The analysis of the physical phenomena in the sense of their mathematical modeling is an important tool in applied science and engineering. These models often can be described in the form of nonlinear partial differential equations (NLPDE) or a system of the same kind with the ability to better simulate these complex systems and their physical situations. Thus, finding accurate solutions to these types of equations capture the interest of many researchers driving them to investigate more in their studies for its solutions. The applicability of these models can be witnessed in many branches of science and engineering including fluid mechanics, nonlinear dynamics, quantum field theory, plasma physics, and solid-state physics. They have enjoyed an intense period over the past few years for numerical and theoretical points of view. One of these types of equations are the nonlinear evolution equations (NLEEs) which has tremendous applications in science. Moreover, studies of finding the soliton solutions to these types of equations attracted huge number of studies in the various fields [1], [2], [3], [4], [5], [6], [7], [8], [9].

Recently, fractional calculus has attracted the attention of many researchers with its ability to provide more realistic models of mathematical type for simulating application NLPDE [[10], [11], [12], [13], [14], [15], [16], [17]]. There has been tremendous research work which has been raised aiming for finding the solutions for these nonlinear fractional models including the classical case. Up until now, several methods have been adopted for solving these models including the Bäcklund transform method [18], Hirota bilinear method [19], inverse scattering method [20], Darboux transformation method [21], general projective Riccati equation method [22], sub-equation method [23], Kudryashov method [24], Jacobi elliptic method [25], modified extended tanh-expansion method [26], solitary wave ansatz method [27], and Riccati equations mapping method [28]. Each of these methods has its advantages and disadvantages which gives some of them the ability to provide better results than the other method in terms of accuracy and efficiency.

Recently, Wazwaz [38], [39] introduced a fifth order non-linear evolution equation as:

and it’s generalized form as:

where $\alpha$ and $\beta$ are parameters. Furthermore, he used the Hereman-Nuseri approach to get the numerous soliton solutions for each model. The search for solitons of NLEEs has sparked a lot of attention in recent decades, because such a class of equations may describe a wide range of phenomena from science to engineering. Solitons are stable nonlinear localized solitary waves that preserve their shape while traveling at a constant velocity and may be classified as dark or bright depending on the governing models; they play an important part in many nonlinear physical phenomena. Soliton propagation has been utilized to explain several important oceanic phenomena, including nonlinear shallow and deep water wave propagation [29], [30], [31], [32], [33], [34], [35], [36], [37]. For instance, dark solitons have lately piqued curiosity in the context of surface gravity water waves: in two seminal studies conducted in water wave tanks, black and gray solitons were discovered on the surface of the water [30], [31].

In this work, based on the work in [38], [39], we are motivated to introduce four generalized forms of this new nonlinear evolution equation (see Eq. (2)) in the conformable sense as follows:

where ${\beta }_1,{\beta }_2,{\beta }_3,$ and ${\beta }_4$ are nonzero real parameters and $\rho$ ( $0<\rho \le 1$) is the fractional order. Therefore, in the study the conformable derivative (CD) will be explored. The fundamental reason for examining this sort of local derivative is that, when compared to other fractional derivative definitions, the CD can meet numerous properties spanning the chain, product, or quotient rules. To exact the dark solitons, singular solitons, periodic wave, and rational form solutions under certain parametric conditions, we employed two novel integrable techniques. These methods are the sub-equation and generalized Kudryashov methods which have been applied many times for solving strong nonlinear models in the field of applied sciences and engineering. The sub-equations method has been used to find soliton solution type different type of equations including space time-fractional equation [40], perturbed nonlinear Schrödinger [23], conformable fractional PDE [41], and others. Also, the generalized Kudryashov method have been used to solve other type of equations including fractional mathematical biology problems [42], coupled sine Gordon equations [43], fractional Boussinsque equations [44], fractional Klein Gordo equation [45], Cahn Hilliard equations [46], Kuramoto-Sivashinsky equation [47], many other equations that can be found in [48], [49], and references therein. It should be emphasized that the discovery of new models that incorporates soliton-type solutions are an intriguing outcome that will also aid future research in numerous fields of applied science and engineering.

The organization of the paper is as follows: In Section 2, we give some preliminaries regarding the fractional calculus that will be used in later sections. In Section 3, a summary for the two proposed methods is provided. Section 4 illustrates the main steps for solving Eqs. (3)–(6) using the two techniques and in Section 5 the results and graphical behavior for some obtained solutions are discussed. Finally, in Section 6 contains a conclusion for the study.

