Abstract: I will present the new method developed in joint work with Killip for proving optimal well-posedness for integrable PDE. I will first discuss this method in the context of the Korteweg-de Vries equation. I will then discuss subsequent developments (joint with Harrop-Griffiths and Killip) that have led to optimal well-posedness results for the integrable nonlinear Schrödinger and the modified Korteweg-de Vries equations.
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