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Exact solutions of a (2+1)-dimensional extended shallow water wave equation |
Feng Yuan(袁丰)1, Jing-Song He(贺劲松)2, Yi Cheng(程艺)1 |
1 School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China;
2 Institute for Advanced Study, Shenzhen University, Shenzhen 518060, China |
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Abstract 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 (3k12+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)k1p1 and a minimum -(3/4)k1p1. 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)k1p1. The width of the contour line is ln[(√2+1)/(√2-1)]. If φ(y)=sn(y, 1/2)/(1+y2), 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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Received: 14 July 2019
Revised: 13 August 2019
Accepted manuscript online:
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PACS:
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02.30.Ik
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(Integrable systems)
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02.30.Jr
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(Partial differential equations)
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05.45.-a
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(Nonlinear dynamics and chaos)
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Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11671219 and 11871446). |
Corresponding Authors:
Jing-Song He
E-mail: hejingsong@szu.edu.cn
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Cite this article:
Feng Yuan(袁丰), Jing-Song He(贺劲松), Yi Cheng(程艺) Exact solutions of a (2+1)-dimensional extended shallow water wave equation 2019 Chin. Phys. B 28 100202
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