Fourth-order dispersive systems on the one-dimensional torus
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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.
KeywordsDispersive system Initial value problem Well-posedness Gauge transform Energy method
Mathematics Subject Classification35G40 47G30 53C44
The author would like to thank Eiji Onodera for invaluable comments and helpful information on the holomomy. The author is also grateful to the referees for careful reading the first version of the manuscript of this paper. The author was supported by JSPS Grant-in-Aid for Scientific Research #23340033.
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