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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{ …

(The same system with slightly different questions has been asked in MSE.) Let $\Omega\subset \mathbb{R}^2$ be some simply connected planar domain. We seek for a mapping $\mathbf{r}:\Omega\rightarrow …

Consider a flat, simply connected surface endowed with the Riemannian metric $g_0=e^{2\Omega(u,v)}\left(\mathbb{d}^2u +\mathbb{d}^2v \right)$, so that $\Omega(u,v)$ is an arbitrary harmonic function. …

This is a cross-post. In the context of two dimensional elasticity theory, when considering deformations of flat membranes into spherical caps, one encounters the following hyperbolic system \begin{al …

Let $\Omega_1$ and $\Omega_2$ be (simply connected) domains on $\mathbb{R}^2$, with coordinates $(x,y)$ and $(X,Y)$ respectively. Given a (smooth) function $Z(X,Y)$ such that $Z\left(\partial \Omega_2 …

Think of a beach ball on an pool of water or sand. Let $\left(\mathcal{M}^2,g\right)$ be a surface homeomorphic to a sphere, endowed with a Riemannian metric $g$, and $\left(\mathcal{N}^2,h\right)$ a …