# Dušan Navrátil

## Lie Symmetry Analysis of the Charney-Hasegawa-Mima equation

The Charney-Hasegawa-Mima equation (CHM) is a partial differential equation that governs the behavior of large-scale waves in rotating fluids, such as Earth's atmosphere and oceans. Specifically, we will consider the CHM equation in the $\beta$-plane model, which takes the form:

$$\frac{\partial}{\partial t}(\Delta u-Fu)+\beta\frac{\partial u}{\partial x}+[u,\Delta u]=0,$$

where $t$ is the temporal coordinate, $x$ and $y$ are spatial coordinates, $u$ is the stream function, and $\beta$/$F$ are constants that depend on the specific situation.

We will focus on the derivation of Lie-point symmetries of the CHM equation and their geometrical interpretation. Lie-point symmetries are important in the study of differential equations, as they provide insight into the underlying structure of the equations and allow us to find exact solutions. By understanding the symmetries of the CHM equation, we can gain a deeper understanding of the behavior of large-scale waves in rotating fluids.