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Ablowitz, Mark J Professor

Positions

Research Areas research areas

Research

research overview

  • Key themes in Professor Ablowitz’ research are the understanding of the nonlinear wave phenomena that arise in physical problems. Mathematical techniques employed are asymptotic approximations, numerical and exact methods to obtain solutions to the underlying equations. Frequently employed are methods to solve certain nonlinear wave equations by the Inverse Scattering Transform (IST). IST allows one to construct general solutions to certain initial-boundary value problems. The problems of interest arise in a variety of physical problems. Physical applications include nonlinear optics, water waves, lattice excitations and Bose-Einstein Condensation (BEC). A special class of solutions are called solitons or solitary waves which are extremely stable localized waves.

keywords

  • Nonlinear wave equations, integrable nonlinear equations and solutions via the inverse scattering transform, solitons and solitary waves, mathematical modeling in fluids and nonlinear optics, mode locked lasers, optical lattices, water waves, differential and integral equations, mathematical physics

Publications

selected publications

Teaching

courses taught

  • APPM 4360 - Methods in Applied Mathematics: Complex Variables and Applications
    Primary Instructor - Spring 2018 / Spring 2019
    Introduces methods of complex variables, contour integration and theory of residues. Applications include solving partial differential equations by transform methods, Fourier and Laplace transforms and Reimann-Hilbert boundary-value problems, conformal mapping to ideal fluid flow and/or electrostatics. Same as APPM 5360.
  • APPM 5360 - Methods in Applied Mathematics: Complex Variables and Applications
    Primary Instructor - Spring 2018 / Spring 2019
    Introduces methods of complex variables, contour integration and theory of residues. Applications include solving partial differential equations by transform methods, Fourier and Laplace transforms and Reimann-Hilbert boundary-value problems, conformal mapping to ideal fluid flow and/or electrostatics. Department enforced prerequisites: APPM 2350 or MATH 2400 and APPM 2360 and a prerequisite or corequisite course of APPM 3310 or MATH 3130 or MATH 3135. Same as APPM 4360.
  • APPM 7300 - Nonlinear Waves and Integrable Equations
    Primary Instructor - Fall 2018
    Includes basic results associated with linear dispersive wave systems, first-order nonlinear wave equations, nonlinear dispersive wave equations, solitons, and the methods of the inverse scattering transform. Department enforced prerequisites: APPM 4350 and APPM 4360.
  • APPM 8300 - Nonlinear Waves Seminar
    Primary Instructor - Spring 2018 / Fall 2018 / Spring 2019 / Fall 2019
    Introduces the core methods in the analysis of nonlinear partial differential and integral equations or systems to graduate students. Provides a vehicle for the development, presentation, and corporative research of new topics in PDE and analysis.

Background

awards and honors

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