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Abdelwahab Bensouilah

Publications and source records attributed to Abdelwahab Bensouilah.

6 recordsLinked to original sources
arXiv2018

A 2D Schrodinger equation with time-oscillating exponential nonlinearity

This paper deals with the 2-D Schrödinger equation with time-oscillating exponential nonlinearity $i\partial_t u+Δu= θ(ωt)\big(e^{4π|u|^2}-1\big)$, where $θ$ is a periodic $C^1$-function. We prove that for a class of initial data $u_0 \in H^1(\mathbb{R}^2)$, the solution $u_ω$ converges, as $|ω|$ tends to infinity to the solution $U$ of the limiting equation $i\partial_t U+ΔU= I(θ)\big(e^{4π|U|^2}-1\big)$ with the same initial data, where $I(θ)$ is the average of $θ$.

math.AP↗
arXiv2018

Scattering in the weighted $L^2$-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity

We investigate the defocusing inhomogeneous nonlinear Schrödinger equation $$ i \partial_tu + Δu = |x|^{-b} \left({\rm e}^{α|u|^2} - 1- α|u|^2 \right) u, \quad u(0)=u_0, \quad x \in \mathbb{R}^2, $$ with $0<b<1$ and $α=2π(2-b)$. First we show the decay of global solutions by assuming that the initial data $u_0$ belongs to the weighted space $Σ(\mathbb{R}^2)=\{\,u\in H^1(\mathbb{R}^2) \ : \ |x|u\in L^2(\mathbb{R}^2)\,\}$. Then we combine the local theory with the decay estimate to obtain scattering in $Σ$ when the Hamiltonian is below the value $\frac{2}{(1+b)(2-b)}$.

math.AP↗
arXiv2018

On stability and instability of standing waves for the nonlinear Schrödinger equation with inverse-square potential

We consider the focusing nonlinear Schrödinger equation with inverse square potential \[ i\partial_t u + Δu + c|x|^{-2} u = - |u|^αu, \quad u(0) = u_0 \in H^1, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d, \] where $d \geq 3$, $c\ne 0$, $c<λ(d)=\left(\frac{d-2}{2}\right)^2$ and $0<α\leq \frac{4}{d}$. Using the profile decomposition obtained recently by the first author \cite{Bensouilah}, we show that in the $L^2$-subcritical case, i.e. $0<α<\frac{4}{d}$, the sets of ground state standing waves are orbitally stable. In the $L^2$-critical case, i.e. $α=\frac{4}{d}$, we show that ground state standing waves are strongly unstable by blow-up.

math.AP↗
arXiv2018

Mass concentration and characterization of finite time blow-up solutions for the nonlinear Schrödinger equation with inverse-square potential

We consider the $L^2$-critical NLS with inverse-square potential $$ i \partial_t u +Δu + c|x|^{-2} u = -|u|^{\frac{4}{d}} u, \quad u(0) = u_0, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d, $$ where $d\geq 3$ and $c\ne 0$ satisfies $c<λ(d) := \left(\frac{d-2}{2}\right)^2$. Using a refined compactness lemma, we extend the mass concentration of finite time blow-up solutions established in the attractive case by the first author in [Bensouilah] to $c<λ(d)$. By means of a simple and short limiting profile theorem, we get the same classification result obtained by Csobo and Genoud in [CsoboGenoud] for $0<c<λ(d)$. It also enables us to extend the classification to $c<λ(d)$.

math.AP↗