Fifth order KdV equations

Fifth order KdV equations

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Soonsik Kwon, Princeton University
Fine Hall 110

We study the fifth-order KdV equations, which arise in the KdV hierarchy. In this talk, we discuss the initial value problem in Sobolev spaces with low regularity. In the linear part the fifth-order equation has stronger dispersion effect and so better smoothing than KdV equation. But it comes with stronger nonlinear parts compared to dispersion. As a result, unlike KdV equation, the fifth-order equation in the hierarchy has too strong low-high frequency interaction. We exploit this to show a negative result. We will discuss both positive and negative results, local well-posedness in the standard sense (existence, uniqueness and continuous dependence of data) for rough data, but ill-posedness in the sense of failure of uniformly continuous dependence on data on a bounded set. If time allows, we will make a comment on analogous problem of the fifth-order modified KdV equation.