User:RotemAssouline/sandbox

The periodic Hunter-Saxton equation can be given a geometric interpretation as the geodesic equation on an infinite-dimensional Lie group, endowed with an appropriate Riemannian metric. In more detail, consider the group $$\mathrm{Diff}(S^1)$$ of diffeomorphisms of the unit circle $$S^1$$. Choose some $$x_0\in S^1$$ and denote by $$G$$ the subgroup of $$\mathrm{Diff}(S^1)$$ consisting diffeomorphisms which fix $$x_0$$:
 * $$G=\{\varphi\in\mathrm{Diff}(S^1) \mid \varphi(x_0)=x_0\}.$$

The group $$G$$ is an infinite-dimensional Lie group, whose Lie algebra consists of vector fields on $$S^1$$ which vanish at $$x_0$$:
 * $$\mathfrak{g}=\left\{ u\frac{\partial}{\partial x} \ \bigg\vert \ u(x_0)=0\right\}.$$

Here $$x$$ is the standard coordinate on $$S^1$$. Endow $$\mathfrak{g}$$ with the homogeneous $$\dot H^1$$ inner product:

\left\langle u\frac{\partial}{\partial x},v\frac{\partial}{\partial x}\right\rangle_{\dot H^1}:=\int_{S^1}u_xv_xdx, $$ where the subscript denotes differentiation. This inner product defines a right-invariant Riemannian metric on $$G$$ (on the full group $$\mathrm{Diff}(S^1)$$ this is only a semi-metric, since constant vector fields have norm 0 with respect to $$\dot H^1$$. Note that $$G$$ is isomorphic to the right quotient of $$\mathrm{Diff}(S^1)$$ by the subgroup of translations, which is generated by constant vector fields).

Let
 * $$U(x,t)=u(x,t)\frac{\partial}{\partial x}$$

be a time-dependent vector field on $$S^1$$ such that $$U(\cdot,t)\in\mathfrak{g}$$ for all $$t$$, and let $$\{\varphi_t\}$$ be the flow of $$U$$, i.e. the solution to:
 * $$\frac{d}{dt}\varphi_t(x)=u(\varphi_t(x),t).$$

Then $$u$$ is a periodic solution to the Hunter-Saxton equation if and only if the path $$t\mapsto\varphi_t\in G$$ is a geodesic on $$G$$ with respect to the right-invariant $$\dot H^1$$ metric.

In the non-periodic case, one can similarly construct a subgroup of the group of diffeomorphisms of the real line, with a Riemannian metric whose geodesics correspond to non-periodic solutions of the Hunter-Saxton equation with appropriate decay conditions at infinity.