10 research outputs found
The dynamics of the 3D radial NLS with the combined terms
In this paper, we show the scattering and blow-up result of the radial
solution with the energy below the threshold for the nonlinear Schr\"{o}dinger
equation (NLS) with the combined terms iu_t + \Delta u = -|u|^4u + |u|^2u
\tag{CNLS} in the energy space . The threshold is given by the
ground state for the energy-critical NLS: . This
problem was proposed by Tao, Visan and Zhang in \cite{TaoVZ:NLS:combined}. The
main difficulty is the lack of the scaling invariance. Illuminated by
\cite{IbrMN:f:NLKG}, we need give the new radial profile decomposition with the
scaling parameter, then apply it into the scattering theory. Our result shows
that the defocusing, -subcritical perturbation does not
affect the determination of the threshold of the scattering solution of (CNLS)
in the energy space.Comment: 46page
On the 2d Zakharov system with L^2 Schr\"odinger data
We prove local in time well-posedness for the Zakharov system in two space
dimensions with large initial data in L^2 x H^{-1/2} x H^{-3/2}. This is the
space of optimal regularity in the sense that the data-to-solution map fails to
be smooth at the origin for any rougher pair of spaces in the L^2-based Sobolev
scale. Moreover, it is a natural space for the Cauchy problem in view of the
subsonic limit equation, namely the focusing cubic nonlinear Schroedinger
equation. The existence time we obtain depends only upon the corresponding
norms of the initial data - a result which is false for the cubic nonlinear
Schroedinger equation in dimension two - and it is optimal because
Glangetas-Merle's solutions blow up at that time.Comment: 30 pages, 2 figures. Minor revision. Title has been change