This is my first question in MathOverflow and I will do my best to format it correctly and make it clear.

I am reading a paper on dispersive wave turbulence which introduces the following family of equations:

$$i\psi_t=|\partial_x|^{\alpha}\psi+|\partial_x|^{-\beta/4}\left(\left||\partial_x|^{-\beta/4}\psi\right|^2|\partial_x|^{-\beta/4}\psi\right)$$

Where $|\cdot|$ denotes the $L^2$ norm in relevant places. This equation is stated to have the dispersion relation $\omega=|k|^{\alpha}$ and becomes an NLS equation when $\alpha=2$ and a water-like dispersion law when $\alpha=1/2$.

I am not terribly comfortable with fractional derivatives, and in my Googling have been unable to find the use of the particular absolute value notation in $|\partial_x|^{\alpha}$. Could anyone help me interpret the equation above or point me towards a solid source?

Many thanks in advance!