1. Introduction
We consider the well-known Hirota-Miwa equation represented in the following form [18, 27]
atn,mjtn+1,m+1j−tn+1,mjtn,m+1j=btn+1,mj−1tn,m+1j+1.(1.1)
Here a and b are constant parameters, the sought function tn,mj depends on three integers j,n,m. Equation (1.1) is the most important discrete integrable equation in three dimensions. Years ago, bilinear equations of the form (1.1) have found applications [19,37] in the context of quantum integrable systems as the model-independent functional relations for eigenvalues of quantum transfer matrices. Universality is a very remarkable property of this equation. As it was observed earlier by many authors (see [38] and the references therein) numerous of known integrable continuous and discrete models can be derived from (1.1) by performing appropriate symmetry reductions, continuum limits etc. Moreover, it is generally accepted that almost all integrable models can be obtained from the Hirota-Miwa equation with the help of a suitable reduction. Initiated by this idea, we studied the problem of integrable boundary conditions for (1.1) in order to derive discrete versions of the integrable systems of exponential type, known as Drinfeld-Sokolov hierarchies [8, 21, 36].
We will interpret the equation (1.1) as an infinite sequence of related quadrilateral equations defined on the flat graph (n,m) and endowed with the additional parameter j. Then we look for boundary conditions that, when imposed at two selected points, say, j = j0 and j = jN reduce the equation (1.1) to an integrable discrete system with a finite set of field variables tn,mj0+1,tn,mj0+2,…,tn,mjN−1.
Let us present some examples of quad systems connected with such kind reductions:
tn,m−1=1,atn,mjtn+1,m+1j−tn+1,mjtn,m+1j=btn+1,mj−1tn,m+1j+1, 0⩽j⩽N−1,tn,mN=1,(1.2)
tn,m−1=1,atn,mjtn+1,m+1j−tn+1,mjtn,m+1j=btn+1,mj−1tn,m+1j+1, 0⩽j⩽N−1,tn,m+1N=tn+1,mN−2,(1.3)
tn+1,m−1=tn,m+11,atn,mjtn+1,m+1j−tn+1,mjtn,m+1j=btn+1,mj−1tn,m+1j+1, 0⩽j⩽N−1,tn,m+1N+1=tn+1,mN−1.(1.4)
In the particular case when a = 1, b = 1 these systems were found in [12]. S.V. Smirnov proved that for a = b = 1 the quad systems (1.2), (1.3) are integrable in the sense of Darboux (see [32]).
In our recent article [17] we studied another reduction of this kind
tn+1,m−1=tn,m+1N−1,atn,mjtn+1,m+1j−tn+1,mjtn,m+1j=btn+1,mj−1tn,m+1j+1, 0⩽j⩽N−1,tn,m+1N=tn+1,m0.(1.5)
Note that for N = 2 the systems (1.4) and (1.5) coincide with each other.
By introducing new variables un,mj=−logtn,mj we can rewrite the equations (1.2)–(1.5) uniformly as follows
ae−un+1,m+1i+un,m+1i+un+1,mi−un,mi−1=bexp(∑j=0i−1ai,jun+1,mj+∑j=i+1N−1ai,jun,m+1j+ai,i2(un+1,mi+un,m+1i)), 0⩽i⩽N−1.(1.6)
Here A = ai,j is a constant matrix. For the cases (1.2) and (1.3) A coincides with the Cartan matrices of the simple Lie algebras AN−1 and BN−1, respectively. Similarly for (1.4) and (1.5) A is the generalized Cartan matrix for the affine Lie algebra DN(2) with N ⩾ 3 and AN−1(1) for N ⩾ 2.
In our article [17] we derived the finite-component lattice (1.5) by imposing a quasi-periodical boundary condition of the form tn,m+1j+N=tn+1,mj on the Hirota-Miwa equation (1.1). We observed that this constraint is consistent with a similar quasi-periodical boundary condition ψn,m+1j+N=λψn+1,mj for the Lax pair (2.1) of the lattice (1.1). In this way we derived immediately the Lax pair for (1.5).
In section 2 of the present article we discuss one more boundary condition for (1.1) having the form tn+1,m−1=tn,m+11 which is used in (1.4) together with the same kind constraint given at the other value of j:tn,m+1N+1=tn+1,mN−1. Unfortunately there is no any appropriate boundary condition of the form
F(ψ0,ψ±1,ψ±2,…)=0
depending also on the shifts of the variable
tn,mj, which being imposed on the system
(2.1) would provide the Lax pair for the semi-infinite lattice
tn+1,m−1=tn,m+11,atn,mjtn+1,m+1j−tn+1,mjtn,m+1j=btn+1,mj−1tn,m+1j+1, j⩾0.(1.7)
Actually in order to derive the Lax pair for (1.7) we need one more Lax pair for (1.1) (see (2.6) below). Indeed by setting the following boundary condition
ψn,m−1=λ−1gn,m+10 and gn,m−1=λψn−1,m0(1.8)
on the combined Lax pair
(2.1),
(2.6) we obtain the desired Lax pair for
(1.7). We refer to
(1.8) as gluing conditions. By imposing in addition to
(1.8) gluing conditions
(2.12) with
NR =
N we obtained in section
2 the Lax pair for
(1.4).
The presence of the constant parameter b allows us to realize a continuum limit passage in the system (1.6). The limit is calculated in section 3. In the case of DN(2), Lax pairs for quadrilateral systems tend to Lax pairs for the corresponding continuous systems. As an illustrative example, the system corresponding to D3(2) is considered.
One of the important applications of Lax pairs is the description of integrals of motion for the corresponding dynamical systems. Usually, for this purpose, asymptotic expansions of the Lax eigenfunctions around singular values of the spectral parameter are used. For discrete operators, the problem of constructing such expansions is complex and remains less studied. In section 4, we found transformations converting the Lax equations to a suitable form and constructed the necessary expansions, which, in turn, allowed us to construct generating functions for local conservation laws. This proves that the proposed Lax pairs are not “fake”.
In the fifth section we presented a higher symmetry for the quad system corresponding to D3(2).
2. Boundary conditions for the Hirota-Miwa equation
The Hirota-Miwa equation (1.1) provides the consistency of the following overdetermined system of the linear equations
ψ1,0j=t1,0j+1t0,0jt0,0j+1t1,0jψ0,0j−ψ0,0j+1, ψ0,1j=ψ0,0j+bt0,1j+1t0,0j−1t0,0jt0,1jψ0,0j−1(2.1)
which however doesn’t define the Lax pair for
(1.1) in the strict sense of the word because the consistency of
(2.1) doesn’t imply
(1.1). Hereinafter, we use the following abbreviation. Instead of the shifted variables
hn+i,m+j we write
hi,j. For example we replace the variables
hn+1,m+1,
hn+1,m,
hn,m+1 and
hn,m respectively with
h1,1,
h1,0,
h0,1 and
h0,0.
In our recent article [17] we observed that the quasi-periodical constraint tn,mj=tn+1,m−1j+N and ψn,mj+N=λψn+1,m−1j imposed on both variables simultaneously reduces the Hirota-Miwa equation (1.1) into an integrable quad system such that the linear system (2.1) generates the Lax pair for this quad system. In other words the two constraints above are compatible. Below in this section we present one more example of a reduction of the equation (1.1) which is compatible with (2.1).
Let us exclude ψj±1 from the system (2.1) and arrive at a discrete equation of the hyperbolic type
ψ1,1j−ψ1,0j−t1,1j+1t0,1jt0,1j+1t1,1jψ0,1j+at0,0jt1,1j+1t1,0jt0,1j+1ψ0,0j=0(2.2)
for which the classical theory of the Laplace invariants can be applied [
1,
29]. Recall that for a hyperbolic type linear equation
f1,1+b0,0f1,0+c0,0f0,1+d0,0f0,0=0(2.3)
the Laplace invariants are defined as follows
K1=b0,0c1,0d1,0, K2=b0,1c0,0d0,1.(2.4)
Equation (2.3) and another equation of the same form
f˜1,1+b˜0,0f˜1,0+c˜0,0f˜0,1+d˜0,0f˜0,0=0(2.5)
are related by linear change of the variables
f=λf˜ if and only if their Laplace invariants coincide:
K1=K˜1 and
K2=K˜2.
Here to study the integrable finite-field reductions of the equation (1.1) we use the method of nonlinear mirror images (see [2–5, 14, 15, 31]). In order to use this method we need in addition to (2.1) one more system of linear equations also associated with the Hirota-Miwa equation (1.1)
g−1,0j=γ0,0j(g0,0j+g0,0j−1), g0,−1j=δ0,0jg0,0j−ρ0,0jg0,0j+1,(2.6)
where
γ0,0j=t−1,0j+1t−1,0jat0,0jt−2,0j+1,
δ0,0j=t0,−1j+1t−1,0j+1t0,0j+1t−1,−1j+1a and
ρ0,0j=t0,−1jt−1,0j+2bt0,0j+1t−1,−1j+1a. It is easily checked that if the function
tn,mj solves
(1.1) then system
(2.6) is consistent. It is important that the systems
(2.1),
(2.6) are not gauge equivalent. In what follows we use also the discrete hyperbolic type equation
g1,1j−g1,0j−t1,0jt−1,1j+1t0,1j+1t0,0jg0,1j+at1,0jt−1,0j+1t0,1j+1t0,0jg0,0j=0,(2.7)
which is a consequence of the
equations (2.6). Our goal is to construct the Lax pair for the quad system
(1.4) by combining the
equations (2.1),
(2.6). We first study the question of when hyperbolic type
equations (2.2) and
(2.7) are connected by a multiplicative transformation for some fixed
j. To answer the question we have to compare the Laplace invariants of these equations. Denote through
K1ψ (
n,m,j) and
K2ψ (
n,m,j) the Laplace invariants for the
equation (2.2) and respectively through
K1g(
n,m,j) and
K2g(
n,m,j) for the
equation (2.7). Due to the formula
(2.4) we have explicit representations
K1ψ=t2,0jt1,1jt1,0jt2,1ja, K2ψ=t1,1j+1t0,2j+1t0,1j+1t1,2j+1a, K1g=t0,1j+1t1,0j+1t0,0j+1t1,1j+1a, K2g=t1,0jt0,1jt0,0jt1,1ja.
