1) ok thanks I'll try to come up with a more clear question and make a new post 2) I see, thanks very much

When I said the equation appears to me not to be separable, I meant that eg. in 2D taking $\phi(x,t) = X(x)T(t)$, we get $X'''' T - 2 X'' T'' + X T''' =0$, and I can't see how to seperate out $T$ and $X$ here. Like you say though, I think this is not important given your Fourier space analysis

2) Using that $u^3 \delta''(u) = 0$, I can also solve the equation in Fourier space by $\hat{\phi}(k) = \Box_k \left[H(k) \delta(k^2)\right]$, because the $\Box_k$ acting on the delta function gives me a factor of $k^2$ additional to the $k^4$ coming from the original PDE. It looks to me like I could potentially continue this process using $u^{n+1} \delta^{(n)}(u) = 0$ to get more and more solutions. Did you use some kind of principle to select only the solution involving $\delta'(k^2)$ and not these higher solutions?

Thanks for a great answer, I can see this is definitely a good way to think about the problem. I have two questions; 1) Suppose $\phi_1,\phi_2 \in L^2$, then I can expand them in a basis of plane waves. Can I think of $a\cdot x \phi_2$ as having an expansion in terms of some basis, (eg. $(a\cdot x) e^{i k \cdot x}$ or similar), such that these basis elements are orthogonal each other and to the plane waves?

@Ryan Budney this is fascinating do you have a reference that makes this definiton? I could imagine interdisciplinary work on this in psychology or neuroscience, that would be really interesting

OK well thanks for a great answer I've got everything I need for this problem now

Thanks for explaining about this point, your post was an interesting read

Thanks for explaining about CO$(n,n)$. What's the name so I can google it?

Here is my proof that $f(u,v)$ solves the equation; $u = a{\cdot}x , v = b{\cdot}y.$ Then $\frac{\partial}{\partial x}\cdot\frac{\partial}{\partial y}f(u,v) = \frac{\partial u}{\partial x}\cdot\frac{\partial v}{\partial y}\partial_u \partial_v f(u,v) = a{\cdot}b\partial_u \partial_v f(u,v) = 0.$ If I apply this to $f(u,v) = u v^2$ I also get zero. Am I making a mistake somewhere?

Oh perfect thanks this is just what I needed. Am I right in thinking that your solution can be generalised to $f(x,y) = h(\,a{\cdot}x ,b{\cdot}y\,)$? And why did you include that $a^2 + b^2 = 1$? I can't see why that condition is needed. Also I'm not familiar with which group $CO$ refers to.

Oh thanks that's really helpful I didn't know it was called that. I've had a quick look and there seems to be a lot of literature on ultrahyperbolic equations! I will spend some more time to look through the literature. Any suggestions for references which construct solutions explicitly would also be greatly appreciated.