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 nonlinear dispersive partial differential equations (PDEs). This area has become exceedingly technical branching out into many different directions in recent decades. With this book, our aim is to provide a gentle introduction to the basic methods employed in this area in a self contained manner and in the setting of a few model equations. However, we should note that these methods are more generally applicable, and play a central role in modern research in nonlinear dispersive PDEs.

We designed this book having in mind a semester-long course in this area for advanced undergraduate and beginning graduate students. For that reason, we restricted the discussion to a few basic equations while providing complete details for each topic covered. We have also included many exercises that supplement and clarify the material that is discussed in the main text. After reading our book, a student should be able to read recent research papers in nonlinear dispersive PDEs and start making contributions.

There are several books, including Cazenave [28, 29, 30], Bourgain [20], Sulem–Sulem [138], Tao [143], and Linares–Ponce [105], which cover a large proportion of this area. In comparison, our book concentrates more on problems with periodic boundary conditions and aims to introduce the wellposedness techniques of model equations, such as the Korteweg de-Vries (KdV) and nonlinear Schrödinger (NLS) equations. The methods we describe also apply to various dispersive models and systems of dispersive equations, such as the fractional Schrödinger equation and the Zakharov system. In cases where the model equations are integrable, such as the periodic KdV and cubic NLS equations, alternative methods based on the symmetries and the structure of the equations have been developed. We refer the interested reader to Pöschel–Trubowitz [123], Kuksin [99], and Kappeler–Topalov [81] for complete integrability and inverse scattering techniques that extend some of the analytical results presented here. However, we should mention that we will not make use of any complete integrability methods in this book.