Full set of lecture notes - School of Mathematics - University of Leeds

1

MATH5492:

ADVANCED DISCRETE SYSTEMS ANDINTEGRABILITY

Frank W NIJHOFF

University of Leeds, School of Mathematics

0

Aknowledgement:

These lecture notes are an adaptation of a text which is intended as a monograph by J.Hietarinta, N. Joshi and F.W. Nijhoff on the same subject.

Lecture 1

Multi-Soliton Solutions

In this lecture we shall construct a particular important class of solutions of the latticeequations that we have encountered and studied in the previous lectures. Much like PDEs,lattice equations, i.e. partial difference equations defined on a grid of points rather than acontinuum space-time, generically admit many solutions. We will encounter different typesof solutions lateron. Among these, the so-called soliton solutions stand out as particularlyimportant, not only because of their intrinsic properties, which make them very compellingfrom a physical point of view, but also because the underlying structure behind these solu-tions is very revealing for the mechanisms at work behind such integrable systems.

Soliton solutions were first found for the famous continuous paradigms of “soliton sys-tems”, such as the nonlinear evolution equations (NLEEs) that we have already met: theKorteweg-de-Vries equation, the nonlinear Schrodinger equation, the sine-Gordon equationand many others. The history behind the discovery of such systems and the solutions thatwe identify with solitons has been cited in many textbooks (cf. e.g. [G.L. Lamb, 1980; M.Ablowitz and H. Segur, 1982; A.C. Newell, 1983; P.G. Drazin, 1989; M.J. Ablowitz and P.A.Clarkson, 1991]), going back to the famous history of John Scott Russell chasing in 1834 onhis horse a real-life soliton (the ”great wave of translation” as he called it) in a canal nearEdinburgh. We will not repeat this somewhat ad nauseam recited story here, nor go intothe controversy that such solutions caused amongst people who studies water wave models(in any of the textbooks cited above you can find a full account of the story). Nonetheless,what we will observe in what follows, is how similar from a mathematical perspective thesolutions are comparing the discrete solitons with their continuous counterparts.

In this Lecture, starting from an Ansatz, we will develop a recursive structure whichat first might seems to be far-fetched. Soon, however, we will apprciate the merits of thisstructure, since not only will it provide us with an infinite class of solutions of the latticeequations, but it will also provide us with various inter-relations between different cognateequations. thus “killing two birds with one stroke”. In a subsequent lecture, we will usethe same basic recurrence relations again, when we generalise the results to the case ofmulti-dimensional equations.

1

2 LECTURE 1. MULTI-SOLITON SOLUTIONS

1.1 Infinite Recurrence Structure

We will start by introducing out of the blue the following matrix:

M = (Mi,j)i,j=1,...,N , Mi,j ≡ρicjki + kj

, (1.1.1)

which will form the core of the structure which we will develop. In (1.1.1) the ci, ki,(i = 1, . . . , N) denote two sets of N nonvanishing parameters, which we may chose at ourliking (apart that we have to assume that ki + kj 6= 0, ∀i, j = 1, . . . , N , in order to avoiddifficulties with the numerators in the matrix M ). These parameters are supposed to beconstant, i.e. they do not depend on the dynamical variables of the lattice (i.e. the discreteindependent variables n,m, . . . ). The dependence on the latter are incorporated entirely inthe functions ρi, which we take of the form:

ρi =

(p+ kip− ki

)n(q + kiq − ki

)m

ρ0i , (1.1.2)

where the ρ0i is another set of N constants to be freely chosen. We refer to the ρi = ρi(n,m)as discrete plane-wave factors.

Let us now introduce for convenience the following notation. Let K denote the diagonalN ×N matrix containing the parameters ki on the diagonal, whilst we introduce a columnvector r containg the entries ρi and the row vector tc containing the entries ci, i.e.

K =

k1k2

. . .

kN

, r =

ρ1ρ2...ρN

, tc = (c1, c2, · · · , cN ) . (1.1.3)

It is easily checked that we have from the definition (1.1.1) immediately the relation:

MK +KM = r tc , (1.1.4)

where significantly the dyadic on the right hand side is a matrix of rank 1.

We will introduce several objects in terms of the matrix M , but the first one we wouldlike to mention is a scalar function of the dynamical variables, which lateron (see e.g. Lecture6) will be called the τ -function1, given in this context by:

τ = τn,m = det (1+M) , (1.1.5)

where 1 is the N ×N unit matrix. We will postpone the properties of the τ -function (1.1.5)until the next Lecture, where we will discuss this issue at length. Here we will use the matrixM first for another purpose.

1The terminology is due to a series of significant papers, (cf. e.g. [T. Miwa,M. Jimbo,E. Date,2000]), inthe early eightees by the Kyoto school of M.Sato and his students, M. Jimbo, T. Miwa, Kashiwara, Date andothers. The so-called τ -function figures already in the work by R. Hirota fomr the early seventies onward.

1.1. INFINITE RECURRENCE STRUCTURE 3

Let us investigate the dynamical propoerties of the matrix M , i.e. the behaviour of Mas we vary (in a discrete fashion) the lattice variables n and m. This follows immediatelyfrom the form of the plane-wave factors (1.1.2), for which we have:

ρi = ρi(n+ 1,m) =p+ kip− ki

ρi ,

and a consequence of the following simple calculation:

Mi,j =ρicj

ki + kj=p+ kip− ki

ρicjki + kj

=

(1

p− ki+

1

ki + kj

)p+ kip+ kj

ρicj

=1

p+ kjρicj +

p+ kip+ kj

Mi,j

⇒ Mi,j(p+ kj) = ρicj + (p+ ki)Mi,j ,

which leads to the matrix relation:

M (p+K)− (p+K)M = r tc , (1.1.6a)

where p +K is short-hand for the diagonal matrix p1 +K (everywhere where we add aconstant multiple of the unit matrix to another matrix, we will conveniently omit the unitmatrix symbol, which is understood to be there).

Conversely, we can do the calculation the other way around, namely by writing:

Mi,j =p− kip+ ki

ρicjki + kj

=

(− 1

p+ ki+

1

ki + kj

)p− kip− kj

ρicj

= − 1

p− kjρicj +

p− kip− kj

Mi,j

⇒ Mi,j(p− kj) = −ρicj + (p− ki)Mi,j ,

leading to the matricial equation:

(p−K)M −M (p−K) = r tc . (1.1.6b)

Obviously, similar relations can be derived in quite thesame fashion for the dynamics interms of the other variable m, basically by interchanging the roles of the parameters p and qand of the variables n and m. Thus, as before denoting the lattice shift in m by the symbol , and using

ρi = ρi(n,m+ 1) =q + kiq − ki

ρi ,

we obtain subsequently

M (q +K)− (q +K)M = r tc , (1.1.6c)

(q −K)M −M (q −K) = r tc . (1.1.6d)


Eqs. (1.1.6a)-(1.1.6d) encode all the information on the dynamics of M , together with eq.(2.3.3) which is basically equivalent to the definition (1.1.1).

Let us now introduce the following quantities:

ui = (1+M)−1Ki r (1.1.7a)

tuj = tcKj (1+M )−1

(1.1.7b)

Ui,j = tcKj (1+M )−1Ki r , (1.1.7c)

for i, j integers (which, if none of the parameters ki is zero can be negative as well). Thus, weobtain an infinite sequence of column vectors ui, of row vectors tuj and a infnite by infinitearray of scalar quantities Ui,j . Furthermore, it is easy to show that the matrix 1+M (seeLecture 6, where we will give an explicit formula for det(1+M)), and hence all quantities(1.1.7)) are well-defined. It is these quantities that play the main role in the constructionthat follows.

We shall now derive, starting from the relations (1.1.6) a system of recurrence relationswhich describe the dynamics for the quantities (1.1.7). Once this recursive structure is inplace, we will then single out specific choices among the latter for which we can derive closedform discrete equations.

First, let us do some simple steps. From the definition (1.1.7a), using the fact that K isa diagonal matrix, we have:

Ki r = (1+M)ui ⇒ Ki r = (1+ M ) ui

⇒ Ki p+K

p−K r = (1+ M) ui

⇒ Ki (p+K) r = (p−K) (1+ M) ui

= (p−K) ui + (p−K) M ui = (p−K) ui +M (p−K) ui + rtcui

where in the last step use has ben made of (1.1.6b). Noting that from (1.1.7c) we have:

Ui,j =tcKj ui =

tujKi r , (1.1.8)

we concluse thatpKj r +Kj+1 r = (1+M) (p−K) ui + Ui,0r .

Multiplying both sides by the inverse matrix (1+M)−1 and identifying the terms on theleft hand side using ((1.1.7a), we thus obtain:

(p−K) ui = (1+M)−1[pKi r +Ki+1 r − Ui,0r

]

= pui + ui+1 − Ui,0u0 . (1.1.9)

Thus, we have obtained a linear recursion relation between the objects ui with the objectsUi,j acting as coefficients. In quite a similar fashion we can derive the relation

(p+K)ui = pui − ui+1 + Ui,0u0 , (1.1.10)

1.1. INFINITE RECURRENCE STRUCTURE 5

in fact by making use of (1.1.6a) in this case. Eq. (1.1.10) can be thought of as an inverserelation to (2.3.3), noting that the ˜-shifted objects are now at the right hand side of theequation. Multiplying both sides of either (2.3.3) or (1.1.6a) from the left by the row vectortcKj and identifying the resulting terms through (1.1.7c), we obtain now a relation purelyin terms of the objects Ui,j , namely:

tcKj (p−K) ui =tcKj

[pui + ui+1 − Ui,0u0

]

⇒ pUi,j − Ui,j+1 = pUi,j + Ui+1,j − Ui,0U0,j (1.1.11)

noting that aUi,0 is just a scalar factor which can be moved to the left of the matrix multi-plication. Thus, we have now obtained a nonlinear recursion relation between the Ui,j andits ˜-shifted counterparts.

In a similar fashion, multiplying eq. (1.1.10) by the row vector tcKj we obtain thecomplementary relation:

pUi,j + Ui,j+1 = pUi,j − Ui+1,j + Ui,0U0,j , (1.1.12)

however this relation can be obtained from the previous relation (2.1.13a) by transpositionprovided we assume that:

Ui,j = Uj,i . (1.1.13)

The latter is, however, not an assumption: from the definition (1.1.7c) one can show easilyshow that the symmetry under exchange of indices holds true, in spite of the fact that theformula for Ui,j does not look at first sight symmetric.

Exercise 1.1.1. Prove the relation (1.1.13) from the definition (1.1.7c).

Exercise 1.1.2. Combine both eqs. (1.1.9) and (1.1.10), using also (2.1.13b), to derive thefollowing algebraic recurrence relation:

K2ui = ui+2 + Ui,1 u0 − Ui,0 u1 . (1.1.14)

Exercise 1.1.3. Show that the latter relation (1.1.14), by using (1.1.8), gives rise to analgebraic recurrence for the objects Ui,j, namely

Ui,j+2 = Ui+2,j + Ui,1 U0,j − Ui,0 U1,j . (1.1.15)

To summarise, starting from a simple matrixM given in (1.1.1) depending dynamicallyon the lattice variables through the plane-wave factors i given in (1.1.2), we have definedan infinite set of objects, namely column and row vectors ui and

tui, and a doubly infinitesequence of scalar functions Ui,j , all related through a system of dynamical (since it involveslattice shift) of recurrence relations. Obviously, all relations that we have derived for the˜-shifts (involving lattice parameter p and lattice variable n) hold also for the -shifts,simply by replacing p by q and interchanging the roles of n and m. Thus, for the scalarobjects Ui,j we have the following set of coupled recurrence relations:

pUi,j − Ui,j+1 = pUi,j + Ui+1,j − Ui,0U0,j , (1.1.16a)

pUi,j + Ui,j+1 = pUi,j − Ui+1,j + Ui,0U0,j , (1.1.16b)

qUi,j − Ui,j+1 = qUi,j + Ui+1,j − Ui,0U0,j , (1.1.16c)

qUi,j + Ui,j+1 = qUi,j − Ui+1,j + Ui,0U0,j . (1.1.16d)


1.2 Closed Form Lattice Equations

Now that we have the full set of rcurrence relations (2.3.8), the next step is derive closed formequations from this system for inividual elements. Let us first, as a warming-up exercise,derive an equations for the variable U0,0. In fact, subtracting (2.3.9b) from (1.1.16a) weobtain

pUi,j − qUi,j − Ui,j+1 + Ui,j+1 = (p− q)Ui,j − (Ui,0 − Ui,0)U0.j . (1.2.1)

On the other hand, taking the -shift of (2.3.9a), let us refer to it as (2.3.9a), and subtracting

from it the ˜-shift of (1.1.16d), i.e. ˜(1.1.16d), we obtain

pUi,j − qUi,j + Ui,j+1 − Ui,j+1 = (p− q)U i,j + (Ui,0 − Ui,0)

U0.j . (1.2.2)

Combining both equations, the terms which have a shift in their second index drop out andwe obtain the equation:

(p+ q)(Ui,j − Ui,j) = (p− q)(Ui,j − U i,j) + (Ui,0 − UI,0)(U0,j − U0,j) . (1.2.3)

Setting now i = j = 0 in the last formula, we see that we get a closed form equations interms of U0,0 which we call u for convenience. This yields, after some trivial algebra theequation:

(p+ q + u− u)(p− q + u− u) = p2 − q2 , (1.2.4)

which is our old friend the lattice potential KdV equation, which we have encountered inLecture 2. So, what we have just shown is that the earlier defined object

u = U0,0 = tc (1+M)−1 r

obeys the lattice equation (1.2.4). This constitutes, in fact, an infinite family of exactsolutions of that nonlinear P∆E with a huge amount of freedom in choosing parameters ofthe solution (namely for fixed integer N , all the ki, ci,ρ

0i , i = 1, . . . , N). The corresponding

solutions are called the N -soliton solutions of the lattice potential KdV equation.However, we can do more. In fact, instead of singling out u = U0,0, we can chose other el-

ements among the Ui,j , or (linear) combinations of them, and then systematically investigatewhat equations these choices satisfy by exploring the system of recurrence relations (2.3.8).For example, from (1.2.1) taking i = 0, j = −1 and introducing the variable v ≡ 1−U0,−1 itis a simple exercise to obtain the following relation:

p− q + u− u =pv − qv

v. (1.2.5)

Alternatively, adding the -shift of (1.1.16a) to (1.1.16d) we get

pU i,j + qUi,j −

U i,j+1 + Ui,j+1 = (p+ q)Ui,j + (Ui,0 − U i,0)U0.j , (1.2.6)

and when taking in (1.2.6) i = 0, j = −1 an easy calculation yields:

p+ q + u− u =pv + qv

v. (1.2.7)

1.3. DERIVATION OF LAX PAIRS 7

Clearly, in (1.2.7), interchanging p and q and the ˜-shift and the -shift should not make adifference, since the left-hand side is invariant under this change. Thus, the right-hand sidemust be invariant as well, leading to the relation:

p(vv − vv

)= q

(vv − vv

). (1.2.8)

Eq. (1.2.8) is P∆E in its own right for the quantity v, for which, by construction, we havean infinite family of solutions, namely given by

v = 1− U0,−1 = 1− tcK−1 (1+M)−1 r .

The P∆E (1.2.8) for the variable v is recognised as being the lattice potential MKdVequation, which we have also encountered before in Lecture 2. Here, we observe how thesetwo variable u solving the lattice potential KdV, and v the lattice potential MKdV, areconnected through the relations (1.2.5) and (1.2.7), which form the discrete analogues of theMiura transformation.

