Related papers: Vortex Collapse for the L2-Critical Nonlinear Schr…
The collapse of marginally bound, inhomogeneous, pressureless (dust) matter, in spherical symmetry, is considered. The starting point is not, in this case, the integration of the Einstein equations from some suitable initial conditions.…
We consider the inhomogeneous nonlinear Schr\"odiger equation (INLS) in $\mathbb{R}^N$ $$i \partial u_t + \Delta u + |x|^{-b} |u|^{2\sigma}u = 0,$$ and show the $L^2$-norm concentration for the finite time blow-up solutions in the…
Rogue waves are extraordinarily high and steep isolated waves, which appear suddenly in a calm sea and disappear equally fast. However, though the Rogue waves are localized surface waves, their theoretical models and experimental…
Adding energy to a system through transient stirring usually leads to more disorder. In contrast, point-like vortices in a bounded two-dimensional fluid are predicted to reorder above a certain energy, forming persistent vortex clusters.…
In this paper, the well-posedness of Cauchy's problem of fractional Schr\"odinger equations with a power type nonlinearity on $n$-dimensional manifolds with nonnegative Ricci curvature is studied. Under suitable volume conditions, the local…
The influence of the shear stress and angular momentum on the nonlinear spherical collapse model is discussed in the framework of the Einstein-de Sitter (EdS) and $\Lambda$CDM models. By assuming that the vacuum component is not clustering…
We mainly consider the focusing biharmonic Schr\"odinger equation with a large radial repulsive potential $V(x)$: \begin{equation*} \left\{ \begin{aligned} iu_{t}+(\Delta^2+V)u-|u|^{p-1}u=0,\;\;(t,x) \in {{\bf{R}}\times{\bf{R}}^{N}}, u(0,…
Point vortex models are presented for the generalized Euler equations, which are characterized by a fractional Laplacian relation between the active scalar and the streamfunction. Special focus is given to the case of the surface…
In this paper, we investigate a universal blow-up bound for the focusing mass-critical nonlinear Schr\"odinger equation for general initial data in $L^2(\mathbb R^d)$, extending previous knowledge for mass near the ground-state threshold…
We study composite solitons, consisting of domain walls and vortex lines attaching to the walls in two-component Bose-Einstein condensates. When the total density of two components is homogeneous, the system can be mapped to the O(3)…
A filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schr\"odinger equation on the two-dimensional torus $\mathbb{T}^2$. The scheme is analyzed in a framework of discrete Bourgain spaces, which allows…
Vortices and axisymmetric vortex rings are considered in the framework of the subcritical nonlinear Schrodinger equations. The higher order nonlinearity present in such systems models many-body interactions in superfluid systems and allows…
Wave function collapse models are considered as the modified theories of standard quantum mechanics at the macroscopic level. By introducing nonlinear stochastic terms in the Schr\"odinger equation, these models make predictions,…
We prove large-data scattering in $H^1$ for inhomogeneous nonlinear Schr\"odinger equations in two space dimensions for all powers $p>0$. We assume the inhomogeneity is nonnegative and repulsive; we additionally require decay at infinity in…
We present a numerical study of solutions to the $2d$ cubic and quintic focusing nonlinear Schr\"odinger equation in the exterior of a smooth, compact and strictly convex obstacle (a disk) with Dirichlet boundary condition. We first…
We analyze the salient features of vortex-ring solitons supported by cylindrically symmetric media with nonlocal thermal nonlinearity. We discover the existence of a maximum allowed topological charge for such vortex solitons to be stable…
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 have found several families of vortex soliton solutions in two-dimensional discrete dissipative systems governed by the cubic-quintic complex Ginzburg-Landau equation. There are symmetric and asymmetric solutions, and some of them have…
We study the periodic non-linear Schrodinger equations with odd integer power nonlinearities, for initial data which are assumed to be small in some negative order Sobolev space, but which may have large L^2 mass. These equations are known…
We consider the non linear focusing wave equation $\partial_{tt}u-\Delta u-u|u|^{p-1}=0$ in large dimensions and for radially symmetric data, in the energy supercritical zone for p large enough. We construct finite time blow up solutions…