Integrability of P Es

Transcription

1 Lecture 3 Integrability of P Es Motivated by the lattice structure emerging from the permutability/superposition properties of the Bäcklund transformations of the previous Lecture, we will now consider the integrability properties of these viewed as partial difference equations (P Es) on the two-dimensional space-time lattice. What we will discover is that the integrability can be given a precise, even algorithmic, meaning. The presence of parameters (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. Furthermore, as we will see in subsequent Lectures, 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. The history of integrable difference equations goes back to seminal papers by Ablowitz & Ladik and by Hirota in the 1970 s, cf. [1,2]. The first was motivated by the search for integrable numerical algorithms through finite-difference approximations. This fits well into the general problem of the analysis of finite-difference P Es arising from numerical studies of PDEs, cf. e.g. the monograph by P.R. Garabedian, Ch. 13. More recently, systematic methods for the construction of integrable nonlinear finite-difference P Es were found, e.g. through the representation theory of infinite-dimensional Lie algebras, [3], or through singular linear integral equations and connections with Bäcklund transformations, cf. [4,5]. 3.1 Quadrilateral P Es We will investigate here partial difference equations (P E s) of the following canonical form (which we will call quadrilateral P Es) f(u, ũ, û, ũ) = 0, (3.1.1) 44

2 3.1. QUADRILATERAL P ES 45 where we adopt the canonical notation of vertices surrounding an elementary plaquette on a rectangular lattice: u := u n,m, ũ = u n+1,m û := u n,m+1, ũ = un+1,m+1 Schematically, this configuration of points is given by: u ũ û ũ The notation is inspired by the one for Bäcklund transformations, which as we have seen in the previous Lecture, give rise to purely algebraic equation as a consequence of their permutability property. However, here we will forget, at first instance, about this connection, and consider lattice equations of the form (3.1.1) in their own right as a partial difference equation (P E) on a two-dimensional lattice. Thus, n and m, play the role of independent (discrete) variables, very much like x and t being the continuous variables for an equation like the KdV equation Even though the form (3.1.1) seems very restrictive, it will turn out that it is in a sense the most elementary form as a model type of equations, and on the other hand remarkably rich. In fact, as we shall see later, P Es of the form (3.1.1) are rich enough to be candidates for discretisations of PDEs of arbitrary order in the spatial and temporal variables. A classification of P Es in the same way as of PDE s, does not yet exist. Nonetheless, we may consider a P E of the form (3.1.1) to have features reminiscent of hyperbolic PDEs. In fact, if the equation f = 0 can be solved uniquely for each elementary quadrilateral, then one may pose initial value problems (IVPs) in ways very similar to hyperbolic type of equations such as the KdV equation itself (i.e. as a discrete nonlinear evolution equation). This can be done as follows. The naive approach would be to consider IVP where we would assign values of the dependenet variable u along horizontal array of vertices in the lattice, i.e. values for u n,0, for all n, considering the variable m to be the temporal discrete variable. It is easy to see, however, that such an IVP would lead to a nonlocal problem if we want to use (3.1.1) as an iteration scheme to find all values u n,m for m > 0. In fact, to calculate any value u n,1 say for given n, we would need to involve all initial values u n,0 with n < n, and furthermore have to assume limiting behaviour as n. This would be a complicated procedure. However, there is nothing that tells us that we should identify the n- and m axes as the spatial and temporal axes respectively. The lattice picture allows us to play other and more

3 46 LECTURE 3. INTEGRABILITY OF P ES natural games, and it is the equation itself that gives the lead in provding is with the natural IVP that we could impose. Thus, changing the perspective slightly, we may tilt the lattice and rather consider initial value data to be imposed on configurations like a sawtooth (or a ladder ), such as: The analogy with hyperbolic PDEs can be further seen from the consideration of the memory each point in the lattice has of the initial values that are involved in its determination upon iteration. In fact, any given point has a backward shadow (the analogue of the so-called lightcone ) of points the values of u on which determine the value at that point, as indicated by the picture: the point at the bottom of the cone being fully determined by the initial values at the top of the cone and only and exclusively by those values! This is very much reminiscent again of what happens in the case of hyperbolic PDEs. Multilinearity In order to have a unique iteration scheme arising from an equation of the form (3.1.1) the equation should be linear in each of the variables around the quadrilateral, i.e. the equation should multilinear. The most general form of a quadrilateral lattice equation which is linear in each dependent variable around the quadrilateral, and which in addition respects reversal symmetry with respect to the shifts - and on the lattice, takes the following form: ) ( ) k 0 uûũ ũ + k 1 (uûũ + uû ũ + uũ ũ + ûũ ũ + k 2 ûũ + u ũ ) ( ) ( +k 3 (uũ + û ũ + k 4 uû + ũ ũ + k 5 u + ũ + û + ũ ) + k 6 = 0, (3.1.2) where k 1,..., k 6 are coefficients (which may depend on additional parameters). Exercise Show that, starting from a multilinear quadrilateral lattice equation (3.1.1) with 16 general coefficients, we arrive at the form (3.1.2) by assuming that the equation remains unchanged when we reverse the - or shifts, i.e. when we replace ũ by ũ (by which we mean the backward shift related to the -shift, see picture), or when we replace û by ṷ (meaning the backward shift related to the -shift, see picture).

