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Geometry of Non-Linear Differential Equations, Backlund Transformations, and Solitons, Part A (Interdisciplinary Mathematics Series No. 12): Robert Hermann

In this Volume, I have embarked on a vast project--the exposition and development of the 19-th century geometric theory of differential equations, particularly in terms of applications to engineering and physics. No those familiar with other branches of mathematics which were extensively worked on in the 19-th century (e.g., complex variables or algebra) might think this is a relatively simple matter of historical research. In fact, it is primarily a matter of new mathematical work, since so much of the material has been completely lost to present day science. My task is to reconstruct it, much in the way that a paleontologist reconstructs a dinosaur from a bone in the foot.

I have used as guides three major works. First, the articles in the Enzyklopdie der Nathematischen issenchaften (or the French version, Encyclopidie des Sciences Nathematiques), particularly the articles by Von eber and Vessiot. Second, E. Cartan's articles and books on "exterior differential systems". (Nuch of his work in this area was, in fact, oriented towards unifying the work of his predecessors by developing it in the context of his own way of looking at differential geometry.) Third, Darboux' "Theorie des Surfaces" which is really much more about the theory of non-linear partial differential equations than it is about "surfaces" in the usual sense.

Although I have been interested in this topic for a long time (for example, see my article "E. Cartan's Geometric Theory of Partial Differential Equations"), the factor compelling me to embark on this almost impossible task is a striking recent development in mathematical physics--the revival of material that was done in the classical period in terms of what is now called the Theory of Non-Linear aves. (See hitham [1].) In particular, recent work on solirons is so strikingly a revival of ideas of the 19-th century that it cannot be a coincidence--in fact, the secret to understanding the nature of elementary particles may have its mathematical roots in this work. I believe that developing this material in the context of modern mathematics will make fundamental contributions to mathematical engineering and physics, as well as stimulating pure mathematical research itself towards working on material that is of far greater intrinsic interest and importance than many of the topics that are fashionable today!

This first volume contains two major topics. First, a reworking of certain general material in Von eber's ency1opedia article, and then a much more specific topic, my own version of very recent work by two mathematical physicists, Frank Estabrook and Hugo ahlquis on a dif?erential geometric setting for the recent work on hOg-linear waves. hat is particularly significant is that they have developed a general setting for two key geometric ideas--"conservation laws" and "prolongation", and shown the very natural way that Cartan's formulation in terms of exterior differential systems leads to these notions.

What has this to do with elementary particles? First, I must say that I believe that 99.98...% of the modern work by physicists on what might be called Fundamental Physics, particularly on finding an appropriate mathematical setting for understanding the experimental elementary particle phenomena, is wrong-headed and even perverse. It is all basically motivated by the wish to take the linear equations and quantum theory with which they are familiar and modify it in some way to account for the observed phenomena, which are strikingly non-linear. They expect mathematics to give them a magical hocus-pocus to do this, but I strongly believe (often by study of my own, part of which has gone into my earlier books) that what they want to do is impossible and that the whole business must be rethought. A truly non-linear theory wilI be completely different than what is now sold to us as Quantum Field and Elementary Particle Theory.

Why do I want to go back to 19-th century ideas? I believe that this was the period when differential equations were thought about geometrically, and most geometrically inspired differential equation theory is intrinsically nonlinear. If 19-th century geometry were developed along the same lines as modern elementary particle physics, the only surfaces to have been investigated in Darboux' treatise would be planes and those which are "perturbations" of planes Notice that two of the most striking and conceptually successful recent ideas in elementary particle physics--gauge (i.e., Yang-Mills) fields and solirons--are closely linked to geometry. In fact, I will develop here Estabrook and Wahlquist's insight into the mathematical nature of solirons in a form that will provide a link between them, involving the theory of connections in fiber bundles. I believe that quantum mechanics for such geometrically inspired objects must be thought out completely anew. I am always horrified to see physicists proudly butchering beautiful mathematical concepts by trying to fit them into the standard ideas of perturbation-Feynman diagram-renormalization-quantta field theory. (Unfortunately, "axiomatic-constructive" quantu field theory is no better, since geometric-Lie group concepts play such a small role in the underlying intuition and esthetics. I just do not understand why a geometric object like a connection should be "quantized" as an "operator-valued distribution", and I think it is rather mindless to set this up as an "axiom".)

Like the small mammals that developed while the dinosaurs reigned, I see certain modest theoretical ideas that are promising for a correct mathematical understanding of the elementary particle phenomena. Evidently, I believe that the solitoh ideas are of this type. Here one encounters purely and superbly non-linear phenomena, that seems to offer a new approach to the traditional "wave-particle" conundrum. Unfortunately, it is unlikely that the striking particle-like properties that have been found in equations of one space dimension will extend to three dimensions. However, something must generalize--perhaps the topological properties mentioned below. Mathematically, these solitons are closely linked to Backlund transformations--surely, they must generalize, and they might even be the key to understanding how Lie groups enter into elementary particle phenomena. In fact, in the last chapter, I present some work lifted from Darboux that shows, for certain linear partial differential equations, how Backlund transformations appear as gauge transformations. It is interesting that two basic equations--the Klein-Gordon and Sine-Gordon equations--appear in a very natural way. I hope this will convince the skeptical reader that there are many goodies buried in the 19-th century literature that have been completely forgotten, but that are very relevant to today's problems in physics.

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