Soliton and Riemann theta function quasi-periodic wave solutions for a $$(2+1)$$ ( 2 + 1 ) -dimensional generalized shallow water wave equation

Chen, Yiren; Song, Ming; Liu, Zhengrong

doi:10.1007/s11071-015-2161-7

Cited by 20 publications

(8 citation statements)

References 48 publications

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“…In addition, several generalized equations about the (2+1)-D SWW equation have been studied, such as the generalized (2+1)-D SWW equation. [27] In this paper, inspired the above results of shallow water wave equations, we study a new integrable nonlinear equation, namely a (2+1)-D extended shallow water wave (eSWW) equation, [28] and further discover new patterns of nonlinear waves due to the appearance of an arbitrary function. This newly introduced eSWW equation [28] is given by a form as…”

Section: Introductionmentioning

confidence: 96%

Exact solutions of a (2+1)-dimensional extended shallow water wave equation*

Yuan¹,

He²,

Cheng³

2019

Chinese Phys. B

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We give the bilinear form and n-soliton solutions of a (2+1)-dimensional [(2+1)-D] extended shallow water wave (eSWW) equation associated with two functions v and r by using Hirota bilinear method. We provide solitons, breathers, and hybrid solutions of them. Four cases of a crucial ϕ ( y ) , which is an arbitrary real continuous function appeared in f of bilinear form, are selected by using Jacobi elliptic functions, which yield a periodic solution and three kinds of doubly localized dormion-type solution. The first order Jacobi-type solution travels parallelly along the x axis with the velocity ( 3 k 1 2 + α , 0 ) on (x, y)-plane. If ϕ ( y ) = sn ( y , 3 / 10 ) , it is a periodic solution. If ϕ ( y ) = cn ( y , 1 ) , it is a dormion-type-I solutions which has a maximum ( 3 / 4 ) k 1 p 1 and a minimum − ( 3 / 4 ) k 1 p 1 . The width of the contour line is ln [ ( 2 + 6 + 2 + 3 ) / ( 2 + 6 − 2 − 3 ) ] . If ϕ ( y ) = sn ( y , 1 ) , we get a dormion-type-II solution (26) which has only one extreme value − ( 3 / 2 ) k 1 p 1 . The width of the contour line is ln [ ( 2 + 1 ) / ( 2 − 1 ) ] . If ϕ ( y ) = sn ( y , 1 / 2 ) / ( 1 + y 2 ) , we get a dormion-type-III solution (21) which shows very strong doubly localized feature on (x,y) plane. Moreover, several interesting patterns of the mixture of periodic and localized solutions are also given in graphic way.

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Section: Introductionmentioning

confidence: 96%

Exact solutions of a (2+1)-dimensional extended shallow water wave equation*

Yuan¹,

He²,

Cheng³

2019

Chinese Phys. B

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“…Then Tian et al presented the Riemann-Hirota method to investigate the solvability of the quasi-periodic wave solutions of many integrable systems [40,41]. Chen et al investigated the quasi-periodic wave solutions and discussed their asymptotic behaviors of some high-dimensional nonlinear equations [42,43]. The one-periodic wave solutions of the modified generalised Vakhnenko equation and a higher-order KdV-type equation were derived by Wang and Chen.…”

Section: Introductionmentioning

confidence: 99%

Resonance Y-type soliton, hybrid and quasi-periodic wave solutions of a generalized (2+1)-dimensional nonlinear wave equation

Zhang

Zhao

2021

Preprint

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In this paper, we consider a generalized (2+1)-dimensional nonlinear wave equation. Based on the bilinear, the N-soliton solutions are obtained. The resonance Y-type soliton and the interaction solutions between M-resonance Y-type solitons and P-resonance Y-type solitons are constructed by adding some new constraints to the parameters of the N-soliton solutions. The new type of two-opening resonance Y-type soliton solutions are presented by choosing some appropriate parameters in 3-soliton solutions. The hybrid solutions consisting of resonance Y-type solitons, breathers and lumps are investigated. The trajectories of the lump waves before and after the collision with the Y-type solitons are analyzed from the perspective of mathematical mechanism. Furthermore, the multi-dimensional Riemann-theta function is employed to investigate the quasi-periodic wave solutions. The one-periodic and two-periodic wave solutions are obtained. The asymptotic properties are systematically analyzed, which establish the relations between the quasi-periodic wave solutions and the soliton solutions. The results may be helpful to provide some effective information to analyze the dynamical behaviors of solitons, fluid mechanics, shallow water waves and optical solitons.

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“…[29,34,35]. And a series of Bäcklund transformations [36,37,38], Lax pairs [38], supersymmetric [39], soliton-like [40]- [43], breather [44,45] and lump-type [46,47,48] solutions are derived.…”

Section: Introductionmentioning

confidence: 99%

Deformed two-dimensional rogue waves in the (2+1)-dimensional Korteweg-de Vries equation

Cao,

Hu,

Cheng

et al. 2020

Preprint

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Within the (2 + 1)-dimensional Korteweg-de Vries equation framework, new bilinear Bäcklund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation. By introducing an arbitrary function φ(y), a family of deformed soliton and deformed breather solutions are presented with the improved Hirota's bilinear method. Choosing the appropriate parameters, their interesting dynamic behaviors are shown in three-dimensional plots. Furthermore, novel rational solutions are generated by taking the limit of obtained solitons. Additionally, two dimensional [2D] rogue waves (localized in both space and time) on the soliton plane are presented, we refer to it as deformed 2D rogue waves. The obtained deformed 2D rogue waves can be viewed as a 2D analog of the Peregrine soliton on soliton plane, and its evolution process is analyzed in detail. The deformed 2D rogue wave solutions are constructed successfully, which are closely related to the arbitrary function φ(y). This new idea is also applicable to other nonlinear systems.

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Soliton and Riemann theta function quasi-periodic wave solutions for a $$(2+1)$$ ( 2 + 1 ) -dimensional generalized shallow water wave equation

Cited by 20 publications

References 48 publications

Exact solutions of a (2+1)-dimensional extended shallow water wave equation*

Exact solutions of a (2+1)-dimensional extended shallow water wave equation*

Resonance Y-type soliton, hybrid and quasi-periodic wave solutions of a generalized (2+1)-dimensional nonlinear wave equation

Deformed two-dimensional rogue waves in the (2+1)-dimensional Korteweg-de Vries equation

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