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 $x_{\pm k}:\mathbb{R}\longrightarrow\mathbb{C}$ and $y_k:\mathbb{R}\longrightarrow\mathbb{C}$, where $k$ is some index. These functions obey the following coupled differential equations: $$ a_1\frac{dx_{\pm k}}{dt}+a_2\frac{dy_{k}}{dt}x_{\pm k}+a_3\frac{dy_{k}^*}{dt}x_{\pm k}+a_4\frac{dy_{k}}{dt}x_{\pm k}+a_5\frac{dy_{k}^*}{dt}x_{\mp k}^*=a_6x_{\pm k}+a_7y_kx_{\mp k}^*+a_8y_k^*x_{\mp k}^* $$ and $$ 2b_1\frac{dy_k}{dt}+4b_2[y_k\frac{dy_k}{dt}+y_k^*\frac{dy_k}{dt}]+b_3[x_k\frac{dx_k^*}{dt}+x_k^*\frac{dx_k}{dt}+x_{-k}\frac{dx_{-k}^*}{dt}+x_{-k}^*\frac{dx_{-k}}{dt}]+b_4[x_k\frac{dx_{-k}}{dt}+x_{-k}\frac{dx_k}{dt}] +b_5[x_k^*\frac{dx_{-k}^*}{dt}+x_{-k}^*\frac{dx_k^*}{dt}]=b_6\,y_k+b_7y_k^2+2b_8y_k^*+2b_7\left|y_k\right|^2-3b_7y_k^{*2}+b_8y_k x_{-k}+b_9x_k^* x_{-k}^* , $$ 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 $$ x_k=\sum_{i=0}^{n}\lambda^ix_k^{(i)}\\ y_k=\sum_{i=0}^{n}\lambda^iy_k^{(i)}\, . $$ After plugging these expansions in and comparing coefficients, the recursion equations are obtained.

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