Related papers: Threshold solutions for the focusing $L^{2}$ -supe…

In this paper we study dynamical properties of blowup solutions to the focusing mass-critical nonlinear fractional Schr\"odinger equation. We establish a profile decomposition and a compactness lemma related to the equation. As a result, we…

We study the defocusing energy-critical inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_tu+\Delta u=|x|^{-b}|u|^{\frac{4-2b}{d-2}}u, \qquad (t,x)\in\R\times\R^d, \] with initial data $u_0\in\dot H_x^1(\R^d)$, where $d\ge 3$ and…

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…

In this paper, we study a class of quasilinear Schr\"{o}dinger equation of the form $$-\varepsilon^2\Delta u+V(x)u-\varepsilon^2(\Delta(|u|^{2\alpha}))|u|^{2\alpha-2}u &=&\lambda|u|^{q-2}u+|u|^{2^*(2\alpha)-2}u,\quad\mbox{in}{\mathbb{R}}^N,…

In this paper we consider a one-dimensional non-linear Schroedinger equation (NLSE) with a periodic potential. In the semiclassical limit we prove that the stationary solutions of the Bose-Hubbard equation approximate the stationary…

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…

This article is concerned with time global behavior of solutions to focusing mass-subcritical nonlinear Schr\"odinger equation of power type with data in a critical homogeneous weighted $L^2$ space. We give a sharp sufficient condition for…

We investigate normalized solutions with prescribed $L^2$-norm for the upper critical fractional Choquard equation \[(-\Delta)^s u+V(\varepsilon x)u=\lambda…

This paper studies for large frequency number $k>0$ the existence and multiplicity of solutions of the semilinear problem $$ -\Delta u -k^2 u=Q(x)|u|^{p-2}u\quad\text{ in }\mathbb{R}^N, \quad N\geq 2. $$ The exponent $p$ is subcritical and…

We consider the following fractional NLS with focusing inhomogeneous power-type nonlinearity $$i\partial_t u -(-\Delta)^s u + |x|^{-b}|u|^{p-1}u=0,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^N,$$ where $N\geq 2$, $1/2<s<1$, $0<b<2s$ and…

This paper is devoted to the $L^2$-constraint variational problem \begin{equation*} We study $L^2$-normalized solutions of the following inhomogeneous fractional Schr\"{o}dinger equation \begin{equation*} (-\Delta)^{s}…

We consider the nonlinear Schr\"odinger equation $iu_t=-\Delta u-|u|^{p-1}u$ in dimension $N\geq 3$ in the $L^2$ super critical range $1+\frac{4}{N}<p<\frac{N+2}{N-2}$. The corresponding scaling invariant space is $\dot{H}^{s_c}$ with…

In this paper, we establish a probabilistic global theory in $H^1$ for the NLS with a Moser-Trudinger nonlinearity posed on compact surfaces. This equation is known to be the two dimensional counterpart to the classical energy-critical…

We study solutions of a semilinear elliptic equation with prescribed mass and Dirichlet homogeneous boundary conditions in the unitary ball. Such problem arises in the search of solitary wave solutions for nonlinear Schr\"odinger equations…

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…

We establish quantitative blow-up criteria below the scaling threshold for radially symmetric solutions to the defocusing nonlinear Schr\"odinger equation with nonlinearity $|u|^6u$. This provides to our knowledge the first generic results…

In this paper, we investigate the following fractional Sobolev critical nonlinear Schr\"{o}dinger (NLS) coupled systems: \begin{equation*} \left\{\begin{array}{lll} (-\Delta)^{s} u=\mu_{1}…

We describe a numerical method for simulating stochastic fluid dynamics near a critical point in the Ising universality class. This theory is known as model H, and is expected to govern the non-equilibrium dynamics of Quantum Chromodynamics…

We consider the $L^2$ critical inhomogeneous nonlinear Schr\"odinger (INLS) equation in $\mathbb{R}^N$ $$ i \partial_t u +\Delta u +|x|^{-b} |u|^{\frac{4-2b}{N}}u = 0, $$ where $N\geq 1$ and $0<b<2$. We prove that if $u_0\in…

This paper is devoted to the analysis of blow-up solutions for the fractional nonlinear Schr\"odinger equation with combined power-type nonlinearities \[ i\partial_t u-(-\Delta)^su+\lambda_1|u|^{2p_1}u+\lambda_2|u|^{2p_2}u=0, \] where…