When transforming nonlinear PDE to its Fourier space, I encounter the following problem:
Consider the equation $u_t=(u^3-u)_{xx}$. Then, when transforming to Fourier space we get
\begin{equation*} \hat{u}_t=-k^2(\hat{u^3-u})=-k^2(\hat{u}*\hat{u}*\hat{u}-\hat{u}). \end{equation*}
On the other hand, if we represent the equation as $u_t=[(3u^2-1)u_x]_x$ we obtain
\begin{equation*} \hat{[(3u^2-1)u_x]_x}=ik[\hat{3u^2u_x}-\hat{u_x}]=\cdots =-k^2(3\hat{u}\ast \hat{u}\ast \hat{u}-\hat{u}). \end{equation*}
Can someone please solve this inconsistency for me? If I derive once more the 3 becomes 9...
Thanks,
Alon.
$$ \widehat{u^2\,u_x}=\hat u\ast\hat u\ast(i\,k\,\hat u)\ne i\,k\,\hat u\ast\hat u\ast\hat u. $$