Consider the one-dimensional viscous Burgers' equation,
$$ u_t+uu_x-\nu u_{xx}=0 \tag{1} $$
In order to solve it numerically using the explicit method of finite differences, one can transform the problem to the 1D heat equation, $u_t-\nu u_{xx}=0$, using the Hopf-Cole transformation. If $\Delta t$ and $\Delta x$ are the time and space steps, then the method would be numerically stable and convergent whenever $r=\nu \Delta t/(\Delta x)^2 \leq 1 / 2$, and the numerical errors would be proportional to the time step and the square of the space step, $O(u)=O(\Delta t)+O(\left(\Delta x\right)^{2})$.
However, what if we wanted to apply the explicit finite difference method directly to the Burgers' equation, without using the Hopf-Cole transformation? We could think about the following discretization:
\begin{equation} \frac{u_{i, j+1}-u_{i, j}}{k}+u_{i, j} \frac{u_{i+1, j}-u_{i, j}}{h}=\nu \frac{u_{i+1, j}-2 u_{i, j}+u_{i-1, j}}{h^{2}}. \end{equation}
Which would the stability and convergence criterium be in this case? Would the numerical error be the same? Would there be any advantage to apply the method directly to the original equation, without using the transformation?
FDM is very tried and tested with the heat equation, and it is linear unlike Burgers' equation. You could rearrange your form into: $$u_i^{j+1}=-\left(\frac kh u_i^j+\frac{\nu k}{h^2}\right)u_{i+1}^j+\left(\frac kh u_i^j+\frac{2\nu k}{h^2}+1\right)u_i^j+\frac{\nu k}{h^2}u_{i-1}^j$$ the problem with it being non-linear is you cannot simply make a substitution and analyse what happens when that value changes as is possible with the heat equation.
you could try giving $u_i^j$ some arbitrary value and say something like $u_{i+1}^j=au_i^j$ and substitute this in and see where that gets you?