Another choice we can consider is the variable U−1,−1, i.e. we can consider (2.3.8) fori = j = −1, leading to

p(U−1,−1 + U−1,−1

)= 1−

(1− U−1,0

)(1− U0,−1) ,

and a similar relation for p replaced by q and the ˜-shift replaced by the -. Using the factthat U−1,0 = U0,−1 = 1 − v and introducing the abbreviation z = U−1,−1 − n

p − mq , the

latter relations reduce to:

p(z − z) = vv , q(z − z) = vv . (1.2.9)

On the one hand, these two equations lead back to the equation (1.2.8) by eliminating thevariable z (considering in addition the ˜- and -shifts of the two relations). On the otherhand, by eliminating the variable v we obtain yet again a P∆E , but now for z which reads:

(z − z)(z − z)(z − z)(z − z)

=q2

p2, (1.2.10)

which we identify with the Schwarzian lattice KdV equation (or cross-ratio equation), alsoencountered before in Lecture 2. Thus, we observe that through the recurrence structureencoded in eqs. (2.3.8) we obtain solutions of various different P∆E s in one stroke. Thus,a infinite family of solutions of eq. eq:dSKdV is given by the formula:

z = z0 +n

p+m

q+ tcK−1 (1+M )−1K−1 r ,

in which z0 is an arbitrary constant.

1.3 Derivation of Lax Pairs

We will now show how the infinte recurrence structure we have developed in section 5.1 canalso be used to systematically derive the Lax pairs for the lattice equations that emerge


from the system. For this we exploit the linear recurrence relations (1.1.9), (1.1.10) for thevectors ui, as well their couterparts involving the -shift (with lattoce parameter q). Thus,recalling the set of linear equations:

(p−K)ui = pui + ui+1 − Ui,0u0 , (1.3.1a)

(p+K)ui = pui − ui+1 + Ui,0u0 , (1.3.1b)

in the spirit of section 5.2 we are going to select special choices of these vectors and derivea closed system of linear relations for them.

As a first example let us select the vectors u0 and u1, and write eqs. (2.3.7) settingi = 0. From (2.3.7a) we immediately get:

(p−K)u0 = pu0 + u1 − U0,0u0 = (p− u)u0 + u1 ,

recalling that U0,0 = u, which is already in the form we want. On the other hand, from(2.3.10a) we obtain:

(p+K)u0 = pu0 − u1 + U0,0 u0 = (p+ u)u0 − u1

⇒ (p2 −K2)u0 = (p+ u) [(p− u)u0 + u1]− (p−K)u1 .

Solving (p−K)u1 from the last relation, we can now write a system of two coupled linearequations giving the discrete evolution of the vectors u0 and u1. Introducing the 2-vector

φ =

(u0

u1

)

(which, in fact, is a 2N -component vector, consisiting of two N -component parts), we cancast the two coupled relations in the following 2×2 block matricial form:

(p−K)φ =

(p− u , 1

K2 − p2 + (p− u)(p+ u) , p+ u

)φ . (1.3.2a)

Obviously, by a similar derivation, we obtain the following matricial equation involving the-shift:

(q −K)φ =

(q − u , 1

K2 − q2 + (q − u)(q + u) , q + u

)φ , (1.3.2b)

which must hold simultaneously. It is easy now to show, by direct computation, that thepair consisting of (1.3.2a) and (1.3.2b) forms a Lax pair for the lattice equation (1.2.4). Infact, in spite of the fact that in principle eqs. (1.3.2) involve 2N×2N matrices, all coefficientsinvolve only scalars (i.e. a scalar quantity multiplying a N×N unit matrix) or the diagonalmatrix K. Thus, this 2N×2N matrix system can be interpreted as a family of N decoupled2×2 matricial systems, acting each on a 2-component vector

φi =

((u0)i(u1)i

), i = 1, . . . , N ,

where (uj)i is the ith component of the N -component vector uj . In each of these equations,

the diagonal matrix is then simpley replaced by its ith entry, ki, and this parameter plays


the role of the spectral parameter in the Lax pair. Thus, each of these N matricial systemstakes the form:

(p− k)φ = L(k)φ , (q − k)φ =M(k)φ , (1.3.3)

with the Lax matrices:

L(k) =

(p− u , 1

k2 − p2 + (p− u)(p+ u) , p+ u

),

M(k) =

(q − u , 1

k2 − q2 + (q − u)(q + u) , q + u

),

and it is straightforward computation along the lines discussed already in Lecture 3 to showthat the resulting Lax equations

L(k)M(k) = M(k)L(k) ,

is satisified iff the quantity u in the entries of the matrix satisfies the nonlinar P∆E (1.2.4),and this holds for any value of the spectral parameter k.

A second example is the derivation of the Lax pair for the lattice potential MKdVequation (1.2.8), which proceeds along similar lines, but with a slightly different choice ofvectors from among the ui. In this case we select the vectors u0 (again) and u−1 instead ofu1. Setting i = −1 in (2.3.7a) we now obtain on the one hand:

(p−K)u−1 = pu−1 + u0 − U−1,0u0 = pu−1 + vu0 ,

recalling that 1− U−1,0 = 1− U0,−1 = v . On the other hand from (2.3.10a) we get

(p+K)u−1 = pu−1 − u0 + U−1,0 u0 = pu−1 − vu0

⇒ (p2 −K2)u−1 = p [pu−1 + vu0]− (p−K)vu0 .

Solving (p−K)u0 from the last relation we get a closed linear system of equstions describingthe dynamics w.r.t. the ˜-shift for u0 and u−1. Introducing now the 2N-component vector

ψ =

(u−1

u0

)

we can cast again this system in a 2N×2N matricial form, namely as follows:

(p−K)ψ =

(p , vK2

v , p vv

)ψ . (1.3.4a)

accompanied, obviously, by the analogous equation for the -shift:

(q −K)ψ =

(q , v

K2

v , q vv

)ψ . (1.3.4b)

As in the previous case, this 2N×2N matricial system decouples into N similarly looking2×2 matricial systems for each of the components of the vectors u0 and u−1, replacing Kby ki and ψ by ψi = ((u−1)i , (u0)i)

t , each of which will take the form:

(p− k)ψ = L(k)ψ , (q − k)ψ = M(k)ψ , (1.3.5)


with Lax matrices:

L(k) =

(p , vk2v , p vv

),

M(k) =

(q , vk2v , q vv

),

and as before the Lax equations

L(k)M(k) = M(k)L(k) ,

yields the required equatio,n (1.2.8) in this case, as consistency condition.As a byproduct, we also obtain the connection between the two Lax pairs (1.3.3) and

(1.3.5), namely by exploiting the algebraic recurrence (1.1.14) (which does not involve thediscrete dynamics). In fact, setting i = −1 in the latter relation we obtain:

K2u−1 = u1 + U−1,1u0 − U−1,0u1 = vu1 + U−1,1u0 ,

where the new object U−1,1 obeys dynamical relations that can be obtained by settingi = −1, j = 0 directly in eqs. (2.3.8). Thus, we can relate the two Lax pair through the2N×2N matricial relation

K2ψ =

(U−1,1 v

K2 0

)φ . (1.3.6)

Eq. :Matgauge again decouples in a set of N matricial 2×2 relations of the form

k2ψ = Uφ , U(k) ≡(U−1,1 vk2 0

), (1.3.7)

which is called a gauge transformation, implying that the Lax matrices for the two Lax pairsare related through the formulae:

U(k)L(k) = L(k)U(k) , U(k)M(k) = M(k)U(k) . (1.3.8)

Finally, let us observe that from the Lax pair (1.3.4) one can easily obtain a Lax pairfor the lattice Schwarzian KdV equation, namely by expliting the relations (1.2.9). In fact,multiplying the vector ψ from the left by the diagonal matrix (1, 1/v) , we obtain

(p− k)

(1 00 1/v

)ψ =

(p vvk2

vvp

) (1 00 1/v

)ψ .

Thus, using (1.2.9) and calling the vector (1, 1/v)ψ := χ we get a 2×2 matricial equationwhich can be expressed entirely in terms of z and z, nemaly:

(1− k

p)χ =

(1 p(z − z)

k2/p2

z − z1

)χ . (1.3.9)

It is an easy exercise to show that the consistency relation between (1.3.9) and its counterpartfor the -shift (making the usual replacements) gives rise the lattice equation (1.2.10) fo thevariable z.


Literature

1. G.L. Lamb, Elements of Soliton Theory, (Wiley Interscience, 1980).

2. M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, (SIAM,1982).

3. R.K. Dodd, J.C. Eilbeck, J.D. Gibbon and H.C. Morris, Solitons and Nonlinear WaveEquations, (Academic Press, 1982).

4. A.C. Newell, Solitons in Mathematics and Physics, (SIAM, 1983).

5. G.R.W. Quispel, F.W. Nijhoff, H.W. Capel and J. van der Linden, On some linear in-tegral equations generating solutions of nonlinear partial differential equations, Physica119A (1983) 101–142.

6. F.W. Nijhoff, G.R.W. Quispel and H.W. Capel, Direct linearisation of nonlineardifference-difference equations, Phys. Lett. 97A (1983) 125-128.

7. G.R.W. Quispel, F.W. Nijhoff, H.W. Capel and J. van der Linden, Linear integralequations and nonlinear difference-difference equations, Physica 125A (1984) 344–380.

8. P.G. Drazin and R.S. Johnson, Solitons. An Introduction, (Cambridge Univ. Press,1989).

9. T. Miwa, M. Jimbo and E. Date, Solitons, Differential Equations, Symmetries andInfinite Dimensional Lie Algebras, (Cambridge University Press, 2000).

10. M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and InverseScattering, LMS Series (Cambridge Univ. Press, 1991).

11. F.W. Nijhoff and H.W. Capel, The discrete Korteweg-de Vries equation, Acta Appli-candae Mathematicae 39 (1995) 133–158.

Lecture 2

Extensions to other classes of

P∆Es

In this Chapter we will generalize the structure displayed in Lecture 1 to other classes ofequations:

• Bilinear integrable equations and ABS type quadrilateral lattice equations;

• Higher-dimensional equations: lattice equations of KP type.

In the first class we have the discrete versions of the famous Hirota discrete-time analogueof the Toda equation, which can be written as:

a τn+1,mτn−1,m + b τn,m+1τn,m−1 + c τ2n,m = 0 , (2.0.1)

[Hirota,1977] where τn,m is the dependent variable as a function of the lattice sites (n,m) ∈Z2, and in which a,b,c are certain constants. Eq. (2.0.1) which is defined on a configurationof five points in the lattice, is an important equation which has appeared in the computationof the magnetic properties of certain fundamental models in Physics, e.g., the famous two-dimensional Ising model of ferromagnetism). It also constitutes an exact discretization ofthe celebrated Toda anharmonic chain equation [Toda, 1967]

d2qndt2

= eqn+1−qn − eqn−qn−1 , (2.0.2)

which has applications in many areas of physics and appears also in developments of puremathematics. Furthermore, we will encounter some more general quadrilateral lattice equa-tions, the so-called NQC equation:

1 + (p− a)Sn,m+1 − (p+ b)Sn+1,m+1

1 + (q − a)Sn+1,m − (q + b)Sn+1,m+1=

1 + (q − b)Sn,m − (q + a)Sn,m+1

1 + (p− b)Sn,m − (p+ a)Sn+1,m(2.0.3)

[Nijhoff, Quispel & Capel, 1983] which “interpolates” between the various P∆Es (latticeKdV, MKdV and SKdV) which we have encountered in Lecture 1. The latter equation is

12

2.1. BILINEAR FORM FROM SOLITON SOLUTIONS 13

closely connected to an integrable P∆E more recently found in the classification of quadri-lateral lattice equations, namely “Q3”:

p(1− q2) (un,mun,m+1 + un+1,mun+1,m+1)− q(1 − p2) (un,mun+1,m + un,m+1un+1,m+1)

= (p2 − q2)

(un,mun+1,m+1 + un+1,mun,m+1 + δ2

(1 − p2)(1− q2)

4pq

), (2.0.4)

[Adler, Bobenko & Suris, 2003]. Finally , we will encounter some equations in three dimen-sions, i.e. depending on three lattice variables, in particular the so-called DAGTE (discreteanalogue of generalized Toda equation)

a τn+1,m,hτn−1,m,h + b τn,m+1,hτn,m−1,h + c τn,m,h+1τn,m,h−1 = 0 , (2.0.5)

[Hirota, 1981] which is an equation defined in a three-dimensional lattice, with independentvariables (n,m, h) ∈ Z3 generalizing the discrete-time Toda equation (2.0.1). This equation,which in the modern literature is often (erroneously) referred to as Hirota-Miwa equationis connected to some nonlinear three-dimensional P∆Es which are discretizations of thefamous Kadomtsev-Petviashvili (KP) equation. One such equation is the following:

p− q + un,m+1,h+1 − un+1,m,h+1

p− q + un,m+1,h − un+1,m,h=p− r + un,m+1,h+1 − un+1,m+1,h

p− r + un,m,h+1 − un+1,m,h(2.0.6)

[Nijhoff, Quispel, Capel & Wiersma, 1985] in which p, q and r denote lattice parametersw.r.t. the three different lattice directions.

2.1 Bilinear Form from Soliton Solutions

As a starting point for our treatment of the τ -function τn,m wahich was already given inLecture 1, cf. (2.1.7), by

τ = det (1+M) = τn,m (2.1.7)

where the matrix M is given as in Lecture 1 by (1.1.1). From the definition (2.1.7), usingthe relation (1.1.6a), we can perform the following straightforward calculation:

τ = det(1+ M

)= det

1+

[(p+K)M + r tc

](p+K)−1

= det(p+K)

[1+M + (p+K)−1r tc

](p+K)−1

= det(1+M)

[1+ (1+M)−1(p+K)−1r tc

]

= τ det1+ (1+M )−1(p+K)−1r tc

from which, using also

r =p+K

p−K r ,

we haveτ

τ= 1+ tc (1+M)−1 (p−K)−1 r = 1 + tu0 (p−K)−1 r , (2.1.8)

14 LECTURE 2. EXTENSIONS TO OTHER CLASSES OF P∆ES

where in the last step we have made use of the famous Weinstein-Aronszajn formula:

det(1+ a tb

)= 1 + tb · a .

The object created on the right-hand side of eq. (2.1.8) is new in the structure of Lecture1, and hence we need to investigate what relations it satisfies.

To do this from a slightly more general perspective, let us now introduce the function:

V (a) ≡ 1− tc (a+K)−1u0 = 1− tu0 (a+K)−1 r = Vn,m(a) . (2.1.9)

From the calculation above, we have immediately:

τ

τ= V (−p) ⇒ V (p) =

T−1p τ

τ, (2.1.10)

setting a = −p, and where we have introduced the notation Tpτ = τ = τn+1,m. In order toderive equations for this function V (a) we need to introduce some further objects, namely:

u(a) = (1+M)−1(a+K)−1r , (2.1.11a)tu(b) = tc (b+K)−1(1+M )−1 , (2.1.11b)

S(a, b) = tc (b+K)−1(1+M )−1(a+K)−1r . (2.1.11c)

We note that:S(a, b) = tc (b +K)−1u(a) = tu(b) (a+K)−1r , (2.1.12)

in which a, b are arbitrary (real or complex valued) parameters.

