1. Introduction
Lately, it has become significant for scholars and researchers to investigate the analytical solutions to NLPDEs with the assistance of computational bundles, which simplify boring and tedious mathematical calculations. NLPDEs arise when one describes the physical mechanisms of natural phenomena in optical fibres, geochemistry, ocean engineering, geophysics, fluid mechanics, and many other fields. As a result, nonlinear phenomena are now, considered by many scholars in this novel field of science.
The nonlinear Schrödinger equation (NLSE), one of NLPDEs, has been investigated in such areas as nonlinear optics and condensed matter physics due to its potential physical applications and mathematical properties. It was found that the NLSE can illustrate the dynamic of nonlinear localized waves generally associated with breather, soliton, and rogue waves. Indeed, rogue waves can be informally thought of as rare events arising from the constructive interference of the surrounding waves. This construction interference can eventuate in waves with heights as large as ninth times that of the surrounding free surface. Multitude real-world consequences of these large spontaneous events exist, which can disrupt and damage human lives, commercial shipping, and many other nautical operations.
The modified nonlinear Schrödinger equation (MNSE) [
1], which models rogue waves in ocean engineering, is given by
where
, and
and
are the wave number and the frequency of the carrier wave, respectively.
The simplified form of this equation was first discovered by Rogister (1971), starting with a Vlasov kinetic explanation for the particle species. Next, it was proposed by Mjølhus (1976) and Mio et al. (1976) on the basis of Hall magnetohydrodynamics (MHD) for cold plasmas, and by Spangler and Sheerin (1982) and Sakai and Sonnerup (1983) from warm-fluid models.
The nonlinear spatio-temporal evolution of gravity water-surface waves can be predicted by the nonlinear Bogoyavlenskii equation. In 1990, the following NPDE was given, later called the Bogoyavlenskii equation [
2], to describe some kinds of waves on the sea surface.
Fractional calculus (FC) was formulated in 1695 after the development of classical calculus. For a long time, FC was regarded as a pure mathematical area without any real applications, but over the last few decades FC was recognized as a useful tool for understanding and modeling several artificial and natural phenomena. Scientific areas where FC concepts are applied include not only physics and mathematics but also finance, human sciences, engineering, economics, chemistry, and biology.
Starting from the earlier stage of FCs, various fractional operators have been presented during the last 300 years. For instance, the Caputo derivative [
3] appeared naturally in the work of Liouville and Abel, and it was proposed by Caputo to solve a practical problem.
The study of fractional-order nonlinear systems of physical models relies on the analysis of wave solutions for nonlinear equations. Recently, numerous and varied techniques have been used to solve nonlinear fractional equations, such as the functional variable method [
4], the trial equation method and the modified trial method [
5], and the bifurcation method [
6].
The objective of the present paper is to apply an interesting method, namely, the
–expansion method [
7], to obtain novel exact solutions for the following nonlinear time-fractional equations:
The conformable time-modified nonlinear Schrödinger equation (CTFMNLSE), given by [
8]
where
,
and
are defined in Equation (
1),
The space–time fractional Bogoyavlenskii equation in the sense of the Jumarie’s modified the Riemann–Liouville derivative defined by [
9]
2. The Basic Idea of the –Expansion Method
Here, we propose the algorithm of the
–expansion procedure [
7] for an NPDE, as follows:
• We assume a common nonlinear PDE of the type:
where
• Convert the nonlinear PDE (
6) into an ODE through
where
d and
c are constants.
• Rewrite Equation (
6) in the following nonlinear ODE form:
• Assume that the general solution of Equation (
8) can be expressed in terms of
as
where
satisfies the following Riccati equation:
in which
The passive constants
or
may be zero, but they cannot be zero at the same time. Note,
are passive constants to be determined in the next step. Additionally, the value of
can be computed through the homogeneous balance principle.
Notice that Equation (
10) results in
• Insert Equation (
9), along with Equation (
10), into Equation (
8). Collect all of the coefficients of the same power of
, and set all of the obtained coefficients to zero. Then, solve the system of algebraic equations on variables
, and
• Based on
and
the solutions of Equation (
10) are given by
Case 3:
where
• By substituting the obtained values of
and the solutions Equations (
13)–(
15) into Equation (
9) with the transformation Equation (
7), the exact traveling-wave solutions of Equation (
6) can be obtained.
3. Application of the –Expansion Method for Equations (1) and (4)
Consider CTFMNLSE (
4), where
is the conformable derivative of order
given by [
10]
in which
is a given function.
If the above limit exists, then is called -differentiable. Assume and and are -differentiable at then we obtain the following results:
∘ where
∘, where
∘
∘
∘
∘
We start with the following traveling-wave assumption:
where
;
; and
, and
are the frequency, phase constant, and wave number, correspondingly. Inserting Equation (
16) into Equation (
4), we obtain the imaginary and real parts as [
8]
and
By integrating the imaginary part and taking the constant equal to zero, we obtain
From Equations (
17) and (
19), we have that
From the above, we have that
Rewrite Equation (
17) in the following form:
or
where
and
.
