Ill-posedness of degenerate dispersive equations

In this paper we provide numerical and analytical evidence that some degenerate dispersive partial differential equations are ill-posed. Specifically we study the K(2, 2) equation u(t) = (u(2))(xxx) + (u(2))(x) and the 'degenerate Airy' equation u(t) = 2uu(xxx). For K(2, 2) our results are computational in nature: we conduct a series of numerical simulations which demonstrate that data which is very small in H-2 can be of unit size at a fixed time which is independent of the data's size. For the degenerate Airy equation, our results are fully rigorous: we prove the existence of a compactly supported self-similar solution which, when combined with certain scaling invariances, implies ill-posedness (also in H-2).