Exercise 2.1.1. Following a similar derivation as the one leading to (1.1.9) and (1.1.10),derive the following relations:

(p−K)u(a) = V (a)u0 + (p− a)u(a) , (2.1.13a)

(p+K)u(a) = −V (a)u0 + (p+ a)u(a) , (2.1.13b)

Exercise 2.1.2. By multiplying (2.1.13) from the left by the row vector tc (b+K)−1 , showthat (2.1.13) leads to

1− (p+ b) S(a, b) + (p− a)S(a, b) = V (a)V (b) . (2.1.14)

It can be shown that as a consequence of (1.1.13) we have that S(a, b) is symmetricunder the exchange of the parameters a and b:

S(a, b) = S(b, a) , (2.1.15)

which is easily seen from the explicit expression (2.1.11c) by expanding the diagonal matrices(a+K)−1 and (b +K)1 in powers of a and b. Using this property and the identity:

V (a)V (b)

˜V (a)V (b)

=V (b)V (a)

V (b)V (a)

2.1. BILINEAR FORM FROM SOLITON SOLUTIONS 15

by inserting (2.1.14) and its counterpart, with p replaced by q and the ˜-shift replaced bythe -shift, as well as the relations with a and b interchanged, we obtain the following closedform equation for S(a, b):

1− (p+ b)S(a, b) + (p− a) S(a, b)

1− (q + b)S(a, b) + (q − a) S(a, b)

=1− (q + a) S(a, b) + (q − b)S(a, b)

1− (p+ a) S(a, b) + (p− b)S(a, b), (2.1.16)

which is eq. (2.0.3) identifying Sn,m = S(a, b). Eq. (2.1.16) is a quite general P∆E whichcontains many special subcases for different choices of the parameters a and b. For eachfixed choice of a, b this quadrilateral P∆E is integrable in the sense of the multidimensionalconsistency property explained in Lecture 3 of the main course, taking into account thatthe role of the parameters a, b (which we can chose to fix the dependent variable S(a, b)) isdifferent from the role of the lattice parameters p and q (which are attached to the latticeshifts).

Making the special choice a = p, b = −p in (2.1.14), we have the relation:

V (p)V (−p) = 1 and similarly, V (q)V (−q) = 1 . (2.1.17)

Furthermore, introducing the variable

W (a) ≡ a− tcK u(a) = a− ct(a+K)−1u1 , (2.1.18)

and multiplying from the left eqs. (2.1.13) by tc we find:

W (a) = (p+ a)V (a)− (p− u)V (a) , (2.1.19a)

W (a) = −(p− a)V (a) + (p+ u)V (a) . (2.1.19b)

Obviously we have a similar set of formulae by replacing in eqs (2.3.22) the ˜-shift by the-shift and p by q, leading to

W (a) = (q + a)V (a)− (q − u)V (a) , (2.1.19c)

W (a) = −(q − a)V (a) + (q + u)V (a) . (2.1.19d)

Eliminating W (a) leads to several relations, for instance by subtracting (2.1.19c) from

(2.1.19a), or ˜(2.1.19d) from (2.1.19b), we obtain the set of relations:

p− q + u− u = (p+ a)V (a)

V (a)− (q + a)

V (a)

V (a)

= (p− a)V (a)

V (a)

− (q − a)V (a)

V (a)

. (2.1.20)

Alternatively, by combining (2.1.19b) with ˜(2.1.19c), or (2.1.19d) with (2.1.19a), one obtains:

p+ q + u− u = (p+ a)V (a)

V (a)+ (q − a)

V (a)

V (a)

= (p− a)V (a)

V (a)+ (q + a)

V (a)

V (a). (2.1.21)


As a direct corollary the equality on the right-hand sides of eqs. (2.1.20) and (2.1.21) actuallyprovide us with a P∆E for each of the variables V (a) (taking the parameter a to be fixed).

Exercise 2.1.3. Show that the P∆E obtained from the equality on the right-hand side of(2.1.20) is the same as the one obtained from (2.1.21).

In particular, choosing a = p in (2.1.20) we get the P∆E for V (p) which takes the form

2pV (p)

V (p)= (p+ q)

V (p)

V (p)+ (p− q)

V (p)

V (p)

. (2.1.22)

Using now (2.1.10) to substitute the variable V (p), and similarly doing the same for V (q)in the analogous equation obtained by interchanging p and q and ˜-shifts and -shifts, weobtain the following to bilinear partial differennce equations for τ , namely

(p+ q) τ˜τ + (p− q) τ

˜τ = 2pτ τ , (2.1.23a)

(p+ q)ττ + (q − p)τ

τ = 2qτ τ , (2.1.23b)

in which the τ˜= τn−1,m and τ

= τn,m−1 denote the τ -function shifted one unit backward

in the respective directions on the lattice.

Let us look at these two P∆Es more closely. Denoting the graphs of the equations(2.1.23a), (2.1.23b) respectively by the following diagrams

τ˜ τ τ

τ˜

τ τ

τ

τ

τ

τ

τ

τ

there is now an issue of the consistency of these two equations. In fact, from the pointof view of the iteration of these two equations on the two-dimensional lattice, we shouldinvestigate that these P∆Es give a unique evaluation of values of the solution function τthroughout the lattice. Considering initial values for τ given on the black dots of the latticeaccording to the the following elementary diagram

2.2. ∗ SOLITON SOLUTION OF Q3 17

τ

there are actually now two way in which we can calculate the value of τ at the vertex in thelower corner. Using both equations (2.1.23) we can either proceed horizontally or vertically

to calculate the valueτ . It is not difficult by explicit computation that the resulting value,

in terms of the values of the initial data, is actually the same!

Exercise 2.1.4. Perform this computation explicitely, and by expressing the value τ interms of the values of the initial data given in the diagram, show that the following equationholds:

(p− q)2

τ τ − (p+ q)2τ

τ˜+ 4pqτ2 = 0 . (2.1.24)

Eq. (2.1.24) is called the discrete-time Toda equation.

2.2 ∗ Soliton solution of Q3

Connection between the NQC and the (Q3)δ=0 equations

We will first establish the connection between (Q3)δ=0, i.e. the equation (2.0.4) in whichwe set the parameter δ = 0, and the equation (2.0.3) for the quantity S which is defined in(2.1.11c). In fact, writing the NQC equation in the following form

[1 + (p− a)S − (p+ b)S

] [1 + (p− b)S − (p+ a)

S

]=

=[1 + (q − a)S − (q + b)S

] [1 + (q − b)S − (q + a)

S

]

and introducing the variable

u0n,m = ρ1/2(a)ρ1/2(b) [1− (a+ b)Sn,m(a, b)] , (2.2.1)

where the quantities ρ(a), ρ(b) behave like the plane-wave factors of Lecture 1, i.e., they aregiven by

ρ(a) =

(p+ a

p− a

)n(q + a

q − a

)m

, ρ(b) =

(p+ b

p− b

)n(q + b

q − b

)m

, (2.2.2)

we obtain an equation for u0 = u0n,m of the following form

P (u0u0 + u0u 0)−Q(u0u0 + u0u 0) = (p2 − q2)(u0u0 + u0u 0) , (2.2.3)


which, in fact, is equivalent to (2.0.4) for δ = 0. In (2.2.3) we have new parameters P andQ, associated with the lattice parameters p and q respectively, given by the formulae:

P 2 = (p2 − a2)(p2 − b2) , Q2 = (q2 − a2)(q2 − b2) . (2.2.4)

In fact, the latter relations imply that (p, P ), (q,Q) are points on a (Jacobi) elliptic curvewith branch points (±a, 0) , (±b, 0), (see Adv. Lecture 3).

Exercise 2.2.1. Show that the original parameters in (2.0.4), let us call them p1, q1 toavoid confusion, are related to the parameters (p, P ) and (q,Q), coming from the solitonstructure, as follows:

p21 =p2 − b2

p2 − a2, P =

(b2 − a2)p11− p21

, q21 =q2 − b2

q2 − a2, Q =

(b2 − a2)q11− q21

. (2.2.5)

Connection between NQC and the (Q3)δ equation

We have shown above that, up to a scaling by discrete exponential factors, solutions ofthe NQC equation (2.0.3) give rise to solutions of the (Q3)δ=0 equation (2.2.3). However,whereas the former equation shows an explicit dependence of the fixed parameters a and b,in the latter those parameters are not visible but enter only through the lattice parametersP and Q on the elliptic curve (2.2.4). In fact, this relation between solutions does not alterin any way if we would change the sign of the parameters a and b. Hence, there are in factfour different NQC solutions: S(a, b), S(−a, b), S(a,−b) and S(−a,−b), which, through arelation of the form (2.2.1), give rise to a solution of the same (Q3)δ=0 equation. Thus, bychanging the sign of the parameters a and b we obtain four different solutions of the latterequation!

The remarkable fact, first observed in the paper [Atkinson, Hietarinta & Nijhoff, 2008]is that these solutions can be combined as a linear combination to obtain a solution of thefull equation Q3 with nonzero value of δ. The result is as follows:

Theorem 2.2.1. The following expression

un,m =

= Aρ1/2(a)ρ1/2(b) [1− (a+ b)Sn,m(a, b)] +Bρ1/2(a)ρ1/2(−b) [1− (a− b)Sn,m(a,−b)] ++Cρ1/2(−a)ρ1/2(b) [1 + (a− b)Sn,m(−a, b)] +Dρ1/2(−a)ρ1/2(−b) [1 + (a+ b)Sn,m(−a,−b)] ,

(2.2.6)

in which Sn,m(±a,±b) are solutions of the NQC equation (2.0.3) with respective parameters±a and ±b, and where ρ(a) and ρ(b) are given by (2.2.2), and where A, B, C, D are constantcoefficients, provides a solution of the Q3 equation in the following form

P (uu+ uu)−Q(uu+ uu) = (p2 − q2)

((uu+ uu) + δ2

4PQ

)(2.2.7)

provided that we have the following condition on the coefficients A,B,C,D:

AD(a+ b)2 −BC(a− b)2 = − δ2

16ab.

2.2. ∗ SOLITON SOLUTION OF Q3 19

The proof of this theorem is quite lengthy and, although it is constructive, it combinesa considerable number of relations. We refer to [Nijhoff, Atkinson & Hietarinta, 2009] forthe full proof of this statement. What is quite remarkable about the formula (2.2.6) is thateven though we are dealing with a highly nonlinear equation, nevertheless it seems a kindof linear superposition formula holds, but in fact this is deceptive, because it relies on somedeep (yet not quite understood) algebraic relations between solutions of the NQC equationfor different parameters a and b.

Exercise 2.2.2. Show that eq. (2.2.7) is equivalent to (2.0.4) by using the connectionbetween the parameters as given in (2.2.5) where as before p1, q1 are the parameters as in(2.0.4), and where the dependent variable u1 is related to the dependent variable of (2.2.7)via the scaling u = (b2 − a2)u1 .

Another connection that follows from the result of Theorem 2.2.1 is that we can writenow solutions of Q3 in terms of the τ -function τ , provided that we extend the lattice from twodimensions to four dimensions ! The reason is that, similar to p and q, we can reinterpretthe a and b as lattice parameters, but associated with some new lattice variables. Thus, ifwe would extend the definition of the ρi as in (1.1.2) to include extra factors depending onnew lattice variables α and β, associated with the parameters a and b respectively, i.e. wewould redefine the ρi to be given by:

ρi =

(p+ kip− ki

)n(q + kiq − ki

)m(a+ kia− ki

)α(b+ kib− ki

)β

ρ0i , (2.2.8)

then in terms of shifts Ta, Tb of the new variables (shifting α 7→ α+1 respectively β 7→ β+1)we would have

Taρi =a+ kia− ki

ρi , Tbρi =b+ kib− ki

ρi

and this would imply similar relations for all the quantities involved, to the ones we havederived earlier for the shifts in the variables n and m. In particular, since the redefenitionof the ρi would work through all the soliton formulae, the S(a, b) would now depend also onthe additional variables α and β and we would have similar relations as (2.1.14) for those

shifts, in particular (replacing p by a and S = TpS by TaS in (2.1.14))

1− (a+ b)TaS(a, b) = (TaV (a))V (b) ⇒ 1− (a+ b)S(a, b) = V (a)T−1a V (b) .

Using the relation (2.1.9), where we replace the shift Tp by Ta and Tb in the right-hand sideof the latter relation, we obtain

1− (a+ b)S(a, b) =T−1a τ

τT−1a

(T−1b τ

τ

)=T−1a T−1

b τ

τ. (2.2.9)

Thus, we can write the solution (2.2.6) as follows:

un,m =

= Aρ1/2(a)ρ1/2(b)T−1a T−1

b τ

τ+Bρ1/2(a)ρ1/2(−b)T

−1a Tbτ

τ+

+Cρ1/2(−a)ρ1/2(b)TaT−1b τ

τ+Dρ1/2(−a)ρ1/2(−b)TaTbτ

τ,


Remark: In the treatment given in [Nijhoff et al., 2009] there is another quantity that isof importance in the solution structure of Q3, namely the quantity

vn,m = (a+ b)Aρ1/2(a)ρ1/2(b)Vn,m(a)Vn,m(b) + (a− b)Bρ1/2(a)ρ1/2(−b)Vn,m(a)Vn,m(−b) +− (a− b)Cρ1/2(−a)ρ1/2(b)Vn,m(−a)Vn,m(b)− (a+ b)Dρ1/2(−a)ρ1/2(−b)Vn,m(−a)Vn,m(−b)

which involves naturally the objects V (±a), V (±b), and this quantity, together with the onegiven by (2.2.6) relates Q3 to solutions of the lattice potential KdV equation. Furthermore,it should be mentioned that soliton solutions to all equations in the list of ABS equations(see Lecture 3 of the core module), except for Q4, can be obtained by suitable limits on theparameters a, b and the coefficients A, B, C and D. The case of Q4 requires a separatetreatment, which is presentend in [Atkinson & Nijhoff, 2010].

2.2.1 Explicit Form of the N-Soliton Solutions

We will now exhibit some explicit formulae for the soliuton solutions in terms of the τ -function, which allow us to investigate their properties. The main thing is now to go backto the definition of the τ -function (2.1.7) and to realise that the matrix M ia actually aCauchy matrix.

First, let us cite the following expansion formula for the determinant of a matrix of theform λ1 +M , where λ is a scalar (a complex number). For arbitrary λ we have thefollowing power series expansion of a N ×N determinant

det (λ1+M) = λN + λN−1N∑

i=1

|Mi,i|+ λN−2∑

i<j

∣∣∣∣Mi,i Mi,j

Mj,i Mj,j

∣∣∣∣

+λN−3∑

i<j<k

∣∣∣∣∣∣

Mi,i Mi,j Mi,k

Mj,i Mj,j Mj,k

Mk,i Mk,j Mk,k

∣∣∣∣∣∣+ · · ·+ det(M ) .

