(First time asking question on this forum so please kindly let me know if this is out of scope/inappropriate etc.)


I have a problem that leads me to the following quadratic system of PDEs:-


$
c\_1 w\_q (\partial\_tw\_q) + c\_2 w\_q (\partial\_xw\_q) + c\_3 (\partial\_{x}w\_q)^2 +c\_4 w\_q (\partial\_{xx}w\_q) + c\_5 (w\_q)^2 + c\_6 w\_q w\_{q-1} = 0\; \forall q \in \mathbb{N}
$


with $w\_q(T,x)=1$ and $w\_0(t,x) = 1$, $t \in [0,T] \subset \mathbb{R}$
and $c\_1 = 1;\;\; c\_2,c\_5 \in \mathbb{R};\;\; c\_3<0;\;\;c\_4,c\_6>0$


I would like to find the series of solutions $w\_q(t,x) \;\forall q \in \mathbb{N}$. 


I would have really liked this to work out into a linear system of PDEs but i'm having no luck at all. I've read something about Quadratic Differential Systems but all the sources seem to be dealing with quadratic systems of ODEs. 


Has anyone encountered anything similar or is my system too exotic to have been considered in the literature? (or just my lack of erudition!) In addition to solutions, would appreciate pointers to relevant materials, keywords or even wild ideas that I should look at.



Quadratic PDE Systems
Anyway, let's comb it a bit. Since we start at $1$ at $T$, I'll assume that $w\_q>0$ at least for $t$ close to $T$. Once $w\_0$ is fixed everywhere, we have no choice but to solve left to right.


Put $W=w\_q^p$. Then $(W)\_{xx}=pw^{p-2}(ww\_{xx}+(p-1)w\_x^2)$, so we can choose the power to combine two second order terms into one ($p=0$ should be understood as $W=\log w\_q$). Then we get
$$
c\_1W\_t+c\_2W\_x+c\_3W\_{xx}+c\_4w\_{q-1}W^{(p-1)/p}=0.
$$
This is almost standard heat except it is written in a skewed coordinate system instead of the rectangular one (which can be trivially adjusted) plus you have an extra term. Since you start at $1$, in any decent function space you can hum the usual Duhamel-Gronwall tune to prove local uniqueness. Of course, your setup may be so exotic that even the unperturbed heat equation has many solutions, so if you want me or anyone else to go beyond this point, pose your problem with full and clear details. 


