For anyone familiar with the BO-ZK model, would you please be able to help me understand this function?
The IVP for the BO-ZK equation is:
$$ \left\{\begin{array}{l} u_{t}+\mathcal{H} \partial_{x}^{2} u+u_{x y y}+u u_{x}=0, \quad(x, y) \in \mathbb{R}^{2}, t>0 \\ u(x, y, 0)=\phi(x, y) \end{array}\right. $$
where $u = u(x, y, t)$ is a real-valued function and H stands for the Hilbert transform.
The general Benjamin-Ono 2-dimensional equation is: $$ u_{t}+\mathcal{H} \partial_{x}^{2} u+u u_{x}=0, \quad x \in \mathbb{R}, t>0 $$
Anyways, I am specifically having trouble pinpointing independent variable and dependent variables regarding their units and the general quantities they represent. I am not studying a math level high enough to understand this question in its entirety, only IVP models, so I am very confused and would appreciate any and all help.
I can link the paper in the comments if anyone needs more context. Sorry to have attached the images in links as well, I do not have the reputation points for that yet.