Areas of Expertise: Partial differential equations and inverse scattering theory

Research Summary:

Professor Zhou studies the 1-D, 2-D inverse scattering theory, using the method of Riemann-Hilbert problems. His current research is concentrated in a nonlinear type of microlocal analysis on Riemann-Hilbert problems. Subjects of main interest are integrable and near intergrable PDE, integrable statistical models, orthogonal polynomials and random matrices, monodromy groups and Painleve equations with applications in physics and algebraic geometry. A number of classical and new problems in analysis, numerical analysis, and physics have been solved by zhou or jointly by zhou and his collaborators.

Collaborators:

Beals, Richard, Yale University

Chen, Pojen

Deift, Percy, Courant Institute, New York University

Fokas, A.S., Imperial College

Its, Alexander, Indiana University-Purdue University in Indianapolis

Kapaev, Alexander, Indiana University-Purdue University in Indianapolis

Kamvissis,Spyridon, Universite de Paris XIII, Ecole Normale Superieure

Kriecherbauer, Thomas, University of Augsburg

Li, Bozang, Institute of Physics, Chinese Academy of Sciences

Lu Wu-Ming, Institue of Theoretical Physics, Chinese Academy of Sciences

McLaughlin, Kenneth T-R, University of Arizona

Mugan, Ugurhan, Bilkent University

Pu, Fu-Cho, Institute of Physics, Chinese Academy of Sciences

Venakides, Stephano, Duke University

Selected Publications:

  1. The Riemann-Hilbert problem and inverse scattering, SIAM J. Math. Anal., 20, No. 4, 966-986 (1989).
  2. Direct and Inverse Scattering Transforms with Arbitrary Spectral Singularities, Comm. Pure Appl. Math. 42, 895-938 (1989).
  3. Inverse scattering transform for the time dependent Schrödinger equation with application to KPI equation, Comm. Math. Phys. 128, 551-564, (1990).
  4. (with P. Deift), Direct and inverse scattering on the line with arbitrary singularities, Comm. Pure Appl. Math. 44, 485-533, (1991).
  5. (with A.S. Fokas), On the solvability of Painleve II and IV, Comm. Math. Phys. 144, 601-622 (1992).
  6. (with A.S. Fokas and Mugan), On the solvability of PI, PIII and PV, Inverse Problems 8, 757-785 (1992).
  7. (with P. Deift), A steepest descent method for oscillatory Riemann-Hilbert problems, AMS Bul. 26, 119-123 (1992).
  8. (with P. Deift), A steepest descent method for oscillatory Riemann-Hilbert problems-asymptotics for MKdV equation, Ann. of Math. 137, 295-368 (1993).
  9. (with R. Beals and P. Deift), IST on the Line. Important Developments in Soliton Theory 1980-1990. Springer-Verlag, 7-32 (1993).
  10. (with P. Deift and A. Its), Long-time Asymptotics for Integrable Nonlinear Wave Equations. Important Developments in Soliton Theory 1980-1990. Springer-Verlag, 181-204 (1993).
  11. (with P. Deift), Long-time asymptotics for the autocorrelation function of the transverse Ising chain at the critical magnetic field , NATO Conference 1991, ASI series B: Phys. 302, 183-202 (1994).
  12. (with P. Deift and S. Venakides), The collisionless shock region for the long-time behavior of solutions of the KdV equation, Comm. Pure Appl. Math. 47, 199-206 (1994).

  13. (with P. Deift), Oscillatory Riemann-Hilbert problems and integrable systems, Advanced Studies in Pure Mathematics 23, Tokyo University Press, 17-26, 1994.
  14. Inverse scattering transform for systems with rational spectral dependence, Jour. Diff. Eq. 115 (2), 277-303 (1995).
  15. (with P. Deift), Asymptotics for the Painleve II equation, Comm. Pure Appl. Math.48 , 277-337 (1995).
  16. (with P. Deift), Long-time asymptotics for integrable systems. Higher order theory, Comm. Math. Phys. 165, 175-191 (1995).
  17. (with P. Deift, S. Kamvissis and T. Kriecherbauer), The Toda rarefaction problem, Comm. Pure Appl. Math. 49, 35-83 (1996).
  18. (with P. Deift and S. Venakides), New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems , IMRN 6, (1997).
  19. Strong regularizing effect of integrable systems, Comm. in PDE, 22, 503--526 (1997).
  20. (with P. Deift and A. Its), A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics, Ann. of Math.146, 149-235 (1997).
  21. (with P. Deift, T. Kriecherbauer, K. McLaughlin and S. Venakides) Asymptotics for polynomials orthogonal with respect to varying exponential weights, IMRN, 16, 759-782 (1997).
  22. (with P. Deift, T. Kriecherbauer, K. McLaughlin and S. Venakides) Strong asymptotics for polynomials orthogonal with respect to varying exponential weights, preprint.
  23. (with P. Deift, T. Kriecherbauer, K. McLaughlin and S. Venakides) Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, preprint.
  24. The L2-Sobolev space bijectivity of scattering and inverse scattering transforms, Comm. Pure Appl. Math., to appear.
  25. (with P. Deift and S. Venakides), An extension of the steepest descent method for Riemann-Hilbert problems-small dispersion KdV , Proc. Natl. Acad. Sc. USA, 95, 450-454 (1998).
  26. (P. Deift), Perturbation theory of near integrable systems on the line. A case study-the Defocusing nonlinear Schrödinger equation, Math. Res. Letts, 4, 761-772 (1997).
  27. (with P. Deift, A. Its,and A. Kapaev), A note on the finite gap integration of the Schlesinger equation and on the elliptic solutions of the Painleve VI equation, Comm. Math. Phys, to appear.
  28. (with W. Liu, W. Zhang, F. Pu, and N. Huang), Nonlinear magnetization dynamics of a ferromagnet with an anisotropy in an external magnetic field, preprint.
  29. (with P. Chen and S. Venakides), Long-time asymptotics for the pure radiation solution of the sine-Gordon equation, Comm. PDE, to appear.
  30. (with P. Deift (Lecturer), T. Kriecherbauer, K. McLaughlin and S. Venakides) Uniform asymptotics for orthogonal polynomials, ICM (1998).

Last modified September, 1998