(2.2.10)

We can now exploit the special form of the matrix M , namely the fact that it is (up toa multiplication by diagonal matrices) a Cauchy matrix. A N × N Cauchy matrix A is amatrix with entries of the form:

Ai,j =1

ki − lj, i, j = 1, . . . , N ,

where the ki, lj are are set of arbitrary numbers, (such that none of the entries in thematrix A becomes singular). There is a famous explicit determinantal formula for Cauchydeterminants which reads:

det(A) =

∏i<j(ki − kj)(lj − li)∏

i,j(ki − lj)(2.2.11)

where i, j run from 1 to N . Noting that the matrix M can be written as a Cauchy matrixof the form A with lj = −kj multiplied from the left by a diagonal matrix with entries ρi

2.3. HIGHER-DIMENSIONAL SOLITON SYSTEMS: THE KP CLASS 21

and from the right by a diagonal matrix with entries cj , we have:

det

(ρicj

ki + kj

)=

(∏

i

ρici2ki

)∏

i<j

(ki − kjki + kj

)2

. (2.2.12)

Noting further that all terms in the expansion (A.3.5) are in terms of determinants with thesame entries as of the matrix M itself, hence all these determinants have the form (2.2.12)(albeit with different selections of labels i, j!), we are now in a position express each of theterms in (A.3.5) in terms of products of the type on the r.h.s. of (2.2.12). Introducing forconvenience the notations

eAj,k ≡(ki − kjki + kj

)2

, eθi =ρici2ki

,

the formula for the τ -function thus takes the form:

τ = 1 +N∑

i=1

eθj +N∑

i<j=1

eθi+θj+Ai,j +

+

N∑

i<j<k=1

eθi+θj+θk+Ai,j+Ai,k+Aj,k + · · · . (2.2.13)

The dependence of τ in eq. (2.2.13) on the discrete dynamical variables n,m enters onlythrough the θi, which in turn depend on n,m via the formula for the plane wave factorsρi, i.e. (1.1.2). Inserting the eplicit formula (2.2.13) into expressions like (2.1.9) and (2.2.9)we obtain the relevant N -soliton solutions for the various P∆E s we have considered so farthrough their expressions in terms of the τ -function.

2.3 Higher-dimensional soliton systems: the KP class

2.3.1 Multidimensional solitons

So far all considerations of the soliton structure were based on the form of the underlyingCauchy matrix (1.1.1). However, that matrix has a special form, whilst the determinantalidentity (2.2.11) suggests that we could explore a more general form of the matrix withtwo types of parameters: ki and lj , not necessarily the same. We will now explore thispossibility, and assume at the same time that the number of parameters ki, i = 1, . . . , N isnot necessarily the same as the number of lj , where j = 1, . . . , N ′. The starting point forthe construction of KP discrete solitons is, as before, a Cauchy matrix M , but this couldbe now a N ×N ′ matrix, not necessarily square, namely

M = (Mi,j)i=1,...,N ;j=1,...,N ′ with Mi,j =ρiσjki + lj

, (2.3.1)

in which ki, li, (i = 1, . . . , N), are two sets of N fixed parameters. This generalises theCauchy matrix considered in Adv. Lecture 1, i.e. eq. (1.1.1), where the two sets of param-eters were chosen to coincide, i.e. ki = li , i = 1, . . . , N . Relaxing the choice of Cauchy


matrix we will see that we can, to a large extent, still implement the scheme developed inLecture 5, except that half of the relations we will lose because of the loss of symmetry be-tween the variables k and l. The dependence on the discrete dynamical variables is throughthe coefficients ρi, σj , (the latter no longer being constants), which are given by:

ρi = ρn,m,h(ki) , σj = σn,m,h(lj) , (2.3.2a)

where we will have to incorprate a dependence on at least three discrete variables n, m,h (each associated with its respective lattice parameter, p, q and r respectively) in thefollowing way:

ρn,m,h(k) =

(1 +

k

p

)n(1 +

k

q

)m(1 +

k

r

)h

ρ0,0,0(k) , (2.3.2b)

σn,m,h(l) =

(1− l

p

)−n(1− l

q

)−m(1− l

r

)−h

σ0,0,0(l) , (2.3.2c)

where ρ0,0,0, σ0,0,0 indicate some initial values independent of the discrete variables n, m,h. From the definition we have immediately the following relation

KM +ML = r ts , r = (ρ1, . . . , ρN )T , ts = (σ1, . . . , σN ′) , (2.3.3)

in whichK is, as before, theN×N diagonal matrix of parameters ki, K = diag(k1, . . . , kN ) ,whereas L is the N ′ ×N ′ diagonal matrix of parameters lj , L = diag(l1, . . . , lN ′) .

Adopting the notation in previous Lectures, whereby the shifts in the variables n, m,and h by one unit are represented on the dependent variables by ˜-, - and the -shifts, itis easy to derive the following relation for the ˜-shift on the matrix M :

M

(1− 1

pL

)=

(1+

1

pK

)M ⇒ M =M +

1

pr ts , (2.3.4)

and similar formulae for the other shifts.

We want to study similar objects as before, namely row and columnvectors and scalars asin (1.1.7). However, sinceM in this case is no longer a square matrix, we need to introducea N ′ ×N matrix of coefficients C to compensate for this deviation, such that the productsCM and MC are now square N ′ ×N ′ respectively N ×N matrices.

The objects we want to study to are the following infinite sets of column- respectivelyrow-vectors

ui ≡ (1+MC)−1Kir , tuj ≡ tsLj (1+CM)−1 , i, j ∈ Z , (2.3.5)

where the ui are N -component column vectors, and the tuj are N ′-component row vectors.Furthermore, as before, we introduce the bi-infinite set of scalar quantities:

Ui,j =tsLj (1+CM)−1CKir = tsLjC ui =

tuj CKir , (2.3.6)

for all i, j integer.


We now use the same recipe as adopted in Lecture 1 to derive the basic recurrencerelations for the vectors ui, and

tuj , leading to

pui = pui + ui+1 − Ui,0 u0 , (2.3.7a)

p tuj = p tuj − tuj+1 +tu0 U0,j , (2.3.7b)

and from these we subsequently derive the double-sided recurrence relation for the Ui,j ,namely:

pUi,j − Ui,j+1 = pUi,j + Ui+1,j − Ui,0U0,j . (2.3.8)

Similar relations to (2.3.7) and (2.3.8) hold for the other lattice shifts, the - and -shifts,associated with the paramaters q and r respectively, hold as well.

Exercise 2.3.1. Prove the relations (2.3.7) as well as (2.3.8).

We observe that the relation (2.3.8) coincides exactly with the relation (2.1.13a) of Lec-ture 5, and the (2.3.7a), up to a factor, coincides with (1.1.9). However, we have now nolonger the analogue of eqs. (2.1.13b) or (1.1.10), nor of the algebraic relations (1.1.15) or(1.1.14), due to the loss of symmetry between the left and right index of the quantities Ui,j .Thus, the relation (1.1.13) no longer holds, as a consequence of the fact that there is nolonger any connection between parameters ki and lj . Effectively this means that we losehalf of the relations needed to obtain closed-form equations, and to compensate that lossof information we need the third variable h and the corresponding relations governing theshifts in that variable, to compensate for this loss of information. Thus, with this extravariable we obtain the additional set of recurrence relations

qUi,j − Ui,j+1 = qUi,j + Ui+1,j − Ui,0U0,j , (2.3.9a)

rU i,j − U i,j+1 = rUi,j + Ui+1,j − U i,0U0,j , (2.3.9b)

and the corresponding set of linear relations

qui = pui + ui+1 − Ui,0u0 , (2.3.10a)

q tuj = q tuj − tuj+1 +tu0 U0,j , (2.3.10b)

rui = rui + ui+1 − U i,0u0 , (2.3.10c)

r tuj = r tuj − tuj+1 +tu0 U0,j . (2.3.10d)

We now combine these relations, eliminating the shifts in the indices i and j to obtainclosed-form P∆E s. There are three cases which run parallel to the cases obtained in thetwo-dimensional case of Lecture 1.

Lattice KP eq.:

As a first example, taking i = j = 0 in eq. (2.3.8) and (2.3.9), and eliminating thevariables U1,0 and U0,1 by combining the relations obtained for different lattice directions, itis straightforward to derive the following three-dimensional lattice equation for the variableu = U0,0

(p− u)(q − r + u− u) + (q − u)(r − p+ u− u) + (r − u)(p− q + u− u) = 0 , (2.3.11)

which we shall refer to as the lattice potential Kadomtsev-Petviashvili equation (LKP).


Exercise 2.3.2. Derive eq. (2.3.11) and show that it can also be written in the equivalentforms:

(p+ u)(q − r + u− u) + (q + u)(r − p+ u− u) + (r + u)(p− q + u− u) = 0 , (2.3.12)

or in either of the equations:

(p− r + u− u)

p− r + u− u=

(q − r + u− u)

q − r + u− u=

(p− q + u− u)

p− q + u− u. (2.3.13)

Lattice MKP eq.:

A second example is obtained by setting i = −1, j = 0 or i = 0, j = −1 in the set ofequations (2.3.8). By combining any two of these equations and eliminating the variablesU1,−1 and U−1,1, we can derive the relations

p− q + u− u =pv − qv

v=pw − qw

w, (2.3.14)

(and similar ones involving a combination of any two of the other lattice shifts), where wehave introduced the variables:

v = 1− U−1,0 , w = 1− U0,−1 . (2.3.15)

By combining (2.3.14) with the other relations obtained by cyclic permutation of the latticevariables and associated parameters, it is straightforward to eliminate the variable u and toderive the following three-dimensional lattice equation

p

(v

v− v

v

)+ q

(v

v− v

v

)+ r

(v

v− v

v

)= 0 , (2.3.16)

which we will refer to as the lattice potential modified KP equation (LMKP).

Exercise 2.3.3. Derive a similar equation for the quantity w = 1 − U0,−1 which, incontrast to the case discussed in Lecture 5, is not the same as v.

Lattice SKP eq.:

A third example is obtained by considering the choice i = j = −1 in eqs. (2.3.8) andderive the relation:

p(z − z) = vw , (2.3.17)

where z = U−1,−1 − np − m

q − hr . By combining (2.3.17) with two similar equations for the

other lattice directions, and substituting the left hand-sides in to the identity:

(vw)

(vw)=

(vw)

(vw)˜(vw)

(vw),

we obtain the following three-dimensional lattice equation for z:

(z − z)(z − z)(z − z)

(z − z)(z − z)(z − z)= 1 . (2.3.18)


This equation, which was for the first time explicitely given by [Dorfman & Nijhoff, 1991] isreferred to as the lattice Schwarzian KP equation, whilst it arises as special parameter caseof a more general equation presented in [Nijhoff, Capel, Wiersma & Quispel, 1985]. It waslater rediscovered and highlighted in a geometric context by Bogdanov and Konopelchenko,and Konopelchenko and Schief pointed out a connection with the classic Menelaus theoremin affine geometry.

2.3.2 Bilinear structure for discrete KP type equations

The τ -function for the lattice KP soliton family can be defined as:

τ = τn,m,h = detN×N(1+MC) = detN ′×N ′(1+CM ) . (2.3.19)

The latter identity is a consequence of the general Weinstein-Aronszajn formula:

detN×N

1+

N ′∑

l=1

ml ctl

= detN ′×N ′

(1+ (ctl ·mk)

), (2.3.20)

where ml = (Mi,l)i=1,...,N and ctl = (Cl,j)j=1,...,N are the N ′ column- resp. row vectorsfrom the matrices M , C respectively.

The elementary shifts on the τ -function are calculated in a similar way as before:

τ = detN×N

(1+ M C

)= det

1+

(M +

1

pr ts

)C

= det(1+MC) det

1+

1

p(1+MC)−1r ts C

= τ

(1 +

1

pts

(1− 1

pL

)−1

C(1+MC)−1 r

)

where we have used the simple (rank 1) version of the Weinstein-Aronszajn formula. Thisleads to the expressions:

τ

τ= 1 + st (p−L)−1C u0 or reversely

τ

τ= 1− tu0C (p+K)−1 r , (2.3.21)

where the latter follows from a similar computation:

τ = detN×N (1+MC) = det

1+

(M − 1

pr ts

)C

= det(1+ M C) det

1− 1

p(1+ M C)−1r ts C

= τ

(1− 1

pts C (1+ M C)−1

(1+

1

pK

)−1

r

).

The relations (2.3.21) suggest the introduction of the quantities, for arbitrary parameter a:

V (a) = 1− tu0C (a+K)−1 r , W (a) = 1− st (a+L)−1C u0 , (2.3.22)


and, henceτ

τ=W (−p) = 1/V (p) . (2.3.23)

Using the linear recursion relations (2.3.7a) and (2.3.10a) for ui by multiplying from the leftby st (a+L)−1C , we obtain

p ts (a+L)−1C u0 − q ts (a+L)−1C u0 = (p− q + u− u) ts (a+L)−1C u0 ⇒ts (p−L) (a+L)−1C u0 − ts (q −L) (a+L)−1C u0 = (p− q + u− u)(1 −W (a))

= (p+ a)(1 − W (a))− u− (q + a)(1− W (a)) + u ,

and hence

p− q + u− u = (p+ a)W (a)

W (a)− (q + a)

W (a)

W (a). (2.3.24)

From the adjoint relations (2.3.7b) and (2.3.10b) for tuj by a similar computation, multi-

plying from the right by C (a+K)−1 r , we obtain:

p− q + u− u = (p− a)V (a)

V (a)

− (q − a)V (a)

V (a)

. (2.3.25)

Setting either a = −p in (2.3.24), or a = p in (2.3.25), and using the relation (2.3.23), wethen get the following expression

p− q + u− u = (p− q)τττ τ

, (2.3.26)

which is the same relation as we got in the two-dimensional case. However, in the three-dimensional case the companion relation is absent, and hence we need the third latticedirections to eliminate the u from the analogous relations in the other directions. This leadsto the bilinear relation:

(p− q)τ τ + (q − r)τ τ + (r − p)τ τ = 0 . (2.3.27)

which we can identify with Hirota’s DAGTE by setting τn,m,h = FN,M,H , with N = 2n,M = 2m, H = 2h, upon which we obtain:

(p− q)FN+1,M+1,H−1FN−1,M−1,H+1 + (q − r)FN−1,M+1,H+1FN+1,M−1,H−1

+(r − p)FN+1,M−1,H+1FN−1,M+1,H−1 = 0 .

Finally, introducing the following variable:

S(a, b) = ts (b+L)−1C (1+MC)−1 (a+K)−1 r . (2.3.28)

where now in general S(a, b) 6= S(b, a) , we need at least three independent discrete variablesto write down closed-form equations. As before a relation of a form similar to holds, namely

1 + (p− a)S(a, b)− (p+ b)S(a, b) = V (a)W (b) , (2.3.29)


and similar relations for the other lattice directions. In a similar way as in the SchwarzianKP case this relation, together with its counterparts in the other lattice directions, yieldsthe following three-dimensional equation Subsequently the following 3D lattice equation canbe derived:

1 + (p− a)S − (p+ b)S

1 + (p− a)S − (p+ b)S=

(1 + (q − a)S − (q + b)S)

(1 + (r − a)S − (r + b)S)

(1 + (r − a)S − (r + b)S)

(1 + (q − a)S − (q + b)S), (2.3.30)

which is a three-dimensional analogue of (2.0.3).

Exercise 2.3.4. Derive the relation (2.3.29) from the soliton structure, by introducing theparameter-dependent vectors

u(a) ≡ (1+MC)−1 (a+K)−1 r , tu(b) ≡ ts (b+ L)−1 (1+CM )−1 , (2.3.31)

and deriving the following relations:

pu(a) = (p− a)u(a) + V (a)u0 ,

p tu(b) = (p+ b) tu(b)−W (b) tu0 .

and by multiplying the first from the left by st (b+L)−1C , or the second from the right byC (a+K)−1 r .Hint: Use can be made of the relations

tsC u(a) = 1− V (a) , tu(b)C r = 1−W (b) .

2.3.3 Multi-soliton formula for discrete KP systems

We can use once again the expansion formula for the τ -function:

τ = det (1+MC) = 1 +

N∑

i=1

|Ni,i|+∑

i<j

∣∣∣∣Ni,i Ni,j

Nj,i Nj,j

∣∣∣∣+ · · ·+ det(N ) . (2.3.32)

where N = MC . We can then exploit the Cauchy matrix structure of M , using thedeterminantal property of a general N ×N Cauchy matrix A as in (2.2.11)

However, since the matrix N is a product of two non-square matrices, the determinantsin the expansion (2.3.32) can no longer be factorized in the usual fashion. For this we needthe following result from the theory of determinants.

