The permutability property of Bäcklund transformations of solutions (of soliton equations), as discussed at the end of the previous chapter, forms one of the main sources of integrable partial difference equations (P∆Es) on the two-dimensional space–time lattice. In this chapter we study these equations in their own right, i.e. ignore their origin, and no longer assume that the dependent variable is a solution of any previously given PDE. In particular, we will consider the “integrability” of these P∆Es.

In general terms the concept of integrability is associated with regularity, solvability and amenability to exact and explicit treatments. These notions will have different mathematical meanings in different contexts, and it is indeed an open problem to give a universal definition of integrability. There are many characteristics associated with integrability, such as the existence of a Lax pair, compatibility structures such as Bäcklund transforms and hierarchies of compatible equations, good behavior of solutions with respect to singularities, regularity with regard to growth properties and complexity of solutions. These aspects of integrability are reflected in various different approaches in constructing and solving P∆Es, such as the τ -function approach of Hirota and the Kyoto school, symmetry methods and inverse methods such as “direct linearization”, some of which will be discussed in later chapters. Many of these features can be posed as definitions of integrability, but they are often restricted to specific classes of integrable systems. It is remarkable that if any two different definitions of integrability can be applied to some joint set of equations, they agree on which equations should be called integrable.

The presence of free parameters in the equations (namely the Bäcklund parameters λ and μ, which we will now reinterpret as lattice parameters) will play a crucial role in the development of the theory in this chapter. Furthermore, as we will see later, the parameters render the P∆Es very rich: since they can be seen to represent the widths of the underlying lattice grid they allow us to recover, through continuum limits, a great wealth of other equations, semi-continuous (i.e. differential-difference type) as well as fully continuous (i.e. partial differential type) ones. The interplay between the discrete and continuous structures will prove to be one of the emerging features of the integrable systems that we study.

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.