Fourth-order dispersive systems on the one-dimensional torus

Abstract

We present the necessary and sufficient conditions of the well-posedness of the initial value problem for certain fourth-order linear dispersive systems on the one-dimensional torus. This system is related with a dispersive flow for closed curves into compact Riemann surfaces. For this reason, we study not only the general case but also the corresponding special case in detail. We apply our results on the linear systems to the fourth-order dispersive flows. We see that if the sectional curvature of the target Riemann surface is constant, then the equation of the dispersive flow satisfies our conditions of the well-posedness.