xii Preface ing, a file containing the code for many of the commands and exam- ples in the textbook can be downloaded from the publisher’s website: www.ams.org/bookpages/stml-54. It is partly through this computer assistance that we are able to make the subject of soliton theory accessible to undergraduates. It serves three different roles: The solutions we find to nonlinear PDEs are to be thought of as being waves which change in time. Although it is hoped that read- ers will develop the ability to understand some of the simplest examples without computer assistance, Mathematica’s ability to produce animations illustrating the dynamics of these waves al- lows us to visualize and “understand” solutions with complicated formulae. We rely on Mathematica to perform some messy (but otherwise straightforward) computations. This simplifies exposition in the book. (For example, in the proof of Theorem 10.6 it is much eas- ier to have Mathematica demonstrate without explanation that a certain combination of derivatives of four functions is equal to the Wronskian of those four functions rather than to offer a more tra- ditional proof of this fact.) In addition, some homework problems would be extremely tedious to answer correctly if the computations had to be computed by hand. Instead of providing a definition of the elliptic function ℘(z k1,k2) that is used in Chapter 4 and deriving its properties, we merely note that Mathematica knows the definition of this function, call- ing it WeierstrassP[], and can therefore graph or differentiate it for us. Although it would certainly be preferable to be able to provide the rigorous mathematical definition of these functions and to be able to prove that it has properties (such as being dou- bly periodic), doing so would involve too much advanced analysis and/or algebraic geometry to be compatible with the goals of this textbook. Of course, there are other mathematical software packages avail- able. If Mathematica is no longer available or if the reader would prefer to use a different program for any reason, it is likely that ev- erything could be equally achieved by the other program merely by appropriately “translating” the code. Moreover, by thinking of the Mathematica code provided as merely being an unusual mathematical notation, patiently doing all computations by hand, and referring to
Previous Page Next Page