Dissipative Petviashvili equation for the two-dimensional Rossby waves and its solutions

The general shallow wave equation can be used to describe the fluid motion which own uniform density and nearly uniform speed. In fact, these problems we face in nature do not satisfy these conditions, for example, the motion system of nonlinear rotation fluid governed by the frictional dissipation as well as the atmosphere system with greater viscosity. In these conditions, we need to amend the general shallow wave equation and consider the dissipative and viscous effect. In this article, starting from the shallow wave equation with dissipative and viscous effects in the horizontal direction, by virtue of plane approximation and quasi-geostrophic approximation of large-scale motion, we derive the dissipative Petviashvili equation to describe the two-dimensional Rossby waves. Based on the ansatz function method, we obtain the exact analytical solutions of dissipative Petviashvili equation and discuss the influence of dissipation on the two-dimensional Rossby waves.