xiv Preface (Readers who have never worked with complex numbers before may wish to consult Appendix B for an overview of the basic concepts.) The n-soliton solutions of the KdV Equation are generalizations of the solitary wave solutions discovered by Korteweg and de Vries based on Russell’s observations. At first glance, they appear to be linear combinations of those solitary wave solutions, although the nonlinearity of the equation and closer inspection reveal this not to be the case. These solutions are introduced and studied in Chapter 5. Although differential operators were introduced in Chapter 1 only in the context of linear differential equations, it turns out that their algebraic structure is useful in understanding the KdV equation and other nonlinear equations like it. Rules for multiplying and factoring differential operators are provided in Chapter 6. Chapter 7 presents a method for making an n × n matrix M depending on a variable t with two interesting properties: its eigen- values do not depend on t (the matrix is isospectral) and its derivative with respect to t is equal to AM − MA for a certain matrix A (so it satisfies a differential equation). This digression into linear algebra is connected to the main subject of the book in Chapter 8. There we rediscover the important observation of Peter Lax that the KdV Equation can be produced by using the “trick” from Chapter 7 applied not to matrices but to a differential operator (like those in Chapter 6) of order two. This observation is of fundamental importance not only because it provides an algebraic method for solving the KdV Equa- tion, but also because it can be used to produce and recognize other soliton equations. By applying the same idea to other types of oper- ators, we briefly encounter a few other examples of nonlinear partial differential equations which, though different in other ways, share the KdV Equation’s remarkable properties of being exactly solvable and supporting soliton solutions. Chapter 9 introduces the KP Equation, which is a generalization of the KdV Equation involving one additional spatial dimension (so that it can model shallow water waves on the surface of the ocean rather than just waves in a canal). In addition, the Hirota Bilinear version of the KP Equation and techniques for solving it are pre- sented. Like the discovery of the Lax form for the KdV Equation, the introduction of the Bilinear KP Equation is more important than it may at first appear. It is not simply a method for producing solu- tions to this one equation, but a key step towards understanding the geometric structure of the solution space of soliton equations.

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