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!


(reference: <https://www.semanticscholar.org/paper/A-one-dimensional-model-for-dispersive-wave-Majda-McLaughlin/75056874558c915a68f9cb53fc0dc989148e6db5>)



Fractional derivative notation in wave turbulence
The fractional derivative $|\partial\_x|^\alpha$ is discussed in [One-dimensional wave turbulence](https://maths.ucd.ie/~dias/ZakharovDiasPushkarev.pdf) by Zakharov, Dias, and Pushkarev. (Zakharov introduced the notation.) As they explain below Eq. 2.1, it is indeed defined via the Fourier transform, such that the Fourier transform of $|\partial\_x|^\alpha\psi(x)$ is $|k|^\alpha\psi(k)$. Their appendix A contains a few more details.


