Non-Linear Equations

When applying the von Neumann analysis to non-linear equations the substitution $y_i \mapsto y_i + \delta y_i$ should be performed before setting $\delta y_i = \delta y \exp(\i k x_i)$. $y_i$ should then be treated as a constant, independent of $x$ (or $i$). The resulting stability condition will then depend on $y$, just as in the case of ODE's.