4 3.2. INTEGRABILITY AND CONSISTENCY-AROUND-THE-CUBE 47 ṷ ũ u ũ u û It is easy to see that the lattice equation (2.2.58) derived in Lecture 2 from the Bäcklund transform of KdV is precisely of the form given above. However, not all equations of the form (3.1.2) will be of interest to us. We will be interested in those equations (for specific choices of the coefficients k 1,...,k 6 ) which have the additional property of being integrable. What we mean by this will be explored in the next section. 3.2 Integrability and Consistency-around-the-Cube We will now consider a class of quadrilateral P Es (3.1.1) in which, apart from the independent discrete variables n, m on which the variable u n,m depends, there are parameters which we associate with these independent variables. We can think of these parameters as being the parameters which measure the width of the grid in the directions associated with n and m, and we refer to them as lattice parameters. Thus, denoting these parameters by p, q, the equations under question will take the form: f(u, û, ũ ũ; p, q) = 0, (3.2.1) and if we demand that the values for u at each vertex can be solved uniquely, implying multilinearity in each of these variables around the quadrilateral, then we are led again to eq. (3.1.2) with the coefficients k 0,..., k 6 depending on the lattice parameters p and q in a specific way. The question is what criteria to use in order to chose that dependence! It is here that we will restrict ourselves to quadrilateral P Es which we regard to be integrable. The question of what is the proper definition integrability, and to answer that question in general is very difficult: a one-fits-all definition (for all the possible type of systems that we would like to regard as being integrable) is possibly not possible to give in a precise mathematical sense. However, if we restrict ourselves here to P Es, and in particular quadrilateral P Es, then we can aspire to be a bit more precise. We will explore the definition of an integrable quadrilateral P E by means of the following example, namely the example of the P E arising from the BTs for the KdV equation, eq. (2.2.58). First, we remark that the presence of the lattice parameters p, q is crucial: whereas normally we would like to consider the parameters to be chosen once and for all and then remain fixed (thus specifying a specifi equation) when solving the equation on the lattice, here we will argue that we should look at (2.2.58) as defining a whole parameter-family of equations, and that it makes sense to look at them altogether with p and q variable. In

5 48 LECTURE 3. INTEGRABILITY OF P ES doing this, however, we must attach each parameter to a specific diecrete variable such as p being associated with the variable n, and q with m. This is also natural from the way in which we derived (2.2.58) from the BT construction: each BT is attached to a parameter (λ, µ,... ), and with each parameter we can build a direction in an infinite-dimensional lattice of BTs. The main point we want to make now is the following: Statement: the infinite parameter-family of P Es represented by the lattice equation, such as (2.2.58) is compatible, i.e. in each quadrilateral sublattice of the infinite-dimensional lattice we can consistently impose a copy of the P E in terms of the relevant discrete variable and associated with a corresponding lattice parameter. Let us illustrate this by means of the example of (2.2.58), in which we will identify (for later convenience) the parameters 4λ := p 2 and 4µ := q 2. What the statement above suggests is that we can embed the equation in a multi-dimensional lattice by conbsidering the dependent variable w, not only to depend on n and m (with associated lattice paameters p and q espectively), but that w may in fact depend on an infinity of lattice variables each associated with its parameter, as follows: w = w n,m,h,... = w(n, m, h,... ; p, q, r,...) and with each of these variables we have a corresponding elementary shift on the lattice: w := w n+1,m,h, ŵ := w n,m+1,h, w := w n,m,h+1... Rewriting the equation (2.2.58) in the form (2.2.50) with the substitutions for λ and µ investigate what happens if we impose a copy of the same equation in all three lattice directions. This would lead to the system of equations: (ŵ w)(w w) = p 2 q 2, (3.2.2a) (w w)(w w) = p 2 r 2, (3.2.2b) (ŵ w)(w ŵ) = r 2 q 2. (3.2.2c) These equations are consistent if the evaluations along the cube are independent of the way of calculating the final point. Imposing initial values: w := a, w =: b, ŵ =: c, w =: d

