NLS In 2D Research Articles

Globally well and ill posedness for non-elliptic derivative Schrödinger equations with small rough data

We show that there exists sc>0 such that the cubic (quartic) non-elliptic derivative Schrödinger equations with small data in modulation spaces M2,1s(Rn) for n≥3 (n=2) are globally well-posed if s≥sc, and ill-posed if s<sc. In 2D cubic case, using the Gabor frame, we get some time-global dispersive estimates for the Schrödinger semi-group in anisotropic Lebesgue spaces, which include a time-global maximal function estimate in the space Lx12Lx2,t∞. By resorting to the smooth effect estimate together with the dispersive estimates in anisotropic Lebesgue spaces, we show that the cubic hyperbolic derivative NLS in 2D has a unique global solution if the initial data in Feichtinger–Segal algebra or in weighted Sobolev spaces are sufficiently small.

Radial standing and self-similar waves for the hyperbolic cubic NLS in 2D

In this note we propose a new set of coordinates to study the hyperbolic or nonelliptic cubic nonlinear Schrödinger equation in two dimensions. Based on these coordinates, we study the existence of bounded and continuous hyperbolically radial standing waves, as well as hyperbolically radial self-similar solutions. Many of the arguments can easily be adapted to more general nonlinearities.

On asymptotic stability in energy space of ground states of NLS in 2D

We transpose work by K. Yajima and by T. Mizumachi to prove dispersive and smoothing estimates for dispersive solutions of the linearization at a ground state of a Nonlinear Schrödinger equation (NLS) in 2D. As an application we extend to dimension 2D a result on asymptotic stability of ground states of NLS proved in the literature for all dimensions different from 2.

Two singular dynamics of the nonlinear Schrödinger equation on a plane domain

We study the cubic, focusing nonlinear Schrodinger equation (NLS) posed on a bounded domain of $ \mathbb{R}^{2} $ with Dirichlet boundary conditions. We describe two types of nonlinear evolutions. First we obtain solutions which blow up with a minimal L 2 norm in .nite time at a .xed point of the interior of the domain. The argument can be performed equally well for the cubic NLS posed on the .at torus $ \mathbb{T}^{2} $ . In the case when the domain is a disc, we also prove that the Cauchy problem is ill posed in the following sense: the .ow map is not uniformly continuous on bounded sets of the Sobolev space Hs, s> 1/3, contrary to what is known on the square (recall that the scale invariant Sobolev space for the cubic NLS in 2D is L 2).