In earlier chapters we explained how to derive discrete Painlevé equations through different perspectives, relying on symmetries, deautonomization through singularities, and geometry. Discrete Painlevé equations were found as recurrence relations of the Painlevé equations in Section 2.3 of Chapter 2, identified through the singularity confinement property in Section 7.1, found as similarity reductions of lattice equations in Chapter 10, and described through a unifying geometric point of view (Sakai, 2001) as mappings on rational surfaces in Section 7.3 in Chapter 7.

Discrete Painlevé equations arise in various mathematical and physical contexts, for example in random matrix theory, in models of quantum gravity, and as correlation functions in statistical mechanics. They also appear as nonlinear equations satisfied by coefficients of linear three-term recurrences governing orthogonal polynomials with semiclassical weights. They are pushing the boundary of what we define to be nonlinear special functions because they are regarded as defining equations for an entirely new class of special functions.

The focus of this chapter lies on the intrinsic properties of discrete Painlevé equations as integrable systems. After starting with a brief history of their discovery, we describe and deduce transformations between solutions, such as Bäcklund or Miura transformations. Then we focus on their associated linear problems and describe the different types that arise. The theory developed by Birkhoff and his school for solving linear difference equations is outlined and we suggest ways in which compatible linear problems could be used to solve nonlinear discrete Painlevé equations. Finally, we describe special solutions of the discrete Painlevé equations along with methods developed for deducing such solutions.

In the literature, there are three different types of discrete Painlevé equations, distinguished by the curves along which they are iterated, i.e. a straight line, a spiral or an elliptic curve in the complex plane, leading to what are respectively called additive-type, multiplicative-type or elliptic-type discrete Painlevé equations. If the equation is iterated by a shift in the independent variable n, say, then these cases are distinguished by the types of coefficient functions z(n) appearing in the equation.

We consider in this chapter the continuum limits of the lattice equations, i.e. the limiting equations (typically differential equations) that we retrieve from the discrete equations by shrinking the lattice grid to a continuous set of values corresponding to spatial and temporal coordinates. Having discussed in Chapter 3 the integrability aspects of quadrilateral P∆Es it is now in order to see what these equations are, and whether they can be identified as discrete analogues or discretizations of PDEs.

In numerical analysis the conventional picture is the one where discrete equations, namely finite-difference schemes, are used as numerical approximations to differential equations. The choice of difference schemes is sought on the basis of a variety of criteria such as numerical stability and speed of convergence. In most cases the continuum limit itself is a reductive procedure: we lose the lattice grid parameters by performing the limit. One major philosophical question in this picture is whether the differential equation is an approximation of a given discrete equation, or conversely, whether it is the finite-difference equation that plays the role of an approximation.

It is our perspective that continuum limits of discrete equations are the degenerations and approximations of the latter. One would expect, therefore, that the continuous equations are less rich in parameters than the discrete ones. In studying continuum limits of integrable discrete systems, however, it turns out that the latter is only partially true. We will observe that in some respects we do not really lose the “richness” of the relevant equations, as long as we keep intact the structure of parameter families of compatible equations. In their most explicit form the discrete equations can be shown to generate entire infinite families of continuous differential equations, which constitute the hierarchies associated with the famous soliton systems.

In practice we should perform continuum limits in such a way that integrability is retained at all levels. We will find that continuum limits are not necessarily unique, because starting from a two-dimensional P∆E, we can separately perform a continuum limit in each direction, leading in first instance to semi-continuous (semi-discrete) equations.

How to take a continuum limit

In a continuum limit involving a step-size parameter, say h, the difference operator ∆hy, defined in (1.9), will tend to a derivative, namely by (1.8), as was explained in Section 1.1.2.

In this chapter we discuss difference equations with one (discrete) independent variable. Their continuous analogues describe, for example, the motion of point particles. In that context the Hamiltonian formulation and the Liouville–Arnold integrability play a central role. In the discrete-time case the situation is more complicated, because we do not have the analogue of the chain rule of time derivative, and furthermore the motion is given by finite time steps rather than by a continuous time flow.

The difference between continuous and discrete systems is underscored by the observation that whereas a one-degree-of-freedom (autonomous continuous time) Hamiltonian system always has one conserved quantity, namely the Hamiltonian (= energy) itself, it turns out that generically one-degree-of-freedom discrete-time systems do not posses a conserved quantity. Surprisingly, there does exist a rather large family of (autonomous) one-degree-of-freedom maps called Quispel–Roberts–Thompson (QRT) maps, which possess a conserved quantity or invariant. They are examples of what we will call integrable maps. There are also extensions to more than one-degree-of-freedom maps (variously called higher-dimensional or multicomponent maps) and to nonautonomous maps (i.e. maps depending explicitly to the discrete independent variable, usually denoted by n). A classification of integrable nonautonomous maps corresponding to Painlevé equations is discussed in Chapter 11.

Integrable systems are rare and it is difficult to find such systems. One efficient way of deriving integrable maps is to start with a 2D integrable lattice equation (such as the ones discussed in Chapter 3) and apply a dimensional reduction to it in order to derive a 1D lattice equation (i.e. an equation in one discrete independent variable), which we interpret as a dynamical map. We will show that this can be done so that the integrability properties of the original 2D lattice, such as the existence of a Lax pair, carry over to the map, and can be used to derive their invariants.

Integrability of maps

There are several properties associated with integrability of maps, for example, the existence of a sufficient number of conserved quantities, symmetries, Lax pair and the behavior around singularities. In this section we will discuss the notion of Liouville integrability and the structures associated with it.

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.