傳播數學知識．促進數學教育

Interview with Prof. Robert Miura

**Interview Editorial Consultant:** Tai-Ping Liu, Yeong-Chuan Kao

**Interviewer:** Tai-Ping Liu (TPL)

**Interviewee:** Robert Miura (RM)

**Date:** January, 2001

**Venue:** Institute of Mathematics, Academia Sinica

Robert M. Miura (September 12, 1938 – November 25, 2018) was born in Fresno County, California. He received his BS in 1960 and MS in 1962 on Engineering from University of California, Berkeley, and MA in1964 and PhD in 1966 on Mechanical and Aerospace Engineering from Princeton University. He was formerly a professor at the University of British Columbia, and also a member of the Mathematical Biology Group, an interdisciplinary research group, at UBC. Between 2001 and 2018, he was faculty member at New Jersey Institute of Technology. Miura discovered an inverse scattering transformation known as the Miura Transformation. This work helped to establish the theory of solitions. He later worked on cell dynamics. In 1995, he was elected a Fellow of Royal Society of Canada, and in 2006, won Steele Prize for contributions to the soliton theory.

TPL: Robert, thank you for taking your time all for this. So, I think we shall begin with the beginning. I understand that you grew up not exactly in the center of civilization. Right? Fresno.

RM: Yes, that’s right. I was born and raised in central California in Selma which is located on Highway 99 near Fresno which you know because you’re from Stanford. This was prior to World War II so my family was in turn during the war. And we went to and we were putting camp in Arkansas and Arizona and then returned to California in 1945.

TPL: So, for how many years you were put in the camp?

RM: We were in camp from 1942 until 1945. So roughly 3 years.

TPL: Do you remember anything in the camp?

RM: Oh, I remember the camps the Arkansas, for example. We were in an area called Jerome, which no longer exists as far as I know there is no camp any more. There is no site. It was a very swampy area. And it was snow there. And I remember I have my first banana there. This is the only impression left. We lived in military camp. For the parents, this period was very difficult because they lost everything they had in California.

TPL: What did they do?

RM: They were farmers. My parents were farmers and they just bought a farm and they bought a truck and a car. They lost everything because of the war. It was very pleasant on the other hand for kids, you know, that was some kind of a vacation in a sense because it was a new adventure I had never seen this kind of things before. I was pretty young at that time. I was probably around 4 when we went into Jerome. And then I can’t remember it was 1 year or 2 years later we then were shift to Arizona to the place called Gila Bend. And of course it was completely different. It was very dry. There, I remember definitely, a fence around the camp. We had a guard of course. But we were allowed to go out of the camp and we were allowed to go between camps. There was another camp called Poston. So there we traveled back and forth by, it wasn’t a bus, a truck basically. There I started the first grade. And then in 1945, we actually came back to California before the War was over in the Pacific. I think the war was gonna to end soon. We were allowed to back to California. And, of course, going back to California was very difficult for everyone because of the prejudice at that time, anti-Japanese and prejudice against Japanese. And so I remember I had in fight in school and I remember when we stayed with my uncle, who fortunately had his local neighbor to care his farm during the war. And so he came back he had his farm back.

TPL: So your parents were not so fortunate?

RM: They lost their farm. And so my parents thought that they labors. I remember the first house we lived in was a two rooms house, a kitchen and a bathroom.

TPL: How many were you?

RM: There were 5 of us, three children. Sister was 2 older than me and my brother was 2 years younger than me. I am the middle child. Then we moved from there to another place where my father rent it. Basically, he was response for the management of the farm. And then we moved to another place where he rent the farm and received part of the revenue, something around 50 percent of revenue. And then in early 1950s, he finally bought a farmer again, his own farm. I was in very small school when he bought the farm. That was a 2 rooms school house. It was very useful for me because I was in the fourth grade then from 5th grade to 7th grade we were in the same classroom. So, when I was in the 5th grad, I could learn materials for the 6th grade from the 7th grade which was very useful, I think. We had a very good time then. We don’t have that anymore. Then they closed that school because it was a so small when I was in 7th grade. And the 8th grade I went to the local town’s school which is larger. And I finished my 8th grade there and my high school there. When I was in high school then my parents moved to the new farm they bought which was in the different district. But I continued to my high school because I didn’t want to leave my friends. I have got to walk I can’t remember how far it was maybe a mile, maybe half mile and a mile to catch the bus because I was out of the zone. I was in the different district. So, I have to walk because of the bus to the high school.

TPL: This was all around Fresno. Right?

RM: This was all around a country in Fresno. Right. That high school was interesting because my graduating class in the elementary school was 60 children. They were all in high school. When I graduated, it was only 28 of us were left. In California, there was a law, I don’t know it’s a law or not, you only have to go to school until age of 16 and after 16 if you want to drop out, then you drop out like many children did. Many other children drop out. It is unfortunate, among those who dropped out, some of them are rather smart.

TPL: They dropped out to joint the farm?

RM: Oh, yes. Mostly, they joined the labors. So, that high school I went to we had 28 children graduated and I was the Valedictorian, which didn’t mean much, as it was such a small school.

