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 $\varphi \left(y\right)$, 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 $(3k_1^2+\alpha ,0)$ on (x, y)-plane. If $\varphi \left(y\right)=sn(y,3/10)$, it is a periodic solution. If $\varphi \left(y\right)=cn(y,1)$, it is a dormion-type-I solutions which has a maximum $\left(3/4\right)k_1{p}_1$ and a minimum $-\left(3/4\right)k_1{p}_1$. The width of the contour line is $ln\left[\right(2+\sqrt{6}+\sqrt{2}+\sqrt{3}\left)/\right(2+\sqrt{6}-\sqrt{2}-\sqrt{3}\left)\right]$. If $\varphi \left(y\right)=sn(y,1)$, we get a dormion-type-II solution (26) which has only one extreme value $-\left(3/2\right)k_1{p}_1$. The width of the contour line is $ln\left[\right(\sqrt{2}+1\left)/\right(\sqrt{2}-1\left)\right]$. If $\varphi \left(y\right)=sn(y,1/2)/(1+y^2)$, 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. (paper)