On the exact solutions of optical perturbed fractional Schrödinger equation

In the present study, the improved sub-equation method is applied to the optical perturbed fractional Schrödinger equation with Beta-derivative and the exact optical solutions are obtained. The generalized hyperbolic and trigonometric function solutions are found by the method. Several novel physical surface structures of the solutions are presented with various appropriate assigned values. The method aids in solving complicated physical phenomena of these dynamical models. Numerical implementations and graphical illustrations verify the theoretical results.


Introduction
Partial differential equations (PDEs) of the nonlinear type are of great interest in modeling and understanding significantly complicated phenomena such as optics, fluid and solid mechanics, propagations of dynamical systems, chemical reactions of fluid dynamics, quantum theory, and so on.Nonlinear PDEs are intensively studied by many researchers, and various methods are obtained.In the literature, many practical mathematical methods have been used to obtain exact solutions of the nonlinear PDEs .Researchers in mathematics have been paying a lot of attention to the new PDE research field of fractional calculus.Many phenomena in physics such as optical fibers, viscoelasticity, plasma, solid mechanics, electromagnetic waves, signal processing, fluid dynamics, biomedical sciences, and diffusion processes can be modeled by fractional PDEs.To get more accurate results for the solutions of PDEs, mathematicians developed several solution methods.The analytical and precise numerical solutions to problems of mathematical models and several physical phenomena can be formally expressed by fractional derivatives.This kind of derivative has been the subject of many recent types of research [7][8][9][10][11].

A brief overview of the Beta-derivative
Conformable fractional derivatives are much easier to use and conform to some standard features that existing fractional derivatives do not have, such as the chain rule.Nevertheless, this fractional derivative has a crucial weakness that is the fractional derivative of any differentiable function at point zero does not fulfill any physical problem and hence cannot be physically comprehended at this time.To improve the conformable derivative's limits, a modified version was built.The derivative is determined by the interval in which the function is differentiated [45].
The term 'Beta-derivative' was originated by Abdon Atangana [21,45,46].Some parts of the suggested version are employed to model various physical phenomena and functioned as fractional derivative restraints.They are a logical extension of the classical derivative even though they are not fractional derivatives [18].It possesses the following properties which the well-known fractional derivatives do not have: x f be a function.The Beta-derivative of ( ) x f is defined by (see, [45]) Some of the important properties of the Beta-derivative are presented as follows (see, [21,45,46]).
For the proof of these statements, we refer to [21,45,46].
¥   be a given function.Then one can show that Atangana's Beta-integral of χ is

Analytical solution methodology
In the present section the basic solution steps of the improved sub-equation method (see, [23,47]) will be presented.We consider the following nonlinear partial differential equation ( ) Here, s is unknown function, F is a polynomial in s and its partial derivatives with respect to s and β is the order of the Beta derivatives of s x t , ( ).Now let us present the solution method with the following steps: Step 1.In the first step we need to select appropriate complex transforms We use the following traveling wave transformations where r, k, m, are nonzero constants.Equation (1) can be reduced to the following nonlinear order differential equation ( ) Step 2. Assume that the solution S ( ) e of equation ( 4) is for the improved sub-equation method.The coefficients d i will be calculated.
Here, N Î +  is determined by considering the homogeneous balance between the nonlinear term and the highest order derivative in the differential equation (4).We obtain the balance term as follows.Let us denote the degree of S ( ) The function ( ) q x satisfies the following second order Riccati equation with v as a constant.
Step 3. The explicit solution for (7) is obtained in [22], by using the generalized exponential function method Step 4.One can obtain a polynomial in terms of ( ) q x substituting (5) and (7) in (4).Next, by collecting the like terms and equating all the coefficients of the polynomial to zero, we get a system of algebraic equations.
Step 5.In the final step, the system of algebraic equations that is obtained in the last step will be solved by the Maple package for d i and v.It will be done by replacing exact solutions for equation (1) with newly obtained values in equation (8).

