Some Results on the Scattering Theory for Nonlinear Schrödinger Equations in Weighted $L^{2}$ Space

Mathematics – Analysis of PDEs

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

28 pages, no figure

Scientific paper

We investigate the scattering theory for the nonlinear Schr\"{o}dinger equation $i \partial_{t}u+ \Delta u+\lambda|u|^\alpha u=0$ in $\Sigma=H^{1}(\mathbb{R}^{d})\cap L^{2}(|x|^{2};dx)$. We show that scattering states $u^{\pm}$ exist in $\Sigma$ when $\alpha_{d}<\alpha<\frac{4}{d-2}$, $d\geq3$, $\lambda\in \mathbb{R}$ with certain smallness assumption on the initial data $u_{0}$, and when $\alpha(d)\leq \alpha< \frac{4}{d-2}$($\alpha\in [\alpha(d), \infty)$, if $d=1,2$), $\lambda>0$ under suitable conditions on $u_{0}$, where $\alpha_{d}$, $\alpha(d)$ are the positive root of the polynomial $dx^{2}+dx-4$ and $dx^{2}+(d-2)x-4$ respectively. Specially, when $\lambda>0$, we obtain the existence of $u^{\pm}$ in $\Sigma$ for $u_{0}$ below a mass-energy threshold $M[u_{0}]^{\sigma}E[u_{0}]<\lambda^{-2\tau}M[Q]^{\sigma}E[Q]$ and satisfying an mass-gradient bound $\|u_{0}\|_{L^{2}}^{\sigma}\|\nabla u_{0}\|_{L^{2}}<\lambda^{-\tau}\|Q\|_{L^{2}}^{\sigma}\|\nabla Q\|_{L^{2}}$ with $\frac{4}{d}<\alpha<\frac{4}{d-2}$($\alpha\in (\frac{4}{d}, \infty)$, if $d=1,2$), and also for oscillating data at critical power $\alpha=\alpha(d)$, where $\sigma=\frac{4-(d-2)\alpha}{\alpha d-4}$, $\tau=\frac{2}{\alpha d-4}$ and $Q$ is the ground state. We also study the convergence of $u(t)$ to the free solution $e^{it\Delta}u^{\pm}$ in $\Sigma$, where $u^{\pm}$ is the scattering state at $\pm\infty$ respectively.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

Some Results on the Scattering Theory for Nonlinear Schrödinger Equations in Weighted $L^{2}$ Space does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Some Results on the Scattering Theory for Nonlinear Schrödinger Equations in Weighted $L^{2}$ Space, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Some Results on the Scattering Theory for Nonlinear Schrödinger Equations in Weighted $L^{2}$ Space will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-194880

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.