Auto-Bäcklund Transformation with the Solitons and Similarity Reductions for a Generalized Nonlinear Shallow Water Wave Equation

Gao, Xin-Yi

doi:10.1007/s12346-024-01034-8

Cited by 33 publications

(7 citation statements)

References 51 publications

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“…Towards system (1), similar to those in Clarkson and Kruskal (1989), Tian (2022, 2024) as well as Gao (2023Gao ( , 2024bGao ( , 2024c, the assumptions…”

mentioning

confidence: 82%

“…Towards system (1), similar to those in Clarkson and Kruskal (1989), Gao and Tian (2022, 2024) as well as Gao (2023, 2024b, 2024c), the assumptions and choice of z y z t ≠ 0 develop into with θ ( x , y , t ), ω ( x , y , t ), ≠ 0, γ ( x , y , t ), κ ( x , y , t ) ≠ 0 and z ( y , t ) ≠ 0 meaning certain real to-be-determined differentiable functions, p ( z ) and q ( z ) denoting two real differentiable functions while the “′” sign representing d/d z .…”

mentioning

confidence: 99%

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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

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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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“…Towards system (1), similar to those in Clarkson and Kruskal (1989), Tian (2022, 2024) as well as Gao (2023Gao ( , 2024bGao ( , 2024c, the assumptions…”

mentioning

confidence: 82%

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

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“…To present these multi-rogue waves, we first consider the case with N = 2. In this case, the f and g functions of the solutions can be obtained from (10) as follows:…”

Section: Multi-rogue Wave Solutionmentioning

confidence: 99%

“…Another subclass of nonfundamental rogue waves consists of higher-order rogue waves, which are obtained when we take N = 1, n 1 > 1 in the rational solution (10) with a real value of p 1 . For instance, if n 1 = 2, we obtain a second rogue wave solutions as follows:…”

Section: Higher-order Rogue Wave Solutionmentioning

confidence: 99%

“…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%

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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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Bilinear-form and similarity-reduction visit to a variable-coefficient generalized dispersive water-wave system concerning Acta Mech. 233, 2527 and 233, 2415

Gao,

Guo,

Shan

2024

Acta Mech

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Auto-Bäcklund Transformation with the Solitons and Similarity Reductions for a Generalized Nonlinear Shallow Water Wave Equation

Cited by 33 publications

References 51 publications

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

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

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

Bilinear-form and similarity-reduction visit to a variable-coefficient generalized dispersive water-wave system concerning Acta Mech. 233, 2527 and 233, 2415

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