Multiple Soliton Solutions for Nonlinear Differential Equations with a New Version of Extended F-Expansion Method

A New Version of the Generalized F-Expansion Method for the Fractional Biswas-Arshed Equation and Boussinesq Equation with the Beta-Derivative

“…In this section, the application steps of the new version of the generalized F-expansion method to obtain the combined and mixed Jacobi elliptic function solutions of differential equations will be given [30][31][32]. With this new method, different and new results can be acquired from the results obtained from other methods.…”

“…With this offered method, it is aimed at finding new and several exact solutions of fractional order differential equations that are not actual in the literature. This method, which has been discussed in some studies in the literature, has been applied to various nonlinear partial differential equations [30][31][32]. There are different F-expansion methods that allow procuring the elliptic function solutions, which are among these exact solutions [33][34][35][36].…”

Dynamics of breather, multi-wave, interaction and other wave solutions to the new (3+1)-dimensional integrable fourth-order equation for shallow water waves

“…Therefore, the solution of NPDEs naturally becomes an important part of nonlinear theory. And, for the time being, many different kinds of the powerful approaches have been derived to handle the NPDEs, such as the extended F-expansion method (Bhrawy et al , 2012; Pandir and Turhan, 2021; Abdou, 2007; Rabie and Ahmed, 2022), Bäcklund transformation (Ma et al , 2021; Yin et al , 2021a, 2021b; Han and Bao, 2021; Du et al , 2018), variational technique (Wang, 2022; Wang, 2023), exp-function approach (Raza and Javid, 2019; Mohyud‐Din et al , 2012; He and Wu, 2006; Wu and He, 2008), general integral method (Shang, 2010; Zayed et al , 2022), trial equation method (Afzal et al , 2019; Raza and Javid, 2019) and so on (Seadawy et al , 2018; Wang et al , 2023a, 2023b, 2023c; Raza et al , 2019; Wang and Liu, 2022; Rezazadeh et al , 2020; Sağlam Özkan et al , 2021; Asjad et al , 2022; Rehman et al , 2022; Seadawy et al , 2021; Rizvi et al , 2020; Wang et al , 2023a, 2023b, 2023c; Kaplan et al , 2016; Hosseini and Ayati, 2016). The purpose of this study is to construct the some novel exact solutions of the new (3 + 1)-dimensional integrable fourth-order nonlinear equation (IFNE), which reads as (Wazwaz, 2021): where Ξ = Ξ ( x , y , z , t ), which is usually used to describe the both right and left travelling waves which is like the Boussinesq equation.…”

Soliton molecules, interaction and other wave solutions of the new (3+1)-dimensional integrable fourth-order equation for shallow water waves

“…Solitons have found applications in fluid physics, elementary particle physics, biomedicine and many other fields, and have explained some phenomena that were difficult to explain in the past. With the development of soliton theory, many and varied effective approaches for solving the NPDEs have been proposed, such as the Kudryashov approach [18,19], extended F-Expansion approach [20][21][22][23], Bäcklund transformation [24][25][26][27], G'/G-expansion technique [28,29], variational technique [30,31], exp-function approach [32][33][34][35], trial equation method [32,36], general integral method [37,38] and so on [39][40][41][42][43][44][45][46][47][48][49]. In this research, we are going to plumb the exact solutions of the new IFNE in the (3+1)-dimensional as [50,51] where , a b and g are the non zero real numbers, x y z t , , , ( ) y y = is like the Boussinesq equation that usually describes the both right and left travelling waves.…”