Mark J. Ablowitz

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Mark Jay Ablowitz (born June 5, 1945 in New York City ) is an American applied mathematician who studies solitons .

Life

Ablowitz studied at the University of Rochester ( Bachelor's degree in 1967) and received his PhD in 1971 from the Massachusetts Institute of Technology with David Benney ( Non-Linear Dispersive Waves and Multiphase Modes ). From 1967 to 1971 he was a teaching assistant at MIT. From 1971 he was first assistant professor , 1975 associate professor and 1976 professor at Clarkson University . From 1979 to 1985 he was head of the mathematics faculty and from 1985 to 1989 dean. From 1989 he was a professor at the University of Colorado in Boulder and until 2000 head of the applied mathematics department there.

In 1977/78 and 1984 he was visiting professor for applied mathematics at Princeton University and in 1984 exchange scientist at the National Academy of Sciences in the Soviet Union. As early as 1979 he was co-director of the joint symposium on soliton theory of the US and Soviet academies of science. In 1985 he was visiting scholar at the Institute of Theoretical Physics at the University of California, Santa Barbara .

He was a Sloan Research Fellow from 1975 to 1977 and a Guggenheim Fellow in 1984 . He is a fellow of the American Mathematical Society . In 1976 he received the Clarkson Graham Research Award. From 1976 to 1979 he was co-editor of the Journal of Mathematical Physics.

He has been married since 1968 and has three children.

plant

He is particularly concerned with the inverse scattering transform (Inverse Scattering Transform, IST), a basic method of solution of certain nonlinear partial differential equation (in one or two spatial dimensions, such as scalar and vector nonlinear Schrödinger equations), which conceptually similar to the Fourier transform in the linear case is, as Ablowitz wrote with Harvey Segur , David J. Kaup , Alan C. Newell in a 1974 essay in which they also coined the term Inverse Scattering Transformation, originally published by Robert Miura , Martin Kruskal , Clifford Gardner and John Greene in 1967 . In the article they also introduced AKPS systems (named after the first letters of the authors), a system of two coupled partial, exactly solvable differential equations in complex, with which, for example, the nonlinear Schrödinger equation can be treated.

A conjecture named after him, A. Ramani and Harvey Segur says that nonlinear partial differential equations can only be solved by the IST if the ordinary differential equations derived from them by reduction have the Painlevé property . With others he showed that the self-dual Yang-Mills equations , which play a central role in the theory of integrable systems, after reduction not only yield most of the known soliton equations, but also nonlinear differential equations that Jean Chazy investigated in 1909 and which also provide connections Studies by S. Ramanujan (1916) have.

Ablowitz also does a lot of research on applications of solitons in optics and quantum optics (dynamics of ultra-short pulses in mode-locked lasers, communication in optical fibers with very high bit rates, non-linear optical waveguides). He also researched dispersive shock waves (DSW) and water waves.

Fonts

  • with Harvey Segur: Solitons and the Inverse Scattering Transform , SIAM Studies in Applied Mathematics, 1981
  • Editor with B. Fuchssteiner, Martin Kruskal Topics in Soliton Theory and Exactly Solvable Nonlinear Equations , World Scientific 1987
  • with PA Clarkson Solitons, Nonlinear Evolution Equations and Inverse Scattering , London Mathematical Society Lecture Notes Series, Volume 149, Cambridge University Press 1991
  • with Athanassios S. Fokas Complex Variables: Introduction and Applications , Cambridge University Press, 1997
  • with M. Boiti, F. Pempinelli, B. Prinari: Nonlinear Physics: Theory and Experiment. II , World Scientific 2003
  • with B. Prinari, AD Trubatch Discrete and Continuous Nonlinear Schrödinger Systems , Cambridge University Press, 2004
  • Nonlinear dispersive waves: Asymptotic Analysis and Solitons , Cambridge University Press 2011
  • with Kaup, Newell, Segur: The inverse scattering transform-Fourier analysis for nonlinear problems , Stud. Appl. Math., Vol. 53, 1974, pp. 249-315
  • with DJ Kaup, AC Newell, H. Segur: Method for Solving the Sine-Gordon Equation , Phys. Rev. Lett., Vol. 30, 1973, pp. 1262-1264

Web links

Individual evidence

  1. ↑ Life data according to American Men and Women of Science , Thomson Gale 2004
  2. ^ Mathematics Genealogy Project
  3. ^ Ablowitz, Kaup, Newell, Segur The inverse scattering transform - Fourier analysis for nonlinear problems , Studies in Applied Mathematics, Volume 53, 1974, pp. 249-315
  4. ^ Ablowitz-Ramani-Segur Conjecture, Mathworld
  5. ↑ M. Ablowitz, A. Ramani, H. Segur, A connection between nonlinear evolution equations and ordinary differential equations of P-type, 2 parts, J. Math. Phys., Volume 21, 1980, pp. 715-721, 1006 -1015
  6. ↑ Like KdV , KP equation , nonlinear Schrödinger equation , Painlevé equations , Darboux-Halphen equations
  7. ↑ Ablowitz, S. Chakravarty, PA Clarkson Reductions of self-dual Yang-Mills fields and classical systems , Phys. Rev. Letters, Vol. 65, 1990, pp. 1985-1987
  8. ^ Ablowitz, Chakravarty, Leon Takhtajan A self dual Yang Mills Hierarchy and its reduction to integrable systems in 1 + 1 and 2 + 1 dimensions , Communications in Mathematical Physics, Volume 158, 1993, pp. 289-314
  9. ↑ Ablowitz, S. Chakravarty, RG Halburd Integrable systems and reductions of the self-dual Yang-Mills equations , Journal of Mathematical Physics, Volume 44, 2003, pp. 3147-3173. Ablowitz, Chakravarty The Generalized Chazy Equation from the Self-duality Equations , Studies in Applied Mathematics, Volume 103, 1999, pp. 75-88. See also Ablowitz, Chakravarty Parametrizations of the Chazy Equations , Preprint 2009, Arxiv
personal data
SURNAME Ablowitz, Mark J.
ALTERNATIVE NAMES Ablowitz, Mark Jay (full name)
BRIEF DESCRIPTION American mathematician
DATE OF BIRTH June 5, 1945
PLACE OF BIRTH New York City