Vtyjkyuk

I am trying to solve non linear coupled first order differential equations. I was looking for an exact result, but I did not find one. So I am leaning towards approximation methods, like perturbation theory.

However, before I calculate approximate solutions, I wanted to ask you guys if you know any exact solutions to the following coupled differential equations:

Let $x_pm k:mathbbRlongrightarrowmathbbC$ and $y_k:mathbbRlongrightarrowmathbbC$, where $k$ is some index. These functions obey the following coupled differential equations:
$$
a_1fracdx_pm kdt+a_2fracdy_kdtx_pm k+a_3fracdy_k^*dtx_pm k+a_4fracdy_kdtx_pm k+a_5fracdy_k^*dtx_mp k^*=a_6x_pm k+a_7y_kx_mp k^*+a_8y_k^*x_mp k^*
$$
and
$$
2b_1fracdy_kdt+4b_2[y_kfracdy_kdt+y_k^*fracdy_kdt]+b_3[x_kfracdx_k^*dt+x_k^*fracdx_kdt+x_-kfracdx_-k^*dt+x_-k^*fracdx_-kdt]+b_4[x_kfracdx_-kdt+x_-kfracdx_kdt]
+b_5[x_k^*fracdx_-k^*dt+x_-k^*fracdx_k^*dt]=b_6,y_k+b_7y_k^2+2b_8y_k^*+2b_7left|y_kright|^2-3b_7y_k^*2+b_8y_k x_-k+b_9x_k^* x_-k^* ,
$$
where all $a_i$ and $b_i$ are complex coefficients.

I have tried several steps, including separating real and imaginary parts but I was utterly unsuccessful.

Thank you in advance!

I haven't found known exact solutions, so I solved these equations myself via perturbation theory and acquired recursion formulas for the solutions for arbitrary accuracy. This is how I proceeded: I multiplied the non-linear terms by a scaling factor $lambda$ and expanded all solutions in terms of this scaling factor
$$
x_k=sum_i=0^nlambda^ix_k^(i)\
y_k=sum_i=0^nlambda^iy_k^(i), .
$$
After plugging these expansions in and comparing coefficients, the recursion equations are obtained.