Nonlocal Integrable Equations in Soliton Theory

This article is to provide a brief overview of the study of nonlocal integrable equations in soliton theory. The concept of nonlocality is explained, a little history of nonlocal dynamics is given, and basic problems of nonlocal differential equations are discussed. With the AKNS matrix spectral problems being taken as examples, a classification of the corresponding nonlocal integrable NLS equations and mKdV equations is presented. The identification of those nonlocal integrable equations is made within the zero curvature formulation, where local and nonlocal group reductions of matrix spectral problems are carefully conducted in pairs. Illustrative integrable models include six couples of scalar nonlocal integrable NLS equations and five couples of scalar nonlocal integrable mKdV equations.