Professional Feature - Mark J. Ablowitz

By Invited Professional Contribution

Early years:

I grew up in the Bronx (NY) in one bed room apartment; my bed was in the adjacent foyer of the living room. My father, a seriously wounded WW II veteran, was a NYC taxi driver. My mother was a sales person in large NYC department store. Neither had more than a high school education yet they felt it was important for me to go to college. They wanted me to be a doctor or a dentist or if need be, a lawyer. But, early on it didn't look promising. I was a mediocre student in elementary school (K-8). I was mainly interested in being on the 'Audio-Visual squad' which allowed me to avoid class to run projectors for movies.

Figure 1: Senior in high

My high school, Evander Childs, was a rough place but I was placed in the 'honor school' which separated out about three-four hundred students out of four thousand. I found math and science interesting. During my senior year I found a way to enter a math competition, a first for Evander. I did very well, competitive with well-known NYC high schools; the chairman of the math department was astonished. The college guidance councilor suggested I apply to colleges with engineering programs. I applied to a few colleges as a Mechanical Engineer (ME), even though I was far from a mechanical type person.

I was accepted to the City College of NY, which was free, and Columbia University which was not too expensive since I got some scholarships and could live at home and commute by subway. The guidance councilor advised me to try to go to a college outside New York City. My parents couldn't afford the tuition much less housing but happily I was offered a generous scholarship to the University of Rochester (U of R). Though I had never visited, and hardly knew where it was, I accepted.

College/graduate school:

In those days the U of R ME program was more like Engineering Physics with a fair amount of math in their courses which suited me. I did particularly well in engineering math courses and a faculty member who taught some of the courses asked me if I had considered going to graduate school in math. After some reflection I decided to apply to two engineering and two math graduate schools. This faculty member also arranged for me to have an interview at the math department where he earned his Phd, MIT; interviews were uncommon at the time.

I was accepted to the graduate schools. I chose MIT because of its strong physical applied math group which was a good match for my background in engineering. David Benney, my advisor, was an expert in the burgeoning area of nonlinear waves. My thesis involved the study of large amplitude multi-periodic waves in nonlinear equations. This extended work of G. Whitham, who was a well-known applied mathematician and this research area, sometimes called Whitham theory which was popular at that time has had numerous 'rebirths' over the years.

Professional career:

After finishing my Phd, rather than taking a postdoctoral position I became an assistant professor in the math department at Clarkson University, in upstate NY. My parents were pleased that I had the title of Doctor but never really understood what a professor did.

Figure 2: Giving a talk in 1979 (US/
USSR Academy Meeting held in Kiev,

The first summer I was at Clarkson we hosted a major nonlinear waves conference with many of the leaders in the field. It was an opportunity to meet one of them, Martin Kruskal, who was one of the discoverers of solitons and new methods of analysis associated with one of the well-known equations in the field: the Korteweg-deVries (KdV) equation. During one afternoon at the conference two colleagues came to a breakout session and said that they just read an article in the library which showed that the so-called nonlinear Schrödinger equation (another well-known equation in the field) had been analyzed by the same methods that applied to the KdV equation. That moment changed the trajectory of my career.

The following year colleagues and I were able to generalize these new methods to a large class of equations including a number of physically significant equations. These equations had many common features such as soliton solutions, infinite number of conservation laws and with deceasing initial data could be linearized via suitable integral equations. We termed the method the 'Inverse Scattering Transform (IST)'. Soon afterwards my graduate student and I extended these methods to nonlinear differential difference and partial difference equations.

Subsequently, colleagues and I found a deep connection between these soliton equations and classical ordinary differential equations studied at the end of the 19th century by P. Painlevé. That work was followed by extending these theories to multi-dimensional equations. Over the years I worked on many applications of solitons to physical problems and showed how IST applies in many settings. Recently we found large new classes of interesting nonlinear nonlocal and fractional nonlinear equations that fit into the IST framework.

Figure 3: Boulder, Colorado, 2022.

At Clarkson I served as Chair of the Math Department and Dean of Science. In 1989 I moved to the University of Colorado, Boulder to restart an applied math effort. We began as a program with four tenure track faculty and two instructors. It evolved into a department that now has some 20 tenure track faculty and more than 10 teaching faculty.

Early in my career a colleague persuaded me that writing monographs on new research was important for the field as well as one's career. I collaborated on three monographs and two textbooks. I was delighted at how well they were received and cited. I have particularly enjoyed creating and describing new methods of applied mathematics. Especially to young researchers I often say: "Combine what you like and what are good at; this will serve you well in the long run".

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