The quintic nonlinear Schr\"odinger equation on three-dimensional Zoll manifolds

Let (M,g) be a three-dimensional smooth compact Riemannian manifold such that all geodesics are simple and closed with a common minimal period, such as the 3-sphere S^3 with canonical metric. In this work the global well-posedness problem for the quintic nonlinear Schr\"odinger equation i\partial_t u+\Delta u=\pm|u|^4u, u|_{t=0}=u_0 is solved for small initial data u_0 in the energy space H^1(M), which is the scaling-critical space. Further, local well-posedness for large data, as well as persistence of higher initial Sobolev regularity is obtained. This extends previous results of Burq-G\'erard-Tzvetkov to the endpoint case

Trigonometric time integrators for the Zakharov system

The main challenge in the analysis of numerical schemes for the Zakharov system originates from the presence of derivatives in the nonlinearity. In this paper a new trigonometric time-integration scheme for the Zakharov system is constructed and convergence is proved. The time-step restriction is independent from a spatial discretization. Numerical experiments confirm the findings

Well-posedness and scattering for the KP-II equation in a critical space

The Cauchy problem for the Kadomtsev-Petviashvili-II equation (u_t+u_{xxx}+uu_x)_x+u_{yy}=0 is considered. A small data global well-posedness and scattering result in the scale invariant, non-isotropic, homogeneous Sobolev space \dot H^{-1/2,0}(R^2) is derived. Additionally, it is proved that for arbitrarily large initial data the Cauchy problem is locally well-posed in the homogeneous space \dot H^{-1/2,0}(R^2) and in the inhomogeneous space H^{-1/2,0}(R^2), respectively.Comment: 28 pages; v3: erratum include