Papers, priprints, and expository notes

 
1. Modified scattering for the cubic Schr\”odinger equation on product spaces and applications. (With Benoit Pausader, Nikolay Tzvetkov, and Nicola Visciglia). Preprint available at arXiv:1311.2275.


2. The weakly nonlinear large box limit of the 2D cubic nonlinear Schr\"odinger equation. (With Erwan Faou and Pierre Germain). Preprint availabe at arXiv:1308.6267


3. Nonlinear resonances with a potential: Multilinear estimates and an application to NLS. (With Pierre Germain and Samuel Walsh). Preprint available at arXiv:1303.4354


4. Long-time strong instability and unbounded orbits for some periodic nonlinear Schr\”odinger equations. Accepted in Archives for Rational Mechanics and Analysis (ARMA). Preprint available at arXiv:1210.7509


5. Scattering for the Zakharov system in 3 dimensions. (With Fabio Pusateri and Jalal Shatah). Communications in Mathematical Physics. September 2013, Volume 322, Issue 3, pp 731-753. Preprint available at arXiv:1206.3473v1.


6. On Scattering for the quintic defocusing nonlinear Schr\”odinger equation on \Rx\T^2, (Joint work with Benoit Pausader). Accepted in Communications in Pure and Applied Mathematics (CPAM). Preprint avaialable at arXiv:1205.6132 (This is the abridged and reader-friendly version. A version with full details is available here).


7. Global well-posedness of the cubic nonlinear Schr\"odinger equation on compact manifolds without boundary. Communications in PDE. Volume 37, Issue 7, 1186--1236 (2012). Preprint available at arXiv:1008.2826v1.


8. A bilinear oscillatory integral estimate and bilinear refinements to Strichartz estimates on closed manifolds.  Analysis & PDE 5-2 (2012) 339--363. Preprint available at arXiv:1008.2827v1.


9. Global and dynamical aspects of the nonlinear Schr\"odinger equations on compact manifolds. Ph.D Thesis.



 

Expository notes:

Most of these notes were loosely written when I was a graduate student at UCLA to accompany talks at the participating analysis seminar:


  1. 1.Discrete and continuous parametrices for variable coefficient wave and Schr\"odinger equations (PDF)

  2. 2.Bourgain's Fourier Truncation (or high-low) method (Fall 2009) (PDF)

  3. 3.Some Scattering Aspects for NLS in Periodic and Non-periodic Regimes (Spring 2009) (PDF)

  4. 4.Notes on the Classical KAM theory (Winter 2009) (PDF)  

  5. 5.Strichartz Estimates on Tori [after J. Bourgain's Fourier Restriction Phenomena... Part I] (Fall 2008)