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**Teichons: solitonlike geodesics on universal Teichmüller space.**
*(English)*
Zbl 1185.58003

Summary: This paper studies \(\text{EPDiff}(S^1)\), the Euler-Poincaré equation for diffeomorphisms of \(S^1\), with the Weil-Petersson metric on the coset space \(\text{PSL}_2(\mathbb R)\setminus\text{Diff}(S^1)\). This coset space is known as the universal Teichmüller space. It has another realization as the space of smooth simple closed curves modulo translations and scalings. \(\text{EPDiff}(S^1)\) admits a class of solitonlike solutions (teichons) in which the “momentum” \(m\) is a distribution. The solutions of this equation can also be thought of as paths in the space of simple closed plane curves that minimize a certain energy. In this paper we study the solution in the special case that \(m\) is expressed as a sum of four delta functions. We prove the existence of the solution for infinite time and find bounds on its long-term behavior, showing that it is asymptotic to a one-parameter subgroup in \(\text{Diff}(S^1)\). We then present a series of numerical experiments on solitons with more delta functions and make some conjectures about these.