Nonlinear waves are ubiquitous throughout the natural world. Some examples are ocean waves, solar wind, vibrational waves in materials, and laser beams. These disparate kinds of phenomena can be described by mathematical models that are based on hyperbolic, elliptic and dispersive partial differential equations and that are surprisingly similar to each other. My research is devoted to understanding the fundamental underlying features of these models and their relationships to physical phenomena.

One of my current research projects is the study of water waves with vorticity. This is a free-boundary problem because the water surface is an unknown. Vorticity indicates the presence of eddies in the water. I study exact waves modeled by the Euler equations without assuming shallow water or small-amplitude approximations. Recently I have proven the existence of many continua of large-amplitude water waves. Some questions under current study include the location of stagnation points, stability properties of the waves, periodic and solitary waves, numerical computation of the waves, and the occurrence of overhanging waves.

Another focus of my research is the instabilities of plasmas for which collisions are rare. Such plasmas occur in various astrophysics phenomena and in hot nuclear reactors. Despite a huge amount of research over several decades, precise analyses of instabilities have been made only in very special situations. The usual model is the coupled Vlasov and Maxwell equations. There are many equilibria, including homogeneous ones, electric ones and magnetic ones. I am developing new general criteria for their stability.

Some other specific areas of my research include the stability of equilibria in semiconductor models, waves in hyperelastic materials, the scattering of fourth-order waves, and the interactions of nonlinear and spatial effects in wave models.

National Science Foundation Grant DMS-0405066: Stability and Existence of Nonlinear Waves; $228,423 (2004-09)

Earlier National Science Foundation grants, continuously funded since 1967, on: linear waves, nonlinear waves, stability, kinetic theory, scattering, evolution equations, etc.

U.S.-France Cooperative Research: Modeling, Analysis and Simulation of Hybrid Quantum Models (1999-2004)

Variational formulations of steady water waves with vorticity, J. Fluid Mech. 548 (2006), p. 151-163, with A. Constantin and D. Sattinger.

Stability of semiconductor states with insulating and contact boundary conditions, Arch. Rat. Mech. Anal.179 (2005), p. 1-30, with Y. Guo.

Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math. 57 (2004), 481-527, with A. Constantin.

Stability of the Camassa-Holm solitons, J. Nonlin. Sci. 12 (2002), 415-422, with A. Constantin.

Instability of traveling waves of the Kuramoto-Sivashinsky equation, Chinese Annals Math. 23B (2002), 267-276, with Guanxiang Wang (in memory of J. L. Lions).

Irving Segal's work in partial differential equations, J. Funct. Anal. 190 (2002), 25-28 (in memory of I. E. Segal).

Exact periodic traveling waves with vorticity, C. R. Acad. Sci. Paris 335 (2002), 797-800. with A. Constantin.

Nonlinear instability of dispersive waves, in: T. Kato's method and principle for evolution equations in math. physics, Yurinsha, Tokyo (2002), 173-178.

Stable and unstable ideal plane flows, Chinese Annals Math. 23B (2002), 149-164, with C. Bardos and Y. Guo (in memory of J. L. Lions).

An inhomogeneous boundary value problem for nonlinear Schrodinger equations, J. Diff. Eqns. 173 (2001), 79-91, with C. Bu.

Time decay for the nonlinear beam equation, Meth. & Applics. of Analysis 7 (2001), 479-488, with S. Levandosky (dedicated to C. S. Morawetz).

Spectral condition for instability, Contemp. Math. 255 (2000), 189-198, with J. Shatah.

Stability of peakons, Comm. Pure Appl. Math. 53 (2000), 603-610, with A. Constantin.

Stability of a class of solitary waves in elastic compressible rods, Phys. Lett. A 270 (2000), 140-148, with A. Constantin.

Magnetically created instability in a collisionless plasma, J. de Maths. Pures et Appliq. 79, 10 (2000), 975-1009, with Y. Guo.