6 3.2. INTEGRABILITY AND CONSISTENCY-AROUND-THE-CUBE 49 w w w w ŵ w ŵ w In fact: w = (p2 q 2 ) wŵ + (q 2 r 2 )ŵw + (r 2 p 2 )w w (r 2 q 2 ) w + (q 2 p 2 )w + (p 2 r 2 )ŵ. (3.2.3) independent of the way in which he value at this vertex is calculated! This property, implying that under relevant initial value problems on the 3-dimensional lattice the iteration of the solution can be performed in an unambiguous way (namely independent of the way by which we perform the calculation of the iterates) will be referred to as the consistency-around-thecube (CAC) property. It is this property that we shall consider to be the main hallmarfk of the integrability of the equation. Remark: We note also, for later reference, that in the above case the formula for w is independent of the value w at the opposite end of the main diagonal across the cube. This property we will refer to as the tetrahedron property. Other examples of integrable quadrilateral P Es: There are many examples of integrable quadrilateral P Es that have been discovered over the years. A number of these belong to the lattice KdV-family of equations, they comprise the following cases: Lattice potential KdV equation: (p q + û ũ)(p + q + u ũ) = p 2 q 2, (3.2.4) and this is, in fact, equivalent to the equation (2.2.58) by the change of (dependent) variable w = u np mq c (c a constant with respect to n and m). We will show in Lecture 4 that this equation reduces to the potential KdV equation after a double continuum limit, so that we rightly regard it as a discretisation of the latter equation. Lattice potential MKdV equation: p(v v ṽ ṽ) = q(vṽ v ṽ) (3.2.5) Solutions of this equation are related to the solutions of the previous one (3.2.4) via the relations p q + û ũ = pṽ q v v, p + q + u ũ = pv + q ṽ ṽ, (3.2.6)

7 50 LECTURE 3. INTEGRABILITY OF P ES which constitute the analogues of the Miura transformation (2.2.42). Lattice SKdV equation: (z ẑ)( z z) p2 = (z z)(ẑ z) q 2 (3.2.7) which is the Schwarzian KdV equation. This equation is invariant under Möbius transformations: z Z = αz + β γz + δ. Solutions of eq. (3.2.7) are related to the ones of (3.2.5) via the relations: p(z z) = vṽ, q(z ẑ) = v v. (3.2.8) Exercise Show by explicit computation that the P Es given by (3.2.4)-(3.2.7) possess the consistency-around-the-cube property. Exercise Show that by using the relations (3.2.6) one can derive (3.2.4) from (3.2.5) and vice versa. 3.3 Lax pair for lattice potential KdV We shall next demonstrate how the CAC property explained in the previous section gives rise to the existence of a Lax pair, i.e. an overdetenrmined linear system of difference equations, the compatibility of which is verified iff the nonlinear lattice equation is satisfied. The idea is the following: Having verified the consistency of the equation around the cube, this tells us that we can add any lattice direction to the original lattice and impose simultaneously the equation in the three two-dimensional quadrilateral sublattices. The main idea is now to consider the additional lattice variable h Z associated with lattice parameter k as an auxiliary virtual variable, whilst acknowledging only the shifts in the orignal lattice variables n and m as the operations of interest. This then suggests that the shift in the third direction: w w should not appear in any of the equations, implying that wherever w appears we should treat it as a new dependent variable w := W. Proceeding in this way, we rewrite (3.2.2b) and (3.2.2c) as follows: (W w)( W w) = k 2 p 2 W = ww + (k2 p 2 ww) W w (W ŵ)(ŵ w) = k2 q 2 W = ww + (k2 p 2 ww) W w, (3.3.1a). (3.3.1b) Noting that eqs. (3.3.1) are both fractional linear in W, we can linearise these equations by the substitution: W = F G,

