## Abstract

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,

## 1 Introduction

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

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. [39], some travelling wave solutions of the integrable generalized NLS system have been obtained in ref. [52], 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. [53].

In addition to these methods, there are three other methods:

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

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

where

**Characteristics:** Let

## 3 Description of strategies

### 3.1 Summary of
Exp
a
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:

Let’s consider a root of Eq. (9) is given in refs [54,56,67,68]:

where

Considering

with the aforementioned achieved results, we gain the nontrivial solitons of Eq. (8).

### 3.2 Explanation of the extended
(
G
′
∕
G
)
-expansion scheme

This portion is about the key steps of the extended

**Step 1:** Let’s assume the below NLPDE:

where

**Step 2:**

By substituting Eq. (15) into Eq. (14), we obtain the nonlinear ODE:

**Step 3:** Suppose the solutions of the Eq. (16) is of the structure:

In Eq. (17),

with

**Step 4:** Let Eq. (17) have solutions in the form given as follows:

**Case 1:** If

**Case 2:** If

**Case 3:** If

**Case 4:** If

**Case 5:** If

where

**Step 5:** Eq. (17) with Eq. (18) is inserted into Eq. (16) and summed up the coefficients of the same power of

**Step 6:**

By manipulating the aforementioned achieved system with the help of MATHEMATICA tool.

**Step 7:**

By inserting the aforementioned achieved results into Eq. (17), we obtain solitons of the nonlinear Eq. (14).

### 3.3 Demonstration of the extended ShGEEM

In this section, we brief the basic steps of the extended ShGEEM:

**Step 1:**

Consider the NLPDE given as follows:

where

Let the following be travelling wave transformations:

By substituting Eq. (25) into Eq. (24), we obtain the following nonlinear ODE given as follows:

**Set 2:**

Consider the solution of Eq. (26) follows:

where

By using the homogeneous balance scheme into Eq. (26), we achieve the value of

Indisputable to ref. [63], we obtain the results from Eq. (28) follows:

and

where

**Step 3:**

Inserting Eq. (27) along natural number

**Step 4:**

By solving the achieved set of algebraic equations with the use of MATHEMATICA tool, we can obtain the values of parameters,

**Step 5:**

By using the gained results and Eqs. (30) and (31), we may obtain solutions of Eq. (24) follows:

and

## 4 Description and mathematical analysis of the model

Consider the following time-fractional integrable generalized NLS system given in ref. [52]:

where

Let’s assume the following travelling wave transformation:

where

By substituting Eq. (35) into Eq. (34), we obtain the imaginary and real parts given as follows:

and

By integrating Eq. (38) once and taking constant of integration equal to zero, we obtain:

and from Eq. (36), we obtain

By substituting Eq. (39) into Eq. (37), we obtain

By using the homogenous balance scheme into Eq. (41), we obtain

## 5 Exact wave solutions

### 5.1 Solution to the
Exp
a
function method

For

By using Eq. (42) into Eq. (41) and solving the system of equations, we achieve the following solution sets:

**Set 1:**

From Eqs. (43), (42), and (35), we obtain

As an explanation, the dynamic properties of Eq. (45) are demonstrated in Figure 1.

**Set 2:**

From Eqs. (35), (42), and (46), we obtain

### 5.2 Solutions to the extended
(
G
′
∕
G
)
-expansion method

For

where

Substituting Eq. (49) with Eq. (18) into Eq. (41) and solving the system for

**Set 1:**

Case 1:

From Eqs. (19), (35), (49), and (50), we obtain

where

Case 2:

From Eqs. (20), (35), (49), and (50), and we obtain

where

Case 3:

From Eqs. (22), (35), (49), and (50), we obtain

where

Case 4:

From Eqs. (23), (35), (49), and (50), we obtain

where

**Set 2:**

Case 1:

From Eqs. (19), (35), (49), and (59), and we obtain

where

Case 2:

From Eqs. (20), (35), (49), and (59), we obtain

where

Case 3:

From Eqs. (22), (35), (49), and (59), we obtain

where

Case 4:

From Eqs. (23), (35), (49), and (59), we obtain

where

**Set 3:**

Case 1:

From Eqs. (19), (35), (49), and (68), we obtain

where

Case 2:

From Eqs. (20), (35), (49), and (68), we obtain