# Von Neumann's stability analysis on non linear and coupled equations

I'd like to know if is possible to make a Von Neumann's stability analysis on an system of coupled equations, featuring quadratics:

\begin{aligned} \frac{\partial u_1}{\partial t}&=D_1\Delta u_1 - s_1\left(\left(\frac{\nabla u}{(k+a)^2}-2u\frac{\nabla a}{(k+a)^3}\right)\nabla(a) + \frac{u}{(k+a)^2}\left(\Delta(a) + 2\frac{\partial^2a}{\partial x \partial y}\right)\right) \\&\quad+ r_1u_1 \left(\frac{n^2}{\beta + n^2}-u_1\right)\\ \frac{\partial a}{\partial t} &= D_a\Delta a + c_1n\frac{u_1^2}{\alpha_1 + u_1^2} - \gamma a\\ \frac{\partial n}{\partial t}&= D_s\Delta n - l_1u_1\frac{n^2}{\beta + n^2} \end{aligned}

My idea was to make different analysis for each of terms of the equations. There are no problems for most of the terms, but I don't know how to deal with the coupled ones and the nonlinear one. Can someone give me hints or link me a paper dealing with these?

Here is the numerical scheme :

\begin{align*} &\frac{A_{i,j}^{k+1}-A_{i,j}^k}{\Delta t}= D_a \frac{A_{i+1,j}^k+A_{i-1,j}^k+A_{i,j+1}^k+A_{i,j-1}^k-4A_{i,j}^k}{{\Delta z}^2}-\gamma A_{i,j}^k+c_1N_{i,j}^k\frac{{U_{i,j}^k}^2}{\alpha_1+{U_{i,j}^k}^2}+c_2N_{i,j}^k\frac{{V_{i,j}^k}^2}{\alpha_2+{V_{i,j}^k}^2} \end{align*}

\begin{align*} &\qquad\frac{N_{i,j}^{k+1}-N_{i,j}^k}{\Delta t}= D_a \frac{N_{i+1,j}^k+N_{i-1,j}^k+N_{i,j+1}^k+N_{i,j-1}^k-4N_{i,j}^k}{{\Delta z}^2}-\frac{{N_{i,j}^k}^2}{\beta + {N_{i,j}^k}^2}\big(l_1{U_{i,j}^k}^2+l_2{V_{i,j}^k}^2\big)\\ \end{align*}

• Von Neumann stability analysis is performed for finite-difference schemes, so discretized equations are kind of the focus of the question - or at least a thorough explanation of what FD-scheme is used for each term. This will highly increase the probability of a good answer from the community; however, maybe someone has particular references for coupled and nonlinear terms handling ready that will suffice you. – Anton Menshov Jun 3 '18 at 15:21
• I just added it. The discretization could certainly be done in a smarter way – nougatine Jun 3 '18 at 16:11

$$d_t\vec{u}+A\vec{u}=\vec{f}$$
where in your case $\vec{u}=(\vec{u}_1,\vec{a},\vec{n})^{T}$ and $\vec{f}=\vec{0}$.
If you have a nonlinear system, i.e. $A=A(u)$, you can always linearise it $A\approx A({\vec{u}_0})$ and see what occurs in the vicinity of a selected solution $\vec{u}_0$, usually the one that makes the system stationary: $$A(\vec{u}_0)\vec{u}_0=\vec{f}$$ and perform the stability analysis.