8 3.3. LAX PAIR FOR LATTICE POTENTIAL KDV 51 leading to F G = wf + (k2 p 2 w w)g, F wg F Ĝ = wf + (k2 q 2 wŵ)g, F ŵg and since at least one of the two functions F or G can be chosen freely, this allows us to split in each of these equations the numerator and denominator to give: { ( F = γ wf + [k 2 p 2 w w]g ) {, ( F = γ wf + [k respectively 2 q 2 wŵ]g ), G = γ (F wg), Ĝ = γ (F ŵg), (3.3.2) in which γ and γ are to be specified later. What happens next is obvious: we introduce the two-component vector ( ) F φ =, G and write (3.3.2) as a system of two 2 2 matrix equations: φ = Lφ, φ = Mφ, (3.3.3a) with the matrices ( w k L = γ 2 p 2 w w 1 w ), M = γ ( w k 2 q 2 wŵ 1 ŵ ) (3.3.3b) How does this linear system work? The consistency relation of the linear problem (3.3.3a) is obtained from the condition that φ = φ, which is the condition that simply expresses that φ must be a proper function of the lattice variables n and m. Calculating the left hand-side of this condition we get on the one hand φ = (Lφ) = L φ = LMφ, whereas the right hand-side would be calculated as φ = (Mφ) = M φ = MLφ. Equating both sides we see that a sufficient condition for the consistency is the matricial equation: LM = ML, (3.3.4) which we will loosely refer to as the Lax equation, but it is sometimes also referred to as a discrete zero-curvature condition (the reason for this terminology will be explained in (3.3.1)). Pictorially the Lax equation is illustrated by the following diagram

9 52 LECTURE 3. INTEGRABILITY OF P ES φ L φ M M φ L φ where the vectors φ are located at the vertices of the quadrilateral and in which the matrices L and M are attached to the edges linking the vertices. Using the explicit form (3.3.3b) of the matrices L and M, and working out the condition (3.3.4) we find ( ) ( ) γγ ŵ k 2 p 2 ŵ w w k 2 q 2 wŵ = 1 w 1 ŵ ( ) ( ) = γ w k γ 2 q 2 w w w k 2 p 2 w w 1 w 1 w and we will chose the γ and γ (which were unspecified so far) such that the relation for the determinants of this equation: γγ det( L) det(m) = γ γ det( M)det(L) (3.3.5) is trivially satisified. Since in this example the determinants of L and M are given by det(l) = p 2 k 2, det(m) = q 2 k 2, respectively, it is clear that the condition (3.3.5) is satisified by simply taking γ = γ = 1. But we will encounter other examples where a nontrivial choice of γ and γ is needed in order to satisfy the determinantal condition (3.3.5). Working out all the entries on both sides of the above matrix products it is straightforward to see that the (1,1)- and (2,2) entry of the matrix equation both yield the same condition on w, namely the equation (2.2.50)! Moreover, the (2,1) entry is trivially, and the (1,2) entry we don t even need to calculate, because having checked three of the entries to be satisfied the final entry must also be satisfied by virtue of the fact that we have aranged the determinantal condition (3.3.5) to be satisfied. In conclusion, we see thus that from the Lax equation (3.3.4) we recover the lattice equation (2.2.58). We observe furthermore, that albeit both Lax matrices L and M depend on the auxiliary variable k the final nonlinear equation for w does not depend on k! Exercise Suppose that the Lax matrices L and M can be expanded in a power series in a small parameter δ and ǫ respectively as follows: L = 1 + δl 1 +, M = 1 + ǫm 1 +,

