Nonlinear Schrödinger equations in inhomogeneous media: wellposedness and illposedness of the Cauchy problem

Wellposedness of Cauchy problem for the Fourth Order Nonlinear Schrödinger Equations in Multi-dimensional Spaces

We study the well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations i∂t u=−ε u+ 2u+ P (( ∂ x u ) |α| 2, ( ∂ x ū ) |α| 2 ) , t ∈R, x ∈Rn, where ε ∈ {−1,0,1}, n 2 denotes the spatial dimension and P(·) is a polynomial excluding constant and linear terms. © 2006 Elsevier Inc. All rights reserved.

The Cauchy Problem for the Nonlinear Schrödinger Equation on a Compact Manifold

We discuss the wellposedness theory of the Cauchy problem for the nonlinear Schrödinger equation on compact Riemannian manifolds. New dispersive estimates on the linear Schrödinger group are used to get global existence in the energy space on arbitrary surfaces and three-dimensional manifolds, generalizing earlier results by Bourgain on tori. On the other hand, on specific manifolds such as sph...

Inhomogeneous Two-Parameter Abstract Cauchy Problem

A ug 2 00 5 NONLINEAR SCHRÖDINGER EQUATION ON FOUR - DIMENSIONAL COMPACT MANIFOLDS

We prove two new results about the Cauchy problem for nonlinear Schrödinger equations on four-dimensional compact manifolds. The first one concerns global wellposedness for Hartree-type nonlinear-ities and includes approximations of cubic NLS on the sphere. The second one provides local wellposedness for quadratic nonlinearities in the case of zonal data on the sphere. Both results are based on...

Ultra-analytic effect of Cauchy problem for a class of kinetic equations

The smoothing effect of the Cauchy problem for a class of kinetic equations is studied. We firstly consider the spatially homogeneous nonlinear Landau equation with Maxwellian molecules and inhomogeneous linear Fokker-Planck equation to show the ultra-analytic effects of the Cauchy problem. Those smoothing effect results are optimal and similar to heat equation. In the second part, we study a m...