Recent papers

1. Global attraction to solitary waves for Klein-Gordon equation with mean field interaction (with Alexander Komech). To appear in Annales de l'Institute Henri Poincaré (Analyse non linéaire). Available online. See also MPI Preprint 66/2007 (On Global Attraction to Quantum Stationary States III. Klein-Gordon equation with mean field interaction).
2. Global attractor for the Klein-Gordon field coupled to several nonlinear oscillators (with Alexander Komech), submitted. See also MPI Preprint 17/2007 (On Global Attraction to Quantum Stationary States II. Several Nonlinear Oscillators Coupled to Massive Scalar Field).

Papers in refereed journals

1. Global attraction to solitary waves in models based on the Klein-Gordon equation (review article) (with Alexander Komech). SIGMA 4 (2008), 010. 1--23. Proceedings of the Seventh International Conference ``Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007; Institute of Mathematics, Kyiv, Ukraine).
2. Global well-posedness for the Schrodinger equation coupled to a nonlinear oscillator (with Alexander Komech), Russ. J. Math. Phys. 14 (2007), no. 2, 164--173. See also MPI Preprint 85/2006.
3. Nonlinear instability of a critical traveling wave in the generalized Korteweg -- de Vries equation (with Scipio Cuccagna and Dmitry Pelinovsky), SIAM J. Math. Anal. 39 (2007), no. 1, 1--33. See also MPI Preprint 67/2006.
4. Global attractor for a nonlinear oscillator coupled to the Klein-Gordon field (with Alexander Komech), Arch. Ration. Mech. Anal. 185 (2007), no. 1, 105--142. See also MPI Preprint 121/2005 (On Global Attraction to Quantum Stationary States I. Nonlinear Oscillator Coupled to Massive Scalar Field).
5. On global attraction to solitary waves for the Klein-Gordon equation coupled to nonlinear oscillator (with Alexander Komech), C. R. Math. Acad. Sci. Paris 343 (2006), no. 2, 111--114.
6. Estimates on Level Set Integral Operators in Dimension Two (with Svetlana Roudenko), J. Geom. Anal. 15 (2005), no. 3, 405--423.
7. Discrete peakons (with Panos Kevrekidis and Jesus Cuevas), Phys. D 207 (2005), no. 3-4, 137--160.
8. Lp-Lq regularity of Fourier integral operators with caustics, Trans. Amer. Math. Soc. 356 (2004), no. 9, 3429--3454.
9. Purely nonlinear instability of standing waves with minimal energy (with Dmitry Pelinovsky), Comm. Pure Appl. Math 56 (2003), no. 11, 1565--1607.
10. Type conditions and Lp-Lp, Lp-Lp' regularity of Fourier integral operators, Contemp. Math. 320 (2003), 91--109.
11. On Lp continuity of singular Fourier Integral Operators (with Scipio Cuccagna), Trans. Amer. Math. Soc. 355 (2003), no. 6, 2453--2476
12. Integral operators with two-sided cusp singularities (with Scipio Cuccagna), Internat. Math. Res. Notices 2000, no. 23, 1225--1242
13. Optimal regularity of Fourier integral operators with one-sided folds, Comm. Partial Differential Equations 24 (1999), no. 7 & 8, 1263--1281.
14. Damping estimates for oscillatory integral operators with finite type singularities, Asymptot. Anal. 18 (1998), no. 3 & 4, 263--278.
15. Sobolev Estimates for Radon Transform of Melrose and Taylor, Comm. Pure Appl. Math 51 (1998), no. 5, 537--550.
16. Integral operators with singular canonical relations, chapter in a book Spectral Theory, Microlocal Analysis, Singular Manifolds (M. Demuth, E. Schrohe, B.-W. Schulze, J. Sjostrand, eds.). Akademie Verlag, Berlin, 1997. pp. 200--248.
17. Oscillatory Integral Operators in Scattering Theory, Comm. Partial Differential Equations 22 (1997), 841--867.

Miscellaneous

1. Book of Practical PDEs (with Alexander Komech). MPI Lecture Note 33/2007.
2. Instability of vacuum in ½D Dirac equation, preprint.
3. Asymptotic Estimates for Oscillatory Integral Operators, PhD. Thesis. Columbia University, New York, 1997.
4. Cotlar-Stein Almost Orthogonality Lemma, lecture note. Columbia University, New York, 1997.

Keywords

AMS Subject Classification

• 37K40 Soliton theory, asymptotic behavior of solutions
• 37L30 Attractors and their dimensions, Lyapunov exponents
• 35B41 Attractors
• 37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics
• 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other quantum-mechanical equations

ACKNOWLEDGMENT: The papers on this site are the product of the research which has been partially supported by the National Science Foundation under Grants DMS-9970330/0296036, DMS-0200880/0434698/0621257, and DMS-0600863.