TheExact Solutionsof theConformableTime-FractionalModified Nonlinear Schrödinger Equation by the Trial Equation Method and Modified Trial Equation Method

In the past ten years, NLSEs have received a great deal of interest from scholars and researchers. Actually, simplified versions of Zakharov’s system give rise to certain NLSHs. For more details, see [1]. NLSEs have applications in different subjects, e.g., quantum mechanics, biology, semiconductor industry, optical communication, energy quantization, quantum chemistry, wave propagation, protein folding and bending, condensed matter physics, solid-state physics, nanotechnology and industry, laser propagation, and nonlinear optics. Lately, the investigation of the CTFNLSE in the form


Introduction
In the past ten years, NLSEs have received a great deal of interest from scholars and researchers. Actually, simplified versions of Zakharov's system give rise to certain NLSHs. For more details, see [1]. NLSEs have applications in different subjects, e.g., quantum mechanics, biology, semiconductor industry, optical communication, energy quantization, quantum chemistry, wave propagation, protein folding and bending, condensed matter physics, solid-state physics, nanotechnology and industry, laser propagation, and nonlinear optics.
Lately, the investigation of the CTFNLSE in the form including numerics, analysis, and applications becomes a significant issue in applied mathematics, and diverse computational methods have been designed to discuss the exact solutions of it. Every method has its own proportion formulas and merits for the application to the administering equation for exploring the exact solutions.
In [2][3][4], the exact solution of CTFNLSE (1) is obtained through different methods such as the first integral method, functional variable method, sine-Gordon method, and direct algebraic method. In [5], Younas et al. presented the CTFMNLSE and studied the exact solutions of it by the generalized exponential rational function method.

Preliminaries
Here, it would be helpful to present some properties and definitions of the conformal derivative and other preliminaries.

Methods and Applications
In this section, we present the first step of the TEM and the MTEM for finding analytical solutions of the CTFMNLSE defined as (2). For more details, see [5]. Suppose a CTFNLPDE where Φ and Γ are an unknown function and a polynomial in its arguments, respectively.
Using a fractional travelling wave transformation where V is the velocity and substituting (7) into (6), we have a NLODE given by where ′ denotes the derivative with respect to ξ.
Here, since Ψ � Ψ(X, τ) in (2) is a complex function, for proceeding, we begin with the following travelling wave assumption: where ξ � η(X − (V/α)τ α ) and ψ � − KX + (X/α)τ α + ζ, and ζ, X, and K are parameters, representing the phase constant, frequency, and wave number, respectively. Substituting (9) into (2), we get real and imaginary parts as follows: Now, integrating the imaginary part of the equation and taking constant equal to zero, one may have From (10) and (12), it can be followed that From the above, it can be followed that Rewrite (10) into the following form: where In the next two sections, we investigate the primary steps for detecting the exact solution of (10) by using the TEM and the MTEM. e exact solution of (12) can be found in a similar way. (6) is reduced to NLODE (8) under transformation (7). Secondly, consider the trial equation

Trial Equation Method (TEM). Firstly, CTFNLPDE
where a i and n are constants, which are derived from the solution of the system and the balancing principle, respectively. Finally, the solution of (17) can be given by the integral form: then integrating (19) with respect to ξ once, we get where n, a i , and integration constant d are to be determined. Now, considering (16) and balancing Λ ″ and Λ 3 , we obtain n � 4. So, the trial equation is in which so we have Now, (18) is rewritten with (23): If we set a 0 � 0 in (24) and integrate this equation, the exact solution of (2) is obtained: If λ 2 > 0, then we have bright and singular solutions, respectively: If λ 2 < 0, the singular periodic solutions appear as where ξ 0 is an arbitrary constant. For more details, see [13][14][15].

Modified Trial Equation Method (MTEM).
In this section, instead of using (17), the trial equation can be chosen as which is seen as the modified trial equation method when the same procedure is applied with the integral form solution of (29) as Similarly, one can solve integral (30) and find the exact travelling wave solution of equation (6).
If we now use trial (29) in (16) for balancing the procedure, we get N − M � 2. Choosing N � 2 and M � 0, the trial equation is

Advances in Mathematical Physics
and we have where a 2 ≠ 0 and b 0 ≠ 0. e corresponding system is Solving the corresponding system, we obtain Using these coefficients in (30), we get When we integrate this equation and use the wave transformation, the exact travelling wave solutions of (2) are obtained as follows.

Numerical Results in Tables and Charts
Here, we let       Table 2: e imaginary part of exact solutions of CTFMNLSE (1) obtained by the TEM with several point sources through arbitrary.          (2) Obtained through the MTEM. Considering the given values in Section 4, we get Tables 3 and 4.

Concluding Remarks
Using the TEM and MTEM, firstly, we found the exact solutions of CTFMNLSE (2), and finally, we presented numerical results in tables and charts.
According to Table 5, we can observe that the differences between solutions Ψ 1,1 and also Ψ 2,1 for fixed X and different values of τ are considerable. It is clear that these differences obtained through the MTEM are less than those of the TEM. In other words, for fixed X, by changing the value of τ, the MTEM results in more minor changes than the TEM. Also, based on Figure 7, we can observe that the results gained by the MTEM have higher accuracy than the TEM. Figure 8 displays the differences between the real part of solutions obtained by the TEM and MTEM for fixed X � 0.062 and α � 0.90. As you can see, the difference obtained among Ψ 1,3 , Ψ 1,4 and Ψ 2,3 , Ψ 2,4 is more minor than the difference obtained among Ψ 1,1 , Ψ 1,2 and Ψ 2,1 , Ψ 2,2 .

Data Availability
No data were used to support this study.