Cauchy-Binet formula: For an arbitrary N × N ′ matrix A and N ′ × N matrix B wehave the following formula for the N ×N determinant of the product:

detN×N (AB) =

0 if N ′ < Ndet(A) det(B) if N ′ = N∑

1≤l1<l2<···<lN≤N ′ det(A(1,2,··· ,N),(l1,··· ,lN )

)det(B(l1,··· ,lN ),(1,2,··· ,N)

)if N ′ > N

in which A(1,2,··· ,N),(l1,··· ,lN ) denotes the matrix obtained by selecting the l1, . . . , lN columnsfrom the matrix A and B(l1,··· ,lN ),(1,2,··· ,N) the matrix obtained by selecting the l1, . . . , lNrows from B.


Applying the Cauchy-Binet formula to the terms in the expansion, and using subsequentlythe Cauchy identity, we can express the τ -function in explicit form.

Literature

1. R. Hirota, The Direct Method in Soliton Theory, (translated from Japanese and editedby A. Nagai, J. Nimmo and C. Gilson), Cambridge Tracts in Mathematics # 155(Cambridge University Press, 2004).

2. E. Date, M. Jimbo and T. Miwa, Method for Generating Discrete Soliton EquationsII, J. Phys. Soc. Japan 51 (1982) 4125-4131; ibid. V, J. Phys. Soc. Japan 52 (1983)766–771.

3. F.W. Nijhoff, G.R.W. Quispel and H.W. Capel, Direct Linearization of NonlinearDifference-Difference Equations, Physics Letters 97A (1983) 125–128.

4. G.R.W. Quispel, F.W. Nijhoff, H.W. Capel and J. van der Linden, Linear IntegralEquations and Nonlinear Difference-Difference Equations, Physica 125A (1984), 344–380.

5. J. Atkinson, J. Hietarinta and F.W. Nijhoff, Soliton solutions for Q3, J. Phys. A:MathTheor. 41 14 (2008) 142001 (11pp).

6. F.W. Nijhoff, J. Atkinson and J. Hietarinta, Soliton Solutions for ABS Lattice Equa-tions: I Cauchy Matrix Approach, J. Phys. A: Math. Theor. 42 40 (2009) 404005(34pp).

7. J. Atkinson and F.W. Nijhoff, A constructive approach to the soliton solutions ofintegrable lattice equations, Commun. Math. Phys. 299 (2010) 283–304.

8. R. Hirota, Discrete Analogue of a Generalized Toda Equation, J. Phys. Soc. Japan 50

(1981) 3785–3791.

9. T. Miwa, On Hirota’s Difference Equations, J. Phys. Soc. Japan 58A (1982) 9–12.

10. F.W. Nijhoff, H.W. Capel, G.L. Wiersma and G.R.W. Quispel, Backlund Transforma-tions and Three-Dimensional Lattice Equations, Physics Letters 105A (1984), 267–272.

11. F.W. Nijhoff, Theory of Integrable Three-Dimensional Lattice Equations, Letters inMathematical Physics 9 (1985), 235–241.

12. F.W. Nijhoff and H.W. Capel, The Direct Linearization Approach to Hierarchies ofIntegrable PDE’s in 2+1 Dimensions. I. Lattice Equations and the Differential- Dif-ference Hierarchies, Inverse Problems 6 (1990), 567–590.

13. I.Ya. Dorfman and F.W. Nijhoff, On a 2+1-Dimensional Version of the Krichever-Novikov Equation, Physics Letters 157A (1991), 107–112.


14. J.J.C. Nimmo and W.K. Schief, Superposition principles associated with the Moutardtransformation: an integrable discretization of a (2+1)-dimensional sine-Gordon sys-tem, Proc. R. Soc. Lond. A 453 (1997) 255–279.

15. L.V. Bogdanov and B.G. Konopelchenko, Analytic-bilinear approach to integrable hi-erarchies. I. Generalized KP hierarchy, J. Math. Phys. 39 (1998) 4683–4700.

16. B.G. Konopelchenko and W. Schief, Menelaus’ theorem, Clifford configurations andinversive geometry of the Schwarzian KP hierarchy, J. Phys. A: Mah. Gen. 25 (2002)6125–6144.

Appendix A

∗ Elliptic Functions & Theta

Functions

The excuse for these notes is the need I felt to collect together a concise number of formulaefor elliptic functions in one coherent notation and from one constructive point of view. Theidea is as much as possible to try to derive all possible identities from one single formula, or–where that is not possible– from a handful of formulae, thereby emphasising the connectionsbetween the various identities. In my view the subject of elliptic functions has suffered fromthe plethora of relations, identities and equations, the internal strcuture of which might belost to all but the initiated. The idea behind these notes is that a lot of the theory of ellipticfunctions can actually be developed from only a handful of key identities, and memorisingthese are in principle sufficient to control the subject. In this respect a student mightcompare this with the situation of the trigonometric functions, where one single identity,e.g. the addition formula

cos(x+ y) = cosx cos y − sinx sin y

plus some information on the periodicities and analytic behaviour around 0, is sufficientto derive all trigonometric identities: addition formulae, differential relations, including thegeometry of the underlying curve (i.e. the circle).

Obviously, the theory behind the elliptic functions is much richer and the number of“interesting” or “useful” relations is substantially larger than in the case of the trigono-metric functions. Even more so, for the working mathematician, to put some order in theabundance of identities. Historically, the subject of elliptic functions has developed throughvarious paths, stemming from the great fathers of the subject (Abel, Weierstrass, Jacobi,Frobenius, and many others), but unfortunately this has also resulted in a rather unsatis-factory abundance of notations which through history have taken their shape. Strangely,some of the notations in the theory of elliptic functions have not been really standardised:whereas the Jacobi functions sn, cn and dn have become standard, the related functionsds, cs, dc, nc and sc, cf. e.g. [Whittaker-Watson], have greatly dropped out of use. Simi-larly, the Weierstrassian functions σ, ζ and ℘ have become standard, but still at the time ofFrobenius, cf. [Frobenius], and well into the beginning of the 20th century various different

30

A.1. SOME DEFINITIONS 31

notations for these functions were in use. The situation with the theta functions is evenmuch less satisfactory, where even today not all agree on the same notation (some peopleusing θ1, θ2, θ3, θ4, whilst others use θ11, θ10, θ00, θ01, and some definitions include factorsπ in the argument), cf. e.g. [Mumford]. When we consider the situation in higher genus(hyperelliptic funcions, Abelian functions), the situation is even worse.

It has become a fine tradition in modern texts to build the theory of elliptic functionsfrom the point of view of geometry. This being very satisfactory and beautiful for theorists(e.g. algebraic geometers) but it is less useful for practicioners (applied mathematicians,physicists, engineers), i.e. to those who really need to work with the formulae and identities.It is somewhat a tragedy that the subject of elliptic functions has been dropped from theregular curriculum, and consequently out of the conscience of the working mathematician,due to the in this respect negative influence of the far-reaching level of abstractness withwhich the subject has become endowed in the course of this century. Elliptic functionsbeing seen nowadays as a subject for specialists only, it is a far cry from the times thatevery mathematics student was supposed to be able to work with elliptic functions with thesame agility as with trigonometric functions. The erroneous impression has been createdthat elliptic functions have no bearing on “real-life problems” and problems of a practicalnature has had the reverse effect that people nowadays only tend to think of those appliedproblems where they are not needed as realistic. Obviously this amounts to a dramaticimpoverishment of the scope of our thnking, which flies in the face of progress: the trulymodern problems in the applied sciences will inevitably require increasingly sophisticatedtools (in particular increasingly more complicated special functions) not only to solve theproblems but even to formulate the problems in the first place.

One of the areas where we have seen some modest revival of elliptic functions is notablyin the theory of integrable systems. Actually, in the last few decades one has realised thatmany of the structures behind integrable systems share deep common features with thebeautiful structure of the theory elliptic functions and elliptic curves. In a sense ellipticfunctions form a microcosm, a paradigm, for the wider theory of integrable systems.

A.1 Some Definitions

We need to give a definition of what an elliptic function is, but the aim of these notes isto avoid in the first place any theoretical groundwork that would trail us in the directionof the general theory of elliptic curves. That theory can be found in many textbooks, andI would consider the monographs by F. Kirwan and by McKean and Moll the best for afirst, yet comprehensive, encounter. (The latter covers in a nice and pedestrian way a greatvariety of ramifications of the theory into other areas of pure mathematics.) The treatment Iprefer here is closest to the one in the book by Akhiezer, namely very much a “constructive”approach highlighting very much the inteconnection between the various functions. In thetreatment given here it is the addition formulae which will play a central role (and whichconnect it also to the main subject of the course, namely Discrete Integrable Systems).

32 APPENDIX A. ∗ ELLIPTIC FUNCTIONS & THETA FUNCTIONS

Analytic and meromorphic functions

We will consider in what follows functions of a single complex variable (in addition toparameters which we consider to be fixed). We recall from complex function theory thata function f : D → C , defined on some open domain D in the complex plane, is calledanalytic or holomorphic if the limit

limz→z0

f(z)− f(z0)

z − z0=: f ′(z0) ,

exists for all z0 ∈ D, in other words if f is differentiable in the complex sense in the domainD. The complex derivative of a function (when it is applicable) follows very similar rules asthe usual derivative of a real-valued function:

df

dz= lim

h→0

f(z + h)− f(z)

h. (A.1.1)

In fact, the usual rules such sum rule, product rule, chain rule apply very similarly tocomplex differentiation as well as to the usual differentiation. The big difference is that inthe complex case, in taking the limit h → 0 the variable h itself is complex valued, andhence is supposed to approach 0 from any direction in the complex plane requiring the limitf ′(z) to be well-defined and unique coming from all directions1. As a consequence, thecondition of complex differentiability on a function f in the complex plane is much morerestrictive than in the real-valued case. In fact, decomposing the function f(z) in its realand imaginary part, and viewed as functions of the two real variables which are the real andimaginary part of z, namely:

f(z) = u(x, y) + iv(x, y) , z = x+ iy ,

it can easily be shown that the complex differentiability at a given point z ∈ C implies thefollowing conditions on the functions u and v, namely:

∂u

∂x=∂v

∂y,

∂u

∂y= −∂v

∂x, (A.1.2)

which are called the Cauchy-Riemann equations. These in turn, imply that both u and vobey Laplace’s equation at the given point (x, y) corresponding to z, namely

∆u = ∆v = 0 , ∆ =∂2

∂x2+

∂2

∂y2. (A.1.3)

Another consequence of f being analytic in a domain D ⊂ C is that in that domain fis automatically not only once complex differentiable, but actually can be differentiated an

1This can be formulated more precisely, by using the strict definition of the complex limit, namely:

limz→z0

g(z) = a

to mean that for all ε > 0 there is a δ > 0 such that for z ∈ D(z0; r) − 0, (D(z0; r) denoting the disk ofradius r in the complex plane centred at z0) for some finite nonzero r, |z−z0| < δ implies that |g(z)−a| < ε .


arbitrary number of times (i.e., not only f ′(z) exists, but als f ′′(z), f ′′′(z) and generally

f (n) = dnfdzn as well), and admits a convergent Taylor series expansion:

f(z) = f(z0) +f ′(z0)

1!(z − z0) +

f ′′(z0)

2!(z − z0)

2 + · · · , z0 ∈ D . (A.1.4)

One of the main consequences of a function being analytic is Cauchy’s theorem, which isone of the centre points of analytic function theory. This theorem states the following:

Theorem: If the function f is analytic in an open domain D ⊂ C, then on any closedcurve Γ which is homotopic to a point2 in D then the contour integral

∫

Γ

f(z)dz = 0 .

A corollary of Cauchy’s theorem is Cauchy’s integral formula which states that for fanalytic in a domain D ⊂ C and Γ a closed curve homotopic to a point in D, then

f(z0) =1

2πi

∫

Γ

f(z)

z − z0dz , (A.1.5)

for z0 ∈ D in the interior of the curve Γ, and where the curve is parametrised in such away that in the contour integral we go around the curve only once in the counter clock-wisedirection. This formula tells us that all values of the analytic function f for points insidethe curve can be found from knowing solely the values of f on the curve! In a similar waywe have an integral formula for all the derivatives of the analytic function f as well.

Thus, analyticity (complex differentiability) is a very strong property. However, it canhappen that complex functions are not everywhere in C differentiable. If a function is ana-lytic everywhere in C then we call the function an entire function, examples being functionslike:

ez , sin(z) =eiz − e−iz

2i, cos(z) =

eiz + e−iz

2.

It can also happen that functions have singular behaviour at isolated points, such as:

1

z,

1

z − z0,

1

(z − z0)n(n integer) .

These isolated points are called poles of the function, and usually these kind of singularitiesare somewhat benign. Worse singular behaviour happens if we consider functions such aslog z,

√z or zp with p noninteger, in which case the maximal domains on which these fuctions

can be regarded as analytic are the ones where we take out from C an entire infinite line(called a branch cut). Furthermore, in these cases the problem of multivaluedness arises, i.e.the problem that there are ambiguities arising in order to define the function as a single-valued object. We refer to the textbooks on analytic function theory for discussions on howto incorporate such functions into the theory.

2Homotopic to a point means that the curve can be continuously deformed (shrunk) to a point, i.e. withthe shrinking curves being all entirely in the domain of interest.


Functions which are analytic apart from the appearance of isolated singularities (poles)are calledmeromorphic functions. If f is meromorphic in a domainD then in a neighborhoodof any regular (nonsingular) point it will admit a Taylor expansion of the form (A.1.4), butmore generally in the neighborhood of a pole z0 it admits the more general Laurent seriesexpansion which is of the form:

f(z) =

∞∑

n=−∞

an(z − z0)n , (A.1.6)

which converges uniformly on any annulus (annular region) surrounding z0 where f is ana-lytic. The coefficients in (A.1.6) are given by

an =1

2πi

∫

Γ

f(ζ)

(ζ − z0)n+1dζ , n = . . . ,−1, 0, 1, 2, . . . .

If an = 0 for n < −p, we say that z0 is a pole of order p, and if an = 0 for n < 0 thefunction is analytic at z0 (in which case the coefficient an for n positive equals f (n)(z0)/n! ).The coefficient a−1 (if it exists) is called the residue of the function f at z0. These values,denoted by Res

z=z0f(z), play a major role in the socalled residue calculus, which is a complex

analytic technique (based on Cauchy’s theorem) to evaluate definite integrals and infinitesums.

Periodic and multiply periodic functions

We will restrict ourselves from now on to meromorphic functions, which as explained aboveare functions having as only singularities poles (and zeroes, which can be considered as beingsingularities at z = ∞). Often it will be useful to consider function, rather than on C, asfunctions on the compactified complex plane C ∪ ∞, where the point z = ∞, which isidentified with the set ζ ∈ C | |ζ| ≥ R , for all R > 0, is added to the complex plane.

Let f be a complex function, then if for any regular point z we have

f(z +Ω) = f(z) , Ω ∈ C fixed and nonzero ,

then f is called periodic with period Ω. Clearly, in that case we have f(z +mΩ) = f(z) forany integer m. If two functions f , g are periodic with the same period Ω, then pointwisesums and differences f ± g, products fg and quotients f/g, and derivatives f ′, are periodicas well with the same period. Any integer multiple of a period Ω is also a period. A periodΩ is called primite if any other period of the function is an integer multiple of the primitiveperiod.