TPL: What did it mean?

RM: Valedictorian was like the top at that academic school. When I was in high school, I have expected I just go to local community college or maybe state college, nothing like university. However, I had a teacher who was a Japanese American who took interest in me and he told me that I should go on to the university. He has gone to U. C. Berkeley. And I think he had got a Master degree there. He told me that I should go to the university. So two of us, my best friend, Carroll Jones, and I, we both have gone to Berkeley and we roomed together for the first year. I went in Engineering. I got into Civil Engineering actually. I immediately changed to Mechanical Engineering because I couldn’t stand the surveying course. I had always been interested in flying, I mean the idea of flying, so I wanted to learn more about aeronautical engineering. So, that was my main interest in mechanical engineering. The first 2 years in the university I had a hard time because coming from such a small school. I didn’t learn how to take notes. I didn’t know how to study even though I can do it very well academically in my high school. I didn’t know how to do these basic things. So, it was very difficult for the first 2 years. The last 2 years, the 3rd and 4th year, was very easy, I mean, because I finally figured out how to do these things.

TPL: Let me interrupt it. You said that your parents didn’t think about the university when you graduated. What’s the background of your parents before they came to California?

RM: My father was born in Fukuoka, Kyushu in Japan. And he came over when he was 12 years old. He was adopted by his uncle who had already immigrated to the United States. So he came basically by himself to the family that adopted him in the United States. My mother’s parents had immigrated to United States and she was born in Oakland, California. So in that sense I am the second and half generation of immigrant, not second and not third. My father’s family had a farmland in Kyushu, rice fields and I am not sure what else they had. The family probably is still there but much of them even sold or converted into housing. It’s unfortunate I don’t quite understand how they do it in Japan for such a premium land as a farmland. My mother’s parents immigrated to Oakland. Then they moved to central California I think it’s because my grandmother had a health problem. So they had to move to warmer climate. That’s why she met my father. They were married there.

TPL: So, your father’s family was a farmer. They came to California farmer again.

RM: Well, my father’s uncle he originally had been a farmer. But when he came to California, he did farming but he also did other things. They ran on hotel one time. They had a store. So they did other things in addition. But my father always likes farming.

TPL: This is typical Japan American at that time?

RM: At that time, yes, very typical.

TPL: Sorry, you’re talking about the university. So you’re ok in Berkeley?

RM: I am ok in Berkeley. But I still didn’t know what I wanted to do. I had no long-term objective. The next thing to do after you graduated is to go to a graduated school. I didn’t know if I wanted to go graduate school or not. So I took a job. I applied to Berkeley for graduated school for master degree. This was a mechanical engineering. I tried to get a summer job so I went to Convair Astronautics in San Diego for a summer job. I worked in the temperature standard laboratory. That was such a boring job. I decided I have to go back to graduated school. It’s just too awful.

TPL: It works out that way.

RM: Works out that way. And at that time I was interested in doing experiment on fluid dynamics. And so I got into a project on gas dynamics which was a hot area at that time. So, I did a project on designing a temperature flow for a low density high speed wind tunnel. That was an interesting project it was clear that although it was interesting doing experiment; it was a very difficult thing to do because of the limitations in instrumentation which we had a that time. I also had problem with my thesis advisors, my Master thesis advisor. You know, he is a very bright guy. I thought I couldn’t judge his temper. Sometimes I would go into his office and he’s very nice and other times he yelled and kicked me out of his office. And I just didn’t like this uncertainty for going to see him. So I decided that for my Ph. D. I would apply to different places. So I applied to Berkeley, Cal. Tech., Princeton and Harvard. I got fellowships at U. C. Berkeley and Princeton. I got admission to Harvard and Cal. Tech., but with partial financial support. Also, at that time, the other reason I got go to Princeton was because I met a new teacher Chang-Lin Tien at Berkeley.

TPL: What year was that?

RM: That would be in 1961. Tien was the teacher for the Mathematical Thermodynamics course. I think that it was his 1st year. He was pretty young, extremely young. He was a very good lecturer. I was much inspired by him. When I was trying transferring from graduated school he said to me “Apply Princeton.” That’s why I applied Princeton. I applied Harvard because my advisor graduated from there. I applied Cal. Tech because Cal. Tech is in California.

TPL: You always support California.

RM: Yes. And so I got a Guggenheim fellowship at Princeton and that was one of the reasons I go to Princeton.

TPL: They don’t have this fellowship. Guggenheim don’t offer fellowship for graduated student anymore.

RM: There are different Guggenheims for different people. One of them is John Simon Guggenheim which is the one for professors. I think I have got a different Guggenheim. I don’t think it was the same one. Just before I left Berkeley, I got married. So we drove by cars to Princeton. It was a very different environment for me because it was a small school and lots of intellectual activities are metaphysical. The hardest thing to do there is trying to keep you focus on your work. So there I started again the experimental work. And I was put on project which was really not a proper project for student, to design a low density high speed nitrogen wind tunnel.

TPL: At which department?