In this section, we summarize some basic definitions, notations and theorems used in this present study.

Definition 2.1 The conformable derivative of a function, $W_1:[0,\infty )\to \mathcal{R}$ of order $\rho$ is defined as [50], [51]

If $W_1$ is $\rho$-differentiable in some $(0,ϵ),$ where $ϵ>0$ and the $\underset{t\to 0^{+}}{\lim }{\mathcal{D}}_t^{\rho }{W}_1\left(t\right)$ exists, then by definition

Theorem 2.2 ([50, 51, 53], Let $\rho \in (0,1]$ and $W_1=W_1\left(t\right),W_2=W_2\left(t\right)$ be $\rho$ differentiable at a point $t>0.$ Then

1. ${\mathcal{D}}_t^{\rho }({\zeta }_1{W}_1+{\zeta }_2{W}_2)={\zeta }_1{\mathcal{D}}_t^{\rho }{W}_1+{\zeta }_2{\mathcal{D}}_t^{\rho }{W}_2,$ $\forall$ ${\zeta }_1,{\zeta }_2\in \mathbb{R}.$

2. ${\mathcal{D}}_t^{\rho }(W_1·W_2)=W_1{\mathcal{D}}_t^{\rho }{W}_2+W_2{\mathcal{D}}_t^{\rho }{W}_1.$

3. ${\mathcal{D}}_t^{\rho }\left(\frac{W_1}{W_2}\right)=\frac{W_2{\mathcal{D}}_t^{\rho }{W}_1-W_1{\mathcal{D}}_t^{\rho }{W}_2}{W_2^2}.$

4. ${\mathcal{D}}_t^{\rho }\left(t^{\nu }\right)=\nu t^{\nu -\rho },$ $\forall \nu \in \mathbb{R}.$

5. ${\mathcal{D}}_t^{\rho }\left(\varpi \right)=0,$ $\varpi$ is constant.

6. If $W_1$ is differentiable, then

Definition 2.3 ([51]). Let $W_1:[{\kappa }_1,\infty )\to \mathbb{R}$ be twice continuously differentiable and $0<\rho \le \frac{1}{2},$ then

Definition 2.4 ([51]). Let $W_1:(-\infty ,{\kappa }_2]\to \mathbb{R}$ be twice continuously differentiable and $0<\rho \le \frac{1}{2},$ then

Theorem 2.5 ([52]). Let $W_1:(0,\infty )\to \mathcal{R}$ be a differentiable and $\rho$ -differentiable function. Let $W_2$ be a differentiable function defined in the range of $W_1.$ Then

Also, using the approach defined in [51] to obtained Definitions 2.3 and 2.4, we acquire the following:

Definition 2.6 Let $W_1:[{\kappa }_3,\infty )\to \mathbb{R}$ be three times continuously differentiable and $0<\rho \le \frac{1}{3},$ then

Definition 2.7 Let $W_1:(-\infty ,{\kappa }_4]\to \mathbb{R}$ be three times continuously differentiable and $0<\rho \le \frac{1}{3},$ then

We later use Definitions 2.6 and 2.7 and Mathematica software package to verify all obtained solutions by putting them back in their original equations.

Here, we present a brief summary of the sub-equation method [54], [55] and modified Kudryashov method [56], [57]. Next, we consider a nonlinear conformable fractional PDE in the form given below:

where ${\mathcal{D}}_t^{*\rho }$ indicates the conformable derivative. Through the use of the fractional wave transform [58]

we reduced Eq. (15) into a nonlinear ordinary differential equation (NODE) as:

where ${\eta }_1,{\eta }_2,\cdots ,$ and $\omega$ are parameters that will be determined afterwards. The solutions of Eq. (17) can be presumed as:

where ${\delta }_i$ $(0\le i\le \Omega )$ are the constants to also be calculated thereafter and $\Omega$ is a positive integer to be found through balancing procedure [59] in Eq. (17). A summary of the two proposed integrable method will be discuss in the following subsections. In the following subsections, a quick summary of the two suggested integrable methods will be discussed.

In this section, we apply the modified Kudryashov and sub-equation methods in the conformable sense to examine various types of new generalized fifth order nonlinear equations.