Different types of phenomenon occurring chemically, biologically, economically and others are represented as a non-linear partial differential equations (NLPDEs). Many different methods are developed to gain the analytical wave solutions of these NLPDEs i.e optical soliton solutions of coupled nonlinear Schrödinger equation have been gained with the use of Kudryashov R function technique [1], some new kind of optical soliton solutions of time-fractional perturbed nonlinear Schrödinger equation have been achieved by using the generalized Kudryashov scheme [2], by applying the modified auxiliary equation technique, optical wave solutions of time-fractional resonant non-linear Schrödinger equation have been obtained [3], new optical wave solutions for the time-fractional perturbed non-linear Schrödinger equation have been achieved by utilizing the improved $tan\left(\varphi \right(\zeta /2\left)\right)$-expansion scheme [4].

We utilize the Sardar sub-equation method for our research work. This method have so many applications in the literature. For example; some solitons solutions of perturbed Fokas-lenells model have been obtained with the use of Sardar sub-equation method [5], new kind of soliton wave solutions of (2+1)-dimensional Sawada-Kotera model have been gained by this method [6], the dark, bright and singular optical wave solutions of higher order non-linear Schrödinger have been gained by using Sardar sub-equation scheme [7], The singular, bright, dark, periodic singular, combined dark-bright solitons and other wave solutions of strain wave equation have been achieved with the use of this model [8].

Our study model is complex three coupled Maccari’s system along with truncated M-fractional derivative. Different types of exact wave solutions of this model have been obtained with the help of different methods in the literature. Instantly, the dark, bright and complex optical wave solutions have been achieved by using extended rational sine-cosine and rational sinh-cosh techniques [9], trigonometric, rational and hyperbolic solutions have been attained with the use of unified solver technique [10], the optical soliton solutions have been gained by utilizing the planar dynamic system scheme [11], some new travelling wave solutions have been obtained by applying the modified trial equation technique [12], the periodic, double periodic, dark, bright, combined soliton solutions etc have been obtained with the help of generalized auxiliary equation mapping technique [13].

Main aim of this research is to discover the new exact wave solutions to the truncated M-fractional complex three coupled Maccari’s system with the help of Sardar sub-equation method.

Paper consists of different sections; In Section 2: we describe the main steps of our concerned method Sardar sub-equation.. In Section 3: we explain the our concerned model and it’s mathematical analysis. In Section 4: we apply the method to gain the new exact wave solutions of our concerned model. In Section 5: we explain the some obtained solutions through 2-D, 3-D and contour graphs. In Section 6: we give the conclusion of our research work.

Here, we explain the fundamental points of Sardar sub-equation method [14]. Let’s consider the non-linear fractional partial differential equation: here $g=g(x,y,t)$ represents a wave function.

Applying the wave transformations given as follow: We obtain a non-linear ODE shown as: Consider Equation (3) posses the results in the given shape: here $\psi \left(\zeta \right)$ fulfill the ODE shown as: here $\sigma$ and $\kappa$ are constants.

Using Equation (4) into Equation (3) with Equation (5) and collecting the coefficients of each power of ${\psi }^i$. Putting co-efficient of each power equal to 0, we gain a sets of algebraic equations in the term $b_i$, $\lambda$, $\mu$. By solving the obtained system of equation, we get the values of parameters..

Case 1: If $\kappa >0$ and $\sigma =0$, then where, $sec{h}_{ab}\left(\zeta \right)=\frac{2}{a{e}^{\zeta }+b{e}^{-\zeta }}$, $csc{h}_{ab}\left(\zeta \right)=\frac{2}{a{e}^{\zeta }-b{e}^{-\zeta }}$

Case 2: If $\kappa <0$ and $\sigma =0$, then where, ${sec}_{ab}\left(\zeta \right)=\frac{2}{a{e}^{\iota \zeta }+b{e}^{-\iota \zeta }}$, ${csc}_{ab}\left(\zeta \right)=\frac{2\iota }{a{e}^{\iota \zeta }-b{e}^{-\iota \zeta }}$

Case 3: If $\kappa <0$ and $\sigma =\frac{{\kappa }^2}{4}$, then where, ${tanh}_{ab}\left(\zeta \right)=\frac{a{e}^{\zeta }-b{e}^{-\zeta }}{a{e}^{\zeta }+b{e}^{-\zeta }}$, ${coth}_{ab}\left(\zeta \right)=\frac{a{e}^{\zeta }+b{e}^{-\zeta }}{a{e}^{\zeta }-b{e}^{-\zeta }}$

Case 4: If $\kappa >0$ and $\sigma =\frac{{\kappa }^2}{4}$, then where, ${tan}_{ab}\left(\zeta \right)=-\iota \frac{a{e}^{\iota \zeta }-b{e}^{-\iota \zeta }}{a{e}^{\iota \zeta }+b{e}^{-\iota \zeta }}$, ${cot}_{ab}\left(\zeta \right)=\iota \frac{a{e}^{\iota \zeta }+b{e}^{-\iota \zeta }}{a{e}^{\iota \zeta }-b{e}^{-\iota \zeta }}$

Assume the M-fractional three coupled non-linear Maccari’s system shown in [15]: where here $E_{{\rm Y}}(.)$ represents truncated Mittag-Leffler function of one parameter given in [16,17].

Let’s consider the following transformations: here ${\theta }_1$, $\sigma$,$\tau$,$\mu$ and $\beta$ are the unknowns and $\vartheta$ is arbitrary constant.

Substituting Equation (22) into Equation (20), yields the real and imaginary parts:

Real parts:

Imaginary parts:

By Equation (24), implies $\beta ={\theta }_1$.

Integrating the fourth equation of system (23), yields

Putting Equation (25) into first three terms of Equation (33), we get

Taking $V={\theta }_2$ U and Taking $W=c$ U, we obtain

Equation (4) changes into given form for m=1. Putting Equation (28) into Equation (27) with Equation (5). By collecting co-efficients of every order of $\psi \left(\zeta \right)$ and taking them equal to 0, we get a set of algebraic equations. By solving the obtained set of equations by Mathematica software, we gain the following sets.

Set 1:

Case 1

Case 2:

Case 3:

Case 4: