It is a nonliear Partial differential equation . you can find it in the appendix.
If you have some good idea about this equation ,ok ,please reply to me .thanks!
If u is a complex valued function, then separate it into real and imaginary parts: u(x)= p(x)+ i q(x). |u|2= p2+ q2 and you have two non-linear differential equations for p and q. They will still be rather nasty I am afraid. MOST non-linear equations can be solved in closed form.