RM: At that time, it was Aerospace And Mechanical Sciences. And now I believe it is called Mechanical and Aerospace Engineering. So when I got there I began to engage in this project. It also was my first experience in meeting people from other countries. My first officemate was in fact from Taiwan. His last name Chiu, Chiu Huei- Huang. (He retired from University of Illinois, Chicago and then went back to Taiwan and had second retirement. He is now in Chicago.) I was amazed that he spoke Japanese also, because I didn’t speak Japanese, but he did. He was educated under the old system. I spent a year, actually almost 2 years on this experimental work. I decided that I didn’t want to continue that. I wanted to switch. So I passed the qualifying exam and at that year I switched then to theoretic work.

TPL: But still in the same department?

RM: Yes, the same department. At that department, I think I went to every faculty member in the gas dynamic lab. I went to Wally Hays. I think I had Wally Hayes for 2 weeks. Then I had Bogdanoff, Sin-I Cheng and Harvey Lam. And then finally my thesis advisor was George Bienkowski. (Few years later he died in a bizarre bicycle accident in Princeton.) He got Ph. D. from MIT and had done his post-doc in Cal. Tech and then came to Princeton. He was very young, only 6 months older than I. Another person who was quite influential on me at that time was a classmate Young-Ping Pao who was a Courant. He and me were at the same class. He graduated 2 years before me. He is very bright, very sharp and went to Courant as a post-doc, in Magneto-Fluid Dynamics Division working with Harold Grad. He was a post-doc elsewhere, but eventually returned to Courant. Unfortunately he later died of stomach cancer.

TPL: So you were changing advisor and searching for your ….

RM: Searching for something that I felt comfortable with. I was assigned Hays when I first arrived at Princeton. When I started with experimental work, Bogdanoff is the natural choice. Then I quitted experiment and tried to do theoretical works with Sin-I Cheng for summer doing 3 dimensional wake flow. I had a hard time in understanding Chen. His accent was so strong even though he had been in the U.S. for 30 years. It was very difficult to understand what he was talking. And then I changed to Harvey Lam. It was OK, except when I finally got the problem to work on, I didn’t believe the problem. I didn’t believe that his idea was correct. He was trying to look at the boundary layer theory using kinetic theory and then continue on Navier-Stokes continuum flow theory outsides. And then how you match one with the other. I just did not believe that was the right thing to do because we don’t even know how go with kinetic theory to continue flow right now with those nice theories.

TPL: Even without boundary.

RM: Even without boundary. So, the question was how did one do the matching. So that’s why I decided, then Bienkowski came to Princeton, that to me it was an opportunity to change advisor to the problem I really wanted to do.

TPL: Which was?

RM: Which was to study the torque for a rotating cylinder in the infinite fluid. I also work on the Couette flow using the kinetic theory. I had a bad experience in my Master thesis from Berkeley, I did experimental work for one whole year when I moved to Princeton. I had these two unpleasant experiences. And so I decided that I am not going to leave Princeton until my Ph. D. thesis is finished. So, I applied a post-doc. One post-doc was from Physics Lab of Courant with Martin Kruskal. So, this was in Sep. 1965. The post-doc the way I got it was as the following. I have applied and I didn’t get it. They did not plan to hire me, they wanted to hire Ben Zinn. What happened was he got the offer from Georgia Tech and right at the same time he got the postdoc. He decided to take the other job instead. He gave up the postdoc. Then they gave the post-doc to me.

TPL: Princeton didn’t know how lucky they were.

RM: Well. So while working on my thesis, I went to talking to Martin Kruskal. He said, well, here are a number of problems you can work on. However, you cannot work on your thesis at day time. When you’re working in lab, you have to working on my problem. You worked your thesis at night. And so I worked on my thesis at night and giving it to the members of committee and getting feedback from them and corrected there. Finally, that was all finished. I’ve finally got my thesis, I think it was about April in 1966. And Kruskal when he asked me to work on this problem first of all he showed me what they had done previously. And what he wanted to know was to decide the conservation laws for Korteweg-de Vries. Actually what I think, I mean, the way I feel about this is he actually gave me a problem not actually worked with me because he was busy preparing to go to Russia and he was learning Russian. He just guaranteed I don’t think he wanted to be bothered. So he gave this problem which was the Korteweg-de Vries equation at that time they knew five conservation laws. The first 2 were obvious. The 3rd was found by, I think it was found by Korteweg and de Vries. The 4th was found by Whitham. Kruskal and Zabusky found 5th one. Then they looked for 6th one they couldn’t find them. So, Kruskal’s idea was maybe they skipped one. Maybe there’s 7th one. So he said, “Look for the 7th one.” And I did that and this is all ….

TPL: How long it took you to find this one?

RM: It wasn’t very long. It just was done algebraically. Algebraically very complicated because you have to do everything exactly. It had very long coefficients I had got to solve. So I found the 7th one. Then I looked back at what they did on the 6th one. They had made an algebraic mistake. So there was the 6th one as well. And I thought maybe I should calculate then I calculated the 8th one and I calculated the 9th one by hand.

TPL: For how long have they looked at the 6th one?

