Bäcklund transformations and explicit solutions of (2+1)-dimensional barotropic and quasi-geostrophic potential vorticity equation

By the Bäcklund transformation method, an important (2+1)-dimensional nonlinear barotropic and quasi-geostrophic potential vorticity (BQGPV) equation is investigated. Some simple special Bäcklund transformation theorems are proposed and used to get explicit solutions of the BQGPV equation. Furthermore, all solutions of a second order linear ordinary differential equation including an arbitrary function can be used to construct explicit solutions of the (2+1)-dimensional BQGPV equation. Some figures are also given out to describe these solutions.