# Fractional derivative notation in wave turbulence

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!

• I guess this is the paper you are referring to? Jun 19, 2021 at 19:37
• Yes it is, I should add it to the question. Jun 19, 2021 at 19:41
• The absolute value of the fractional derivative is defined with respect to the Fourier transform, so that the Fourier transform of $|\partial_x|^\alpha f(x)$ is $|k|^\alpha f(k)$. Jun 19, 2021 at 19:42
• I see, so you would say they are not working with the explicit form of the equation but rather using it as a stepping stone to the transformed equation? Jun 19, 2021 at 19:44
• Do you know of a reasonably succinct source where I could read up on this? Jun 19, 2021 at 19:47

The fractional derivative $$|\partial_x|^\alpha$$ is discussed in One-dimensional wave turbulence 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.