Research Interest:
My research work deals with the theory of dispersive partial differential equations, especially the Schrödinger-type equations. These types of equations arise naturally in many physical settings, for example, in quantum mechanics, in fractional quantum mechanics, in nonlinear optics, in water waves model, and in the theory of Bose-Einstein condensation. Some topics that I am currently interested in: Strichartz estimates, local and global well-posedness, finite time blow-up, asymptotic completeness (energy scattering), stability/instability of standing waves, and probabilistic well-posedness.
Selected Publications:
- (with N. Rougerie and D.-T. Nguyen) Blow-up of two dimensional attractive Bose-Einstein condensates at the crittical rotational speed, Annales de l’Institut Henri Poincaré C, 2023. [arXiv] [Journal]
- (with S. Keraani) Long time dynamics of non-radial solutions to inhomogeneous nonlinear Schrödinger equations, SIAM Journal on Mathematical Analysis Vol. 53 (2021), No. 4, 4765-4811. [arXiv] [Journal]
- (with A. H. Ardila and L. Forcella) Sharp conditions for scattering and blow-up for a system of NLS arising in optical materials with χ^3 nonlinear response, Communications in Partial Differential Equations Vol. 46 (2021), no. 11, 2134-2170. [arXiv] [Journal]
- On the instability of standing waves for 3D dipolar Bose-Einstein condensates, Physica D: Nonlinear Phenomena Vol. 419 (2021), ID: 132856. [pdf][Journal]
- Dynamics of radial solutions for the focusing fourth-order nonlinear Schrödinger equations, Nonlinearity Vol. 34 (2021), No. 2, pp. 776-821. [pdf][Journal]
- (with S. Keraani and M. Majdoub) Long time dynamics for the focusing nonlinear Schrödinger equation with exponential nonlinearities, Dynamics of Partial Differential Equations Vol. 17 (2020), No. 4, pp. 329-360. [arXiv][Journal]
- Existence, non-existence and blow-up behavior of minimizers for the mass-critical fractional nonlinear Schrödinger equations with periodic potentials, Proceedings of the Royal Society of Edinburgh Section A: Mathematics Vol. 150 (2020), No. 6, pp. 3252–3292. [pdf][Journal]
- Global Strichartz estimates for the fractional Schrödinger equations on asymptotically Euclidean manifolds, Journal of Functional Analysis Vol. 275 (2018), No. 8, pp. 1943-2014. [arXiv] [Journal]
- Strichartz estimates for the fractional Schrödinger and wave equations on compact manifolds without boundary, Journal of Differential Equations 263 (2017), No. 12, pp. 8804-8837. [arXiv] [Journal]