# Soliton solutions for the elastic metric on spaces of curves

## TL;DR

This paper studies a reparametrization-invariant Sobolev metric on the space of curves, showing that geodesics can be interpreted as soliton solutions, with implications for shape analysis and discretization.

## Contribution

It extends the metric to Lipschitz curves, proves well-posedness of the geodesic equation, and reveals soliton solutions as geodesics, a novel finding in this context.

## Key findings

- Geodesics are soliton solutions of the geodesic equation.
- Piecewise linear curves form a totally geodesic submanifold.
- The metric is well-posed on Lipschitz curves.

## Abstract

In this article we investigate a first order reparametrization-invariant Sobolev metric on the space of immersed curves. Motivated by applications in shape analysis where discretizations of this infinite-dimensional space are needed, we extend this metric to the space of Lipschitz curves, establish the wellposedness of the geodesic equation thereon, and show that the space of piecewise linear curves is a totally geodesic submanifold. Thus, piecewise linear curves are natural finite elements for the discretization of the geodesic equation. Interestingly, geodesics in this space can be seen as soliton solutions of the geodesic equation, which were not known to exist for reparametrization-invariant Sobolev metrics on spaces of curves.