Bäcklund transformation and some different types of N‐soliton solutions to the (3 + 1)‐dimensional generalized nonlinear evolution equation for the shallow‐water waves

Abstract

The (3 + 1)‐dimensional generalized nonlinear evolution equation is investigated based on the Hirota bilinear method. N‐soliton solutions, bilinear Bäcklund transformation, high‐order lump solutions, and the interaction phenomenon of high‐order lump solutions for this equation are obtained with the help of symbolic computation. Besides, some different types of periodic soliton solutions are studied. Analysis and graphical simulation are presented to show the dynamical characteristics of some different types of N‐soliton solutions are revealed. Many dynamic models can be simulated by nonlinear evolution equations, and these graphical analyses are helpful to understand these models. Compared with the published studied, some completely new results are presented in this paper.

Superposition Formula of arbitrary functions to a (3+1)-demensional Boiti-Leon-Manna-Pempinelli equation

“…Different selection of n i , M values can yeild various solutions to Eq. ( 1), such as the lump solutions [27]- [29], the lump-type solutions [30], the high-order lump-type solutions [31]- [35], the localized solutions [36,37], etc. When n i = 1, the solutions of Eq.…”

Resonant Y-type soliton, X-type soliton and some novel hybrid interaction solutions to the (3+1)-dimensional nonlinear evolution equation for shallow-water waves

“…Searching for the exact solutions to NEEs has always been a research hotspot for many scholars. Up to now, many effective and powerful approaches have been proposed, such as the Bäcklund transformation technique [9][10][11][12], variational technique [13,14], Sardar-subequation approcah [15,16], Darboux transformation approach [17][18][19], Kudryashov approach [20,21], exp-function approach [22,23], tanhfunction method [24,25], (G′/G)-expansion technique [26,27], unified solver approach [28,29] and so on [30][31][32][33][34][35][36]. In this study, we are going to look into the (3+1)-dimensional NEE as [37]:…”

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.…”