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

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.