A multiply periodic function is a function that has more than one primitive periods,namely Ω1,Ω2, . . . ,Ωn, with

f(z +m1Ω1 + · · ·+mnΩn) = f(z) ,

with the Ωi all independent (meaning that the only nontrivial integral linear combinationof these periods adding up to zero is the one with all coefficients equal to zero). The set ofall points of the form m1Ω + · · ·+mnΩn , with the mi integer, is called the period lattice.


An interesting question is whether functions exist with more than one nontrivial inde-pendent period. For n = 1 we know many examples, such as

f(z) = e2πiz/Ω ,

which is manifestly periodic with primitive period Ω. The answer is given by the followingstatement:

Proposition: 1. There does not exist a nonconstant function with n ≥ 3 primitive periods;2. there exist nonconstant functions with n = 2 given primitive periods iff the ratio of theseperiods is not real.

This statement is rephrasing of a theorem of Jacobi. A proof can be found in [Akhiezer].

Elliptic functions and elliptic curves

Definition: A complex function f(z) is called an elliptic function if it is meromorphic anddoubly periodic, i.e. it admits two independent primitive periods.

At least one of the two prinitive periods Ω1,Ω2 of an elliptic function should be complexsince the ratio Ω2/Ω1 should be nonreal. Hence the form a parallellogram of the form:

Ω1

Ω2

O

and thus the complex plane can be tesselated by all the parallelograms formed by the periodlattice obtained by translating this parallelogram over integer multiples of the two periods.The collection of the following statements are known as Liouville’s theorem.

The theory of elliptic functions provides a number of general results which we list herewithout proof:

1. elliptic functions are fully characterised (up to a constant multiplicative factor) bytheir poles and zeroes, as well as their periods;


2. the sum of the residues with respect to all the poles inside a single parallelogram ofthe period lattice is zero;

3. there does not exist a nonconstant elliptic function that is regular in a period paral-lelogram;

4. the number of poles of an elliptic function in a period parallelogram counting multi-plicity cannot be less than 2.

The elliptic functions are closely connected to a family of complex algebraic curves, calledelliptic curves. These are curves which, in appropriate coordinates, can be cast in the form:

w2 = R(z) , (A.1.7)

where R is a polynomial of order p = 3 or p = 4 in z, i.e.

R(z) = αz3 + βz2 + γz + δ or R(z) = αz4 + βz3 + γz2 + δz + ǫ .

Although we can consider (A.1.7) for real variables, it is more useful to consider this equationfor complex values of w and z. The elliptic functions arise as the functions in terms of whichone can naturally parametrise such curves. We can compare this to the case where the orderp of the polynomial R is p ≤ 2 in which case we call the curve rational, and in which casethe curve can be parametrised in terms of trigonometric functions. The simplest exanple ofthe latter is the case of the cicle:

w2 + z2 = 1 ⇔ w = cos(t) , z = sin(t) .

The elliptic functions are, thus, closely related to the geometry of the above mentionedelliptic curves. Although this would be the starting point for most treatments, it is not theone adopted in this course. We will take a constructive point of view where we will startwith a definition of the functions involved in terms of explicit formulae.

A.2 Theta Functions

We define the theta functions with characteristic a, b = 0, 1 and modulus τ as follows:

θab(x|τ) =∑

n∈Z

exp

[πiτ(n+

a

2)2 + 2πi(n+

a

2)(x+

b

2)

](A.2.1)

The sum on the r.h.s. being uniformly convergent for all |x| ≤ R and all R > 0 wheneverℑ τ > 0, (i.e. τ is in the Siegel half-plane τ ∈ H), the theta function is an entire function ofx ∈ C. It is doubly quasi-periodic with the periodicity relations being given by:

θab(x+ 1|τ) = eπiaθab(x|τ) , θab(x+ τ |τ) = e−πi(τ+2x+b)θab(x|τ) . (A.2.2)

The resummation n↔ −n in (A.2.1) leads to the relation

θab(−x|τ) = θ−a,−b(x|τ) = eπi(τa2−2ax)θab(x− aτ |τ) = eπiabθab(x|τ) , (A.2.3)

where the last two steps hold for b integer. Thus, it is immediately clear that θ00, θ01, θ10are all even functions of its argument, but that θ11 is an odd function:

θ00(−x|τ) = θ00(x|τ) , θ10(−x|τ) = θ10(x|τ) , θ01(−x|τ) = θ01(x|τ) , θ11(−x|τ) = −θ11(x|τ) .

A.2. THETA FUNCTIONS 37

Fundamental Addition Formulae

We will now use the series (A.2.1) to obtain expressions for the products of two thetafunctions, namely

θab(x|τ)θa′b′(y|τ) =

=∑

n,m∈Z

exp

πiτ [(n+

a

2)2 + (m+

a′

2)2] + 2πi[(n+

a

2)(x+

b

2) + (m+

a′

2)(y +

b′

2)

=∑

n,m∈Z

exp

1

2πiτ [(n +m+

a+ a′

2)2 + (n−m+

a− a′

2)2]

+πi[(n+m+a+ a′

2)(x + y +

b+ b′

2) + (n−m+

a− a′

2)(x − y +

b− b′

2)]

This invites the change of summation variables from n, m to N = n+m resp. M = n−m.However, since N +M = 2n ∈ 2Z and N −M = 2m ∈ 2Z implying that either both N andM are even integers, or that they are both odd. Thus these replacements in the double sumlead to ∑

n,m∈Z

∑

N,M∈Z

N,M even

+∑

N,M∈Z

N,M odd

Replacing N , M in the first sum by 2N , 2M and in the second sum by 2N + 1, 2M + 1 weget

. . . =∑

N,M∈Z

exp

2πiτ [(N +

a+ a′

4)2 + (M +

a− a′

4)2]

+2πi[(N +a+ a′

4)(x+ y +

b+ b′

2) + (M +

a− a′

4)(x − y +

b− b′

2)]

(A.2.4)

+∑

N,M∈Z

exp

2πiτ [(N +

a+ a′ + 2

4)2 + (M +

a− a′ + 2

4)2]

+2πi[(N +a+ a′ + 2

4)(x+ y +

b+ b′

2) + (M +

a− a′ + 2

4)(x − y +

b− b′

2)

= θAB(x+ y|2τ)θA′B′(x − y|2τ) + θA+1,B(x+ y|2τ)θA′+1,B′(x − y|2τ) (A.2.5)

with new characteristics:

A =a+ a′

2, B = b+ b′

A′ =a− a′

2, B′ = b− b′

Clearly for a, b, a′, b′ ∈ Z2 to have integer characteristics we must take either a = a′ = 0 ora = a′ = 1, whilst b and b′ can take on all values in Z2. This leads to the eight equations:

θ00(x|τ)θ00(y|τ) = θ00(x+ y|2τ)θ00(x − y|2τ) + θ10(x+ y|2τ)θ10(x − y|2τ)θ01(x|τ)θ00(y|τ) = θ01(x+ y|2τ)θ01(x − y|2τ) + θ11(x+ y|2τ)θ11(x − y|2τ)θ00(x|τ)θ01(y|τ) = θ01(x+ y|2τ)θ01(x − y|2τ)− θ11(x+ y|2τ)θ11(x − y|2τ)θ01(x|τ)θ01(y|τ) = θ00(x+ y|2τ)θ00(x − y|2τ)− θ10(x+ y|2τ)θ10(x − y|2τ)


as well as

θ10(x|τ)θ10(y|τ) = θ10(x + y|2τ)θ00(x− y|2τ) + θ00(x + y|2τ)θ10(x− y|2τ)θ11(x|τ)θ10(y|τ) = θ11(x + y|2τ)θ01(x− y|2τ) + θ01(x + y|2τ)θ11(x− y|2τ)θ10(x|τ)θ11(y|τ) = θ11(x + y|2τ)θ01(x− y|2τ)− θ01(x + y|2τ)θ11(x− y|2τ)θ11(x|τ)θ11(y|τ) = −θ10(x+ y|2τ)θ00(x− y|2τ) + θ00(x+ y|2τ)θ10(x− y|2τ) .

(A.2.6)

These identities will serve as the starting point from which all key identities for ellipticfunctions will be derived. Fortunately, these relations are not all independent, and neitherare the four theta functions θ00, θ10, θ01, θ11. In fact, these functions are related throughshifts over half periods, i.e. shifts over 1

2 , andτ2 in the argument. They are given by the

relations

θa,b+1(x|τ) = θab(x+1

2|τ) , θa+1,b(x|τ) = e

14πiτ+πi(x+ b

2)θab(x+

τ

2|τ) . (A.2.7)

Using these relations the fore-last equation in the list can be written as:

∣∣∣∣θ11(x|2τ) θ11(y|2τ)θ01(x|2τ) θ01(y|2τ)

∣∣∣∣ = θ11

(x+ y

2− 1

2| τ)θ11

(x− y

2| τ)

(A.2.8)

which is (A.2.17) for N = 2, whilst the others can be deduced from (A.2.8) by appropriateshifts in the arument (shifts over τ and 1). Furthermore, taking y = x + 1 we obtain from(A.2.8) the famous Landen transform relations, in particular:

θ11(x|τ) = 2e−πix+ 12πiτ θ11(x|2τ)θ11(x− τ |2τ)

iθ11(− 12 |τ)

(A.2.9)

connecting elliptic functions of one set of period to the ones with one of the periods halved.Below we present a general proof of the identities (A.2.8) and (A.2.21) which allows us

to identify the proportionality factor CN (τ) in (A.2.17). Below we show that from (A.2.6)all well-known identities between elliptic functions of the same modulus (periods) can bederived, either in the Jacobian representation or in the Weierstrass representation. Also allfive-term theta-function relations can be found, such as:

2θ00(x)θ00(y)θ00(z)θ(w) =

= θ00(X)θ00(Y )θ00(Z)θ00(W ) + θ01(X)θ01(Y )θ01(Z)θ01(W )

+ θ10(X)θ10(Y )θ10(Z)θ10(W ) + θ11(X)θ11(Y )θ11(Z)θ11(W ) (A.2.10)

in which all theta functions are of the same modulus τ and where we have abbreviated:

X =1

2(x+ y + z + w) , Y =

1

2(x + y − z − w)

Z =1

2(x− y + z − w) , W =

1

2(x− y − z + w)


using the oddness of the θ11 one actually has a closed-form equation for this function, namely:

θ11(x+ y)θ11(x− y)θ11(z + w)θ11(z − w) + θ11(x+ z)θ11(x− z)θ11(w + y)θ11(w − y)

+ θ11(x + w)θ11(x− w)θ11(y + z)θ11(y − z) = 0 (A.2.11)

The theta function relation (A.2.11) constitutes the key identity which we will use for theconstitution of the Weierstrass family of elliptic functions in the next section.

Product Formulae

One of the most important type of identities in combinatorics are the ones that relate infinitesums to infinite products, e.g. the famous Rogers-Ramanujan identities. Many of theseidentities have an origin in the representation theory of infinite-dimensional Lie algebras.One of the most fundamental of such relations is the Jacobi triple product relation, whichcan be most easily formulated as:

∑

n∈Z

qn(n−1)/2(−z)n =∞∏

j=1

(1− qj−1z)(1− qjz−1)(1− qj) , |q| < 1 . (A.2.12)

Proof: The proof is elementary. If we call the r.h.s. ϑ(z; q), it is easy to see that itconverges uniformly for all z within a finite radius whenever |q| < 1. By reordering thefactors it is clear that the following q-difference equation holds:

ϑ(z; q) = −zϑ(qz; q)

Thus, if we assume the expansion

ϑ(z; q) =∑

n∈Z

cnzn

we obtain the recursion relation cn = −qn−1cn−1 for the coefficients, which leads to cn =(−1)nqn(n−1)/2c0 upon iteration. The coefficient c0 = c0(q) can be determined by consider-ing the series and product for z = iq1/2, in which case we get

∞∏

j=1

(1 + q2j−1)(1− qj) = c0(q)∑

k∈Z

(−1)kq2k2

,

whilst if we take z = q1/2 we get

∞∏

j=1

(1− qj−1/2)2(1− qj) = c0(q)∑

k∈Z

(−1)kq12k2

,

a comparison of which leads to the relation

c0(q4)

c0(q)=

∏∞j=1(1− q4j−2)2(1− q4j)∏∞j=1(1 + q2j−1)(1 − qj)

= 1


where the last equality is a consequence of the surprising relation

∞∏

j=1

(1 − qj) =

∞∏

k=1

(1 − q2k−1)(1− q2k) =

∞∏

k=1

(1− q2k−1)(1 + qk)(1− qk)

⇒∞∏

k=1

(1 − q2k−1)(1 + qk) = 1 .

Thus it follows that c0(q) = c0(0) = 1, since if q = 0 the l.h.s. of (A.2.12) reduces to 1− z,whilst the r.h.s. only gets contributions from j = 0, 1. Thus, we obtain the required identity.Q.E.D.

Eq. (A.2.12) is a confirmation of the fact that ϑ(z; q) is an entire function of z with simplezeroes at z = qj , j ∈ Z. We mention that the product

η(q) = q1/24∞∏

j=1

(1− qj) (A.2.13)

which enters in the coefficient of (A.2.12) plays an important role in combinatorics and isknown as the Dedekind η-function. A classic result by Euler relates the infinite product in(A.2.13) to the infinite sum

∑k∈Z

(−1)k exp((3k2 + k)/2), whilst its inverse serves as thegenerating function of partitions.

It is clear that by takingz = e2πix , q = e2πiτ

the function ϑ(z; q) reduces to the θ11 function (apart from a factor). Thus, the tripleproduct relation (A.2.12) directly yields the various expressions for the theta functions interms of infinite products.

Theta functions of rational characteristic

We introduce now so-called theta function of rational characteristic, i.e. when the charac-teristics (the labels of the theta functions) are given as fractions of a natural number N ,namely we introduce:

ϑj(x) =∑

n∈Z

exp

[Nπiτ(n+

1

2− j

N)2 + 2πi(n+

1

2− j

N)(x+

1

2)

](A.2.14)

for N positive integer and j = 0, 1, . . . , N , which obeys the periodicity conditions

ϑj(x+ 1) = −e−2πi(j/N)ϑj(x) , ϑj(x+Nτ) = −e−πi(Nτ+2x)ϑj(x) . (A.2.15)

Collecting such theta function in a matrix of the form

Θ(x) =

ϑ0(x1) ϑ0(x2) . . . ϑ0(xN )ϑ1(x1) ϑ1(x2) . . . ϑ1(xN )

......

...ϑN−1(x1) ϑN−1(x2) . . . ϑN−1(xN )

(A.2.16)


where x = (x1, . . . , xN ), the following remarkable identity holds

det(Θ(x)) = CN (τ)θ11

(x1 + · · ·+ xN

N− N − 1

2| τ) N∏

i<j=1

θ11

(xi − xjN

| τ)

(A.2.17)

Proof: The proof of eq. (A.2.17) is remarkably simple. Defining f(x) = f(x1, . . . , xN ) =det(Θ(x)), whilst g(x) = g(x1, . . . , xN ), denoting the product of theta functions on the r.h.s.of eq. (A.2.17), we use the periodicity conditions (A.2.15) to establish that

f(. . . , xj +N, . . . ) = (−1)Nf(. . . , xj , . . . ) ,

f(. . . , xj +Nτ, . . . ) = −e−πi(2xj+Nτ)f(. . . , xj , . . . )

On the other hand, using the periodicity conditions (A.2.2) for θ11 we find exactly the sameperiodicity conditions for the functions g(x1, . . . , xN ) in all its arguments:

g(. . . , xj +N, . . . ) = (−1)Ng(. . . , xj , . . . ) ,

g(. . . , xj +Nτ, . . . ) = −e−πi(2xj+Nτ)g(. . . , xj , . . . )

Noting further that

ϑj(x) = eπiτ(j2/N)−2πi(x+ 1

2)j/Nθ11(x− jτ |Nτ) , (A.2.18)

it follows that the zeroes of ϑj(x) appear at x = jτ mod(ΛN ), where ΛN = n+Nmτ |n,m ∈Z is the lattice generated by 1, Nτ . This implies that ϑ0(x),. . . ,ϑN−1(x) do not havesimultaneously a zero, the only zero occurring if xi−xj = 0mod(ΛN ), for any i, j = 1, . . . , N ,i.e. if (xi − xj)/N = 0mod(Λ1). Concludingly, both f(sx) and g(sx) are doubly periodicentire functions with the same periodicities in each argument and having the same zeroes,hence by Liouville’s theorem they must be proportional to eachother. Q.E.D.