Regular solutions of the Vlasov-Poisson-Fokker-Planck system, Discrete & Cont. Dyn. Sys. 6 (2000), 751-772, with K. Ono.

Relativistic unstable periodic BGK waves, Comput. and Appl. Math. 18 (1999), 87-122, with Y. Guo.

Decay of the linearized Boltzmann-Vlasov system, Trans. Th. Stat. Phys. 28, 135-156, with R. Glassey.

Unstable oscillatory-tail waves in collisionless plasmas, SIAM J. Math. Anal. 30 (1999), 1076-1114, with Y. Guo.

Robustness of instability for the two-dimensional Euler equations, SIAM J. Math. Anal. 30 (1999), 1343-1354, with S. Friedlander and M. Vishik.

Perturbation of essential spectra of evolution operators and the Vlasov- Poisson-Boltzmann system, Discrete & Cont. Dyn. Sys. 5 (1999), 457-472, with R. Glassey.

Stability and instability in the kinetic theory of plasmas, Mathemática Contemporanêa 15 (1999), 249-258.

Unstable BGK solitary waves and collisionless shocks, Comm. Math. Phys. 195 (1998), 267-293, with Y. Guo.

Nonlinear instability in an ideal fluid, Annales de l'IHP (Anal. NL) 14 (1997), 187-209, with S. Friedlander and M. Vishik.

Existence and blow up of small-amplitude nonlinear waves with a negative potential, Discrete & Cont. Dyn. Sys. 3 (1997), 175-188, with K. Tsutaya.

Stability, instability and regularity of nonlinear waves, in: Nonlinear Waves, T. Ozawa, ed., Gakuto Int'l Series, Gakkotosho, Tokyo, 1997, p. 451-468.

Breathers as homoclinic geometric wave maps, Physica D 99 (1996), 113-133, with J. Shatah.

The relativistic Boltzmann equation, in: Quantization, Nonlinear PDEs and Operator Algebras, W. Arveson et al., eds., P.S.P.M. 59 (1996), Amer. Math. Soc., p. 203-209.

Asymptotic stability of the relativistic maxwellian via fourteen moments, Transport Th. Stat. Phys. 24 (1995), 657-678, with R. Glassey.

Instability of periodic BGK equilibria, Comm. Pure Appl. Math. 48 (1995), 861-894, with Y. Guo.

Microlocal dispersive smoothing for the Schrodinger equation, Comm. Pure Appl. Math. 48 (1995), 769-860, with W. Craig and T. Kappeler.

Global finite-energy solutions of the Maxwell-Schrodinger system, Comm. Math. Phys. 170 (1995), 181-196, with Y. Guo and K. Nakamitsu.

Book: Partial Differential Equations: An Introduction, Wiley and Sons, New York, 1992

Book: Nonlinear Wave Equations, NSF-CBMS Research Monograph, Amer. Math. Soc., Providence, 1989.

Nonlinear stability and instability of relativistic Vlasov-Maxwell systems, preprint, to appear in Comm. Pure Appl. Math., with Zhiwu Lin.

Linear stability and instability of relativistic Vlasov-Maxwell systems, preprint, to appear in Comm. Pure Appl. Math., with Zhiwu Lin.

Stability properties of steady water waves with vorticity, preprint, to appear in Comm. Pure Appl. Math., with A. Constantin.

NSF Postdoctoral Fellowship, 1962-63

Fulbright Fellowship, 1967

Guggenheim Fellowship, 1971

Fellow of the Japan Society for the Promotion of Science, 1972

Principal lecturer, C.B.M.S., George Mason U., 1989

Harrison S. Kravis University Professorship, Brown U., 1993-94

L. Herbert Ballou University Professorship, Brown U., 1995-present

Institut Henri Poincare Prize, 1997

Fellow of the Society of Industrial and Applied Mathematics

Fellow of the American Mathematical Society

Fellow of the American Academy of Arts and Sciences

 

 

American Mathematical Society

Society for Industrial and Applied Mathematics

Association of Mathematical Physics

Royal Physiographic Society of Lund