I'm studying the non linear Schrödinger equation:
$$A_t=i A_{xx} - i \vert A \vert^2 A$$
In the problem sheets that I'm working on my teacher writes that this equation has solutions $A=Q e^{i \Omega t}$ with $\Omega=-i Q^2$. We want to find the equation governing small perturbations in space and time. In order to do so we consider:
$$ A(x,t)=[Q+r(x,t)] e^{i(\Omega t + \phi(x,t))}$$
Afterwards I am asked to write the equations for $\partial_t r$ and $\partial_t \phi$. I am confused right now. I see clearly that I can write a equation which involves those two terms and also the spatial derivatives of $r$ and $\phi$ but, how could I get two separate equations? Would I find them just by taking the real and the imaginary part of the full equation?
Effectively I had to consider the real and imaginary parts of the equation. Considering $A=Re^{i \theta}$ we can express the original equation as the following system of 2 non linear equations:
$$R_t = -2 R_x \theta_x - R ~\theta_{xx}$$ $$R ~ \theta_t = R_{xx} -R(\theta_x)^2 - R^3 $$
Considering the given perturbation we have $R=Q+r$ and $\theta=\Omega t + \phi$, so the final equations are:
$$r_t = - 2 r_x \phi_x - (Q+r) \phi_{xx}$$ $$(Q+r)(\Omega + \phi_t) = r_{xx} - (Q+r)(\phi_x)^2 - (Q+r)^3$$