10 3.4. CLASSIFICATION OF QUADRILATERAL P ES 53 and we expand the shifted variable by Taylor expansion L = L + ǫ t L +, M = L + δ x M +, by expansion we obtain in dominant order (namely terms proportional to ǫδ) the following matricial equation: t L 1 x M 1 + [ L 1, M 1 ] = 0, (3.3.6) where [ L, M ] = LM ML denotes the usual matrix commutator bracket. Eq. (3.3.6) arises also in differential geometry and it is from there that it has the interpretation as a zero-curvature condition of differential manifolds (with the relevant interpretation of the matrices L and M). 3.4 Classification of Quadrilateral P Es In a beautiful paper by V. Adler, A. Bobenko and Yu. Suris, cf. ref. [7], the problem of classifying all quadrilateral lattices which are integrable in the sense of the CAC property discussed in section 3.2 was considered. Since this is in our view an important result, it is useful to reproduce the whole list of resulting equations here. Theorem ([7]): Consider quadrilateral P Es of the general form: Q(u, ũ, û, ũ; p, q) = 0, using the notation indicated in the following diagram (renaming the lattice parameters in order to avoid confusion with previously used notation), u p ũ q q û p ũ subject to the following restrictions: a) Linearity: Q is multilinear in its arguments, i.e., it is linear in each vertex-variable u,ũ, û, ũ; b) Symmetry: Q is invariant under the group D 4 of symmetries of the square, generated by the interchangements: Q(u, ũ, û, ũ; p, q) = ±Q(u, û, ũ, ũ; q, p) = ±Q(ũ, u, ũ, û; p, q) ; (3.4.1) c) Tetrahedral Condition: in the consistency check, the evaluation of the point on the cube given by ũ is independent of u.

11 54 LECTURE 3. INTEGRABILITY OF P ES Q-list: Q 1 : p(u û)(ũ ũ) q(u ũ)(û ũ) = δ 2 pq(q p) (3.4.2a) Q 2 : p(u û)(ũ ũ) q(u ũ)(û ũ) + pq(p q)(u + ũ + û + ũ) = = pq(p q)(p 2 pq + q 2 ) (3.4.2b) Q 3 : p(1 q 2 )(uû + ũ ũ) q(1 p 2 )(uũ + û ũ) = ( = (p 2 q 2 ) (ûũ + u ũ) + δ 2 (1 p2 )(1 q 2 ) ) 4pq Q 4 : p(uũ + û ũ) q(uû + ũ ũ) = pq qp ( ) = 1 p 2 q 2 (ûũ + u ũ) pq(1 + uũû ũ) where P 2 = p 4 γp 2 + 1, Q 2 = q 4 γq H-list: (3.4.2c) (3.4.2d) H 1 : (u ũ)(û ũ) = p 2 q 2 (3.4.3a) H 2 : (u ũ)(ũ û) = (p q)(u + ũ + û + ũ) + p 2 q 2 (3.4.3b) H 3 : p(uũ + û ũ) q(uû + ũ ũ) = δ 2 (p 2 q 2 ) (3.4.3c) A-list: A 1 : p (u + û)(ũ + ũ) q(u + ũ)(û + ũ) = δ 2 pq(p 2 q 2 ) (3.4.4a) A 2 : p(1 q 2 )(uû + ũ ũ) q(1 p 2 )(uũ + û ũ) + (p 2 q 2 )(1 + uũû ũ) = 0 (3.4.4b) The Q-equations are related through the following coalescence diagram, i.e. displaying how the equations relate to eachother through certain limits on the parameters: Q 3 (Q 3 ) δ=0 Q 4 Q 1 Q 2 (Q 1 ) δ=0 whilst the H-equations and A-equations are related through Miura type Bäcklund transforms, (J. Atkinson, 2008).

12 3.5. LATTICE KDV EQUATION 55 Remarks: It is in itself remarkable that the list of equations found is so short. In fact, all equations in the list Q are in fact special subcases of the last equation, Q4, which was discovered by V. Adler in The latter equation can also be expressed as: [ ] A[(u b)(û b) (a b)(c b)] +B [(u a)(ũ a) (b a)(c a)] (ũ b)( ũ b) (a b)(c b) [ (û a)( ũ a) (b a)(c a) + ] = = ABC(a b) (3.4.5) cf. [9]. Here the extra parameters c, C are related through C 2 = 4c 3 g 2 c g 3 and to (a, A) and (b, B) through the relations: A(c b) + B(c a) = C(a b), a + b + c = 1 ( ) 2 A + B. 4 a b A third alternative form of Adler s equation reads as follows: ( ) ( ) ( ) sn(α) vṽ + v ṽ sn(β) v v + ṽ ṽ sn(α β) ṽ v + v ṽ ( ) +ksn(α)sn(β)sn(α β) 1 + vṽ v ṽ = 0, (3.4.6) (J. Hietarinta, 2005) which is somewhat simpler. In (3.4.6) the sn denote the Jacobi elliptic functions, cf. Appendix A, of modulus k, i.e. sn(α) = sn(α; k), etc. It should be noted that all previous examples presented in section 3.2 can be recognised as special subcases of the equations in the lists Q,H and A, and which can be obtained by degeneration of Adler s equation in either of its forms. All these equations possess Lax pairs, but it was only recently that explicit solutions were found for most of the new cases including of Adler s equation, cf. e.g. [12]. Adler s equation is the discrete analogue of a famous soliton equation discovered in 1980 by I. Krichever and S. Novikov, which reads: u t = u xxx 3 u 2 xx (4u3 g 2 u g 3 ), (3.4.7) 2 u x and which generalises the Schwarzian KdV equation. 3.5 Lattice KdV Equation We finish this Lecture by deriving the lattice analogue of the KdV equation, in contrast to the lattice potential KdV equation (2.2.58) (or equivalently (3.2.4)): (w n,m w n+1,m+1 )(w n,m+1 w n+1,m ) = p 2 q 2. (3.5.1) We recall that the difference between the continuous KdV equation (2.2.38) and its potential version (2.2.46), as discussed in Lecture 2, was simply that its solutions were connected

