On Optical Solitons of the Fractional (3+1)-Dimensional NLSE With Conformable Derivatives

In this article, the fractional (3+1)-dimensional nonlinear Shrödinger equation is analyzed with kerr law nonlinearity. The extended direct algebraic method (EDAM) is applied to obtain the optical solitons of this equation with the aid of the conformable derivative. Optical solitons are investigated for this equation with the aid of the EDAM after the nonlinear Shrödinger equation transforms an ordinary differential equation using the wave variables transformation.


INTRODUCTION
Over the past few decades, there have been many studies on optical solitons [1][2][3][4][5][6][7][8][9][10]. The nonlinear wave process can be viewed in several scientific fields, such as optical fiber, quantum theory, plasma physics, fluid dynamics [11,12], etc. Solitons are one pulse forms which are created due to the proportion between nonlinearity and wave stage speed dispersal impacts in the system. The envelope soliton, which holds both fast and slow vibrations, performs for nonlinearity proportions with the wave group dispersal impacts in the physical systems. The envelope soliton is controlled by a small field adjusted wave package whose dynamics are controlled via the nonlinear Schrödinger equation (NSE) [1][2][3][4][5][6][7][8][9][10][11][12]. The analytical solutions of these NPDEs plays a significant part in the analysis of nonlinear phenomena. Over the past few decades, numerous methods were developed to obtain analytical solutions of NPDEs such as the inverse scattering method [13], the Sinecosine function method [14], the tanh-expansion method, and the Kudryashov-expansion method [15], etc.
There has also been considerable interest and significant theoretical improvements in fractional calculus, applied in many fields, and in fractional differential equations and its applications [16][17][18][19][20][21][22][23][24][25]. Nonlinear fractional partial differential equations (FPDEs) are a special type of NPDEs. Several studies have discussed these equations. Additionally, FPDEs are significant in several analyses because of the iterative reporting and the probability explanation process in water wave hypothesis, nonlinear optics, fluid dynamics, plasma physics, optical fiber, quantum mechanics, signal processing, and so on. Several researchers have investigated the wave solutions of NPDEs with the aid of some mathematical algorithms. Besides, one advantage of the conformable fractional derivative is that it is easy to apply [26][27][28][29][30][31][32][33][34].
The conformable derivative of order α ∈ (0, 1) is defined as the following expression [28] A few properties for the conformable derivative are given by [28,31].
Recently, there have been about the conformable model of fractional computations [25][26][27][28][29][30][31][32][33]. The (3+1)-dimensional dependent NLSE is given by: where F is a real-valued function and has the fluency of the complex function F( q 2 )q : C → C. When the F( q 2 )q is k times continuously differentiable, the following situation can be written, For Kerr law nonlinearity, Equation (1.2) is converted to In (1.3), the first expression describes the evolution condition, the second expression, describes the dispersal in x, y, and z directions while the third expression describes nonlinearity. Solitons are the consequence of an attentive adjust between dispersal and nonlinearity.
In this work, we analyze the fractional (3+1)-dimensional nonlinear Shrödinger's equation with the aid of a conformable derivative operator to find solitons using the extended direct algebraic method (EDAM) [8,26].
This method is a powerful in solving nonlinear evolution equations and it can be applied to solve the above mentioned equations. This has led to the innovation of many modern techniques to solve these equations. There are several advantages and disadvantages of this modern method. Although a closed type soliton solution can be found with the aid of this process, the disadvantage of this method is that this technique cannot calculate the conserved quantities of nonlinear evolution equations.

DESCRIPTION FOR THE EXTENDED DIRECT ALGEBRAIC METHOD
Suppose the general nonlinear partial differential equation, where q = q(x, y, z), U is a polynomial in q = q(x, y, z, t) and the x, y, z, t define the partial fractional derivatives.
• Assume the traveling wave transformation: With the aid of (2.2) wave transformation, Equation (2.1) is changed into an ordinary differential equation for v(φ) : where the sub-indices define the ordinary derivatives with respect to φ.
where a M = 0 and G(φ) can be satisfied as follows: where f , g, h are arbitrary constants.

SOLUTIONS OF TIME FRACTIONAL (3+1)-DIMENSIONAL NLSE WITH KERR LAW NONLINEARITY USING CONFORMABLE DERIVATIVES
Now, suppose the wave variable transform: q(x, y, z, t) = e i(a(x cos ξ +y cos κ+z cos χ )+w t α α ) v(φ), φ = x cos ξ + y cos κ + z cos χ + Q t α α , Suppose the solution of Equation (3.2) is expressed as a finite series. We can write this solution as follows, where G(φ) satisfies Equation (2.5), φ = x cos ξ + y cos κ + z cos χ + Q t α α and a k for k = 1, M are values to be described.
We can write the solution of Equation (3.3) in the following form:

1) When
< 0 and f = 0, the singular periodic solutions are obtained as follows

5)
When g = 0 and f = h, the singular periodic solution is obtained as follows 6) When g = 0 and f = −h, the combined soliton solution is obtained as follows

7)
When g 2 = 4hf , the rational solution is obtained as follows 8) When h = 0 and g = 0, the singular soliton is obtained as follows )),
We obtained the sum of solutions found for the fractional (3+1)-dimensional NLSE with kerr law nonlinearities via the conformable fractional derivative operator. In addition, we presented some graphics of solutions in Figures 1, 2.

CONCLUSION
In this article, the EDAM is applied to find new soliton solutions for the (3+1)-dimensional NLSE with kerr law  Frontiers in Physics | www.frontiersin.org nonlinearities, with the aid of the conformable fractional derivative operator. The dark, bright, and combined optical solitons are obtained. There are 12 different situations in these solutions. The existence of solutions obtained from these functions are all stipulated through limitation states that are also listed in addition to the solutions. Some interesting figures are also presented in Figures 1, 2