Related papers: On scattering for NLS: from Euclidean to hyperboli…
We consider the cubic nonlinear Schr\"odinger equation with long-range linear potentials in one space dimension, and prove the modified scattering in the energy space for the associated final state problem with a prescribed small asymptotic…
We study the problem of propagation of analytic regularity for semi-linear symmetric hyperbolic systems. We adopt a global perspective and we prove that if the initial datum extends to a holomorphic function in a strip of radius (=width)…
We prove decay with respect to some Lebesgue norms for a class of Schr\"odinger equations with non-local nonlinearities by showing new Morawetz inequalities and estimates. As a byproduct, we obtain large-data scattering in the energy space…
In this paper, we investigate the global well-posedness and scattering theory for the defocusing energy supcritical inhomogeneous nonlinear Schr\"odinger equation $iu_t + \Delta u =|x|^{-b} |u|^\alpha u$ in four space dimension, where $s_c…
We consider the radial defocusing nonlinear Schr\"odinger equations $iu_t+\Delta u=|u|^{p}u$ with supercritical exponent $p>4$ in four space dimensions, and prove that any radial solution that remains bounded in the critical Sobolev space…
We consider the defocusing $\dot H^1$-critical nonlinear Schr\"odinger equation in all dimensions ($n\geq 3$) with a quadratic potential $V(x)=\pm \tfrac12 |x|^2$. We show global well-posedness for radial initial data obeying $\nabla…
We study solutions to the linear wave equation on the cosmological region of Schwarzschild-de Sitter spacetimes. We show that all sufficiently regular finite-energy solutions to the linear equation possess a particular finite-order…
We consider the focusing inhomogeneous biharmonic nonlinear Schr\"odinger equation in $H^2(\mathbb{R}^N)$, \begin{equation} iu_t + \Delta^2 u - |x|^{-b}|u|^{\alpha}u=0 \end{equation} when $b > 0$ and $N \geq 5$. We first obtain a small data…
In this paper, we consider the wave equation in space dimension 3 with an energy-supercritical, focusing nonlinearity. We show that any radial solution of the equation which is bounded in the critical Sobolev space is globally defined and…
This paper is a continuation of our previous study on the long time behavior of solution to the nonlinear Schr"odinger equation with higher order anisotropic dispersion (4NLS). We prove the long range scattering for (4NLS) with the…
We investigate an energy-subcritical defocusing nonlinear Schr\"odinger equation in $\mathbb R^3$ subject to a lower order nonlinear trapping potential and a spatially dependent nonlinear damping: \begin{equation*} i\partial_t u + \Delta u…
We consider the focusing inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_t u + \Delta u + |x|^{-b}|u|^\alpha u = 0\qtq{on}\R\times\R^N, \] with $\alpha=\tfrac{4-2b}{N-2}$, $N=\{3,4,5\}$ and $0<b\leq…
We study dynamics of the 4$d$ energy-critical nonlinear Schr\"odinger equation at the ground state energy. Previously, Duyckaerts and Merle [Geom. Funct. Anal. (2009)] proved that any radial solution with kinetic energy less than that of…
We consider the Schr\"odinger equation with a general interaction term, which is localized in space, for radially symmetric initial data in $n$ dimensions, $n\geq5$. The interaction term may be space-time dependent and nonlinear. Assuming…
We revisit a result by Cuccagna, Kirr and Pelinovsky about the cubic nonlinear Schr\" odinger equation (NLS) with an attractive localized potential and a time-dependent factor in the nonlinearity. We show that, under generic hypotheses on…
We prove small-data global existence to semi-linear wave equations on hyperbolic space of dimension greater than or equal to three, for nonlinearities that have the form of a sufficiently high integer power of the solution. We also prove…
The appearance of a fundamental long-time asymptotic regime in the two space one time dimensional hyperbolic nonlinear Schr\"odinger (HNLS) equation is discussed. Based on analytical and extensive numerical simulations an approximate…
In this note, we use an elementary argument to show that the existence and unitarity of radiation fields implies asymptotic partition of energy for the corresponding wave equation. This argument establishes the equipartition of energy for…
We consider a dispersive equation of Schr{\"o}dinger type with a non-linearity slightly larger than cubic by a logarithmic factor. This equation is supposed to be an effective model for stable two dimensional quantum droplets with LHY…
We study solutions of the Newtonian $n$-body problem which tend to infinity hyperbolically, that is, all mutual distances tend to infinity with nonzero speed as $t \rightarrow +\infty$ or as $t \rightarrow -\infty$. In suitable coordinates,…