Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms - Mathematics > Differential GeometryReport as inadecuate




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Abstract: We study an equation lying `mid-way- between the periodic Hunter-Saxton andCamassa-Holm equations, and which describes evolution of rotators in liquidcrystals with external magnetic field and self-interaction. We prove that it isan Euler equation on the diffeomorphism group of the circle corresponding to anatural right-invariant Sobolev metric. We show that the equation isbihamiltonian and admits both cusped, as well as smooth, traveling-wavesolutions which are natural candidates for solitons. We also prove that it islocally well-posed and establish results on the lifespan of its solutions.Throughout the paper we argue that despite similarities to the KdV, CH and HSequations, the new equation manifests several distinctive features that set itapart from the other three.



Author: Boris Khesin U Toronto, Jonatan Lenells U Cambridge, Gerard Misiolek U Notre Dame

Source: https://arxiv.org/







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