RM: I don’t think they have looked at. I think they had tried to do the calculation. They couldn’t get it. So they thought the 6th one does not exist.

TPL: Did Whitham try the 6th one?

RM: Well, I don’t know. I don’t think so. I don’t think that he pursued the Korteweg-de Vries equation seriously. But he found the 4th one. He didn’t look for the 5th one. So after finding these then of course I decided there must be infinitely many of them. It was very nice. When I was a graduate student, Kruskal offered two courses. He is a very confusing lecturer I find because whenever he is explaining something, in a middle of his presentation he will inject some remarks. Unless you know what he is doing, it very difficult to follow the path he’s going along. For me as a student, it’s very difficult to follow what he is explaining about. However, working with him by one on one, it’s fantastic, excellent, you know, I can ask all kind of crazy questions, stupid questions and he will answer all of them and very nicely. I really like to work with him. It was probably the most exciting thing in my research career. What was happening was I would come early in the morning. And he would come around 3 or 4 in the afternoon and we then would work until maybe 6 or 7 o’clock. And he would stay at work very late and then he would go home. There is a funny story. There was a guy from the Plasma Physics Lab who was working on computing center problem. He had a problem that he was working on. So he asked Kruskal, he said, “I have this problem and I want to discuss with you. Can I talk to you about it?” So Kruskal said, ”Why don’t you make an appointment with me. How about 1 o’clock in the morning?” One o’clock in the morning at his house. So he went to his house and talked with him for about an hour. And so it’s 2 o’clock in the morning. So he thought he’s going to leave. And he saw there’s a graduated student waiting for 2 o’clock in the morning. That’s supposed to be a true story.

TPL: I understood that you have the nonlinear transformation of Riccati type.

RM: Yes. That was later.

TPL: So the first few conservation laws was found by brute force.

RM: Yes, brute force. So, as I said we had this conjecture that there are infinitely many conservation laws, I tried to prove that there are infinitely many. We just couldn’t do it. So I had drawn the huge tables of the conserved densities and the fluxes and I looked it. It was like numerology I was looking at how are they related to each other. As I showed this table to a friend of mine, he was a post-doc. He was a student of Goldberger, physicist who became president of Cal. Tech eventually. A friend of mine, his name is David Weily, and I showed him this table. And he was looking at it and we were trying to figure out. One day, he had been with a dentist and he had a brilliant idea. What he did is he just took the functional derivative of one of the conserved densities and it turned out that was the previously conserved density. So if you know the high one then you come down. But there is no way to go back up. That was the problem. Well, it’s a very nice little result. But he saw the connection. This all happened within roughly a half year. Then in the summer of 1966, Kruskal and Norman Zabusky ran this Nonlinear School on Mathematics and Physics in Munich at the Max Plank Institute. And by then of course Kruskal had come back from Russia. And so I went to that conference. It was the first conference I attended. I had met Zabusky earlier but I didn’t know him very well. I showed him all these conservation laws. And when I was there he said, “Do you know the Fermi-Pasta-Ulam problem? This is where all of these came from. There was another equation, not just Korteweg-de Vries.” Kruskal and Zabusky had derived another equation from the Fermi-Pasta- Ulam problem. That equation has different nonlinearity.

TPL: This is by different scaling?

RM: No, by different force term, a higher degree force term. And he said, ”The result is this modified Korteweg-de Vries equation is just u square.” And he said, “Maybe that one has lots conservation laws.” And so I was in Max Plank Institute, instantly I would go to the library. I was calculating the conservation laws. And I found there are many of them also and in fact I couldn’t finish calculating. So I thought maybe it’s true that if I just put u to the up ux for p is any integer, then it will work. It turned out that it didn’t work. I could only do it for p equals 1 and 2. But for p equals to 3,4,5…etc., I can only find three. And I couldn’t find anymore. But that particular case was where the transformation came from. When we were there, Peter Lax came and he said, “Cathleen Morawetz had proved that there are only 9 conservation laws of the Korteweg-de Vries equation.” Which is exactly the number I calculated. And I said, “I don’t believe that.” He said that they had proved that this was true. He didn’t have the proof at hand. What he showed was something has changed algebraically. I had never seen what they did. I asked Cathleen for several times, she never showed me what they did. So I don’t know what they found. All I know was that the claim was that Peter said that there were only 9 and Cathleen said that in fact something had happened that there were only 9. So when I got that from Munich, ….

TPL: So of course Morawetz knew of your result.

RM: They knew the conservation laws that we had, the conservation laws of the 9th. So when I came back from Munich, I was going on vacation in Canada. This is an interesting connection with Canada. I had a friend who was a mathematical physicist. We’ve got to be regular friends from graduated school. So we went to their cabin or his parent’s cabin in Peterborough Ontario. For two weeks, I basically calculated number 10th. I can’t make mistakes. The 10th one was very long. And I found the 10th one. I confirmed that I hadn’t made any algebraic mistake. So I was actually convinced by them that we had actually infinitely many. So Kruskal and I were working on this problem. We had a work going to use WKB method. We finally figured out how to do it. We were sitting there in his office. It was probably about 6 or 7 o’clock in the evening. We were sitting. We looked on the board and we were admiring at and the phone rang. Cliff Gardner has called and said, “I just figure out how to show you the infinite many conservation laws.” And we were just finding that we had this result on the board. But what he had was very different from we what had. His method is just beautiful. It’s very nice. Basically, like a generation function.

