How to solve this system of nonlinear ODE's? We have $a,b,c,d$, functions of $x$, such that
$a_x=-d-4acd$
$b_x=c+4bcd$
$c_x=b+4abc$
$d_x=-a-4abd$
I've already discovered that $(ab+cd)_x=0$, which I believe may be helpful.
I've tried taking second derivatives of the above in an attempt to reduce the number of variables, but this didn't lead anywhere for me.
Edit: The boundary conditions are that $a\rightarrow\pm a_0$ as $x\rightarrow\pm\infty$, where $a_0$ is a constant, and similarly for $b,c,d$.

You can simplify your system through a sequence of variable transformations. First, based on your discovery and that of @Michael, it seems reasonable to define
\begin{equation}
x = a b,\quad y = a c,\quad z = c d,\quad w = b d
\end{equation}
and write the system in terms of those variables. Then, we introduce
\begin{equation}
A = x + z,\quad B = x-z,\quad C = y+w,\quad D = y-w
\end{equation}
to yield $A'=0$, which you found. @Michael ' s observation then boils down to $C = B^2 + k$, with $k$ a constant. The only two remaining free variables $B$ and $D$ can be used to obtain the following second order equation for $B$:
\begin{equation}
B'' = 4 B (1 +2 k +2 B^2),
\end{equation}
which should be straightforward to analyse.
