Abdelwahab Bensouilah

Upon the consideration of a superconducting ring interrupted by a barrier and the use of Wilson–Sommerfeld quantization rule along with some fundamental facts of electrodynamics and characteristic properties of the superconductive state, we give a short proof of Josephson's voltage-frequency relation and we establish, in a simple manner, the depend...

The paper reconsiders the issue of the regularity of the Duhamel part of the solution to the L²-critical high-order NLS already studied by the authors in [4]. This model includes the mass critical 4D fourth order NLS. The improvement is due to the use of a more sophisticated space involving a nonlinear term and taking profit of a suitable trilinear...

This paper deals with the 2-D Schr\"odinger equation with time-oscillating exponential nonlinearity $i\partial_t u+\Delta u= \theta(\omega t)\big(e^{4\pi|u|^2}-1\big)$, where $\theta$ 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 $u_{\omega}$ converges, as $|\omega|$ tends to infin...

We investigate the defocusing inhomogeneous nonlinear Schr\"odinger equation $$ i \partial_tu + \Delta u = |x|^{-b} \left({\rm e}^{\alpha|u|^2} - 1- \alpha |u|^2 \right) u, \quad u(0)=u_0, \quad x \in \R^2, $$ with $0<b<1$ and $\alpha=2\pi(2-b)$. First we show the decay of global solutions by assuming that the initial data $u_0$ belongs to the weig...

We consider the focusing nonlinear Schrödinger equation with inverse square potential i∂tu + ∆u + c|x| −2 u = −|u| α u, u(0) = u 0 ∈ H 1 , (t, x) ∈ R + × R d , where d ≥ 3, c = 0, c < λ(d) = d−2 2 2 and 0 < α ≤ 4 d. Using the profile decomposition obtained recently by the first author [1], we show that in the L 2-subcritical case, i.e. 0 < α < 4 d...

In this paper, we prove a refined version of a compactness lemma and we use it to establish mass-concentration for the focusing nonlinear Schrödinger equation with an inverse-square potential.

We investigate the initial value problem for a defocusing nonlinear Schr\"odinger equation with weighted exponential nonlinearity $$ i\partial_t u+\Delta u=\frac{u}{|x|^b}(e^{\alpha|u|^2}-1); \qquad (t,x) \in \mathbb{R}\times\mathbb{R}^2, $$ where $0< b <1$ and $\alpha=2\pi(2-b)$. We establish local and global well-posedness in the subcritical and...