We discuss a short-time existence theorem of solutions to the initial value problem for a fourth-order dispersive flow for curves parametrized by the real line into a compact Kähler manifold. Our equations geometrically generalize a physical model describing the motion of a vortex filament or the continuum limit of the Heisenberg spin chain system. Our results are proved by using so-called the energy method. We introduce a bounded gauge transform on the pullback bundle, and make use of local smoothing effect of the dispersive flow a little.
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Hiroyuki Chihara, Eiji Onodera, A Fourth-Order Dispersive Flow into Kähler Manifolds. Z. Anal. Anwend. 34 (2015), no. 2, pp. 221–249DOI 10.4171/ZAA/1537