Related papers: Solitons and scattering for the cubic-quintic nonl…
We consider the three-dimensional cubic nonlinear Schr\"odinger system \begin{equation*} \begin{cases} i\partial_tu+\Delta u+(|u|^2+\beta |v|^2)u=0,\\ i\partial_tv+\Delta v+(|v|^2+\beta |u|^2)v=0. \end{cases} \end{equation*} Let $(P,Q)$ be…
We study the cubic-quintic NLS in three space dimensions. It is known that scattering holds for solutions with mass-energy in a region corresponding to positive virial, the boundary of which is delineated both by ground state solitons and…
We consider the nonlinear Schr\"odinger equation in three space dimensions with a focusing cubic nonlinearity and defocusing quintic nonlinearity and in the presence of an external inverse-square potential. We establish scattering in the…
We consider the nonlinear Schr\"odinger equation in three space dimensions with combined focusing cubic and defocusing quintic nonlinearity. This problem was considered previously by Killip, Oh, Pocovnicu, and Visan, who proved scattering…
In this paper, we consider the following three dimensional defocusing cubic nonlinear Schr\"odinger equation (NLS) with partial harmonic potential \begin{equation*}\tag{NLS} i\partial_t u + \left(\Delta_{\mathbb{R}^3 }-x^2 \right) u = |u|^2…
We consider a 3d cubic focusing nonlinear Schr\"odinger equation with a potential $$i\partial_t u+\Delta u-Vu+|u|^2u=0,$$ where $V$ is a real-valued short-range potential having a small negative part. We find criteria for global…
We consider the cubic-quintic nonlinear Schr{\"o}dinger equation in space dimension up to three. The cubic nonlinearity is thereby focusing while the quintic one is defocusing, ensuring global well-posedness of the Cauchy problem in the…
The purpose of this work is to study the 3D focusing inhomogeneous nonlinear Schr\"odinger equation $$ i u_t +\Delta u+|x|^{-b}|u|^2 u = 0, $$ where $0<b<1/2$. Let $Q$ be the ground state solution of $-Q+\Delta Q+ |x|^{-b}|Q|^{2}Q=0$ and…
In this paper, we consider the scattering problem for a class of $N$-coupled systems of the cubic nonlinear Schr\"odinger equations in three space dimensions. We prove the scattering of solutions that have a mass-energy quantity less than…
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,…
We consider the problem of identifying sharp criteria under which radial $H^1$ (finite energy) solutions to the focusing 3d cubic nonlinear Schr\"odinger equation (NLS) $i\partial_t u + \Delta u + |u|^2u=0$ scatter, i.e. approach the…
We consider the cubic defocusing nonlinear Schr\"odinger equation in one dimension with the nonlinearity concentrated at a single point. We prove global well-posedness in the scaling-critical space $L^2(\mathbb{R})$ and scattering for all…
We consider a class of nonlinear Schr\"odinger equations with potential \[ i\partial_t u +\Delta u - Vu = \pm |u|^\alpha u, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^3, \] where $\frac{4}{3}<\alpha<4$ and $V$ is a Kato-type potential. We…
We undertake a comprehensive study of the nonlinear Schr\"odinger equation $$ i u_t +\Delta u = \lambda_1|u|^{p_1} u+ \lambda_2 |u|^{p_2} u, $$ where $u(t,x)$ is a complex-valued function in spacetime $\R_t\times\R^n_x$, $\lambda_1$ and…
We study the scattering problem for the nonlinear Schr\"odinger equation $i\partial_t u + \Delta u = |u|^p u$ on $\mathbb{R}^d$, $d\geq 1$, with a mass-subcritical nonlinearity above the Strauss exponent. For this equation, it is known that…
We characterize the soliton solutions of the nonlinear Schroedinger equation on the half line with linearizable boundary conditions. Using an extension of the solution to the whole line and the corresponding symmetries of the scattering…
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…
We consider the linear vector Schr\"odinger equation subjected to quadratic constraints. We demonstrate that the resulting nonlinear system is closely related to the Ablowitz-Ladik hierarchy and use this fact to derive the N-soliton…
We establish soliton-like asymptotics for finite energy solutions to the Schr\"odinger equation coupled to a nonrelativistic classical particle. Any solution with initial state close to the solitary manifold, converges to a sum of traveling…
We use multiscale perturbation theory in conjunction with the inverse scattering transform to study the interaction of a number of solitons of the cubic nonlinear Schroedinger equation under the influence of a small correction to the…