Related papers: Threshold solutions for the intercritical inhomoge…
We consider the blowup rate for blowup solutions to $L^2$-critical, focusing NLS with a harmonic potential and a rotation term. Under a suitable spectral condition we prove that there holds the "$\log$-$\log$ law" when the initial data is…
We consider a nonlinear Schr\"odinger equation with double power nonlinearity, where one power is focusing and mass critical and the other mass sub-critical. Classical variational arguments ensure that initial data with mass less than the…
In this paper, we consider blowup of solutions to the Cauchy problem for the following biharmonic nonlinear Schr\"odinger equation (NLS), $$ \textnormal{i} \, \partial_t u=\Delta^2 u-\mu \Delta u-|u|^{2 \sigma} u \quad \text{in} \,\, \R…
For the 3D focusing cubic nonlinear Schrodinger equation, Scattering of $H^1$ solutions inside the (scale invariant) potential well was established by Holmer and Roudenko~\cite{HR2} (radial case) and Duyckaerts, Holmer and…
In this paper, we study the $H^s$-stability of the log-log blowup regime (which has been completely described in a series of recent works by Merle and Raphael) for the focusing mass-critical nonlinear Schr\"odinger equations…
We consider a class of power-type nonlinear Schr\"odinger equations for which the power of the nonlinearity lies between the mass- and energy-critical exponents. Following the concentration-compactness approach, we prove that if a solution…
We are concerned with the global behavior of the solutions of the focusing mass supercritical nonlinear Schr{\"o}dinger equation under partial harmonic confinement. We establish a necessary and sufficient condition on the initial data below…
We study the $L^2$-critical damped NLS with a Stark potential. We prove that the threshold for global existence and finite time blowup of this equation is given by $\|Q\|_2$, where $Q$ is the unique positive radial solution of $\Delta Q +…
We consider the $L^2$-supercritical nonlinear Schr\"{o}dinger equation with a repulsive Dirac delta potential in one dimensional space. In a previous work, we clarified the global dynamics of even solutions with the same action as the…
In this paper, we consider the Schr\"odinger equation with a mass-supercritical focusing nonlinearity, in the exterior of a smooth, compact, convex obstacle of $\R^{d}$ with Dirichlet boundary conditions. We prove that solutions with…
We consider the focusing energy-critical nonlinear Schr\"odinger equation $iu_t+\Delta u = - |u|^{\frac4{d-2}}u$ in dimensions $d\geq 5$. We prove that if a maximal-lifespan solution $u:I\times\R^d\to \C$ obeys $\sup_{t\in I}\|\nabla…
In this paper, we study the dispersion-managed nonlinear Schr\"odinger (DM-NLS) equation $$ i\partial_t u(t,x)+\gamma(t)\Delta u(t,x)=|u(t,x)|^{\frac4d}u(t,x),\quad x\in\R^d, $$ and the nonlinearity-managed NLS (NM-NLS) equation: $$…
In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schr\"odinger equation $$ \gamma \Delta ^2 u -\Delta u + \alpha u=|u|^{2 \sigma} u, \quad u \in H^2(\R^N), $$ under the constraint $$ \int_{\R^N}|u|^2 \,…
In the present work we examine multi-hump solutions of the nonlinear Schr{\"o}dinger equation in the blowup regime of the one-dimensional model with power law nonlinearity, bearing a suitable exponent of $\sigma>2$. We find that families of…
In this paper, we study the ground state standing wave solutions for the focusing bi-harmonic nonlinear Schr\"{o}dinger equation with a $\mu$-Laplacian term (BNLS). Such BNLS models the propagation of intense laser beams in a bulk medium…
In this paper, we are concerned with solutions to the following nonlinear Schr\"odinger equation with combined inhomogeneous nonlinearities, $$ -\Delta u + \lambda u= \mu |x|^{-b}|u|^{q-2} u + |x|^{-b}|u|^{p-2} u \quad \mbox{in} \,\, \R^N,…
We study the focusing inhomogeneous nonlinear Schr\"odinger equation $$ i\partial_t u + \Delta u = -|x|^b |u|^{p-1}u ,\quad (t,x)\in (0,\infty)\times\mathbb{R}^N, $$ with $b>0$ and $p>1$. Due to the spatial growth of the nonlinearity,…
We study the existence of standing waves, of prescribed $L^2$-norm (the mass), for the nonlinear Schr\"{o}dinger equation with mixed power nonlinearities $$ i \partial_t \phi + \Delta \phi + \mu \phi |\phi|^{q-2} + \phi |\phi|^{2^* - 2} =…
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 investigate the focusing and defocusing energy-critical stochastic nonlinear Schr\"odinger equation, subject to random perturbations in the form of either additive or multiplicative (Stratonovich) noise. We establish local well-posedness…