Although strangely enough the fundamental relation (A.2.17) seems not to have appearedin any of the classic textbooks on elliptic functions, (see list of references), expressionsin terms of theta functions of one argument as products of theta functions of differentmoduli do occur in the text (cf. e.g. [Tannery & Molk], nos. 134-140), and these areknown as transformation rules of order N . They can seen to be simple consequences of thefundamental relation (A.2.17), as follows. Let us take

xj = x+ (j − 1) , j = 1, . . . , N

and use the periodicity relations (A.2.15) as well the relation (A.2.18) from which we havethat

ϑj+1(x) = eπiτ/N−2πi(x+1/2)/Nϑj(x−τ) ⇒ ϑj(x) = eπiτj2/N−2πi(x+1/2)(j/a)Nθ11(x−jτ |Nτ)

and using also ϑN−j(x) = ϑj(−x− 1) we get that

CN (τ)

[∏

k<l

θ11

(k − l

N|τ)]

θ11(x|τ) = (A.2.19)

= deteπiτj

2/N−2πi(jk/N)(−1)k+jj,k=0,...,N−1

e−πi(N−1)xN−1∏

j=0

θ11(x − jτ |Nτ)


This equation generalises the Landen transform (A.2.9) to a general Nth order transforma-tion between theta functions of modulus τ and theta functions of modulus Nτ .

A further consequence of (A.2.17), which in fact can be considered to be an ellipticanalogue of the famous VanderMonde determinant formula, we find using Cramer’s rulefrom linear algebra:

[Θ−1(u−Nx) ·Θ(u−Ny)

]ij= Ψκ(xi − yj)

∏Nk=1 θ11(xk − yj)∏k 6=i θ11(xk − xi)

, (A.2.20)

where κ = u−(∑N

i=1 xi

)− N−1

2 , and

Ψκ(x) =θ11(x+ κ)

θ11(x)θ11(κ),

the θ11 functions on the r.h.s. all being of modulus τ . The theta function relation (A.2.11)as well as the relation (A.2.20), the determinant of which leads to the famous Frobeniusformula, (A.3.15) below, constitutes the key identities for the Weierstrass family of functions,cf. section 4 below.

The general proof given above of (A.2.17) does not provide us with the proportionalityconstant CN . A constructive proof can be given for low values of N , as is clear from thecase N = 2. Proceeding along similar lines for N = 3 we obtain the formula:

θ11

(x+ y + z

3| τ)θ11

(x− y

3| τ)θ11

(x− z

3| τ)θ11

(y − z

3| τ)

=

= −eπi/3ϑ1(0)

∣∣∣∣∣∣

ϑ0(x) ϑ0(y) ϑ0(z)ϑ1(x) ϑ1(y) ϑ1(z)ϑ2(x) ϑ2(y) ϑ2(z)

∣∣∣∣∣∣(A.2.21)

Proof: This case reveals truly the complex combinatorics that is sitting behind eq. (A.2.17).In this case the theta functions of rational characteristic from (A.2.14) we will be using are:

ϑ0(x) =∑

n∈Z

exp

[2πi(n+

1

2)(x +

1

2) + 3πiτ(n+

1

2)2]

(A.2.22a)

ϑ1(x) =∑

n∈Z

exp

[2πi(n+

1

6)(x +

1

2) + 3πiτ(n+

1

6)2]

(A.2.22b)

ϑ2(x) =∑

n∈Z

exp

[2πi(n− 1

6)(x +

1

2) + 3πiτ(n− 1

6)2]

(A.2.22c)


We now investigate the product

θ11

(x1 + x2 + x3

3| τ)θ11

(x1 − x2

3| τ)θ11

(x1 − x3

3| τ)θ11

(x2 − x3

3| τ)

=

=∑

n∈Z4

exp

2πi

[(n0 +

1

2)(x1 + x2 + x3

3+

1

2) + (n12 +

1

2)(x123

+1

2)

+(n13 +1

2)(x133

+1

2) + (n23 +

1

2)(x233

+1

2)

]

+πiτ

[(n0 +

1

2)2 + (n12 +

1

2)2 + (n13 +

1

2)2 + (n23 +

1

2)2]

=∑

n∈Z4

exp

2πi

[(x1 +

3

2)(n0 + n12 + n13

3+

1

2) + (x2 +

1

2)(n0 − n12 + n23

3+

1

6)

+(x3 −1

2)(n0 − n13 − n23

3− 1

6) +

1

2(n12 − n13 + n23

3+

1

6)

]

+1

3πiτ

[(n0 + n12 + n13 +

3

2)2 + (n0 − n12 + n23 +

1

2)2

+(n0 − n13 − n23 −1

2)2 + (n12 − n13 + n23 +

1

2)2]

where for notational convenience we use as summation variables the integers n0, nij , (i, j =1, 2, 3), constituting a four-fold sum over the vector of integers n = (n0, n12, n13, n23), andwhere we abbreviate xij ≡ xi − xj . The last way of rewriting the product invites a changeof summation variables, namely

N1 = n0 + n12 + n13

N2 = n0 − n12 + n23

N3 = n0 − n13 − n23

N0 = n12 − n13 + n23

⇒

N1 +N2 +N3 ∈ 3ZN1 −N2 +N0 ∈ 3ZN1 −N3 −N0 ∈ 3ZN2 −N3 +N0 ∈ 3Z

which means that the new summation variables actually are represented as:

Ni = −αi + 3mi , mi ∈ Z , αi ∈ Z3

with the αi are subject to the conditions

α1 + α2 + α3 = 0 (mod 3) , α1 + α2 + α0 = 0 (mod 3)

α1 − α3 − α0 = 0 (mod 3) , α2 − α3 + α0 = 0 (mod 3) (A.2.23)


In terms of these new variables αi,mi, (i = 0, 1, 2, 3) the sum above can be rewritten as

. . . =∑

α∈Z43

′∑

m∈Z4

exp

2πi(x1 +

1

2+ 1)(m1 +

1

2− α1

3) + 3πiτ(m1 +

1

2− α1

3)2

2πi(x2 +1

2)(m2 +

1

2− α2 + 1

3) + 3πiτ(m2 +

1

2− α2 + 1

3)2

2πi(x3 +1

2− 1)(m3 +

1

2− α3 + 2

3) + 3πiτ(m3 +

1

3− α3 + 2

3)2

2πi(1

2)(m0 +

1

2− α0 + 1

3) + 3πiτ(m0 +

1

2− α0 + 1

3)2

=∑

α∈Z43

′ ϑα1(x1 + 1)ϑα2+1(x2)ϑα3+2(x3 − 1)ϑα0+1(0)

where the prime in the sum over the αi denotes that we perform a restricted sum under thecondition (A.2.23). Noting that ϑα(0) = 0 for α = 0 (mod3) in the present case, in the sumover α0 we have only contributions from α0 = 0, 1, leading to

. . . = ϑ1(0) [ϑ0(x1 + 1)ϑ1(x2)ϑ2(x3 − 1) + ϑ1(x1 + 1)ϑ2(x2)ϑ0(x3 − 1)

+ϑ2(x1 + 1)ϑ0(x2)ϑ1(x3 − 1)]

+ϑ2(0) [ϑ0(x1 + 1)ϑ2(x2)ϑ1(x3 − 1) + ϑ1(x1 + 1)ϑ0(x2)ϑ2(x3 − 1)

+ϑ2(x1 + 1)ϑ1(x2)ϑ0(x3 − 1)]

where use has been made of the condition (A.2.23), and when we make use of the periodicityconditions

ϑ0(x± 1) = −ϑ0(x) , ϑ1(x± 1) = e±πi/3ϑ1(x) , ϑ2(x± 1) = e∓πi/3ϑ2(x)

together with the fact that ϑ2(0) = e−πi/3ϑ1(0), an easy calculation demonstrates that werecover eq. (A.2.21). QED.

A.3 The Weierstrass Family of Elliptic Functions

Here, we collect some useful formulae for the Weierstrass elliptic functions. The Weierstrasssigma-function is basically the θ11 apart from a multiplicative factor and a scaling in itsargument. It is the latter theta function for which we have a closed form addition formula,namely (A.2.11)), so understandibly this function plays the main role in the theory.

Thus, we define the so-called Weierstrass σ-function to be given by

σ(z|2ω1, 2ω2) = 2ω1 exp

(η1z

2

2ω1

)θ11(x|τ)θ′11(0|τ)

, τ =ω2

ω1, z = 2ω1x . (A.3.1)

The exponential prefactor and scaling are motivated in order to make the function auto-morphic under the action of the group SL(2,Z), i.e. invariant under linear transformationsacting on the period lattice:

(ω1

ω2

)7→

(α βγ δ

) (ω1

ω2

)(A.3.2)

A.3. THE WEIERSTRASS FAMILY OF ELLIPTIC FUNCTIONS 45

The relations between the Weierstrass elliptic functions are given by

ζ(z) =σ′(z)

σ(z), ℘(z) = −ζ′(z) , (A.3.3)

where σ(z) and ζ(z) are odd functions and ℘(z) is an even function of its argument. Werecall also that the σ(z) is an entire function, and ζ(z) is a meromorphic function havingsimple poles at ωkl, both being quasi-periodic, obeying

ζ(x + 2ω1,2) = ζ(x) + 2η1,2 , σ(x+ 2ω1,2) = −σ(x)e2η1,2(x+ω1,2) , (A.3.4)

in whichηi ≡ ζ(ωi) , i = 1, 2 .

The relation between periods and the ηi is given by

η1ω2 − η2ω1 =πi

2.

The function ℘(z) is doubly periodic, and it is only the ℘ function that is truly ellipticaccording to the definition of an elliptic function.

As a consequence of the product formula for the theta functions we obtain the followingproduct formula foe the Weierstrass sigma-function:

σ(z) = z∏

(k,ℓ) 6=(0,0)

(1− z

ωkℓ) exp

[z

ωkℓ+

1

2(z

ωkℓ)2], (A.3.5a)

with ωkl = 2kω1 + 2ℓω2. Taking the logarithmic derivatives we obtain from (A.3.5a) thefollowing double-sum expansions for the Weierstrass ζ- and ℘ functions:

ζ(z; 2ω1, 2ω2) =1

z+

∑

(k,ℓ) 6=(0,0)

[1

z + ωk,ℓ− 1

ωk,ℓ+

z

ω2k,ℓ

], (A.3.5b)

℘(z; 2ω1, 2ω2) =1

z2+

∑

(k,ℓ) 6=(0,0)

[1

(z + ωk,ℓ)2− 1

ω2k,ℓ

]. (A.3.5c)

From a computational point of view, the most important property of these elliptic func-tions is the existence of a number of functional relations, the most fundamental being thethree-term relation for the σ-function

σ(x + y)σ(x− y)σ(a+ b)σ(a− b) = (A.3.6)

= σ(x+ a)σ(x − a)σ(y + b)σ(y − b)− σ(x + b)σ(x− b)σ(y + a)σ(y − a) ,

which is a direct consequence of the similar relation for the θ11 functions (A.2.11). Fromthis functional equation one can derive by differentiation:

ζ(x) + ζ(y) + ζ(z)− ζ(x + y + z) =σ(x + y)σ(x+ z)σ(y + z)

σ(x)σ(y)σ(z)σ(x + y + z). (A.3.7)


In fact, all identities for the Weiertrass functions as functions of their arguments can bederived from these functional relations.

Now it will be convenient to express the addition formulae in terms of a function Φκ(x)(with κ some complex number) given by

Φκ(x) ≡σ(x + κ)

σ(x)σ(κ), (A.3.8)

which is similar to the function Ψ used in the previous section3.Eq. (A.3.6) can also be cast into the following form

Φκ(x)Φκ(y) = Φκ(x+ y) [ζ(κ) + ζ(x) + ζ(y)− ζ(κ+ x+ y)] , (A.3.9)

The following three-term relation for σ(x) is a consequence of (A.3.7) and this equation canbe cast into the following convenient form

Φκ(x)Φλ(y) = Φκ(x− y)Φκ+λ(y) + Φκ+λ(x)Φλ(y − x) , (A.3.10)

which is obtained from the elliptic analogue of the partial fraction expansion, i.e. eq. (A.3.9).From (A.3.7) various other addition formulae can be derived by simple manipulations,

taking into account that these are functional identities valid for all values of the arguments(away from the singularities of the functions). Thus, by differentiation we can derive:

ζ(x + y)− ζ(x) − ζ(y) =1

2

℘′(x) − ℘′(y)

℘(x) − ℘(y). (A.3.11)

Furthermore, we have the relation

℘(x) + ℘(y) + ℘(x+ y) =1

4

(℘′(x) − ℘′(y)

℘(x)− ℘(y)

)2

. (A.3.12)

Exercise: Derive eqs. (A.3.11) and (A.3.12).

Eq. (A.3.12) is the well-known addition formula for the Weierstrass ℘ function. Finally,we get the following remarkable addition formula:

σ(x − y)σ(x + y)

σ2(x)σ2(y)= ℘(y)− ℘(x) , (A.3.13)

and the fundamental 3-term relation form σ (A.3.6) can be easily seen to arise as theconsistency condition for the functional relation (A.3.13), by rewriting the trivial identity

(℘(x) − ℘(y))(℘(a)− ℘(b)) = (℘(x) − ℘(a))(℘(y)− ℘(b)).− (℘(x) − ℘(b))(℘(y)− ℘(a))

in terms of σ-functions. Thus, we have come full circle: from the three-term relation for σwe have derived a sequel of addition formulae at the end of which we recover the three-termrelation itself!

3In the literature sometimes a different form for Φk(x) is introduced, namely

Φ(x; κ) :=σ(x− κ)

σ(x) σ(κ)eζ(κ)x ,

which is oftern referred to as the Lame funtion. We prefer to work with the form given in (A.3.8).


Higher degree addition formulae

We finish the list of addition formulae by mentioning a number of higher order additionformulae, which are most conveniently expressed in terms of the function Φ. The bilinearrelation (A.3.10) for the function Φ can be generlised to multilinear equations by simpleiteration. Thus we obtain e.g. the trilinear relation:

Φκ(x)Φλ(y)Φµ(z) = Φκ+λ+µ(x)Φλ(y − x)Φµ(z − x)

+Φκ(x− y)Φκ+λ+µ(y)Φµ(z − y) + Φκ(x− z)Φλ(y − z)Φκ+λ+µ(z) ,(A.3.14a)

The latter relations can be iterated to higher degree relations, the simplest of which isof the form which in the limit z → y yields the relation

Φκ(x)Φλ(y)Φµ(y)− Φκ+λ+µ(x)Φλ(y − x)Φµ(y − x) =

= Φκ(x− y)Φκ+λ+µ(y) [ ζ(λ) + ζ(µ) + ζ(y) − ζ(κ+ λ+ µ+ y) + ζ(κ+ x− y)− ζ(x − y) ] ,

(A.3.14b)

which can be conveniently rewritten as

ζ(κ) + ζ(λ) + ζ(µ) + ζ(x) + ζ(y)− ζ(κ+ λ+ µ+ x+ y) =

=Φκ(x)Φλ(x)Φµ(x)Φκ+λ+µ(y)− Φκ(y)Φλ(y)Φµ(y)Φκ+λ+µ(x)

Φκ+λ+µ(x + y) (℘(x)− ℘(y)). (A.3.14c)

In the double limit y, z → x yields the relation

Φκ(x)Φλ(x)Φµ(x) =1

2Φκ+λ+µ(x)

[ζ(κ) + ζ(λ) + ζ(µ) + ζ(x) − ζ(κ+ λ+ µ+ x)]

2

− [℘(κ) + ℘(λ) + ℘(µ) + ℘(κ+ λ+ µ+ x)− ℘(x)] . (A.3.14d)

In a similar fashion we can derive increasingly higher degree multilinear relations (all ofwhich can be easily proven by induction). These relations all follow from general ellipticdeterminant type relations, mostly due to Frobenius and Stickelberger, which we will explorenow.

Elliptic Determinantal Identities

Possibly the most important and surprising identity in the whole theory is the elliptic versionof the well-known Cauchy determinantal identity, which reads

det (Φκ(xi − yj)) = Φκ(Σ)σ(Σ)

∏k<ℓ σ(xk − xℓ)σ(yℓ − yk)∏

k,ℓ σ(xk − yℓ), where Σ ≡

∑

i

(xi − yi) ,

(A.3.15)and which is due to Frobenius, [Frobenius,1882].

Proof: A proof of the elliptic Cauchy formula (A.3.15) can be given by induction usingthe following general block determinantal formula:

∣∣∣∣∣∣∣∣

A b

ct d

∣∣∣∣∣∣∣∣= d1−N det

dA− b ct

, (A.3.16)


where A is an N × N matrix and where b and c a N -component column- and row vectorrespectively, and where d 6= 0. Applying this formula to a (N + 1)× (N + 1) matrix of theFrobenius form, namely

det (Φκ(xi − yj))i,j=1,...,N+1

and exploiting the addition formula (A.3.9) and (A.3.10) the induction step can be veri-fied. Furthermore for N = 2, the determinantal formula is readily verified using the sameequations. QED.

Corollary: The elliptic Cauchy determinantal formula (A.3.15) is closely related to an-other identity found by Frobenius and Stickelberger, [Frobenius & Stickelberger,1880], whichis of the form

∣∣∣∣∣∣∣∣∣

0 1 . . . 11... (ζ(xi − yj))1

∣∣∣∣∣∣∣∣∣= −σ(Σ)

∏k<ℓ σ(xk − xℓ)σ(yℓ − yk)∏

k,ℓ σ(xk − yℓ). (A.3.17)

Exercise: By taking the limit κ→ 0 in (A.3.15) derive the formula (A.3.17).

From (A.3.15), by expanding along one of its rows or columns, an elliptic form of theLagrange interpolation formula is obtained in the form

N∏

i=1

σ(ξ − xi)

σ(ξ − yi)=

N∑

i=1

Φ−Σ(ξ − yi)

∏Nj=1 σ(yi − xj)

∏Nj=1

j 6=i

σ(yi − yj), (A.3.18)

for Σ 6= 0, where

Σ ≡N∑

i=1

(xi − yi) . (A.3.19)

When Σ = 0 we recover the following formula

N∏

i=1

σ(ξ − xi)

σ(ξ − yi)=

N∑

i=1

[ζ(ξ − yi)− ζ(x− yi)]


∏Nj=1

j 6=iσ(yi − yj)

, (A.3.20)

in which x denotes any one of the zeroes xi. Note that in this case the left hand side isa meromorphic function on the elliptic curve as a consequence of Abel’s theorem. Using(A.3.18) it can be easily verified that eq. (A.3.20) is independent of the choice of x. In fact,this follows from the key property that

N∑

i=1


∏Nj=1

j 6=iσ(yi − yj)

= 0, (A.3.21)

whenever∑

i(xi − yi) = 0 . This latter relation (A.3.21) is nothing else than a rewriting of(A.3.18).


Finally, we give the expression for the inverse of the elliptic Cauchy matrix, namely[(Φκ(x· − y·))

−1]ij= Φκ+Σ(yi − xj)

P (yi)Q(xj)

Q1(yi)P1(xj), (A.3.22)

(with Σ as before), in terms of the elliptic polynomials

P (ξ) =

N∏

k=1

σ(ξ − xk) , Q(ξ) =

N∏

k=1

σ(ξ − yk) ,

andP1(xj) =

∏

k 6=j

σ(xj − xk) , Q1(yi) =∏

k 6=i

σ(yi − yk) . (A.3.23)

Equation (A.3.22) can be derived using (A.3.18) and (A.3.20).Finally, there are a number of alternative determinantal formulae that can be expressed

in terms of the ℘ function rather than in terms of the function Φ, and we give them withoutproof. Most celebrated is the Frobenius-Stickelberger formula:

∣∣∣∣∣∣∣∣∣∣∣∣

1 ℘(x0) ℘′(x0) · · · · · · ℘(n−1)(x0)1 ℘(x1) ℘′(x1) · · · · · · ℘(n−1)(x1)...

......

. . ....

......

.... . .

...

1 ℘(xn) ℘′(xn) · · · · · · ℘(n−1)(xn)

∣∣∣∣∣∣∣∣∣∣∣∣

=

= (−1)12n(n−1)1!2! · · ·n!

σ(x0 + x1 + · · ·+ xn)∏n

i<j=0 σ(xi − xj)

σn+1(x0)σn+1(x1) · · ·σn+1(xn). (A.3.24)

Denoting the Frobenius-Stickelberger matrix by P(x0, x1, . . . , xn) = P(x), given by thematrix associated with the left-hand side of (A.3.24), we have the following factorisationformula:

[P(x) · P(y)−1

]i,j

=1

σn+1(xi)ΦΣ(xi − yj)σ

n+1(yj)

∏nl=0 σ(xi − yl)∏l 6=j σ(yj − yl)

, (A.3.25)

in which Σ ≡ Σnl=0(xl − yl) . Thus, we can derive the determinantal formula (A.3.15) from

(A.3.24) and vice versa.

Landen Transforms

We finish by mentioning a number of identities which relate Weierstrass functions of differentperiods together. The famous Landen transforms relate elliptic functions of a given periodto those of twice (or vice versa half) the period. They are obtained from the fundamentalrelation (A.2.9) for the θ functions. Translating them to the Weierstrass family by using thecorrespondence (A.3.2) we obtain the following relations

σ(z) = e−12(e1+e2)z

2+η2zσ(z)σ(z − 2ω2)

σ(−2ω2)(A.3.26a)

η(z) = −(e1 + e2)z + η2 + ζ(z) + ζ(z − 2ω2) (A.3.26b)

℘(z) = e1 + e2 + ℘(z) + ℘(z − 2ω2) (A.3.26c)


in which σ(z) = σ(z|2ω1, 2ω2) , σ(z) = σ(z|2ω1, 4ω2) , and ζ(z), ℘(z) the correspondingWeierstrass functions with periods 2ω1, 2ω2, respectively ζ(z) and ℘(z) the correspondingWeierstrass functions with periods 2ω1, 4ω2.

A.4 Half-period functions and the elliptic curve

Since the Weierstrass functions are periodic or quasi-periodic with primitive period Forconvenience let us introduce as a third half period ω3 = −ω1 − ω2, for which η3 ≡ ζ(ω3) =−η1 − η2 . We then have also the relations:

η1ω2 − η2ω1 =πi

2⇒ η2ω3 − η3ω2 =

πi

2, η3ω1 − η1ω3 =

πi

2.

Introducing now the three functions:

Wi(x) ≡ Φωi(x)e−ηix , i = 1, 2, 3 (A.4.1)

we can derive several relations from the addition formulae (A.3.9) and (A.3.10) for Φ as wellusing the periodicity conditions (A.3.4).

The following properties hold for the functions Wi:

1. The Wi(x) are periodic with period 2ωi, but picking up a minus sign w.r.t. the otherperiods:

Wi(x+ 2ωi) =Wi(x) , Wi(x+ 2ωj) = −Wi(x) , i, j, k = 1, 2, 3 cyclic.

2. The Wi are odd functions w.r.t. their argument:

Wi(−x) = −Wi(x) .

3. The addition formula:

Wi(x)Wj(z) +Wj(y)Wk(x) +Wk(z)Wi(y) = 0 , x+ y + z = 0 . (A.4.2)

follows by applying (A.3.10) together with the periodicity relations (A.3.4).

4. The following relations are a consequence of (A.3.13) together with the periodicity(A.3.4)

W 2i (x) = ℘(x)− ei , ei ≡ ℘(ωi) , (A.4.3)

which means that for fixed constants e1, e2, e3 the (W1,W2,W3) are coordinates onthe intersection of three complex cylinders.

5. From (A.3.9) we obtain the relation

Wi(x)Wj(x)

Wk(x)= ηk + ζ(x) − ζ(x + ωk) = −1

2

℘′(x)

℘(x)− ek, i, j, k = 1, 2, 3 cyclic .

(A.4.4)

6. Using the periodicity (A.3.4) we have

Wi(x)Wi(x+ ωi) = − eηiωi

σ2(ωi). (A.4.5)

A.4. HALF-PERIOD FUNCTIONS AND THE ELLIPTIC CURVE 51

Exercise: Verify, by direct computation, the properties 1-6 of the functions Wi, using theaddition formulae and periodicity relations of the Weierstrass functions.

We are now in a position to derive the most important property, nemaly the equation forthe elliptic curve. In fact, combining (A.4.3) and (A.4.4), we get

W1(x)W2(x)W3(x) = −1

2℘′(x) , (A.4.6)

and taking the square of this relation, using again (A.4.3), we obtain:

(℘′(x))2 = 4(℘(x)− e1)(℘(x) − e2)(℘(x) − e3) . (A.4.7)

Using the fact thate1 + e2 + e3 = 0 ,

and taking as (complex) coordinates (z, w) = (℘(x), ℘′(x), we can cast (A.4.7) in the form:

w2 = R(z) = 4z3 − g2z − g3 , (A.4.8)

which is the standard form of the so-called Weierstrass curve, which is one of the standardforms of an elliptic curve. The constants g2, g3 are given in terms of the ei through theformulae:

g2 = −4(e1e2 + e1e3 + e2e3) , g3 = 4e1e2e3 , (A.4.9)

and are called the moduli of the curve. The roots ei, i = 1, 2, 3, of the cubic R(z) are calledthe branch points of the curve, and they are expressed through the formulae ei = ℘(ωi) interms of the Weierstrass elliptic function.

We finish by mentioning the relations:

℘′(x) = −σ(2x)σ4(x)

,1

2

℘′′(x)

℘′(x)= ζ(2x)− 2ζ(x) , (A.4.10)

as well as the so-called duplication formulae:

σ(2x) = 2σ(x)σ(x + ω1)σ(x + ω2)σ(x + ω3)

σ(ω1)σ(ω2)σ(ω3), (A.4.11a)

2ζ(2x) = ζ(x) + ζ(x + ω1) + ζ(x+ ω2) + ζ(x + ω3) , (A.4.11b)

4℘(2x) = ℘(x) + ℘(x+ ω1) + ℘(x+ ω2) + ℘(x+ ω3) . (A.4.11c)

Exercise: Derive eqs. (A.4.10) from (A.3.13) by setting y = x + ǫ and taking the limitǫ→ 0 , using σ(ǫ) ∼ ǫ in this limit. Subsequently, derive (A.4.11a) by using (A.4.6) and theothers by differentiation.

Finally, we mention that by eliminating the derivatives from (A.3.12) using the relationof the elliptic curve (A.4.7), we obtain the purely “discrete” addition formula for the ℘function, namely

(XY +XZ + Y Z +

g24

)2− (4XY Z − g3)(X + Y + Z) = 0 (A.4.12)


in which

X = ℘(x) , Y = ℘(y) , Z = ℘(z) such that x+ y + z = 0 .

The relation (A.4.12) can be viewed as a discretization of the formula for the elliptic curve:if we set z = ε and take the limit ε→ 0, the dominant contribution in the relation (A.4.12)goes over into the defining equation for the Weierstrass curve in standard form.

Literature

1. N.I. Akhiezer, Elements of the Theory of Elliptic Functions, AMS Transl. of Math.Monographs Series col. 79, (American Mathematical Society, 1980)

2. A. Erdelyi e.a., Higher Transcendental Functions, volume 2, (McGraw-Hill, New York,1953).

3. H. Hancock, Lectures on the Theory of Elliptic Functions, (Dover Publs., New York,1958)

4. D. Mumford, Lectures on Theta Functions, (Birkauser Verlag, 1983).

5. H. McKean and V. Moll, Elliptic Curves, (Cambridge University Press, 1999).

6. F. Kirwan, Complex Algebraic Curves, LMS Student Texts 23 (Cambridge UniversityPress, 1992).

7. E.T. Whittaker and G.N. Watson, A Course in Modern Analysis, (Cambridge Univer-sity Press, 4th ed., 1927, reprinted 1988).

8. H. Abel, Oeuvres Completes, (Christiania, 1881).

9. C.G.J. Jacobi, Fundamenta Nova, (Konigsberg, 1829).

10. K. Weierstrass, Werke, I, (1894), pp. 1–49, (1895), pp 245–255, 257–309.

11. C. Briot and J.C. Bouquet, Theorie des fonctions elliptiques, (Paris, 1875).

12. J. Tannery and J. Molk, Fonctions Elliptiques, (Paris, 1893-1902).

13. G.H. Halphen, Fonctions Elliptiques, (Paris, 1886).

14. G. Frobenius and L. Stickelberger, Zur Theorie der elliptischen Functionen, J. ReineAngew. Math. 83 (1877), 175–179.

15. G. Frobenius and L. Stickelberger, Uber die Addition und Multiplication der elliptis-chen Functionen, J. Reine Angew. Math. 88 (1880), 146–184.

16. G. Frobenius, Uber die elliptischen Functionen zweiter Art, J. Reine Angew. Math.93 (1882), 53–68.

Contents

1 Multi-Soliton Solutions 1

1.1 Infinite Recurrence Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Closed Form Lattice Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Derivation of Lax Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Extensions to other classes of P∆Es 12

2.1 Bilinear Form from Soliton Solutions . . . . . . . . . . . . . . . . . . . . . . . 132.2 ∗ Soliton solution of Q3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 Explicit Form of the N -Soliton Solutions . . . . . . . . . . . . . . . . 202.3 Higher-dimensional soliton systems: the KP class . . . . . . . . . . . . . . . . 21

2.3.1 Multidimensional solitons . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.2 Bilinear structure for discrete KP type equations . . . . . . . . . . . . 252.3.3 Multi-soliton formula for discrete KP systems . . . . . . . . . . . . . . 27

A ∗ Elliptic Functions & Theta Functions 30

A.1 Some Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31A.2 Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36A.3 The Weierstrass Family of Elliptic Functions . . . . . . . . . . . . . . . . . . 44A.4 Half-period functions and the elliptic curve . . . . . . . . . . . . . . . . . . . 50

53

Documents

Full set of lecture notes - School of Mathematics - University of Leeds