TPL: Where did Gardner come from? He has been keeping in watch on you and that project.

RM: Cliff Gardner was a member of the research member of Plasma Physics Lab. His office was next to mine. It was a small office next to mine. Actually, initially he had a bigger office. I am not sure why he moved to a small office from a bigger one. When Kruskal went to Russia, Kruskal told me that if I had questions on conservation laws, go ask Cliff Gardner. I had never met him. So the first time I had questions about conservation laws, I went to Gardner’s office. I knocked on his door, I walked in and I said, “Martin Kruskal told me that if I have questions about conservation laws, I can come and talk to you about it.” He said, ”I don’t know anything about conservation laws.”

TPL: Is that roughly before you found the number 7?

RM: No. Exactly I found number 7 because it was after Kruskal had left. Kruskal went to Russia in January 1966. So that was my first time with Cliff Gardner. He is a wonderful guy. He is probably the smartest guy I know. I mean Kruskal is very smart. But they are in different way. Gardner has a mind of looking at things in very funny ways. But he also told me something very interesting. I said, ”How do you find these things?” And he said,” Well, 95 maybe 99 percent of this stuff garbage“ But he’s just trying on 100 percent garbage.

TPL: Let us back to the infinitely many and the 9th conservation laws. I mean did you have any hint or any understanding that you believe infinitely many. Or you just don’t ….

RM: Just Believe. Just belief. That’s why when Peter Lax came and told me that there were only 9. It was hard for me to believe that the sequence would suddenly just stop at 9.

TPL: But also you don’t believe in authority or anything. All right? You just followed what you saw.

RM: Yes, maybe. I thought he was reporting on someone else, not his own result.

TPL: But he was not objecting to that conjecture.

RM: No. He wasn’t objecting to this conjecture. I think he firmly believed that the result with Cathleen Morawitz that there were only 9. That’s the way he reported. I believe that’s what he thought.

TPL: At least it didn’t occur to him that number 9 ought to be wrong.

RM: Yes, that’s right.

TPL: I guess this soliton theory many people consider to be the most important theory in the mathematical science in the whole 20th century. So that you don’t believe on anyone else’ supposedly understanding or what not. It’s a key thing for this.

RM: Yes, I mean it wasn’t stated as a theorem or anything. It was just reported what they had. They were just like a conversation. But the whole soliton idea was Zabusky and Kruskal’s idea. That was a very interesting story, too.

TPL: Does this can be traced up to Fermi, Ulam?

RM: Right. What happened was when Zabusky was a post-doc in the Princeton physics lab worked with Kruskal. The way he got on to the Fermi-Pasta-Ulam problem was because they were just, I think, reading a book of the collection of mathematical problems written by Ulam. One of the problems he listened there was the Fermi-Pasta- Ulam problem. So that’s how Kruskal and Zabusky were working on that. And Zabusky, I am not sure exactly what his background was, but he knew about the hodograph transformation. And he added several theory to the equation. And the quadratic degree equation came out with this as a result of them trying to come up with a simple description of the problem of waves on the both direction of the Boussinesq equation. So they looked at the solution with the propagation on only one direction. That’s why it was called Korteweg-de Vries. And after they got, they did all kind of things. And the very important thing they did was the numerical work on that problem. Then they looked back at the Fermi-Pasta-Ulam. And they realized that one of the things that Fermi, Pasta and Ulam had not done. They only looked at the energy, the linear energy and the Fermi-Pasta-Ulam string, of the PDE. What they found was the recurrence of the energy. If you put all the energy into one mode of this nonlinear string, then the energy in that one mode would cascade into higher modes. Then they eventually will come back. This is that recurrence. There’s exact some question about what the problem actually was because I spoke with Ulam about it and he said that it was the PDE he was all thinking about. But Pasta told me that he was never thinking about PDE. He’s always thinking about the discrete problem. So, there is some disagreement. I don’t know who is right. Maybe they were both right, just thinking about differently. But Pasta really firmed about that they’re not looking at PDE, they’re looking at the discrete problem, the lattice problem. It depends on how you look at it. You’ve gonna talk about the breaking of the solutions or you have to talk something about discrete system. Very different from each other.

TPL: I understood that Fermi was very surprised at the numerics.

RM: Yes, I am not sure that he was the one that suggested the numerics initially. That was one the first computers worked Los Alamos out there and they wanted to do something on the distribution of solution of nonlinear equation in this discretization problem. He was the one who decided to work on and tried to understand how energy in different modes distribute.

TPL: That is not what happened. All right?

RM: No, in fact, they never plotted the solution describing this. They never plotted the actual solution. All they plotted was their Fourier modes. They completely missed the phenomena of the waves crossing each other and coming out.

TPL: So the only thing they understood was that it was something unusual.

RM: Yes.

