2008/07/29 by Hiroyuki Chihara, Chihara, Hiroyuki, Eiji Onodera +1
Mathematics · Physics and Astronomy · #35Q53 (Secondary) #47G30 #53G44 (Primary) 58J40 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Nonlinear Waves and Solitons #math.AP #math.DG #msc:35Q53 #msc:47G30 #msc:53G44 #msc:58J40
paper · pdf · doi:10.48550/arxiv.0807.4591
13 pages, no figure
arxiv created 2008/07/29 · openalex publication_date 2008/07/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We discuss a short-time existence theorem of solutions to the initial value problem for a third order dispersive flow for closed curves into a compact almost Hermitian manifold. Our equations geometrically generalize a physical model describing the motion of vortex filament. The classical energy method cannot work for this problem since the almost complex structure of the target manifold is not supposed to be parallel with respect to the Levi-Civita connection. In other words, a loss of one derivative arises from the covariant derivative of the almost complex structure. To overcome this difficulty, we introduce a bounded pseudodifferential operator acting on sections of the pullback bundle, and eliminate the loss of one derivative from the partial differential equation of the dispersive flow.