5) �(x, t) = �(̟ )e ιφ(x,t) .www.nature.com/scientificreports/Solution of equation of ( 7) is, Here, a 0 , a n , and b n are unknown parameters to be found.The function φ(̟ ) is defined as Step: 3  We apply the homogeneous balance technique on (7) to attain the value of N.
Step:4 Substituting (7) with equation ( 8) into ( 6), then we attain the system of algebraic equations.The system is solved by utilizing Mathematica software, and then the achieved solution of ( 8) is put into (7) by using (5).Finally, the solution of ( 4) is attained.

Solitary wave structure
We consider the travelling wave transformation for FDSW (2) as follows, Using ( 9) to ( 2) and then we get , From (10), we have Putting the value of into (11) and integrating one time then we get, Now we have to apply the balancing technique on (13) then we get N = 1 .Utilizing N = 1 in (7) then we get, where a 0 , a 1 , and b 1 are unknown constants to be find.The solution of (2) is discussed as, When equations ( 14) and ( 15) are putting into equation (13), we arrive at a system of algebraic linear equations.By solving these equations simultaneously, we obtain the following set of solitary wave solutions.set-1 Putting ( 16) into ( 14) then solution of ( 2) is,

Set-4
Substituting ( 25) into ( 14) then solution of ( 2) is, When equations ( 28) and ( 15) are putting into equation (13), we arrive at a system of algebraic linear equations.By solving these equations simultaneously, we obtain the following set of solitary wave solutions.

Set-3
Putting Eq. ( 35) into ( 14) then solution of (2) is, Vol  38) into ( 14) then solution of ( 2) is, When equations ( 41) and ( 15) are putting into equation ( 13), we arrive at a system of algebraic linear equations.By solving these equations simultaneously, we obtain the following set of solitary wave solutions.

Set-1
Putting ( 42) into ( 14) then solution of ( 2) is, When equations ( 45) and ( 15) are putting into equation (13), we arrive at a system of algebraic linear equations.By solving these equations simultaneously, we obtain the following set of solitary wave solutions.

Set-1
Substituting ( 46) into ( 14) then solution of ( 2) is, When equations ( 49) and ( 15) are putting into equation (13), we arrive at a system of algebraic linear equations.By solving these equations simultaneously, we obtain the following set of solitary wave solutions. Set-1 where κ and ω are the wave number and frequency of perturbation.Putting (76) into (75), the dispersion rela- tion (DR) is acquired as from (77), one can see that the real component is negative for all values of κ then any superposition of the results will appear to decay.So, the dispersion is stable.

Conclusion
In this work, we have successfully achieved some fresh and further general traveling wave solutions to the nonlinear fractional Drinfel' d-Sokolov-Wilson (FDSW) model with beta derivative.The solutions attained by using the GERF method for the proposed model are competent to examine the scientific model of gravity water waves in shallow water.It is capable of investigating plasma waves in the seaside oceans and breaking down the unidirectional spread of long waves in oceans and harbors.The proposed method is not only more powerful than previous approaches but has also introduced novel solutions that have not been reported before. ( .) = 0.