There has been a surge of interest in discrete integrable systems in the last two decades. The term “discrete integrable systems” (DIS) combines two aspects: discreteness and integrability. The subtle concept of integrability touches on global existence and regularity of solutions, explicit solvability, as well as compatibility and consistency – fundamental features, which form the recurrent themes of this book. On the other hand, by discrete systems we mean mathematical models that involve finite (as opposed to infinitesimal) operations. In a sense, discrete systems are essential to an understanding of integrability, and this book serves to provide an introduction to integrability from the perspective of discrete systems.

Discrete integrable systems include many types of equations, such as recurrence relations, difference equations and dynamical mappings as well as equations that contain a mixture of derivative and difference operators. Integrable systems have appeared throughout the history of mathematics without being recognized as integrable. An example is the equation that arises from the geometric collinearity theorem of Menelaus of Alexandria in the second century. Other examples are found in the defining equations of classical and nonclassical special functions. Important physical models, such as the equations of motion of the Euler top, are also integrable. The elliptic billiard is a classic example of a DIS, and the corresponding geometric result is the Poncelet's porism. It is, however, only in the second half of the twentieth century that the subject of integrable systems has grown as a discipline in its own right, and only in the last roughly two decades that the field of DIS has come to prominence as an area in which numerous breakthroughs have taken place, inspiring new developments in other areas of mathematics.

The number of integrable systems is large and growing, and the list of them includes a large number of examples that have application to physics and other scientific fields. In many cases, it turns out that one and the same discrete equation may be interpreted in different ways: as a dynamical map, as a difference equation and as an addition formula. This means that in the study of a given DIS a large number of branches of mathematics come together. Moreover, integrable equations are not necessarily isolated objects, but have many mathematical interconnections between each other. Highlighting these interconnections is another aim of this book.