I would like to find the solution to this nonlinearly coupled PDE:
This is an equation involving quantum mechanics and found in this paper. (Technically these E's are operators, but you can simply treat them as functions of z and t. Also $E^+$ is the conjugate of E so $E^+E = E^2 = E(z, t)^2$)
According to this paper, with some approximations, this has an analytical solution of the form $$E_{1,2}(z, t) = E_{1,2}(0,t') \exp(i \eta z |E_{2,1}(0, t')|^2)$$ where $t' = t-z/v_g$
If I am reading the paper correctly, the approximations are that $\beta \to 0$ and $F \to 0$ - but I'm not entirely sure. In my attempts to find the solution, I first tried to solve it as an ode (where the time derivatives are zero).
$$ E_1'(z) = -k E_1(z) + (i \eta)|E_2|^2 E_1 $$ $$ E_2'(z) = -k E_2(z) + (i \eta)|E_1|^2 E_2 $$ But I'm struggling to even work this out. Any ideas for how I can proceed?
EDIT:
One of the answers suggests a method at arriving at the solution, and here I'm showing my handwritten attempt at getting a solution using this method. I was successful in finding a solution (maybe with mistakes?) but ended up with an answer that does not match what is described in the text.
The most straightforward procedure to arrive at the stated result is:
a) Change to comoving coordinates, i.e. write $E_{1,2}(z,t) = \tilde{E}_{1,2}(z,t')$ with $t'=t-z/v_g$;
b) Write the complex functions $\tilde{E}_{1,2}$ in terms of modulus and argument, i.e. write $\tilde{E}_{1,2}(z,t') = r_{1,2}(z,t')\, e^{i\, \omega_{1,2}(z,t')}$, where $r_{1,2}$ and $\omega_{1,2}$ are real-valued;
c) Substitute this into the equations and set $F_{1,2} = 0$ and $\beta = 0$;
d) Separate the real and imaginary parts in each equation, such that your equations are of the form $A + B i = 0$, $C + D i = 0$.
e) Solve the four equations $A = 0$, $B = 0$, $C = 0$, $D = 0$ for the four functions $r_{1,2}$ and $\omega_{1,2}$.