This current research is about some new optical solitons to the time-fractional integrable generalized (2+1)-dimensional nonlinear Schrödinger (NLS) system with novel truncated M-fractional derivative. The obtained results may be used in the description of the model in fruitful way. The novel derivative operator is applied to study the aforementioned model. The achieved results are in the form of dark, bright, and combo optical solitons. The achieved solutions are also verified by using the MATHEMATICA software. The obtained solutions are explained with different plots. Modified integration methods, function, extended -expansion, and extended sinh-Gordon equation expansion method are applied to achieve the results. These exact solitons suggest that these methods are effective, straight forward, and reliable compared to other methods.
Fractional calculus [1,2,3, 4,5,6, 7,8,9, 10,11,12] has become very popular due to its many applications in different areas of sciences. Many models have been made in the area of physical sciences and engineering that are representing the different phenomenon. For example, mostly naturally occurring phenomena are modeled in the form of nonlinear Schrödinger equations [13,14,15]. To determine the exact solutions of the models, a lot of schemes have been developed. Instantly, the modified extended expansion scheme  has been applied to discuss the Biswas–Arshed model. Some different wave solutions of the perturbed Gerdjikov-Ivanov equation are gained with the help of the semi-inverse variational method . Various solitons of the new coupled evolution equation were explained . Distinct solitons are investigated by applying the sine-Gordon equation method . Two types of soliton solutions have been obtained by using -expansion and generalized Kudryashov methods in ref. . New kinds of general solutions have been achieved in ref. . The Sardar subequation method is used to gain the optical and some other wave solutions in ref. . Three new types of wave solutions have been gained with the help of the modified -expansion method in ref. . Different types of optical soliton solutions have been collected by using the collective variable method in ref. . Similarly, other methods have been applied; generalized exponential rational function method [25,26,27], Liu’s extended trial function method , generalized unified method , sine-Gordon expansion method , enhanced modified simple equation method , unified method , extended function method , Lie symmetry method , symbolic computational method, Hirota’s simple method and long wave method , Jacobi elliptic function expansion method , Elzaki transform decomposition method , -expansion method and adomian decomposition method , extended modified auxiliary equation mapping method , simplest equation method and Kudryashov’s new function method , modified simple equation method , modified Kudryashov simple equation method , first integral method , Bäcklund transformation method , extended Jacobi elliptic function expansion method , improved -expansion method, improved -expansion method , and many more [47,48,49, 50,51].
Our concerning model is time-fractional integrable generalized (2+1)-dimensional nonlinear Schrödinger system. Different types of exact solitons have been found by various techniques as follows: optical soliton solutions have been calculated by using the extended modified auxiliary equation mapping method in ref. , some travelling wave solutions of the integrable generalized NLS system have been obtained in ref. , and various optical wave solutions have been achieved of this system with the help of Kudryashov method and it is modified form given in ref. .
In addition to these methods, there are three other methods: function method, extended -expansion method, and extended sinh-Gordon equation expansion method (ShGEEM). These methods have been used to explain the many different models: the Tzitzéica like equations are investigated for their exact solitons by using the function method . New kind of optical wave solutions of two nonlinear Schrödinger equations were searched by utilizing two analytical methods . Similarly, this is used to investigate the roots of the other many nonlinear Schrödinger equations [56,57]. By using the extended -expansion method, different optical solitons of the Biswas–Milovic equation are generated in ref. ; bell-shaped, kink-shaped, and periodic type solitons of the Pochhammer–Chree equations are derived with the help of this method ; and discrete and periodic type solitons of the Ablowitz–Ladik lattice system are found . Similarly, the extended ShGEEM has been applied to determine the various wave solutions of different models in refs [61,62, 63,64].
The main task of this study is to research some new exact soliton solutions of the truncated M-fractional integrable generalized (2+1)-dimensional NLS system based on the function method, the extended -expansion method, and extended ShGEEM.
This paper is organized as follows: Section 2 describes the truncated M-fractional derivative and its characteristics. Section 3 presents the demonstration and mathematically treatment of model. Section 4 gives the mathematical analysis of the time-fractional integrable generalized NLS system. Section 5 obtains a large number of exact wave solutions. Section 6 makes a conclusion.
2 Truncated M-fractional derivative
Definition. Let , then the truncated M-fractional derivative of of order is shown :
where is a truncated Mittag-Leffler function of one parameter that is defined as ref. :
Characteristics: Let , and -differentiable at a point , then by ref. :
3 Description of strategies
3.1 Summary of function method
In this section, we demonstrate this method.
Suppose a nonlinear partial differential equation (NLPDE):
This NLPDE shown in Eq. (8) changed in to nonlinear ordinary differential equation (ODE):
with the use of following wave transformations:
Considering in Eq. (12) to be equal to 0, a system of algebraic equations is achieved as follows:
with the aforementioned achieved results, we gain the nontrivial solitons of Eq. (8).
3.2 Explanation of the extended -expansion scheme
This portion is about the key steps of the extended -expansion scheme .
Step 1: Let’s assume the below NLPDE:
where is a wave profile and depend on and and . Let the following travelling wave transformations:
Step 3: Suppose the solutions of the Eq. (16) is of the structure:
In Eq. (17), and are undetermined that are found later. Note that . By applying the homogenous balance principle into Eq. (16), we find the value of . The function satisfy the Riccati differential equation given as follows:
with , , , and are constants.
Step 4: Let Eq. (17) have solutions in the form given as follows:
Case 1: If and , then
Case 2: If and , then
Case 3: If and , then
Case 4: If and , then
Case 5: If and , then
where , and are the constants.
Step 5: Eq. (17) with Eq. (18) is inserted into Eq. (16) and summed up the coefficients of the same power of . By taking each coefficient equal to 0, we gain the system of algebraic equations involving , , and other parameters.
By manipulating the aforementioned achieved system with the help of MATHEMATICA tool.
3.3 Demonstration of the extended ShGEEM
In this section, we brief the basic steps of the extended ShGEEM:
Consider the NLPDE given as follows:
where is a wave profile and depends on and and .
Let the following be travelling wave transformations:
Consider the solution of Eq. (26) follows:
where , , are unknown parameters that are found later and is a new function of that satisfy the below equations:
Inserting Eq. (27) along natural number with Eq. (28) into Eq. (26) to achieve the algebraic expressions in ( ). Now take the each coefficient of equal to zero, to gain the set of algebraic equations containing and .
By solving the achieved set of algebraic equations with the use of MATHEMATICA tool, we can obtain the values of parameters, , and .
4 Description and mathematical analysis of the model
Consider the following time-fractional integrable generalized NLS system given in ref. :
where is the complex-valued wave function and is the real-valued wave function. In Eq. (34), are the parameters.
Let’s assume the following travelling wave transformation:
where and , represent the speed of soliton and wave number, respectively, while and show the frequency of the soliton.
By integrating Eq. (38) once and taking constant of integration equal to zero, we obtain:
and from Eq. (36), we obtain
By using the homogenous balance scheme into Eq. (41), we obtain .
5 Exact wave solutions
5.1 Solution to the function method
For , Eq. (11) reduces into:
As an explanation, the dynamic properties of Eq. (45) are demonstrated in Figure 1.
5.2 Solutions to the extended -expansion method
For , Eq. (17) becomes:
where , and are undetermined parameters.
where is given in Eq. (59).
where is given in Eq. (59).
where is given in Eq. (68).