We can obtain the precise solution of Equation (
19) in a similar way.
Balancing
in Equation (
22) with
, we obtain the balancing number
or
Suppose that the solution of Equation (
22) can be given by
in which
are constants to be determined later.
Inserting Equation (
23) with Equation (
10) into Equation (
22) and collecting all of the terms with the same powers of
and then equating every coefficient of the polynomial to zero, we obtain a system of algebraic equations, as follows:
Solving the above system of nonlinear algebraic equations, we have
Inserting Equation (
24) into Equation (
23), we obtain
Therefore, we obtain three types of traveling-wave solutions, as follows:
Case 3:
where
The
Figure 1 and
Figure 2 display the plots of Equation (
30), for
and two different cases
and
, respectively.
The
Figure 3 and
Figure 4 display the plots of Equation (
30), for
and two different cases
and
, respectively.
The
Figure 5 and
Figure 6 display the plots of Equation (
30), for
and two different cases
and
, respectively.
The
Figure 7 and
Figure 8 display the plots of Equation (
30), for
and two different cases
and
, respectively.
Notice that [1] and [3] are the imaginary and real part of Equation (
30), and [2], [4], [5], and [6] display the
z-axis orientation, the contour plot, the 2d for
, and the 2d for
, of the imaginary part of [1], respectively.
By considering Equation (
25), we obtain three types of traveling-wave solutions, as follows:
Case 3:
where
In
Figure 9,
Figure 10,
Figure 11 and
Figure 12, the imaginary parts of Equation (
34) are displayed for two different cases
and
, and different values of
respectively.
By considering Equation (
26), we obtain three types of traveling-wave solutions, as follows:
Case 3:
where
In
Figure 13 and
Figure 14, the real part of Equation (
38) is displayed for two different cases
and
, and different values of
respectively.
3.1. Modulation Instability Analysis
Here, we deduce the modulation instability of Equation (
4) for the special case
through employing the criterion of linear stability analysis in [
11].
Assume the steady-state solutions of Equation (
4) for the special case
which is given by
in which
is the normalized optical power.
By inserting Equation (
42) in Equation (
4), and by linearizing, we obtain
Assume that the solution to Equation (
43) is as follows:
in which
w is the frequency of perturbation and
is the normalized wave number.
By inserting Equation (
44) into Equation (
43) and splitting the coefficients of
and
and solving the determinant of the coefficient matrix, we have
By solving Equation (
45) for
w, we obtain
If
for any
,
then the steady-state is stable if the perturbations are small. In other words, small changes in the system input do not result in a large change in the system output. In addition, the steady-state solution is unstable if
here,
w is imaginary, and the perturbation grows exponentially.
Therefore, the growth rate of the modulation stability gain spectrum
is defined as
3.2. Discussion
The imaginary and real parts of solutions of Equation (
4) are obtained by the
-expansion method, whose diverse point sources, though arbitrary, and
are displayed in
Table 1,
Table 2 and
Table 3. As you can see, for fixed
by changing
, the
expansion method results in minor changes for solutions
. In other words, the mentioned solutions are stable against small perturbations. However, overall, for
the governing method cannot properly describe all of the real parts of the solutions of Equation (
4).
4. Application of the –Expansion Method for Equation (5)
Consider the time fractional Bogoyavlenskii Equation (
5), where
is the Jumaries-modified Riemann-Liouville derivative of order
given by [
12]
Some important properties of Jumarie’s derivative are
Through the following transformations:
integrating the second equation of Equation (
5) once, and inserting the constant of integration to zero, we obtain
Now, setting the second equation of Equation (
48) into the first equation, integrating the gained result with respect to
, and substituting the integration constant to zero, we obtain
We now obtain the balancing number
Suppose the solution of Equation (
49) can be given by
in which
are constants to be determined later.
Letting equation Equation (
50) with Equation (
10) into Equation (
49), collecting all terms with the same powers of
, and next equating every coefficient of the polynomial to zero, we obtain the following system of algebraic equations:
Solving the system of nonlinear algebraic equations, we have
Inserting Equation (
51) in Equation (
50), we obtain
Inserting Equation (
52) in Equation (
50), we obtain
Therefore, we obtain three types of traveling-wave solutions, as follows:
Case 3:
where
Inserting Equation (
53) in Equation (
50), we obtain
Therefore, we obtain three types of traveling-wave solutions, as follows:
Case 3:
where
Inserting Equation (
54) in Equation (
50), we obtain
Therefore, we obtain three types of traveling-wave solutions, as follows:
Case 3:
where
Inserting Equation (
55) in Equation (
50), we obtain
Therefore, we obtain three types of traveling-wave solutions, as follows:
Case 3:
where