Ph.D. 1971, Massachusetts Institute of Technology
Nonlinear phenomena (shock and dispersive waves, solitons, dynamical systems), asymptotic and perturbation methods for ordinary and partial differential equations, chaotic interactions of nonlinear waves in fiber optics.
Professor Haberman was the author of textbooks on ordinary and partial differential equations in science and engineering and on mathematical modeling in mechanical vibrations, population dynamics, and traffic flow. His research involved various areas of physical applied mathematics: hydrodynamic stability, solitons for nonlinear dispersive waves, slowly varying bifurcations in nonlinear ordinary differential equations, and transitions such as caustics and shocks in partial differential equations. He investigated the change in action (adiabatic invariant) resulting from the slow passage of a separatrix and has extended this work to the slow passage of a separatrix for bifurcations of Hamiltonian systems. His work usually involved singular perturbation techniques: the methods of matched asymptotic expansions (boundary layers) and multiple scales (averaging).
MATLAB codes for his popular book Applied Partial Differential Equations have been archived on GitHub.
In the later years of his career, he used dynamical systems and singular perturbation methods to study chaotic collisions of nonlinear solitary wave in various different problems in fiber optics. He analyzed the trapping of light due to interaction of a nonlinear pulse with an optical defect. He introduced a universal separatrix map that explains the fractal structure in solitary wave interactions for dispersive nonlinear partial differential equations.
This research from the later years of his career appeared in journals such as SIAM Journals of Applied Mathematics and Applied Dynamical Systems, Physical Review Letters, Physica D, Chaos, and the Journal of Nonlinear Science.