Soliton theory based on water waves, plasmas, optical fibers etc., was developed in 1960-1970. This is significant branch of applied mathematics as well as mathematical physics. It has significant uses in non-linear optics, fluid mechanics, plasmas etc. This theory is applied in various natural sciences, including telecommunication, biology, chemistry, mathematics and many fields of physics including fluid dynamics, condensed matter, plasma physics etc. Distinct kinds of naturally occurring phenomenon are shown as a non-linear fractional partial differential equations (NLFPDEs). Distinct schemes are made to attain exact results of non-linear partial differential equations i.e modified direct algebraic technique [1], modified Khater scheme [2], Kudryashov technique [3], novel $(G^{\prime }/G)$-expansion method [4], extended mapping scheme [5].

In our study we use three simple, useful and significant methods named as modified simplest equation method, Sardar sub-equation method and generalized Kudryashov method. There are various uses of these methods. Instantly; some solitary wave solutions of BBM and Chan-Hilliard equations by utilizing modified simplest equation method [6], exact solitons of Boussinesq and coupled Boussinesq equations have been attained by using this technique in [7]. Some solitons solutions of perturbed Fokas-lenells model have been obtained with the help of Sardar sub-equation technique [8], new kind of solitons of (2+1)-dimensional Sawada-Kotera (SK) model have been gained by this method [9], some exact wave solitons of new Hamiltonian Ampltude equation have been attained [10], the dark, bright and singular optical solitons of higher order non-linear Schrödinger have been gained by utilizing Sardar sub-equation scheme [11], the singular, bright, dark, periodic singular, combined solitons and other solutions of strain wave model have been achieved with the use of this technique [12]. Similarly, the bell, anti-bell, dark, kink, flat kink and other wave solutions of Fokas-Lenelles by applying generalized Kudryashov technique [13], exact solitons of KdV-Burger model are obtained in [14] etc.

Our mathematical physical important model is the (1+1)-dimensional non-linear simplified Modified Camassa-Holm model (SMCHM). Different types of exact solitons of our concerned model have been achieved with the use of different methods. Likely, some distinct kind of travelling wave solitons of SMCHM have been attained with the help of generalized $(G^{\prime }/G)$-expansion scheme in [15], solitary wave solitons are gained through Exp-function technique in [16], different types of wave solutions have been gained by utilizing the Riccati-Bernoulli sub-ODE scheme in [17] etc.

The purpose of our research is to discover the new types of M-fractional wave solitons to the (1+1)-dimensional non-linear simplified Modified Camassa-Holm model by utilizing modified simplest equation, Sardar sub-equation and generalized Kudryashov techniques.

The paper have distinct sections; In section 2: we explain the our concerned model and it’s mathematical analysis. In section 3: we describe the modified simplest equation technique and it’s application to obtain the soliton solutions. In section 4: we explain the Sardar sub-equation technique and apply it to gain the new soliton solutions of our concerned model. In section 5: we describe the generalized Kudryashov technique and it’s application to obtain the soliton solutions. In section 6: we explain the some obtained solutions through 2-dimensional, 3-dimensional and contour plots. In section 7: we give a conclusion of our research work.

Let’s assume the non-linear M-fractional simplified Modified Camassa-Holm model from the family of important equations known as the modified $\beta$-equations explained by Wazwaz [18]. here g=g(x,t) represents the wave profile while $\Omega$ and $\theta$ are nonzero parameters. Putting $\beta =2$ in the Eq.(1), we get

This form is called modified Camassa-Holm model. Furthermore simplified form is of Eq.(2) given in [19]. here where $E_{{\rm Y}}(.)$ is the truncated Mittag-Leffler (ML) function of one parameter shown in [20,21].

Assume the wave transformation given as follows: where $\mu$ and $\delta$ are the nonzero constants. By substituting the Eq.(5) into Eq.(3) we attain

The fundamental points of this scheme are given as:

Step 1:

Consider a NLPDE: here $h=h(x,y,t)$ represents wave profile.

Consider the wave transformation:

Inserting Eq. (8) into Eq. (7), we gain a NLODE:

Step 2:Consider Eq.(9) has below shape solution: where $b_j(j=1,2,3,...,m)$ are undetermined and $b_m\ne 0$.

A new profile $\psi \left(\xi \right)$ fulfil below ODE: where $\omega$ is a parameter.

Eq.(11) have solutions for different cases of $\omega$:

if $\omega <0$,

if $\omega >0$,

If $\omega =0$,

Step 3:

Putting Eq.(10) into Eq.(9) with Eq.(11) and collecting the coefficients of every order of ${\psi }^j$. Substituting co-efficients of equal to 0, we obtain a set of equations involving $b_i$, $\lambda$, $\mu$. Manipulating the gained set of equations, we gain results for parameters.

Step 4:

Inserting Eq.(9) of which $b_j$, $\lambda$, $\mu$ has been obtained into Eq.(10), we attain solutions of Eq.(7).

Here, we explain the fundamental points of Sardar sub-equation method [22]. Assume the nonlinear fractional PDE: here $g=g(z,t)$ is a wave profile.

Substituting a wave transformation shown as:

We obtain a non-linear ODE shown as:

Consider Eq.(50) posses the results in the given shape: here $\psi \left(\zeta \right)$ fulfill the ODE shown as: here $\sigma$ and $\kappa$ are constants.

Using Eq.(51) into Eq.(50) along Eq.(52) and collecting the coefficients of every power of ${\psi }^i$. Inserting co-efficient of every power equal to 0, we achieve a set of equations having $b_i$, $\lambda$, $\mu$. Solving the achieved set of equation, we gain the results for unknowns.

Case 1: if $\kappa >0$ and $\sigma =0$, we have 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 $\sigma =0$, we have 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 $\sigma =\frac{{\kappa }^2}{4}$, we have 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 $\sigma =\frac{{\kappa }^2}{4}$, we have where, ${tan}_{ab}\left(\zeta \right)=-\iota \frac{a{e}^{\iota \zeta }-b{e}^{-\iota \zeta }}{a{e}^{\iota \zeta }+b{e}^{-\iota \zeta }}$, ${cot}_{ab}\left(\zeta \right)=\iota \frac{a{e}^{\iota \zeta }+b{e}^{-\iota \zeta }}{a{e}^{\iota \zeta }-b{e}^{-\iota \zeta }}$

The basic steps of this technique are given as [23,24]:

Step 1:

Take a nonlinear PDE: here q is a wave profile.

Assume the transformation given as:

Inserting Eq. (84) into Eq. (83), we gain the nonlinear ODE:

Step 2:

Consider the roots of Eq. (85) is of form: here ${\alpha }_0$ and ${\alpha }_j$, $(j=1,2,3,...,m)$ are undetermined parameters and $\psi$ is a new wave profile of $\xi$ that is a root of the general Riccati equation shown as: where a, b and c are the constants. The roots of Eq.(87) are given in the below cases [25]:

Case 1: If all a, b and c are nonzero, we have $\psi \left(\xi \right)$ is shown by

Case 2:If a=0 and $c\ne 0$, we have

Case 3:If b=0 and $c\ne 0$, we have