I am studying the nonlinear Schrodinger equation $$A_t+iA_{xx}+i|A|^2A=0$$ for $A=ae^{i\theta}$ a complex valued function, with $a,\theta$ real. I am trying to figure out what sets the signs of the phase dislocations, and in particular the frequency singularities.
We consider an example where $A_0 = sech(x)e^{-ic_0x^2}$ where $c_0$ is a constant that sets the chirp of the packet. I solve the governing equation numerically. The packet focuses, and eventually splits.
Locations where $a\to 0$ will then correspond to phase dislocations, and frequency singularities, where we define the frequency as $\omega=-\theta_t$. One can see this by noting that $$\theta_t -\theta_x^2 = -\frac{a_{xx}}{a},$$ where we have ignored the nonlinear term as it's dominated by the curvature term as $a\to 0$.
The first plot 
shows $|A|$ and $\omega$. The second plot shows a sketch of why the phase (which has been reduced to show the point) is becoming large, as $a$ is going to 0.
My question is, what sets the sign of these singularities? That is, why is the frequency going to $+\infty$ on the right side, and $-\infty$ on the left?
