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(Too long to be a comment.) $u(x,0)$ is not enough, you need to set some condition(s) on the $x$ coordinate. For instance, let's say that $u(0,t)=u(L,t)=0$ for some given $L$. In that case a natural a …

(Too long to be a comment.) A hand-waving, physicist calculation of a specific example for $d=3$ goes as follows. It may be put in more rigours terms for higher but not for lower dimensions. Let's tak …

Bäcklund transformations may be used also in ODE to solve non-linear problems; for instance, it's well known that for the equation $$ \frac{\mathrm{d}^2\omega}{\mathrm{d}t^2}=\sin\omega \tag{*}\label{ …

Consider the equation $$ \begin{equation} \frac{\partial^2f}{\partial x\partial y}=f \end{equation} $$ on a Jordan domain (i.e. the interior of a simple, closed curve on the plane). The equation is hy …

I have the following PDE in two dimensions $$ 2\partial_x\partial_y\sqrt{1-u^2}+\left(\partial^2_x-\partial^2_y \right)u=0, $$ with $u=u(x,y)$ with values between $-1$ and $1$, or alternatively $$ 2\p …

I need to find solutions to the Beltrami equation $$ \frac{\partial w}{\partial\overline{{z}}}=e^{i\varphi(z)}\frac{\partial w}{\partial z} $$ for $w=w(z,\overline{z})$ and $\varphi(z)$ some given, r …