Extractions of some new travelling wave solutions to the conformable Date-Jimbo-Kashiwara-Miwa equation

In this paper, complex and combined dark-bright characteristic properties of nonlinear Date-Jimbo-Kashiwara-Miwa equation with conformable are extracted by using two powerful analytical approaches. Many graphical representations such as 2D, 3D and contour are also reported. Finally, general conclusions of about the novel findings are introduced at the end of this manuscript.


Introduction
In the last two decades years, applied mathematics and physics nonlinear phenomena play an important role in soliton theory, the calculation of analytical and numerical solutions, especially the travelling wave solutions of nonlinear equations in mathematical physics [1]. Thus, deeper investigation of the analytical solutions to the nonlinear evolution equations with the help of newly developed and improved approaches have been considered as one of the important study area in 4 2 6 2 0, xxxxy xxy x xxx y xy xx yyy where  and  are non-zero and   ,, u u x y t  is the wave-amplitude function, which describes long water waves. In case of 1 in Eq (4), gives (5), gives   sin . w w   (6) Solving Eq (6) via separation of variables, we obtain     sin sec ( ), w h    (7)     cos tan ( ). w h    (8) Suppose that the nonlinear fractional differential equations given in the more general form;   2 , , , , 0, x t P u u u u   (9) where and    (11) Applying the homogeneous balance principle between the highest power nonlinear term and highest derivative in the nonlinear ordinary differential equation (NODE), we determine the value of n . Putting Eq (11) (10), we get the new travelling wave solutions to the Eq (9).

The structure of IBSEFM
In this subsection of the paper, we will mention about general structures of the IBSEFM [29,30].
Step 1: Suppose that the following fractional differential equation, where ,  are real constants and can be determined later. This transformation reduces Eq (12) into NODE as following; Step 2: Let consider the trial solution form of Eq (14) as following We can determine the general form of Bernoulli differential equation for F  according to Bernoulli theory as , 0, 0, We put account the homogeneous balance principle to determine the relation between , and n m M constants.
Step 3: Equating all the coefficients of   Step 4: We have two situations depend on and bd according to solution of Eq (16).
We get the analytical solutions to Eq (14) via software program by using complete discrimination system for polynomial of   F  .

Application of SGEM
In this section of the paper, we apply SGEM to the Eq (1) to investigate some analytical solutions such as exponential and complex.
First of all, we transform Eq (1) into a NODE by the following wave transformation where ,, k m c are non-zero,  is conformable derivative order. Putting Eq (20) into Eq (1) and after some simple calculating, we reach the following NODE   where , V U   and also both integral constants are zero.
With the help of balance principle for Eq (10), we find 2 n  . For this value, Eq (10) can be written as and its second derivation  Putting Eq (22) and its second derivation into Eq (21), we find a trigonometric algebraic equation.
where 2 ,, Bm  are real constants and non zero.
Bm  are real constants and non zero.
in which , , , km are real constants and non zero.
in which , , , km are real constants and non zero.

Application of IBSEFM
This section applies IBSEFM to the governing model Eq (1) for obtaining some new complex analytical solutions. Let's consider the following wave transformation into Eq (1) Using this relationship, we can find many new complex solutions for governing model as following cases.
gives the following complex solution to the governing model        (40)

Discussion, comparision and physical explanations
This paper finds entirely new complex analytical solutions for governing model with the help of two powerful approaches such as SGEM and IBSEFM. These solutions have some more important physical features. The hyperbolic secant (bright soliton) arises in the profile of a laminar jet, the hyperbolic tangent (dark soliton) arises in the calculation of magnetic moment, the hyperbolic sine (periodic wave solution) arises in the gravitational potential, and the hyperbolic cotangent (singular soliton) arises in the Langevin function for magnetic polarization [76]. In this sense, Eq (24) is a dark soliton solution. Eqs (25)(26)(27)(28)(29) are used to explain the combined dark-bright soliton solutions. It is estimated that these solutions may be related to such physical meanings. When we compare these solutions in [21], one can see that these solutions entirely new complex dark, mixed dark-bright and dark soliton solutions to the governing model.
In IBSEFM, if we consider more values of 3, 2 Mm  as 6 n  , we obtain another new solution for the governing model as 2

Conclusions
This paper studies on the nonlinear Date-Jimbo-Kashiwara-Miwa equation with conformable which defines to explain wave propagation. By using SGEM and IBSEFM, we reach the some new dark, bright, singular solitons and complex wave solutions. All the found wave solutions in this study are entirely new and they have satisfied the nonlinear Date-Jimbo-Kashiwara-Miwa equation with conformable. Under the suitable chosen of the values of parameters, we plotted 2D, 3D and contour simulations of the wave solutions. From these Figures (1-35), it may be observed that wave solutions to the studied nonlinear model show the estimated wave propagations.