We consider the behavior of nonlinear KdV-type equations that admit quasilinear dynamics in the sense that the nonlinear flow cannot be simply treated as a perturbation of the linear flow, even for small initial data.

We treat two problems in particular. First we study the local dynamics of KdV-type equations with nonlinearities involving two spatial derivatives. A key obstruction to well-posedness arises from to the Mizohata condition. This leads to an additional integrability requirement for the solution in the absence of a suitable null structure. In this case we prove local well-posedness for large, low-regularity data in translation-invariant spaces.

Second we explore the global dynamics of the modified Korteweg de-Vries equation. We establish modified asymptotic behavior without relying on the integrable structure of the equation. This approach has the advantage that it can be used for a wide class of short-range perturbations of the mKdV. To give a thorough description of the asymptotic behavior we prove an asymptotic completeness result that relates mKdV solutions to the 1-parameter family of solutions to the Painlevé II equation.