Definitions of Fourier Transform: $$ \hat{v}(k,t) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-ikx}\ v(x,t) \ dx $$ $$ v(x,t) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{ikx}\ \hat{v}(k,t) \ dk $$ Using these definitions, I was able to derive the fact that the transform takes partial derivatives with respect to $x$ and simply converts them to the transform of the function multiplied by a factor of $ik$. Conversely, the transform of the partial with respect to $t$ is simply the partial with respect to $t$ of the partial derivative. I was wondering however how the transform would act on the partial with respect to $x$ squared. I.e. what is the transform of: $$ \left(\frac{\partial u}{\partial x}\right)^2 $$ Plugging into the definition, we have: $$ \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-ikx}\ \left(\frac{\partial u}{\partial x}\right)^2 \ dx $$ Following the lead of my first two calculations, I would like to use integration by parts to solve this however what would the antiderivative of the partial squared be? Is there possibly a simpler way of figuring this out? Thanks for any help!
Note, this is part of me trying to solve a nonlinear PDE. My thought was that maybe using either the $1$D (these definitions) or $2$D (in space and time) Fourier transform might help. However, I think the nonlinearities might preclude either of these methods from being fruitful. If this is true, are there any other analytic techniques that might aid me? The equation I want to solve is: $$ u_t = u_{xxxx}-3u(u_x)^2-\frac{3}{2}u^2u_{xx}+\frac{1}{2}u_{xx}+F $$ where $F$ may or may not be a function of $u$ and its derivatives. I was hoping to be able to solve the equation with $F=0$ and then introducing it as an unknown function and seeing how it affects the solution.
EDIT: After thinking about this a little longer, I realized I may just be able to use the Convolution Theorem. Taking $ \left(\frac{\partial u}{\partial x}\right)^2 =\frac{\partial u}{\partial x}\cdot \frac{\partial u}{\partial x}$, would the transform of this object simply be the convolution of it with itself: $$ \left(\frac{\partial u}{\partial x}*\frac{\partial u}{\partial x}\right)(x,t)=\int_{-\infty}^{\infty} \frac{\partial u}{\partial x}(\xi,t)\frac{\partial u}{\partial x}(x-\xi,t) \ d\xi $$ My broader questions still stump me a bit though. Even with this, I am unsure how this will help me in solving the PDE.