I am studying the dynamics of thin liquid films. In this context, I obtained the following set of equations: \begin{equation} \frac{\partial y}{\partial t}+\frac{\partial }{\partial x}\left(y^3\frac{\partial^3 y}{\partial x^3}+\frac{y}{2}r\right)=0, \end{equation} \begin{equation} \frac{\partial y}{\partial t}+\alpha(1+\beta y)^2\frac{\partial r}{\partial x}=0. \end{equation}
The next step for me is to solve them numerically. I started by searching for similar equations in the literature, but could not find any. Moreover, advanced books on numerical analysis (Randall J. Leveque, N Trefethen etc.) barley address solving non-linear equations. Any reference for books, names of similar equations, code, will be very helpful.
Edit: $r=r(x,t)$, $y=y(x,t)$ are unknown functions which their solution I am looking for.