On KP-II type equations on cylinders

In this article we study the generalized dispersion version of the Kadomtsev-Petviashvili II equation, on \T \times \R and \T \times \R^2. We start by proving bilinear Strichartz type estimates, dependent only on the dimension of the domain but not on the dispersion. Their analogues in terms of Bourgain spaces are then used as the main tool for the proof of bilinear estimates of the nonlinear terms of the equation and consequently of local well-posedness for the Cauchy problem.Comment: 32 page

Scattering for nonlinear Schrodinger equation under partial harmonic confinement

We consider the nonlinear Schrodinger equation under a partial quadratic confinement. We show that the global dispersion corresponding to the direction(s) with no potential is enough to prove global in time Strichartz estimates, from which we infer the existence of wave operators thanks to suitable vector-fields. Conversely, given an initial Cauchy datum, the solution is global in time and asymptotically free, provided that confinement affects one spatial direction only. This stems from anisotropic Morawetz estimates, involving a marginal of the position density.Comment: 26 pages. Some typos fixed, especially in Section