Mathematical analysis of the analytical solutions obtained with the improved subequation method
In the research field of optics, fractional differential equations have been applied to model several phenomena, including waveguide optics, nonlinear optics, and light propagation in disordered media.For instance, the dynamics of light waves in disordered media have been discussed using the fractional Schrödinger equation.Since it takes into account the long-range correlations that are inherent in such media, this equation provides a more accurate depiction of the wave dynamics than conventional wave equations.Now we consider the time fractional perturbed Schrödinger equation (see, [50]) in terms of Beta-derivative i s t s ass i bs c s s h s s 0, 0, 1 9 where s(x, t) display complex wave in (9).Here b, c, h represent distribution coefficients, in particular, b is a thirdorder distribution, c and h are nonlinear distributions.
Using the traveling wave transformation (2), as well as equation (3) in equation ( 9), these equations are transformed into the ordinary differential equation below.
Letting b = 0 and c + 2h = 0 in the imaginary part of (10), we get Thus, the following equation needs to be solved.
By the help of (6), and balancing between S ( ) x  and S 3 ( ) x , we find 1. + =  = Thus, from equation (5), the solution of equation ( 11) is where d −1 , d 0 , d 1 are constants to be calculated.By substituting equation (12) along with equation (7) into equation (11), the algebraic equation of θ(ξ) is obtained.In this equation, all coefficients of the powers of θ(ξ) are set to zero.We derive the following system of algebraic equations: Solving this system by symbolic computation software gives the following four sets of coefficients: Case 1.
. ) gives the set of hyperbolic solutions of equation (11) as where where for v < 0 and Beta-derivative illustrates the physical properties of β derivative values for s x t , 1 ( ).Using Case 2, equation (12) gives the set of trigonometric solutions of equation (11) as shows the physical properties of β derivative values for s x t , 3 ( ).Using Case 3, equation (12) gives the set of hyperbolic solutions of equation (11) as Using Case 4, equation (12) gives the set of hyperbolic solutions of equation (11) as for v < 0 and Beta-derivative

Results and discussions
In the present research, the well-known Beta-derivatives are applied to obtain the optical solitons for the governing equation with the nonlinear perturbed fractional Schrödinger equation.The solutions are obtained under necessary constraints in order to avoid trivial solutions and singularities.The Maple implementations and graphical illustrations of the physical behavior of the solutions that are obtained by the Beta-derivatives are presented.
As free parameters are assigned specific values, fractional impacts for four distinct values have been graphically displayed for some of the found solutions.Each picture illustrates fractional effects for β = 0.3, β = 0.5, β = 0.75 and β = 0.99 respectively.
First of all, s 1 (x, t) has been drew and as displayed in figure 1, some appropriate hyperbolic behaviors of the obtained analytical optical solutions have been shown.In this figure, the hyperbolic solutions of s 1 (x, t) is plotted for some β values when k = c = a = 1, v = − 1.It is obvious from the graphical presentation that the shape of the wave seems similar for different values of the fractional order using the β derivative.In figure 2, the graphs corresponding to some suitable values of the trigonometric solution found as s 3 (x, t) have been presented.It has been taken as k = c = a = v = 1.It is clear from the graphical presentation that the shape of the waves appears different using the above fractional order values.
It is possible to find exact solutions to fractional differential equations in a systematic way using the improved sub-equation method, a potent analytical tool.With the distinctiveness and restrictions of the method, hyperbolic and trigonometric solutions have clearly appeared.As a result, significantly newer and more understandable solutions have been found.The physical meaning of fractional derivatives has been the subject of several investigations.Fractional derivative is an emerging field of research that has drawn numerous researchers.

Conclusions
In this paper, a nonlinear fractional differential equation with Beta-derivative is solved using the improved subequation method.The time-fractional perturbed Boussinesq equation effectively produced several novel exact solutions.A comprehensive variety of options, including trigonometric and hyperbolic ones, are used for the improved sub-equation method analysis of the model mentioned so far.It is clear that our research is novel and hasn't been investigated previously in the literature.It is because the solutions of equation (9) are found by employing Beta-derivative and the presented method.The originality of our results is also confirmed by the fact that the solutions for the equation in [50] are produced utilizing Beta-derivative.
The results of this work contribute to the body of literature by skillfully justifying a variety of nonlinear systems.The answers that are described here have not, to our knowledge, been discovered before.The results also proved using Maple software, a symbolic program computing system, to be a useful mathematical tool that enhances, accelerates, and simplifies the recommended method.Attention should be paid to the potential that the proposed method can be used to solve many nonlinear evolution problems in mathematical physics.The findings of the study might influence how various physical issues are understood.

Figure 2 .
Figure 2. When k = c = a = v = 1, the trigonometric solutions of s 3 (x, t) are illustrated for some β values.