13 56 LECTURE 3. INTEGRABILITY OF P ES through taking a derivative, u = w x. However, on the lattice there are many ways in which can do the analogue of taking the derivative, for instance we can replace it by taking a difference involving two neighbouring vertices, like n w n,m w n+1,m w n,m or m w n,m w n,m+1 w n,m, but these are not the only choices. Alternatively we can take a difference between vertices farther away or across diagonals, such as: w n,m = w n+1,m+1 w n,m or w n,m = w n,m+1 w n+1,m, or a host of other choices. As long as a well-chosen continuum limit reduces these to a derivative they can be justifiably be considered to be the discrete analogues of the operation of taking a partial derivative. The most sensible way to make a choice is to look at the equation at hand and then decide what would be for the given equation the most natural choice to apply in that case. In the case of the lattive potential KdV equation (3.5.1) the natural choice of the discrete analogue of the derivative seems to be either one of the choices given above, namely a difference across the diagonal. This suggest that as lattice KdV variables we would take either one of the two choices: Q = w w or R = ŵ w. (3.5.2) It is then straightforward from (3.2.4) (using the notation in terms of - and shifts) to derive the following equation for Q: Q Q = a Q a Q R R = a R ã R, (3.5.3) where a p 2 q 2. The equations for Q and R are simply related by the fact that QR = a. A Lax pair for either of the equations (3.5.3) can be easily derived, by starting from the Lax pair for the potential KdV equation. In fact, rather than writing it as a first order matricial system, one may derive from the Lax pair a three-point linear equation in terms of one of the components of the vector φ on which the Lax matrices act. In this way we obtain the following scalar Lax pair: remembering that QR = a. ϕ = Q ϕ + Λ ϕ, (3.5.4a) ϕ = ϕ + R ϕ, (3.5.4b) Exercise Derive the Lax pair (3.5.4) from the set of equations (3.3.2) (with γ = γ = 1) by setting ϕ = G and identifying the spectral variable by Λ = k 2 q 2. Exercise Show that the consistency condition of the Lax pair (3.5.4) viewed as an overdetermined discrete linear system, leads to the lattice KdV equation (3.5.3). 3.6 Singularity confinement In the seminal paper of B. Grammaticos et al. another point of view on integrability was developed than the one discussed in section 3.2, namely based on the issue of well-posedness

14 3.6. SINGULARITY CONFINEMENT 57 of initial value problems on the lattice. It is based on a phenomenon happening in integrable discrete systems, which nowadays is called singularity confinement and which describes what happens with singularities of the solutions of discrete system in the space of initial values. Proposition: In an integrable lattice equation of KdV type singularities induced by initial data do not propagate. As an example let us consider the lattice KdV equation (3.5.3), (setting for convenience the parameter a = 1 w.l.o.g.) R n+1,m+1 R n,m = 1 1. R n+1,m R n,m+1 We will now study how the the initial data progresses, when one hits a singularity. The initial data is given at the solid line on Figure 3.1, a, b, 0, c, d. When one proceeds from these initial values one obtains infinity at two places, then one 0 at the next level and finally two ambiguities of the type. A more detailed analysis with the initial value 0 1 = ε (small) yields the following values at the subsequent iterations at the first stage, and on the next 1 = b + 1 ε 1 a, 2 = c + 1 d 1 ε, s = a f, t = d + 1 g 1 2, 0 2 = ε ( = ε + b c a 1 ) ε d b c a 0 1 d f 1 2 g s 0 2? 1? 2 t Figure 3.1: Propagation of singularities in a 2D map. Here n grows in the SE direction and m in the SW direction.

