Preface xv The wedge product of a pair of vectors in a 4-dimensional space is introduced in Chapter 10 and used to motivate the definition of the Grassmann Cone Γ2,4. Like elliptic curves, this is an object that was studied by algebraic geometers before the connection to soliton theory was known. This chapter proves a finite dimensional version of the theorem discovered by Mikio Sato who showed that the solution set to the Bilinear KP Equation has the structure of an infinite dimensional Grassmannian. This is used to argue that the KP Equation (and soliton equations in general) can be understood as algebro-geometric equations which are merely disguised as differential equations. Some readers may choose to stop at Chapter 10, as the connection between the Bilinear KP Equation and the Pl¨ ucker relation for Γ2,4 makes a suitable “finale”, and because the material covered in the last two chapters necessarily involves a higher level of abstraction. Extending the algebra of differential operators to pseudo-differen- tial operators and the KP Equation to the entire KP Hierarchy, as is done in Chapter 11, is only possible if the reader is comfortable with the infinite. Pseudo-differential operators are infinite series and the KP Hierarchy involves infinitely many variables. Yet, the reader who persists is rewarded in Chapter 12 by the power and beauty of Sato’s theory which demonstrates a complete equivalence between the soliton equations of the KP Hierarchy and the infinitely many algebraic equations characterizing all possible Grassmann Cones. A concluding chapter reviews what we have covered, which is only a small portion of what is known so far about soliton theory, and also hints at what more there is to discover. The appendices which follow it are a Mathematica tutorial, supplementary information on complex numbers, a list of suggestions for independent projects which can be assigned after reading the book, the bibliography, a Glossary of Symbols and an Index. Acknowledgements I am grateful to the students in my Math Capstone classes at the College of Charleston, who were the ‘guinea pigs’ for this experiment, and who provided me with the motivation and feedback needed to get it in publishable form. Thanks to Prudence Roberts for permission to use Terry Toedte- meier’s 1978 photo “Soliton in Shallow Water Waves, Manzanita- Neahkahnie, Oregon” as Figure 9.1-6 and to the United States Army

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