Could you please help with the following problem? I have a set of two coupled ODE for $a$ and $b$ waves [wave is a general form of solution $a(z)=A(z)\exp(\imath \beta z)$]. The equations in question are as follows:
$a_z(z) = \imath \beta a(z) + f[a(z),b(z)]$
$b_z(z) = - \imath \beta b(z) + g[(a(z),b(z)]$
$z$ is a direction of propagation, wave $a$ propagates from left to right (from $z=0$ to $z=L$) and wave $b$ propagates from right to left (from $z=L$ to $z=0$). $\beta$ is a complex constant with positive real and imaginary parts. $f[a(z),b(z)]$ and $g[a(z),b(z)]$ are lightly perturbing polynomial functions (i.e. $|f[a(z),b(z)]| < |\beta a(z)|$ and $|g[a(z),b(z)]| < |\beta b(z)|$). $a$ is defined at the left boundary $a(0)=a_0$ and freely goes through the right boundary, and $b$ is defined at the right boundary $b(L)=b_0$ and freely goes through the left boundary.
Could you advice how to solve this system numerically? Especially how to make sure the $a$ and $b$ waves do not get reflected from the boundaries and go through them freely?
Thank you in advance!