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, exact and numerical 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. Applications include water waves, nonlinear optics, lattice dynamics and Bose-Einstein Condensation (BEC). A special class of solutions are called solitons or solitary waves which are robust stable localized waves.
keywords
Nonlinear wave equations, integrable nonlinear equations and solutions via the inverse scattering transform, solitons and solitary waves, mathematical modeling in water waves, nonlinear optics, topological waves, inverse problems, differential and integral equations, mathematical physics
ON THE METHOD OF SOLUTION OF THE DIFFERENTIAL-DELAY TODA EQUATION.
Physics Letters A: General Physics, Nonlinear Science, Statistical Physics, Atomic, Molecular and Cluster Physics, Plasma and Fluid Physics, Condensed Matter, Cross-disciplinary Physics, Biological Physics, Nanosciences, Quantum Physics.
413-418.
1993
ON THE METHOD OF SOLUTION TO THE 2+1 TODA EQUATION.
Physics Letters A: General Physics, Nonlinear Science, Statistical Physics, Atomic, Molecular and Cluster Physics, Plasma and Fluid Physics, Condensed Matter, Cross-disciplinary Physics, Biological Physics, Nanosciences, Quantum Physics.
293-298.
1992
NOTE ON SOLUTIONS TO A CLASS OF NONLINEAR SINGULAR INTEGRODIFFERENTIAL EQUATIONS.
Physics Letters A: General Physics, Nonlinear Science, Statistical Physics, Atomic, Molecular and Cluster Physics, Plasma and Fluid Physics, Condensed Matter, Cross-disciplinary Physics, Biological Physics, Nanosciences, Quantum Physics.
215-218.
1987
APPM 4350 - Methods in Applied Mathematics: Fourier Series and Boundary Value Problems
Primary Instructor
-
Fall 2023 / Fall 2024
Reviews ordinary differential equations, including solutions by Fourier series. Physical derivation of the classical linear partial differential equations (heat, wave, and Laplace equations). Solution of these equations via separation of variables, with Fourier series, Fourier integrals, and more general eigenfunction expansions. Same as APPM 5350.
APPM 4360 - Methods in Applied Mathematics: Complex Variables and Applications
Primary Instructor
-
Spring 2018 / Spring 2019 / Spring 2020 / Spring 2021 / Spring 2022
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 / Spring 2020 / Spring 2021
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 5430 - Methods in Applied Mathematics: Applications of Complex Variables
Primary Instructor
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Fall 2019 / Fall 2021
Reviews basic ideas of complex analysis, including solutions of ODEs and PDEs of physical interest via complex analysis; conformal mapping, including Schwarz-Christoffel transformations and generalizations; computational methods; Riemann-Hilbert problems; topics in asymptotic methods. Department enforced prerequisite: APPM 4360 or APPM 5360.
APPM 7300 - Nonlinear Waves and Integrable Equations
Primary Instructor
-
Fall 2018 / Fall 2020
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 / Fall 2020 / Fall 2021 / Spring 2022 / Fall 2023 / Fall 2024
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.
