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We consider the question of global existence of small, smooth, and localized solutions of a certain fractional semilinear cubic NLS in one dimension, $$i\partial_t u - \Lambda u = c_0{|u|}^2 u + c_1 u^3 + c_2 u \bar{u}^2 + c_3 \bar{u}^3, \qquad \Lambda = \Lambda(\partial_x) = {|\partial_x|}^(1/2)$$, where $c_0\in\mathbb{R}$ and $c_1,c_2,c_3\in\mathbb{C}$. This model is motivated by the two-dimensional water waves equations, which have a somewhat similar structure in the Eulerian formulation, in the case of irrotational flows. We show that one cannot expect linear scattering, even in this simplified model. More precisely, we identify a suitable nonlinear logarithmic correction, and prove global existence and modified scattering of solutions.

We prove global well-posedness in $H^{1}({\mathbb{T}}^{3})$ for the energy-critical defocusing initial-value problem \begin{eqnarray*}(i\partial_{t}+\Delta)u=u|u|^{4},\quad u(0)=\phi.\end{eqnarray*}

We consider the question of global existence of small, smooth, and localized solutions of a certain fractional semilinear cubic NLS in one dimension, $$i\partial_t u - ��u = c_0{|u|}^2 u + c_1 u^3 + c_2 u \bar{u}^2 + c_3 \bar{u}^3, \qquad ��= ��(\partial_x) = {|\partial_x|}^(1/2)$$, where $c_0\in\mathbb{R}$ and $c_1,c_2,c_3\in\mathbb{C}$. This model is motivated by the two-dimensional water waves equations, which have a somewhat similar structure in the Eulerian formulation, in the case of irrotational flows. We show that one cannot expect linear scattering, even in this simplified model. More precisely, we identify a suitable nonlinear logarithmic correction, and prove global existence and modified scattering of solutions.

In this paper we present a method to study global regularity properties of solutions of large-data critical Schrodinger equations on certain noncompact Riemannian manifolds. We rely on concentration compactness arguments and a global Morawetz inequality adapted to the geometry of the manifold (in other words we adapt the method of Kenig-Merle to the variable coefficient case), and a good understanding of the corresponding Euclidean problem (in our case the main theorem of Colliander-Keel-Staffilani-Takaoka-Tao). As an application we prove global well-posedness and scattering in $H^1$ for the energy-critical defocusing initial-value problem (i\partial_t+\Delta_\g)u=u|u|^{4} on the hyperbolic space $H^3$., Comment: 43 pages, references added

In this paper we present a method to study global regularity properties of solutions of large-data critical Schrodinger equations on certain noncompact Riemannian manifolds. We rely on concentration compactness arguments and a global Morawetz inequality adapted to the geometry of the manifold (in other words we adapt the method of Kenig-Merle to the variable coefficient case), and a good understanding of the corresponding Euclidean problem (in our case the main theorem of Colliander-Keel-Staffilani-Takaoka-Tao). As an application we prove global well-posedness and scattering in $H^1$ for the energy-critical defocusing initial-value problem (i\partial_t+��_\g)u=u|u|^{4} on the hyperbolic space $H^3$., 43 pages, references added

We consider the Schr\"{o}dinger map initial-value problem in dimension two or greater. We prove that the Schr\"{o}dinger map initial-value problem admits a unique global smooth solution, provided that the initial data is smooth and small in the critical Sobolev space. We prove also that the solution operator extends continuously to the critical Sobolev space., Comment: 60 pages

We prove global well-posedness and scattering in $H^1$ for the defocusing nonlinear Schr\"{o}dinger equations \begin{equation*} \begin{cases} &(i\partial_t+\Delta_\g)u=u|u|^{2\sigma}; &u(0)=\phi, \end{cases} \end{equation*} on the hyperbolic spaces $\H^d$, $d\geq 2$, for exponents $\sigma\in(0,2/(d-2))$. The main unexpected conclusion is scattering to linear solutions in the case of small exponents $\sigma$; for comparison, on Euclidean spaces scattering in $H^1$ is not known for any exponent $\sigma\in(1/d,2/d]$ and is known to fail for $\sigma\in(0,1/d]$. Our main ingredients are certain noneuclidean global in time Strichartz estimates and noneuclidean Morawetz inequalities.

We prove global well-posedness and scattering in $H^1$ for the defocusing nonlinear Schr��dinger equations \begin{equation*} \begin{cases} &(i\partial_t+��_\g)u=u|u|^{2��}; &u(0)=��, \end{cases} \end{equation*} on the hyperbolic spaces $\H^d$, $d\geq 2$, for exponents $��\in(0,2/(d-2))$. The main unexpected conclusion is scattering to linear solutions in the case of small exponents $��$; for comparison, on Euclidean spaces scattering in $H^1$ is not known for any exponent $��\in(1/d,2/d]$ and is known to fail for $��\in(0,1/d]$. Our main ingredients are certain noneuclidean global in time Strichartz estimates and noneuclidean Morawetz inequalities.

We consider the Schr��dinger map initial-value problem in dimension two or greater. We prove that the Schr��dinger map initial-value problem admits a unique global smooth solution, provided that the initial data is smooth and small in the critical Sobolev space. We prove also that the solution operator extends continuously to the critical Sobolev space., 60 pages