Dispersive Partial Differential Equations

The authors note that this book is intended for “beginning graduate students in mathematics with some background in real and complex analysis who are interested in pursuing research in dispersive partial differential equations (PDEs).*rdquo; They add that after one finishes the book, “…a student should be able to read recent research papers in nonlinear dispersive PDEs and start making contributions.” I completely agree with the statement: it’s just a matter of the time required to work out all of the estimates in the book!

The first chapter is a gentle recollection of necessary tools from analysis. Then, wasting no time, chapter two delves right into basic properties of linear dispersive PDEs. One of my favorite parts of the book comes at the end of this chapter, with an application to the Talbot effect; it’s remarkable that the profile of linear dispersive PDEs depends on algebraic properties of time!

The rest of the book is certainly not easy-going, but one has a sense of great pride after working through the estimates. Chapter three is devoted to well-posedness techniques, including the energy method, oscillatory integral method, and restricted norm method. The next chapter then presents a classical method of Bourgain and of Colliander-Keel-Staffilani-Takaoka-Tao called the I-method. Finally, the last chapter discusses applications of the previous methods, in particular to the Talbot effect for nonlinear equations.

I very much liked the emphasis on the KdV and cubic NLS equations rather than presenting results in full generality. The exercises in each chapter, while not at all trivial, tremendously enhance one’s understanding of the material. The focus on periodic boundary conditions sets this book apart from related ones in the area, and yet the authors do a nice job discussing related well-posedness results and estimates on the real line as well.

While certainly not meant for students without any analysis background, a thorough work-through of this book certainly brings one to the frontier of the theory of nonlinear dispersive PDEs.

Eric Stachura is currently a Visiting Assistant Professor of Mathematics at Haverford College.