In the Shallow Water: Auto-Bäcklund, Hetero-Bäcklund and Scaling Transformations via a (2+1)-Dimensional Generalized Broer-Kaup System

Gao, Xin-Yi

doi:10.1007/s12346-024-01025-9

Cited by 36 publications

(6 citation statements)

References 85 publications

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“…In [32], Qawasmeh and Alquran obtained the following solutions: In this article we attained the following solutions. From Family I, we get the following solution:…”

Section: Comparison Of the Resultsmentioning

confidence: 61%

“…The (2+1)-dimensional dispersive long wave (DLW) equation and the (1+1)-dimensional Phi-four equation, are significant mathematical models describing long gravity waves with small amplitudes, long wave propagation in oceans and seas, sediment transport, quantum field theory, phase transitions of matter, the structure of optical solitons [31][32][33] etc. Therefore, various researchers have explored soliton and other solutions to the DLW and Phi-four equations using analytical and semi-analytical approaches, such as Khater et al put in use the sech-tanh expansion procedure [31], Qawasmeh and Alquran [34] used the / ¢ ( ) G G -expansion method to determine soliton solutions to the (2+1)-dimensional DLW equation.…”

Section: Introductionmentioning

confidence: 99%

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Analytic soliton solutions to the shallow water dispersive long gravity wave equations: the first integral approach in nonlinear physics

Hussain,

Akbar,

İlhan

2024

Phys. Scr.

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In this article, we investigate the (2+1)-dimensional dispersive long water wave equation and the (1+1)-dimensional PHI-four equation, which describe the behavior of long gravity waves with small amplitudes, long wave propagation in oceans and seas, coastal structures and harbor design, effects of wave motion on sediment transport, quantum field theory, phase transitions of matter, ferromagnetic systems, liquid-gas transitions, and the structure of optical solitons. We use the first integral technique and obtain new and generic solutions for the models under consideration. By setting definite values for the associated parameters, various types of richly structured solitons are generated. The solitons include kink, flat kink, bell-shaped, anti-bell-shaped, and singular kink formations. These solutions allow for a profound understanding of the behavior and properties of the phenomena, offering new insights and potential applications in the associated field. The first integral technique is simpler, directly integrates the models, and the solutions offer clear insights into the underlying phenomena without requiring intermediate steps, making it widely applicable to various other models, including nonlinear equations and those that are challenging to solve using other standard techniques.

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“…In [32], Qawasmeh and Alquran obtained the following solutions: In this article we attained the following solutions. From Family I, we get the following solution:…”

Section: Comparison Of the Resultsmentioning

confidence: 61%

Section: Introductionmentioning

confidence: 99%

Analytic soliton solutions to the shallow water dispersive long gravity wave equations: the first integral approach in nonlinear physics

Hussain,

Akbar,

İlhan

2024

Phys. Scr.

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“…Nonlinear evolution equations (NLEEs) are a class of partial differential equations in mathematical physics that contain the time variable t. They are derived from a large number of nonlinear phenomena in different backgrounds, such as mechanics, physics, and engineering [7,8]. Because of their important position in electromagnetic fluid dynamics, quantum mechanics, and nonlinear optics, they have attracted great attention from the fields of mathematics, physics, and even engineering [9][10][11]. Methods of solving nonlinear development equations are important components of nonlinear science with strong interdisciplinary aspects and integration [12,13].…”

Section: Introductionmentioning

confidence: 99%

Oceanic Shallow-Water Investigations on a Variable-Coefficient Davey–Stewartson System

Chen,

Wei,

Song

et al. 2024

Mathematics

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In this paper, a variable-coefficient Davey–Stewartson (vcDS) system is investigated for modeling the evolution of a two-dimensional wave-packet on water of finite depth in inhomogeneous media or nonuniform boundaries, which is where its novelty lies. The Painlevé integrability is tested by the method of Weiss, Tabor, and Carnevale (WTC) with the simplified form of Krustal. The rational solutions are derived by the Hirota bilinear method, where the formulae of the solutions are represented in terms of determinants. Furthermore the fundamental rogue wave solutions are obtained under certain parameter restrains in rational solutions. Finally the physical characteristics of the influences of the coefficient parameters on the solutions are discussed graphically. These rogue wave solutions have comprehensive implications for two-dimensional surface water waves in the ocean.

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“…Shallow-water special cases of system (1) have been seen in Ying and Lou (2000), Li and Zhang (2004), Ma et al (2015), Zhao and Han (2015), Kassem and Rashed (2019), (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, 2022Cheng et al, , 2023aCheng et al, , 2023bCheng et al, , 2024Feng et al, 2023;Gao, 2024aGao, , 2024dGao et al, 2023c;Shen et al, 2023cShen et al, , 2023dWu et al, 2023b;Zhou et al, 2023aZhou et al, , 2023bZhou et al, , 2024.…”

mentioning

confidence: 99%

“…(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).…”

mentioning

confidence: 99%

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

Gao

2024

HFF

Self Cite

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Ramos and Garcia Lopez (2024) have absorbingly studied the shallow water, i.e., on certain multiple soliton solutions and lump solutions for two integrable shallow water wave equations (Wazwaz, 2022), Gramian solutions and solitonic interactions for a (2þ1)-dimensional Broer-Kaup-Kupershmidt system in the shallow water (Li et al., 2022), soliton solutions for a (2þ1)-dimensional Korteweg-de Vries equation describing the shallow water waves (Khuri, 2023) as well as blowup in finite time of the numerical solutions for a (1þ1)-dimensional, bidirectional, nonlinear wave model for the small-amplitude waves in shallow water, depending upon the relaxation time, linear and nonlinear drift, power of the nonlinear advection flux, viscosity coefficient, viscous attenuation, etc. (Ramos and Garcia Lopez, 2024).Enthusiasm for the shallow water has been warmed by Wazwaz (2022), Li et al. (2022), Khuri (2023) and Ramos and Garcia Lopez (2024), and consequently this shallow-wateroriented Letter aims to deal with a (2þ1)-dimensional generalised modified dispersive waterwave system for the nonlinear and dispersive long gravity waves travelling along two horizontal directions in the shallow water of uniform depth (Gao et al., 2023a;Liu et al., 2023

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In the Shallow Water: Auto-Bäcklund, Hetero-Bäcklund and Scaling Transformations via a (2+1)-Dimensional Generalized Broer-Kaup System

Cited by 36 publications

References 85 publications

Analytic soliton solutions to the shallow water dispersive long gravity wave equations: the first integral approach in nonlinear physics

Analytic soliton solutions to the shallow water dispersive long gravity wave equations: the first integral approach in nonlinear physics

Oceanic Shallow-Water Investigations on a Variable-Coefficient Davey–Stewartson System

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

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