Solving a system of coupled nonlinear equations numerically

Question

I have the following system of two coupled equations for the variables $A(r, t)$ and $B(r, t)$: $$\begin{split} \partial_tA&=\partial^2_rA+\left\vert A\right\vert^2A+A+B\cdot A\\ \partial_tB&=\left\vert A\right\vert^4 \end{split} $$ When reducing those equations to linear equations (by setting $\vert A\vert^2=0$ and $\vert A\vert^4=A$, I can create a matrix system: $$\begin{pmatrix}\partial_tA\\\partial_tB\end{pmatrix}=\begin{pmatrix}\partial_r^2+1&B\\1&0\end{pmatrix}\begin{pmatrix}A\\B\end{pmatrix}$$ but I do not know how to do that for my example above.
Furthermore, how can I then solve the system stepwise using an implicit method, i.e. how to get a system like the following $$\begin{pmatrix}A_{n+1}\\B_{n+1}\end{pmatrix}\vec{x}=\begin{pmatrix}A_n\\B_n\end{pmatrix}$$, or is that not possible?