Related papers: On the multi-bubble blow-up solutions to rough non…
We study the focusing NLS equation in $\mathbb{R}^N$ in the mass-supercritical and energy-subcritical (or intercritical) regime, with $H^1$ data at the mass-energy threshold $ \mathcal{ME}(u_0)=\mathcal{ME}(Q)$, where $Q$ is the ground…
In this paper we study dynamical properties of blowup solutions to the focusing intercritical (mass-supercritical and energy-subcritical) nonlinear fourth-order Schr\"odinger equation. We firstly establish the profile decomposition of…
This work is concerned with a coupled system of focusing nonlinear Schr\"odinger equations involving general power-type nonlinearities in the energy-critical setting for dimensions $3\leq d\leq 5$ in the radial setting. Our aim is to…
We study the asymptotic dynamics for solutions to a system of nonlinear Schr\"odinger equations with cubic interactions, arising in nonlinear optics. We provide sharp threshold criteria leading to global well-posedness and scattering of…
This paper is devoted to the analysis of blow-up solutions for the nonlinear Schr\"{o}dinger equation with combined power-type nonlinearities \[ iu_{t}+\Delta u=\lambda_1|u|^{p_1}u+\lambda_2|u|^{p_2}u. \] When $p_1=\frac{4}{N}$ and…
We study the focusing mass-critical rough nonlinear Schroedinger equations, where the stochastic integration is taken in the sense of controlled rough path. We obtain the global well-posedness if the mass of initial data is below that of…
We present a numerical study of a derivative nonlinear Schr\"odinger equation with a general power nonlinearity, $|\psi|^{2\sigma}\psi_x$. In the $L^2$-supercritical regime, $\sigma>1$, our simulations indicate that there is a finite time…
In this paper we develop two different types of criteria for the finite time blow-up solutions to the combined nonlinear Schr\"odinger equation in 1D. The first one is a negative energy criterion developed for triple combined nonlinearity…
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 construct solutions of Schr\"odinger equations which are asymptotically self-similar solutions as time goes to infinity. Also included are situations with two bubbles. These solutions are global, with non-zero $L^2$ norms, and are…
We consider the energy supercritical defocusing nonlinear Schr\"odinger equation $i\partial_tu+\Delta u-u|u|^{p-1}=0$ in dimension $d\ge 5$. In a suitable range of energy supercritical parameters $(d,p)$, we prove the existence of $\mathcal…
We consider the $L^{2}$-critical quintic focusing nonlinear Schr\"odinger equation (NLS) on ${\bf R}$. It is well known that $H^{1}$ solutions of the aforementioned equation blow up in finite time. In higher dimensions, for $H^{1}$…
We study the scattering behavior of global solutions to stochastic nonlinear Schr\"odinger equations with linear multiplicative noise. In the case where the quadratic variation of the noise is globally finite and the nonlinearity is…
We consider the 1D nonlinear Schr\"odinger equation (NLS) with focusing \emph{point nonlinearity}, $$i\partial_t\psi + \partial_x^2\psi + \delta|\psi|^{p-1}\psi = 0$$ where $\delta=\delta(x)$ is the delta function supported at the origin.…
We consider the following critical fractional Schr\"{o}dinger equation \begin{equation*} (-\Delta)^s u+V(|y'|,y'')u = u^{2_s^*-1},\quad u>0,\quad y =(y',y'') \in \mathbb{R}^3\times\mathbb{R}^{N-3}, \end{equation*} where $N\geq 3,s\in(0,1)$,…
We consider the Schr\"odinger equation in dimension two with a fixed, pointwise, focusing nonlinearity and show the occurrence of a blow-up phenomenon with two peculiar features: first, the energy threshold under which all solutions blow up…
In this paper, we consider the focusing mass-critical nonlinear fourth-order Schr\"odinger equation. We prove that blowup solutions to this equation with initial data in $H^\gamma(\mathbb{R}^d), 5\leq d \leq 7,…
We construct radially symmetric self-similar blow-up profiles for the mass supercritical nonlinear Schr\"odinger equation $i\partial_t u + \Delta u + |u|^{p-1}u=0$ on $\mathbf{R}^d$, close to the mass critical case and for any space…
We consider a mass critical nonlinear Schr\"{o}dinger equation with a real-valued potential. In this work, we construct a minimal mass solution that blows up at finite time, under weaker assumptions on spatial dimensions and potentials than…
We construct a two-parameter continuum of type II blow up solutions for the energy-critical focusing NLS in dimension $ d = 3$. The solutions collapse to a single energy bubble in finite time, precisely they have the form $ u(t,x) = e^{i…