Bilinear form, auto-Bäcklund transformations, Pfaffian, soliton, and breather solutions for a (3 + 1)-dimensional extended shallow water wave equation

Abstract

Study of the water waves remains central to fluid physics, ocean dynamics, and engineering. In this paper, a (3 + 1)-dimensional extended shallow water wave equation is investigated via symbolic computation. Bilinear form and two kinds of the bilinear auto-Bäcklund transformations with some solutions are given via the Hirota method. The Nth-order Pfaffian solutions are worked out by means of the Pfaffian technique, where N is a positive integer. N-soliton solutions are exported through the Nth-order Pfaffian solutions. By virtue of the asymptotic analysis, elastic and inelastic interactions between the two solitons on some periodic backgrounds are discussed. Interaction among the three solitons is illustrated graphically. The higher-order breather solutions are investigated via the complex parameter relation. Elastic and inelastic interactions between the two breathers on the periodic backgrounds are depicted graphically. Hybrid solutions consisting of the solitons and breathers are obtained. Interaction between the one soliton and one breather on a periodic background is presented.

An exploration of the (3+1)-dimensional negative order KdV-CBS model: Wave solutions, Bäcklund transformation, and complexiton dynamics

“…To uncover soliton solutions, diverse approaches have been developed. Researchers have employed various mathematical techniques, such as Raza et al [16] employed the unified techniques to find solition solutions, Zhao [17] utilized the Kadomtsev-Petviashvili hierarchy reduction method to retrieve dark soliton solutions, Hirota bilinear and Darboux transformation(DT) methods were applied by Gao et al [18] and Hang Yin and Lu ¨ [19] respectively, to find soliton solutions, Cheng et al [20] employed auto-Ba ¨cklund transformations, and Shen et al [21] used DT to obtain multi-pole solitons. Moreover, several alternative techniques have been investigated in the literature [22][23][24].…”

Letter to the Editor on HFF 32, 138 (2022); 32, 2282 (2022); 33, 965 (2023) and 34, 1189 (2024) for the recent shallow-water studies

“…(2022), Gao et al (2023a) as well as Liu et al (2023). Additionally, fluids from all over the Solar System have been observed and discussed (Lainey et al , 2024; Neish et al , 2024; Cheng et al , 2022, 2023a, 2023b, 2024; Feng et al , 2023; Gao, 2024a, 2024d; Gao et al , 2023c; Shen et al , 2023c, 2023d; Wu et al , 2023b; Zhou et al , 2023a, 2023b, 2024).…”

Letter to the editor: Discussing some Korteweg-de Vries-directional contributions in fluid mechanics, atmospheric science, plasma physics and nonlinear optics concerning HFF 33, 3111 and 32, 1674

“…Special cases of equation ( 1) have been seen in fluid mechanics, atmospheric science, plasma physics and nonlinear optics (Brugarino, 1989;Kapadia, 2001;Zhang et al, 2013;Chen et al, 2020aChen et al, , 2020b. By the way, other nonlinear models in fluid mechanics, atmospheric science, plasma physics and nonlinear optics have also been presented (Chen et al, 2024a(Chen et al, , 2024bPeng et al, 2024;Cheng et al, 2022Cheng et al, , 2023aCheng et al, , 2023bGao, 2023aGao, , 2024bGao, , 2024cGao et al, 2023aGao et al, , 2023bWu and Gao, 2023;Wu et al, 2023aWu et al, , 2023bShen et al, 2023aShen et al, , 2023bShen et al, , 2023cZhou and Tian, 2022;Zhou et al, 2023aZhou et al, , 2023bFeng et al, 2023).…”