TPL: But who had the clear conception that they are actually solitons?

RM: That was done by Kruskal and Zabusky. And that was published in a Physical Review Letter. What they’ve done at that paper was to take the Fermi-Pasta-Ulam problem. Since it goes backwards. Normally we think of from the numerical point of view. We think of taking a continuous equation, discretizing it and solve the discrete equation. Their idea was they had a discrete equation on the lattice that was corresponding to PDE. So you just do Taylor’s series and you calculate out the terms and the question is what are the actual terms that are necessary to make you work on this problem. That’s what they did and they found that they got basically Boussinesq equation. But in fact if you look at the waves from one direction, you get the Korteweg-de Vries. So the Physical Review paper that they wrote clearly demonstrates soliton. The only thing different about the Physical Review paper is that soliton is finite periodic. And so you see the behavior but it’s not perfectly.

TPL: Let me change a subject. This is really exciting to hear this. Then after that you went to Courant. All right?

RM: Well, there were a lot of things happened still before I went to Courant. Let me tell you some more. So we found the infinitely many conservation laws. I can’t remember if we found the infinitely many conservation laws before or after I found the transformation between Korteweg-de Vries and modified Korteweg-de Vries.

TPL: Although the paper of this nonlinear transformation preceded the paper on the infinitely many conservation laws.

RM: Right. I am not sure. I can’t remember. So what happened was I had the conjecture that there were infinitely many conservation laws of the Korteweg-de Vries, infinitely many conservation laws for the modified Korteweg-de Vries. And none of the others. For other p (i.e. p>2) there are only finite ones. So I made this conjecture that somehow the solutions for the quadratic degrees and the solutions for the modified quadratic degrees will be related because of their infinitely many conservation laws. That’s not the reason to believe that’s true, but in fact it’s the conjecture I had. Maybe it’s not easy to find them. So then the question was: if that’s true, then how to find the transformation between that two? Initially, it’s a hard problem. The only way I can think to do with was that must be true of the conservation laws, one of them must map to the conservation laws of the other one. So the first step I have got to do was try to relate them, the conservation laws and their curls.

TPL: which you knew…

RM: It doesn’t work because that’s not the right dimensionality. And I thought maybe I should skip some or something. And so I skipped them and that worked. The only trouble was that the transformation I had was complex. It had complex coefficients there. I couldn’t figure that one out. And so the first thing I did was, immediately I went to the next door, I showed Cliff Gardner. My derivation was certain mapping these conservation laws and he said, “Well, I don’t know this transformation will work or not. Prove it to me.” So right there, I had to prove to him. I had to go through it and do the calculation. They had not done yet. This was a conjecture actually. I had not done the calculation so I should show him the differentials of the solution and put into Korteweg-de Vries and this would get the other equation. So actually I worked out the calculations. I was actually surprised that it worked. And then I went to Kruskal, he at that time was talking with a student. He was talking and dinning with Joe Su when I came in and I showed him this. He quitted talking to Joe and started talking to me about this problem. I showed him this transformation and he was very exciting. So that’s the transformation. The trouble of the transformation is that it takes you from one nonlinear equation that you can’t solve to another nonlinear equation you can’t solve. That’s a good criticism. I can’t remember, but I think Kruskal was the one who came back and he said that this equation was linearizable. It wasn’t a nonlinear equation. It was in Bernoulli form so you can linearize the equation. The other thing you needed, however, was it looked a little bit like the Schrodinger equation, but did not have eigenvalues of it. For the eigenvalues you use Galilean transformation. Use Galilean transformation, you get a constant from the equation, which is the eigenvalue. This equation was unusual. First of all, there was a problem of the complex coefficients. Then you can get rid of by changing the sign of the quadratic degree equation and the sign of the modified degree equation. Then you get rid of “i” for both of the equations. Then the resulting equation you can transform into a linear equation by the standard transformation for the equation of Bernoulli type. And then you get into a time independent Schrodinger equation with an eigenvalue by Galilean transformation. Now you have this eigenvalue problem. And the problem was that is the inverse scattering problem we’re trying to solve. You want to know what the coefficient is. The potential. We don’t know how to do that. A part of this I should mention has a lot to do just the physical layout of the processes. This is an interesting thing because what happened was there was a common area with the blackboard and Kruskal and I just sat there, we could wrote on the large blackboard bigger than his office. We sat there, wrote thing on the board and people passed by. One person passed by before he went into his office. That was John Green. He looked on this problem and John Green said, “That’s an inverse scattering problem that you are trying to solve.” He’s trained in nuclear physics and so he knew about the inverse scattering problem. And he found in Landau- Lifshitz book an exact solution to the inverse scattering problem. I think he showed it to me and actually I found the hyperbolic secant problem solution of this equation. Then Cliff ’s passed by. He didn’t say anything to us. All he did was he looked at the equation and he went to the physics library. Then he found a Courant Institute report written by Kay and Moses. As far I know that that report was never published. But he remembered the report and that problem actually solved this inverse scattering problem on the doubly infinite line. The classical problem is on the half infinite line, but Kay and Moses actually did on doubly infinite line. That was what we finally realized what to do with the whole theory with the inverse scattering theory. However, there’re a lot of problems because we’re trying to solve the problem where the initial data for the potential. Then you want to know how was that potential evolved in time of the Korteweg-de Vries. We needed to know something about if the potential evolves with the Korteweg-de Vriess, how the eigenvalues evolve or how the wave function evolves. And so lot of that was done by Green because for some reason he knew how to calculate these things.