Evidently we have coincidence of the first invariants K1ψ (n,m,j) = K1g(n + 1,m,j−1) without any additional assumption on the function tj = tj(n,m). Now if we assume the coincidence also of the second invariants K2ψ (n,m,j) = K2g(n + 1,m, j−1) then we obtain the equation
t1,1j+1t0,2j+1t0,1j+1t1,2j+1=t2,0j−1t1,1j−1t1,0j−1t2,1j−1
which is easily solved
t1,0j−1=a^(n)b^(m)t0,1j+1.
Here â(n) and b^(m) are arbitrary functions different from zero. However, the freedom in choosing of the factors is deceptive, since they are eliminated by an appropriate point transformation of the restricted system. Therefore it is reasonable to focus on such a choice
t1,0j−1=t0,1j+1.(2.8)
In a manner similar to that applied in [17] we can show that the constraint (2.8) generates the gluing conditions of the form
ψ0,0j−1=λ−1g0,1j, g0,0j−1=λψ−1,0j,(2.9)
where
λ is an arbitrary constant.
Presence of the gluing conditions indicates the consistency of the boundary condition with the integrability property of the equation (1.1). Now we are ready to construct the Lax pair for (1.4). Let us impose boundary conditions of the form (2.8) at the endpoints NL and NR of the segment [NL,NR]:
t1,0NL−1=t0,1NL+1, t1,0NR−1=t0,1NR+1.(2.10)
Due to the reasonings above they generate two pairs of gluing conditions
ψn,mNL−1=λ−1gn,m+1NL, gn,mNL−1=λψn−1,mNL,(2.11)
gn,mNR=ψn,m−1NR−1, ψn,mNR=gn+1,mNR−1.(2.12)
The gluing conditions allow immediately to derive a finite closed subsystem of the combined system (2.1), (2.6) for the eigenfunctions ψj, gj with j ∈ [NL,NR−1], indeed we have
ψ1,0j=α0,0jψ0,0j−ψ0,0j+1, NL⩽j⩽NR−2,(2.13)
ψ1,0NR−1=α0,0NR−1ψ0,0NR−1−g1,0NR−1,(2.14)
ψ0,1NL=ψ0,0NL+β0,0NLλ−1g0,1NL,(2.15)
ψ0,1j=ψ0,0j+β0,0jψ0,0j−1, NL+1⩽j⩽NR−1,(2.16)
g0,0NL=γ1,0NL(g1,0NL+λψ0,0NL),(2.17)
g0,0j=γ1,0j(g1,0j+g1,0j−1), NL+1⩽j⩽NR−1,(2.18)
g0,0j=δ0,1jg0,1j−ρ0,1jg0,1j+1, NL⩽j⩽NR−2,(2.19)
g0,0NR−1=δ0,0NR−1g0,0NR−1−ρ0,1NR−1ψ0,0NR−1,(2.20)
where
α0,0j=t1,0j+1t0,0jt0,0j+1t1,0j, β0,0j=bt0,1j+1t0,0j−1t0,1jt0,0j.
The obtained system of the equations (2.13)–(2.20) can be rewritten in a compact form
AΦ1,0=BΦ, RΦ0,1=SΦ,
where Φ is a column-vector Φ = (
ψNL,
ψNL+1,…,
ψNR−1,
gNL,
gNL+1,…,
gNR−1)
T and
A,
B,
R,
S are matrices. Finding inverse matrices, we can present the system in the usual form:
Φ1,0=A−1BΦ, Φ0,1=R−1SΦ.
However the more convenient way is to express function g1,0j from equation (2.18) consecutively by using equation (2.17) for determining g1,0NL As a result we obtain
g1,0j=∑k=NLj(−1)j−kat1,0kt−1,0k+1t0,0kt0,0k+1g0,0k+(−1)j+1−NLλψ0,0NL,(2.21)
where
NL ⩽
j ⩽
NR−1. Similarly, we can express
g0,1j+1 from
(2.19) and use
(2.20) for finding
g0,1NR−1
g0,1j=1δ0,1jg0,0j+∑k=j+1NR−1abk−jt0,0jt−1,0k+1t0,1k+1t−1,1j+1t0,0kt0,0k+1g0,0k+bNR−jt0,0jt0,0NR−1t−1,1j+1t0,0NRψ0,0NR−1,(2.22)
where
NL ⩽
j ⩽
NR−1. Now by combining
(2.13)–
(2.20),
(2.21) and
(2.22) we find the final form of the Lax pair for the quad system
(1.4):
ψ1,0j=t1,0j+1t0,0jt0,0j+1t1,0jψ0,0j−ψ0,0j+1, NL⩽j⩽NR−2,(2.23)
ψ1,0NR−1=λ(−1)NR−NL−1ψ0,0NL+t0,0NR−1t1,0NRt1,0NR−1t0,0NRψ0,0NR−1+∑k=NLNR−1(−1)NR−kat1,0kt−1,0k+1t0,0kt0,0k+1g0,0k,(2.24)
g1,0j=∑k=NLj(−1)j−kat1,0kt−1,0k+1t0,0kt0,0k+1g0,0k+(−1)j+1−NLλψ0,0NL, NL⩽j⩽NR−1,(2.25)
ψ0,1NL=ψ0,0NL+λ−1bNR−NL+1t0,0NL+1t0,0NR−1t0,1NLt0,0NRψ0,0NR−1+λ−1∑k=NLNR−1abk−NL+1t0,1NL+1t0,1k+1t−1,0k+1t0,1NLt0,0kt0,0k+1g0,0k,(2.26)
ψ0,1j=ψ0,0j+bt0,1j+1t0,0j−1t0,0jt0,1jψ0,0j−1, NL+1⩽j⩽NR−1,(2.27)
g0,1j=at−1,0j+1t0,1j+1t0,0j+1t−1,1j+1g0,0j+∑k=j+1NR−1abk−jt0,0jt−1,0k+1t0,1k+1t−1,1j+1t0,0kt0,0k+1g0,0k+bNR−jt0,0jt0,0NR−1t−1,1j+1t0,0NRψ0,0NR−1, NL⩽j⩽NR−1.(2.28)
We rewrite the Lax pair (2.23)–(2.28) of system (1.4) under the condition NL = 0 and NR = N as
Φ1,0=FΦ, Φ0,1=GΦ,(2.29)
where Φ = (
ψ0,
ψ1,…,
ψN−1,
g0,
g1,…,
gN−1)
T,
F=(α0,00−10…00…00α0,01−1…00…0……………………(−1)N−1λ0…α0,0N−1(−1)N1γ1,00(−1)N−11γ1,01…−1γ1,0N−1−λ0…01γ1,000…0λ0…0−1γ1,001γ1,01…0……………………(−1)Nλ0…0(−1)N−11γ1,00(−1)N−21γ1,01…1γ1,0N−1),
G=(10…bN+1λt0,11t0,0N−1t0,10t0,0Nabλ(t0,11)2t−1,01t0,10t0,00t0,01ab2λt0,11t0,12t−1,02t0,10t0,01t0,02…abNλt0,11t0,1Nt−1,0Nt0,10t0,0N−1t0,0Nbt0,12t0,00t0,01t0,111…000…0……………………0…bt0,1Nt0,0N−2t0,0N−1t0,1N−1100…00…0bNt0,00t0,0N−1t−1,11t0,0Nat−1,01t0,11t0,01t−1,11abt0,00t−1,02t0,12t−1,11t0,01t0,02…abN−1t0,00t−1,0Nt0,1Nt−1,11t0,0N−1t0,0N0…0bN−1t0,01t0,0N−1t−1,12t0,0N0at−1,02t0,12t0,02t−1,12…abN−2t0,01t−1,0Nt0,1Nt−1,12t0,0N−1t0,0N……………………0…0b(t0,0N−1)2t−1,1Nt0,0N00…at−1,0Nt0,1Nt0,0Nt−1,1N).
Here α0,0j=t1,0j+1t0,0jt0,0j+1t1,0j, γ0,0j=t−1,0j+1t−1,0jat0,0jt−2,0j+1.
In the particular case N = 2 we have
at0,00t1,10−t1,00t0,10=b(t0,11)2,at0,01t1,11−t1,01t0,11=bt1,00t0,12,at0,02t1,12−t1,02t0,12=b(t1,01)2.(2.30)
The Lax pair for (2.30) is of the form (2.29) where F and G are 4 × 4 matrices
F=(t1,01t0,00t0,01t1,00−100−λt1,02t0,01t0,02t1,01at1,00t−1,01t0,00t0,01−at1,01t−1,020t0,01t0,02−λ0at1,00t−1,01t0,00t0,010λ0−at1,00t−1,01t0,00t0,01at1,00t−1,02t0,01t0,02),(2.31)
G=(1b3λt0,11t0,01t0,10t0,02abλ(t0,11)2t−1,01t0,00t0,10t0,01ab2λt0,11t−1,02t0,12t0,10t0,01t0,02bt0,00t0,12t0,01t0,111000b2t0,00t0,01t−1,11t0,02at0,11t−1,01t0,01t−1,11ab0,00t−1,02t0,12t−1,11t0,01t0,020b(t0,01)2t−1,12t0,020at−1,02t0,12t0,02t−1,12).(2.32)
Below in (4.16) the system is written in a more familiar way. For the particular case a = b = 1 the Lax pairs presented in this section have been found earlier in [12].
3. Evaluation of the continuum limit
Let us briefly discuss the continuum limit in the exponential type quad system (1.6) with arbitrary constant matrix A. To this end we assume that a = 1 + o(δ2), b =−δ2 + o(δ2), when δ → 0. We assume that smooth functions vj(x,y) exist such that vj(x,y)=un,mj, where x = nδ, y = mδ and 1 ⩽ j ⩽ N. Then evidently we have
un+1,mj=vj+δvxj+δ22vxxj+o(δ2),(3.1)
un,m+1j=vj+δvyj+δ22vyyj+o(δ2),(3.2)
un+1,m+1j=vj+δ(vxj+vyj)+δ22(vxxj+2vxyj+vyyj)+o(δ2).(3.3)
We substitute (3.1)–(3.3) into (1.6) and after some transformation we obtain a relation vx,yi=exp(∑j=1Naijvj)+o(1), for δ → 0, 1 ⩽ i ⩽ N showing that quad system (1.6) goes in the continuum limit to an exponential type system in partial derivatives
vx,yi=exp(∑j=1Naijvj), 1⩽i⩽N.(3.4)
Due to the fact that the generalized Cartan matrix A = {ai,j} is degenerate, the system (3.4) admits reducing of the order. For the reduced system the Lax representation is given in terms of the Cartan-Weyl basis in [8, 21].
Let us concentrate now on the continuum limit for the quad systems corresponding to DN(2) at the level of the Lax pairs. Our formulas below differ from those used in [8], since in [8] the coefficient matrix {ai,j} in the system (3.4) denotes the Cartan transposed matrix. This leads to the fact that the system (3.4), corresponding to the algebra DN(2) in our work coincides with the system corresponding to the algebra CN−1(1) in [8]. Up to this mismatch, the continuum limit completely coincides with the Drinfeld-Sokolov system.
Below we illustrate in detail the continuum limit for D3(2), since in the general case DN(2) it is evaluated in a similar way. Let us first change the variables Φ=σΦ˜ in the system (2.31), (2.32). Here σ = diag(1,δ,δ3,δ2) is a diagonal matrix. We also replace tn,mj=e−un,mj. As a result we arrive at the system
Φ˜10=F˜Φ˜, Φ˜0,1=G˜Φ˜,(3.5)
where
F˜=(eu1−u1,01+u1,00−u0−δ00−ξδ3eu2−u1,02+u1,01−u1aδ2eu0−u1,00+u1−u−1,01−aδeu1−u1,01+u2−u−1,02−ξδ0aeu0−u1,00+u1−u−1,010−ξδ20−aδeu0−u1,00+u1−u−1,01aeu1−u1,01+u2−u−1,02),
G˜=(1−δ3ξeu0,10−u0,11−u1+u2−aδξeu0+u0,10+u1−2u0,11−u−1,01aδ2ξeu0,10+u1+u2−u0,11−u−1,02−u0,12−δeu1−u0+u0,11−u0,121000δ2eu−1,11−u0−u1+u2aeu1−u0,11+u−1,11−u−1,01−aδeu−1,11+u1+u2−u0−u−1,02−u0,120−δeu−1,12+u2−2u10aeu2+u−1,12−u−1,02−u0,12).
Here ξ = λδ−4. Below we consider ξ and δ as independent parameters. It is easily observed that due to the representations (3.1)–(3.3) potentials F˜ and G˜ admit asymptotic expansions of the form
F˜=E+δR+o(δ), G˜=E+δS+o(δ), for δ→0,(3.6)
where
E is the unity matrix.
We assume that eigenfunction Φ˜ is represented as follows Φ˜n,m=Ψ(x,y) where x = nδ, y = mδ and Ψ is a smooth function of the variables x and y. Then we can write
Φ˜n+1,m=Ψ(x,y)+δΨx(x,y)+o(δ),Φ˜n,m+1=Ψ(x,y)+δΨy(x,y)+o(δ), δ→0.(3.7)
Now evidently formulas (3.6) and (3.7) imply a system of the linear PDE
Ψx=RΨ, Ψy=SΨ,(3.8)
where
R=(vx0−vx1−1000vx1−vx20−1−ξ0vx1−vx0000−1vx2−vx1),
S=(00−ξ−1e2v0−2v10−e2v1−v0−v2000000−e2v1−v0−v20−e−2v1+2v200)
The consistency condition of the system (3.8) leads to a system of the partial differential equations
vx,y0−vx,y1=e2v0−2v1−e−v0+2v1−v2,vx,y1−vx,y2=e−v0+2v1−v2−e−2v1+2v2,(3.9)
which doesn’t coincide with the continuum limit of the quad system
(2.30), having the form
vx,y0=e2v0−2v1,vx,y1=e−v0+2v1−v2,vx,y2=e−2v1+2v2,(3.10)
as might be expected by virtue of the formula
(3.4), but system
(3.9) can be rewritten as a reduction of
(3.10) obtained by introducing new variables
w0 =
v0−
v1,
w1 =
v1−
v2:
wx,y0=e2w0−e−w0+w1,wx,y1=e−w0+w1−e−2w1.(3.11)
The latter belongs to the class of the generalized Toda lattices, studied in [8]. Under appropriate linear transformation the Lax pair (3.8) for the system (3.11) is brought to the standard form [8,21]:
φx=fφ, φy=gφ,(3.12)
where
f=(−wx000−ζ−ζ−wx1000−ζwx1000−ζwx0),
g=(0−ζ−1ew1−w00000−ζ−1e2w10000−ζ−1ew1−w0−ζ−1e2w0000).
4. Formal asymptotics of the Lax operators eigenfunctions around singular values of λ and local conservation laws of the quad systems
The asymptotic behavior of the system of differential equations with respect to a parameter in the vicinity of the singular value of this parameter is an important characteristic of the system. A detailed presentation of the methods of studying these asymptotics can be found in Wasow’s famous book [34]. In the theory of integrability, the mentioned asymptotics find applications in solving the scattering problem, in describing integrals of motion, and constructing symmetries of nonlinear equations (see [39]). For the discrete equations with a parameter such a problem is rather difficult. Some particular cases are studied in [13, 16, 26], which are not fit in the case of (2.29). Hence we use here a suitable scheme suggested in [17]. Below we briefly explain the algorithm.
Let us consider a system of the discrete linear equations
Yn+1=fnYn, fn=∑j=−1∞fn(j)λ−j,(4.1)
where
fn(j)∈Ck×k for
j ⩾ −1 are matrix valued functions. In order to identify the matrix structure of the potential we divide the matrices into blocks as
A=(A11A12A21A22),(4.2)
where the blocks
A11,
A22 are square matrices. Here we assume that in
(4.1) the coefficient
fn(−1) is of one of the forms
fn(−1)=(000A22), detA22≠0,(4.3)
or
fn(−1)=(A11000), detA11≠0.(4.4)
Now our goal is to bring (4.1) to a block-diagonal form
ϕn+1=hnϕn(4.5)
where
hn is a formal series
hn=hn(−1)λ+hn(0)+hn(1)λ−1+hn(2)λ−2+⋯(4.6)
with the coefficients having the block structure
hn(j)=((hn(j))1100(hn(j))22).(4.7)
To this end we use the linear transformation Yn = Tnφn assuming that Tn is also a formal series
Tn=E+Tn(1)λ−1+Tn(2)λ−1+⋯,
where
E is the unity matrix and
Tn(j) is a matrix with vanishing block-diagonal part:
Tn(j)=(0(Tn(j))12(Tn(j))210).
After substitution of Yn = Tnφn into (4.1) we get
Tn+1hn=(∑j=−1∞fn(j)λ−j)Tn,(4.8)
where
hn=ϕn+1ϕn−1. Let us replace in
(4.8) the factors by their formal expansions:
(E+Tn+1(1)λ−1+⋯)(hn(−1)λ+hn(0)+⋯)=(fn(−1)λ+fn(0)+⋯)(E+Tn(1)λ−1+⋯).
By comparing coefficients at the powers of λ we derive a sequence of equations
hn(−1)=fn(−1)(4.9)
Tn+1(k)hn(−1)+hn(k−1)−fn(−1)Tn(k)=Rnk, k⩾1.(4.10)
Here Rnk denotes terms that have already been found in the previous steps.
To find the unknown coefficients Tn(j), we must solve linear equations, that look like difference equations. However due to the special form of the coefficient fn(−1) these equations are linear algebraic and therefore are solved without “integration”. In other words Tn(j) and hn(j) are local functions of the potential since depend on a finite number of the shifts of the functions fn(−1), fn(0), fn(1), etc. Indeed, equation (4.10) obviously implies
(0(Tn+1(k))12A2200)+(p00q)−(00A22(Tn(k))210)=Rnk
where
p=(hn(k−1))11,
q=(hn(k−1))22. Evidently this equation is easily solved and the searched matrices
Tn(k) and
hn(k−1) are uniquely found for any
k ⩾ 1.
Suppose now that a system of equations of the form
Yn+1,m=(fn,m(−1)λ+fn,m(0)+⋯)Yn,m, Yn,m+1=Gn,m(λ)Yn,m,(4.11)
where
Gn,m(
λ) is analytic at a vicinity of
λ = ∞, is the Lax pair for the nonlinear quad system
F([un,mj])=0,
i.e.
F depends on the variable
un,mj and on its shifts with respect to the variables
j,n,m.
Assume that function fn,m(−1) has the structure (4.3). Then due to the reasonings above there exists a linear transformation Yn,m↦ϕn,m=Tn,m−1Yn,m which reduces the first equation in (4.11) to a block-diagonal form (4.5)–(4.7). It can be checked that this transformation brings also the second equation of (4.3) to an equation of the same block structure
ϕn,m+1=Sn,mϕn,m,
where
Sn,m is a formal power series
Sn,m=Sn,m(0)+Sn,m(1)λ−1+Sn,m(2)λ−2+⋯
and
Sn,m(j)=((Sn,m(j))1,100(Sn,m(j))22).
Since the compatibility property of linear systems is preserved under a change of the variables, we have the relation
Sn+1,mhn,m=hn,m+1Sn,m
which implies due to the block-diagonal structure that
(Dn−1)log det(S)ii=(Dm−1)log det(h)ii, i=1,2.(4.12)
Here Dn and Dm are shift operators with respect to n and m: Dnun,m = un+1,m, Dmun,m = un,m+1. By evaluating and comparing the coefficients at the powers of λ we derive the sequence of the local conservation laws.
The Lax pairs considered in the article have also the second singular point λ = 0, so we briefly discuss the Lax pair represented as
Yn+1,m=Fn,mYn,m,Yn,m+1=(gn,m(−1)λ−1+gn,m(0)+gn,m(1)λ+⋯)Yn,m,(4.13)
where
Fn,m =
Fn,m(
λ) is analytic at a vicinity of
λ = 0. We request that here the term
gn,m(−1) has the block structure
(4.4). In this case the block-diagonalization is performed in a way very similar to one recalled above for the point
λ = ∞.
4.1. System corresponding to affine Lie algebra DN(2)
Note that the procedure of formal diagonalization of the Lax pair provides an effective way to construct infinite series of the local conservation laws. Procedure of finding of the formal diagonalization of a linear system (4.1) satisfying (4.3) or (4.4) is purely algorithmic. However, in order to apply the method to an arbitrary linear system we must transform it to an appropriate form and this step may lead to some difficulties.
In the case related to DN(2) we overcome these difficulties by applying the linear transformations
Φ=HY, (or Φ=H¯Y)
converting the Lax
equations (2.29) to the suitable form
(4.11) (or, respectively,
(4.13)), where the factor
H is lower (or,
H¯ is upper) block-triangular matrix
H=(H110H21H22), (or H¯=(H¯11H¯120H¯22)).(4.14)
Here we use the block representation (4.2) where the blocks Aij are square matrices of the size (N−1) × (N−1). Note that (4.14) H and H¯ have the same block structure. Moreover, the blocks H11, H22, H¯11 and H¯22 are diagonal matrices, some entries of which depend on the spectral parameter λ. Factors H and H¯ are effectively found (see, for example, (4.17), (4.18)). Below we illustrate all of the computations with the example.
Example 4.1.
Let us briefly discuss the quad system
{at0,00t1,10−t1,00t0,10=b(t0,11)2,at0,01t1,11−t1,01t0,11=bt1,00t0,12,at0,02t1,12−t1,02t0,12=b(t1,01)2(4.15)
corresponding to the algebra
D3(2). Its Lax pair reads as
Φ1,0=FΦ, Φ0,1=GΦ, (4.16)
where the potentials
F=(t0,00t1,01t1,00t0,01−100−λt0,01t1,02t1,01t0,02at0,00t0,00t0,01−at1,01t0,01t0,02−t0,01λ0at1,00t0,000t0,02λ0−at1,00t0,02t0,00t0,01at1,01t0,01), G=(1b3t0,01t0,11λt0,10t0,02ab(t0,11)2λt0,00t0,10t0,01ab2t0,11t0,12λt0,10t0,01t0,02bt0,00t0,12t0,01t0,111000b2t0,00t0,01t0,02at0,11t0,01ab0,00t0,12t0,01t0,020b(t0,01)2t0,020at0,12t0,02)
are not in a suitable form for the application of formal diagonalization. Therefore we have to do the transformation Φ =
HY, with the block-triangular factor
H:
H=(10000ξ00(t−1,01)2t0,02t0,01t−1,02+at0,01t−2,02t−1,02t−1,01ξ10−t−1,01t0,02t0,01−t−1,02ξ01), ξ=λ.(4.17)
The new variable Y solves the system of equations
Y1,0=fY, Y0,1=gY,
where the potential
g is analytic at
ξ = ∞ and
f is given by
f=f(−1)ξ+f(0)+f(1)ξ−1,
the factor
f (−1) has the block-diagonal structure
f(−1)=(0−100−100000000000),
matrices
f (0) and
f (1) are of the form
f(0)=(α0,000000α0,01+1γ1,00+1γ1,0100−α0,00t0,01(α0,01+1γ1,00+1γ1,01)00at1,01t0,02α0,00α0,01t0,01000),
f(1)=(00001γ1,00(α0,00+α−1,01+1γ0,01)01γ1,00t−1,01−1γ1,01t−1,0200000000),
where
α0,0j=t1,0j+1t0,0jt0,0j+1t1,0j,
γ0,0j=t−1,0j+1t−1,0jat0,0jt−2,0j+1 .
