Exact solutions of a (2+1)-dimensional extended shallow water wave equation *Project supported by the National Natural Science Foundation of China (Grant Nos. 11671219 and 11871446). (October 2019) - Yuan 袁, Feng 丰; He 贺, Jing-Song 劲松; Cheng 程, Yi 艺
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 ( 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.