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{U_{i,j}^{k+1}-U_{i,j}^k}{\Delta t}= D_1 \frac{U_{i+1,j}^k+U_{i-1,j}^k+U_{i,j+1}^k+U_{i,j-1}^k-4U_{i,j}^k}{{\Delta z}^2} + r_1 U_{i,j}^k\bigg(\frac{{N_{i,j}^k}^2}{\beta + {N_{i,j}^k}^2} - U_{i,j}^k\bigg)\\ &\qquad\qquad\qquad-s_1\bigg( \frac{U_{i+1,j}^k-U_{i-1,j}^k+U_{i,j+1}^k-U_{i,j-1}^k}{2\big(k_1+A_{i,j}^k\big)^2\Delta z} -2U_{i,j}^k\frac{A_{i+1,j}-A_{i-1,j}^k+A_{i,j+1}^k-A_{i,j-1}^k}{2\Delta z(k_1+A_{i,j}^k)^3} \bigg)\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\frac{A_{i+1,j}^k-A_{i-1,j}^k+A_{i,j+1}^k-A_{i,j-1}^k}{2\Delta z}\\ &+ \frac{s_1U_{i,j}^k}{(k_1+{A_{i,j}^k}^2)}\times \bigg( \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} + \frac{A_{i+1,j+1}^k-A_{i+1,j-1}^k-A_{i-1,j+1}^k+A_{i+1,j+1}^k}{2{\Delta z}^2}\bigg) \\ \end{align*}
\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*}