15 58 LECTURE 3. INTEGRABILITY OF P ES Then at the next step we can resolve the ambiguities:? 1 = s = c + 1 d 1 a 1/f + O(ε)? 2 = t = b 1 a + 1 d + 1/g + O(ε) Thus the singularity is confined. Note that if the lattice equation is deformed, e.g. by taking: R n+1,m+1 R n,m = 1 R n+1,m λ R n,m+1 with λ 1, the above fine cancellation would no longer happen, and singularities would again occur at? 1 and? 2, and would persist throughout! In that case the singularities are no longer confined to a finite number of iteration steps, and we conclude that the corresponding map is not integrable. Remark: Another point of view on 2D singularity confinement is provided by the requirement of ultra-local singularity confinement (R. Sahadevan and H. Capel, 2003): One only assumes a singularity at one point and requires regularity at all other points. Referring to Figure 3.2 we have initial values are at black disks, the values at open circles are determined from them. The initial values can be chosen so that u 11 =, and the requirement is that u 12, u 21 are finite and ambiguity at u 22 can be resolved using ǫ-analysis. For example for the lattice potential KdV in the form the initial value u 01 = u 10 implies u n+1,m+1 = u n,m 1 u n,m+1 u n+1,m, u 11 =,, u 12 = u 01, u 21 = u 10,, u 22 =? u 02 u 12 u 22 u 01 u 11 u 21 u 00 u 10 u 20 Figure 3.2: Setting for ultra-local singularity confinement.

16 3.6. SINGULARITY CONFINEMENT 59 and a detailed analysis reveals that if u 10 = u 01 + ǫ then u 11 = 1 ǫ + u 00, u 12 = u 01 + ǫ + ǫ 2 (u 02 u 00 ) + O(ǫ 3 ), u 21 = u ǫ 2 ( u 20 + u 00 ) + O(ǫ 3 ), u 22 = 1 u 11 u 12 u 21 = 1 ǫ + u 1 00 ǫ + ǫ 2 (u 02 + u 02 2u 00 ) + O(ǫ 3 ) = u 02 + u 20 u 00 + O(ǫ) This resolves the singularity, and recovers initial value u 00 which was temporarily submerged to order ǫ 2. Literature 1. M.J. Ablowitz and F.J. Ladik, A nonlinear difference scheme and inverse scattering, Stud. Appl. Math. 55 (1976) ; On the solution of a class of nonlinear partial difference equations, ibid. 57 (1977) R. Hirota, Nonlinear partial difference equations I-III, J. Phys. Soc. Japan 43 (1977), , E. Date, M. Jimbo and T. Miwa, Method for generating discrete soliton equations I-V, J. Phys. Soc. Japan 51 (1982) , 52 (1983) , B. Grammaticos, A. Ramani and V. Papageorgiou, Do integrable mappings have the Painlevé property?, Phys. Rev. Lett. bf 67 (1991) F.W. Nijhoff, G.R.W. Quispel and H.W. Capel, Direct linearization of nonlinear differencedifference equations, Phys. Lett. 97A (1983) ; G.R.W. Quispel, F.W. Nijhoff, H.W. Capel and J. van der Linden, Linear integral equations and nonliner difference-difference equations, Physica 125A (1984) F.W. Nijhoff and H.W. Capel, The discrete Korteweg-de Vries equation, Acta Applicandae Mathematicae 39 (1995) F.W. Nijhoff and A.J. Walker, The discrete and continuous Painlevé VI hierarchy and the Garnier systems, Glasgow Math. J. 43A (2001) V.E. Adler, A.I. Bobenko and Yu.B. Suris, Classification of integrable equations on quadgraphs, Commun. Math. Phys. 233 (2003) F.W. Nijhoff, Lax pair for the Adler (lattice Krichever-Novikov) system, Phys. Lett. 297A (2002), V.E. Adler and Yu.B. Suris, Q4, Intl. Math. Res. Notes (2004). 11. J. Hietarinta, Searching for CAC-maps, J. Nonlin. Math. Phys. 12 Suppl. 2 (2005) J. Atkinson, J. Hietarinta and F.W. Nijhoff, Seed and soliton solutions for Adler s lattice equation, J. Phys A: Math. Theor. (40 (2007) F1-8.