Nonlinear systems are generic in the mathematical representation of physical phenomena. It is unusual for one to be able to find solutions to most nonlinear equations. However, a certain physically significant subclass of problems admits deep mathematical structure that further allows one to find classes of exact solutions. Solitons are a particularly important subclass of such solutions. Solitons are localized waves that, in an appropriate sense, interact elastically with each other. They have proved to be extremely interesting to physicists and engineers due, in part, to their localized and stable nature.

This broad field of study is sometimes called “soliton theory” or “integrable systems.” This field has witnessed numerous important developments, which have been studied intensively worldwide over the past 30 years. Some of the directions that researchers have pursued include the following: direct methods to find solutions; studies of the underlying analytic structure of the equations; associated Painlevè-type solutions and relevant generalizations; tests to locate integrable systems; studies of the underlying geometric structures inherent in integrable systems; Bäcklund and Darboux transformations, which can be used to produce new classes of solutions; and so on.

In principle, one would like to be able to solve the general initial-value problem associated with these special nonlinear soliton systems. Depending on the boundary conditions under consideration, sometimes this is feasible.