TPL: So by now there were all four of you.

RM: All four of us were working on this problem. And everyday new result, that was just amazing. Everyday something new has happened. That was extremely exciting.

TPL: People amaze, I read that someone who made a remark that such an important revolution, for its small group of people and at a single location.

RM: Yes. But I think it was just the combination of Kruskal and me and Green and Gardner and we all also had made some major contribution to the whole project. It’s just amazing.

TPL: That was what the idea comes from in fact you are the early one the infinitely many conservation laws and nonlinear transformation. That must be the period that you have really fun memory of it.

RM: 2 years I was there as a post-doc.

TPL: What you said is just all happened within 2 years from the beginning to the end?

RM: Yes. And in fact even more because most of the papers were written while I was still at Princeton. Although it’s very interesting because, none of them, we were under no pressure to publish, that was probably not good for me in my early career because after that I never thought this. Need to publish, you know, really quickly. Once we started writing papers then of course they all came out. There was another result which I thought actually a really nice result. Which was about the squares of the eigenfunctions satisfy very nice equation. I mean you can write down formulas for the sum of the eigenvalues. After all these inverse scattering were done before I went to Courant. When I went to Courant, I actually moved to Greenwich village in July or August of 1997. And for the first month, I was back to Princeton to work. Then I started working at Courant. And Peter Lax had stayed the whole summer in Stanford with Ralph Phillips. They’re going to classical mechanical theory, not the quantum mechanical theory. So when I went to Courant, I went to Peter Lax’s office and I said, ”I want to show you some results that we’ve got in summer.” And I said maybe you already know and I then explained the inverse scattering theory. He said that he’s not getting to talk about that. Maybe he didn’t feel very good. But actually before that he has proved, he came down to Princeton, I think that was the first time I met him, he came down to Princeton and he has shown the exact passage of solitons to each other. They read this paper and he showed in that paper that if you take two solitary waves and you let them run through each other, they come out unchanged. That was a very nice paper. However, he gave me a reprint. And with it I went through details, really detailed. And I found tons of mistakes. But every answer was right. So he had made compensated errors that was giving him a right answer.

TPL: He was in a rush maybe.

RM: Yes. I think there must be a calculation he actually carried out all the details. When he wrote the paper, he didn’t remember all the details exactly. But he knew the answers. So I looked at them and I showed him all the correction. In the printed version all are corrected. So what I had was the preprint. Li: The Lax pair was also in that paper?

RM: That was the Lax pair. But the Lax pair I believe that it wasn’t in that paper but in another related paper. I had a hard time in understanding the Lax pair at that time. I didn’t know very much about the Poisson bracket.

TPL: Now could we talk about how you got into mathematical biology.

RM: Sure. So I continued working on Korteweg-de Vries when I was at Courant. And I also worked on some classical mathematical physics problems. I guess I was mainly focusing on Korteweg-de Vries. So I was at Courant for four years. Then I went to Vanderbilt University and I was writing the survey paper on Korteweg-de Vries. When I was teaching calculus when I was at Washington Square College, there was a professor of biology came to my class. He came up to me after class, I asked him, “Why are you in the class?” He said that he had taken in calculus when he was a student. But he had forgotten most of it and he was finding that he couldn’t read papers without knowing calculus. So he was starting to take in calculus over again. So I said, ” That’s fine.” Then he showed me his lab. He was working on the self-cycle phenomena. I thought that was interesting but I didn’t know anything about modeling and biology. Also, there was a group of people at Courant, Frank Hoppenstead and other people, who were working population dynamics. That also was interesting. But I decided that I was not working in that area. At that time what was happening was that more and more pure mathematicians started coming to this field of inverse scattering. And in fact what was happening was that it was going in this direction of algebraic geometry. I didn’t have a clue of what was that all about. So at that time I then moved to British Columbia as a visitor for one year. Don Ludwig was holding a college workshop.

TPL: That’s the person who wrote the paper with Joe Keller on geometric optics.

RM: Yes. Also he wrote a big part of second volume of Courant-Hilbert. And he had gone to British Columbia the year before. And he was running his workshop in the summer time. I decided at that time that I would switch field. When I was in Vanderbilt, I had already talked a lot with two guys who are physiologist. So I learned the ecology from them. Things might be superficial in that difficult area. So when I went to British Columbia, I decided to spend the first year learning about ecology. I went to their seminars. I went to their colloquia. And I decided after that I said, “There is not a data.” So that when people made up a model. They made up models that no basis on the data. They just made up by look. I mean that reasoning have to go behind it. That wasn’t clear to me that they wanted to make models that were biological development should do lots of description of them. So I decided instead I would work in brain research. The reason was because I gave a talk on solitons and there was a young post-doc Henry Tuckwell who was a student from University Chicago and got his Ph. D. In biophysics. He had worked partly with Jack Cowan at Chicago who was a very famous biologist, but I think his official advisor was Luigi Ricciardi on geology from Italy who actually was at Chicago at that time. But anyway, when I gave my talk and I talked about waves ran into each other, he came up to me afterwards and he said, “Is it possible to relate this with memory because you preserving information?” And I said, “I don’t think so because solitons occur in conserved systems but the brain is a dissipative system.” That was served as the end of that conversation.

