Chin. Phys. B, 2019, Vol. 28(10): 100202    DOI: 10.1088/1674-1056/ab3e65
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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
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.

Keywords:  (2+1)-dimensional extended shallow water wave equation      Hirota bilinear method      dormion-type solution
Received:  14 July 2019      Revised:  13 August 2019      Accepted manuscript online:
 PACS: 02.30.Ik (Integrable systems) 02.30.Jr (Partial differential equations) 05.45.-a (Nonlinear dynamics and chaos)
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