Preface xiii the suggested supplemental readings on elliptic curves, it should be possible to fully benefit from reading this book without any computer assistance at all. Book Overview Chapters 1 and 2 introduce the concepts of and summarize some of the key differences between linear and nonlinear differential equations. For those who have encountered differential equations before, some of this may appear extremely simple. However, it should be noted that the approach is slightly different than what one would encounter in a typical differential equations class. The representation of linear differential equations in terms of differential operators is emphasized, as these will turn out to be important objects in understanding the special nonlinear equations that are the main object of study in later chapters. The equivalence of differential equations under a certain simple type of change of variables is also emphasized. The computer program Mathematica is used in these chapters to show animations of exact solutions to differential equations as well as numerical approx- imations to those which cannot be solved exactly. Those requiring a more detailed introduction to the use of this software may wish to consult Appendix A. The story of solitons is then presented in Chapter 3, beginning with the observation of a solitary wave on a canal in Scotland by John Scott Russell in 1834 and proceeding through to the modern use of solitons in optical fibers for telecommunications. In addition, this chapter poses the questions which will motivate the rest of the book: What makes the KdV Equation (which was derived to explain Russell’s observation) so different than most nonlinear PDEs, what other equations have these properties, and what can we do with that information? The connection between solitary waves and algebraic geometry is introduced in Chapter 4, where the contribution of Korteweg and de Vries is reviewed. They showed that under a simple assumption about the behavior of its solutions, the wave equation bearing their name transforms into a familiar form and hence can be solved using knowledge of elliptic curves and functions. The computer program Mathematica here is used to introduce the Weierstrass ℘-function and its properties without requiring the background in complex anal- ysis which would be necessary to work with this object unassisted.
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