TPL: What made you think the brain is a dissipative system?

RM: Because a lot of the energy mechanisms in the brain and a little bit about Hudgkin-Huxley and those two told me that it was a dissipative system and so they are not conserved systems. Then what happened was he continued to put papers on the brain in my mailbox. And I looked at them then I put them aside. Finally, one day he came and put a paper in my mailbox it has a target pattern. I can’t remember that was Nancy Kopell or Lou Howard coming to talk. And they were looking at the reaction-diffusion equation and there were target patterns. I said, “Maybe it’s the same kind of phenomena.” So I got Henry Tuckwell in and I said, “Maybe there’s something to do with this reaction-diffusion system.” So we started working on this problem that came with the model and that was the reactiondiffusion model. The difficulty was that when we first started doing this problem, initially we thought it was an electrical phenomena. Well, I should mention what the problem was. The problem was there’s something called spreading epilepsy depression that was a psychological depression. In 1944 there was a Brazilian physiologist who had a Ph. D. student named Carlburg. And he was studying epilepsy in cats. What he was doing was he simulated the brain of the cat and they observed that there was a very fast epilepsy wave to go out. However, then there will be a very slow wave that follows it. Extremely slow, it was traveling like 1 to 10 mm per minute. So we decided to model this. The first thing that we did was trying to model it from electrical point of view. But that was too fast, all the waves were too fast. So then we thought that it must be chemical phenomenon. So we modeled it as chemical phenomenon and we concluded the model that showed it traveled about 1 mm per minute. So we can work with the reasonable model this conjecture as to mechanism which had not a literature before then. This phenomenon was discovered in 1944, it had been studied by groups of people over the world and no one had a real good idea how are that works. And we feel that we have some understanding about that work. People are still working on today.

TPL: In deriving this model, is your background not only your collaborator but you yourself knew some chemical process?

RM: Yes. So the way that the model was constructed was based on something I had learned called plasma physics. It was called two-bag model with trees, electrons and proton separately. Although they are occupying same space, their trees are different.

TPL: You have a mathematical background but those backgrounds in chemistry and so forth are as important. All right?

RM: Yes. You need to know science. I think anytime you do modeling you need to know the science. I was really experienced about this because we teach modeling courses but I think modeling courses by themselves were not useful. The way I think you learn modeling is that you go to a subject like physics or chemistry whatever. And you learn how they do it. My background of engineering will very be helpful because I learned a lot of modeling. So when I do modeling now, I know what the steps should be. The most important step is that you understanding the background, you understanding the science.

TPL: So do you suggest that in the mathematics department we should teach mathematics and if students interested in science, they should go another department to take? And vice versa, if in the engineering or other department they should teach their subjects. If they learn math, they should come to the math department.

RM: Yes, that’s right. Most of my students, Ph. D. students or graduated students, for example, when they come in, they all know no biology. So one of things I ask them to do is to take a course in either ecology or physiology or in zoology, etc. so they learn the background. They are math major. They have never learned anything about biology.

TPL: Do you would rather have math major to come in than biology major to come in?

RM: Yes, this was what we discussed at the night on the bus. I feel that a mathematical student can learn biology easier than a biological student can learn mathematics. So for me it’s more desirable to have mathematical students come in. Then had them learning biology and they are doing well.

TPL: In your career that you have changed several directions and one of the characteristics is that you are not afraid to talk to people in other field. You started with changing your advisors several times.

RM: I should say what the problem was. One of the things I didn’t like as a student in engineering was that I felt, this is just not like the people in Princeton there, people in Princeton much better, once in Berkeley when I was in undergraduate. Several professors were intolerant. You couldn’t ask them lots of questions and expect to get reasonable answers. You would be littered. They would say that you don’t know that. They would say something to you that would put you walking. And as I went more and more advanced in mathematics I found less and less of that. Kruskal, for example, is very good about that. One of the reason why I relaxed working with him was I can ask almost anything and he would answer.

TPL: No question is stupid because you never ask stupid questions.

RM: No, no. I did it. I really asked stupid questions. I think it’s really important to ask stupid questions because lots of people know thing because of good in their area and they believe that this is true. They don’t know why it’s true they just believe that’s true. So if you ask them to explain something then they don’t explain because they don’t know why it’s true. So I asked stupid questions. The worst part is: There is no stupid questions, but stupid answers.

TPL: Ok, I see. So I feel that we have explored you sufficiently. Thank you very much. That’s really exciting.

- Tai-Ping Liu is a faculty member at the Institute of Mathematics, Academia Sinica.