# Nonlinear first order system of ODEs

While solving some physical problem, I have obtained the following system of differential equations with boundary conditions:

$$\left\{\begin{matrix} \frac{d\phi_1}{dz}=\frac{m^2}{\lambda}- \lambda\phi_1^2-\alpha\phi_2^2 \\ \frac{d\phi_2}{dz}=-2\alpha\phi_1\phi_2 \\ \phi_2(\pm\infty)=0 \\ \phi_1(\pm\infty)=\pm\frac{m}{\lambda} \end{matrix}\right.$$

where $\phi_1(z),\phi_2(z)$ are just functions of real variables, $m,\lambda,\alpha\in \mathbb{R}$

As far as I know, to solve this problem I should solve the system of differential equations and after that use boundary conditions.

I see that it's easy to solve this system when $\lambda=\alpha$:

Just sum this two equations to obtain $$\frac{d(\phi_1+\phi_2)}{dz}=\frac{m^2}{\lambda}-\lambda(\phi_1+\phi_2)$$ This DE easily solved $$(\phi_1+\phi_2)=\frac{m}{\lambda}\tanh(mz-m\lambda C_1)$$ where $C_1$ is an integration constant. After that I can find exact solutions for $\phi_1$ and $\phi_2$. And boundary conditions are satisfied automaticaly.

But I would like to obtain the solution for any $\alpha$ and $\lambda$ at least by quadrature.

My attempt was to reproduce the aproach as in case $\lambda=\alpha$:

I've obtained

$$\frac{1}{\sqrt{\alpha}}\frac{(\sqrt{\alpha}\phi_1+\lambda\phi_2)}{dz}=\frac{m^2}{\lambda}-(\sqrt{\lambda}\phi_1+\sqrt{\alpha}\phi_2)^2$$

So, this attempt was a fail.

Also I have noticed that these two equations look like as Riccati equation

And I have no any ideas how to solve it. Any help will be appreciated.