$$ \frac{\partial u}{\partial t} + u^m \left( \frac{\partial u}{\partial z} \right)^2 + u^n \frac{\partial^2 u}{\partial z^2} + u^p \frac{\partial u}{\partial z} = 0$$
I'm trying to solve this system by numerical methods using an implicit method (my discretization is: a backward time differentiation in time and a central differentiation in space), as you can see the acquired system will be non-linear and I don't know how to approach the non-linearity that arises in this problem. My first intuition was to use Jacobi's method but the term $2 U_{i-1} U_{i+1}$ causes difficulty. I was wondering how can I solve this problem. Any help would be much appreciated.