Fourier Transform of a function squared.

[Dustinsfl]

Consider \(u_t = -u_{nxxx} - 3(u^2)_{nx}\).

The Fourier Transform is linear so taking the Inverse Fourier transform of the Fourier Transform on the RHS we have
\begin{align}
-\mathcal{F}^{-1}\left[\mathcal{F}\left[u_{nxxx} - 3(u^2)_{nx}\right]\right] &= -\mathcal{F}^{-1} \left[\mathcal{F}\left[(ik)^3u\right]\right] - 3\mathcal{F}^{-1}\left[\mathcal{F} \left[(ik)u^2\right]\right]\\
&= ik^3\mathcal{F}^{-1}\left[\mathcal{F}(u)\right] - ik\mathcal{F}^{-1}\left[\mathcal{F}(u^2)\right]
\end{align}

1. Is the above reduction correct?
2. Can \(\mathcal{F}(u^2) = \mathcal{F}(u\cdot u)\) be further reduced?

 

[Jhoneine]

Yes, the reduction is correct. You cannot further reduce \(\mathcal{F}(u^2) = \mathcal{F}(u\cdot u)\).