According to the reasoning above we can apply the formal diagonalization algorithm and find the series T, h and S. Here we illustrate the first few coefficients
h=(0−100−100000000000)ξ+(α0,000000α0,01+1γ1,00+1γ1,0100000at1,01t0,020000)+(00001γ1,00(α0,00+α−1,01+1γ1)00000000000)ξ−1++(00000α−1,00γ1,00(α0,00+α−1,01+1γ0,00+1γ0,01)0000−at1,01t0,01(α0,01+1γ1,00+1γ1,01)at1,01α0,00t0,02(α0,01+1γ1,00+1γ1,01)00at1,02t0,01−at1,02α0,00t0,02)ξ−2+⋯,
T=(1000010000100001)+(000000000α−1,00t−1,01(α−1,01+1γ0,00+1γ0,01)000−t−1,00t0,02t0,0000)ξ−1++(001γ1,00t−1,01−1γ1,01t−1,020000α−1,00t−1,01[(α−1,01+1γ0,00+1γ0,01)2+α−2,00γ0,00]000−α−1,00α−1,01t−1,02(α−1,01+1γ0,00+1γ0,01)000)ξ−2+⋯,
S=(1000010000at0,11t0,01ab0,00t0,12t0,01t0,02000at0,12t0,02)+(0ab−1,01(t0,11)2t0,00t0,01t0,10−ab2t−1,02t0,11t0,12t0,10t0,01t0,02+b3t0,01t0,11t0,10t0,0200bt0,00t0,12t0,01t0,1100000000000)ξ−1+(ab0,11t0,10(t0,11(t−1,01)2t0,02t0,00(t0,01)2t−1,02+at0,11t−2,02t0,00t−1,02−bt−1,01t0,12(t0,01)2)000000000S332S34200S432S442)ξ−2+⋯,
where
S332=abt0,11t0,01t0,10(b3t0,00t0,12t0,02−(at0,11)2t−2,02t0,00t−1,02−a(bt0,00t−1,02t0,12−t−1,01t0,11t0,02)2(t0,01)2t0,00t−1,02t0,02),
S342=ab2t0,12t0,01t0,10(b3t0,12(t0,00)2(t0,02)2+2abt0,12t0,11t−1,01t0,00(t0,01)2t0,02−a(t0,11t−1,01)2(t0,01)2t−1,02−a2(t0,11)2t−2,02t−1,02t0,02−ab2t−1,02(t0,00t0,12)2(t0,01t0,02)2),
S432=abt0,11t0,12t0,10(at0,11t−1,01(t0,01)2t0,00+b2t0,02−abt0,12t−1,02(t0,01)2t0,02),
S442=ab2(t0,12)2t0,10t0,02(at0,11t−1,01(t0,01)2+b2t0,00t0,02−abt0,12t0,00t−1,02t0,02(t0,01)2).
By virtue of the formulas (4.12) we derive local conservation laws:
- 1.
(Dn−1)(b4t0,00t0,12t0,10t0,02−a2b(t0,11)2t−2,02t0,00t0,10t−1,02−ab(bt0,00t0,12t−1,02−t−1,01t0,11t0,02)2t0,00t0,10(t0,01)2t−1,02t0,02)=(Dm−1)(t0,00t1,02t1,00t0,02+a2t1,00t−2,02t0,00t−1,02+a(t0,00t1,01t−1,02+t1,00t−1,01t0,02)2t0,00t1,00(t0,01)2t−1,02t0,02),
- 2.
(Dn−1)(b22(t0,10)2(a2(t0,11)2t−2,02t0,00t−1,02−b3t0,00t0,12t0,02+a(bt0,00t−1,02t0,12−t−1,01t0,11t0,02)2t0,00(t0,01)2t−1,02t0,02)2−a2bt−2,00(t0,11)2t0,00t0,10t−1,00−abt−1,00t0,10(at0,11t−2,01t−1,01+t−1,00t−1,01t0,11t0,02t0,00t0,01t−1,02+at−1,00t0,01t0,11t−2,02t0,00t−1,01t−1,02−bt−1,00t0,12t0,01)2)=(Dm−1)(a2t1,00t−2,00t0,00t−1,00+12(t0,00t1,02t1,00t0,02+a2t1,00t−2,02t0,00t−1,02+a(t0,00t1,01t−1,02+t1,00t−1,01t0,02)2t0,00t1,00(t0,01)2t−1,02t0,02)2at−1,00t1,00(t1,00t−1,00t−1,01t0,02t0,00t0,01t−1,02+at1,00t−2,01t−1,01+at−1,00t1,00t0,01t−2,02t0t−1,01t−1,02+t1,01t−1,00t0,01)2),
- 3.
(Dn−1)(−abt2,00t0,00t1,00t0,10(at−1,01t0,11t0,02−ab0,00t−1,02t0,12+b2t0,00(t0,01)2t0,01t0,02)2−bt1,00t0,10(a2t1,00t0,11t−2,02t0,00t−1,02+at1,00t0,11(t−1,01)2t0,02t0,00(t0,01)2t−1,02+at1,01t0,11t−1,01(t0,01)2−ab1,00t−1,01t0,12(t0,01)2−abt0,00t1,01t−1,02t0,12(t0,01)2t0,02+b2t0,00t1,01t0,02)2−b22(t0,10)2(b3t0,00t0,12t0,02−a2(t0,11)2t−2,02t0t−1,02−a(bt0,00t0,12t−1,02−t0,11t−1,01t0,02)2t0,00(t0,01)2t−1,02t0,02)2)=(Dm−1)(a2t0,00t3,00t1,00t2,00+at0,00(t2,01)2(t1,01)2t2,00+at2,00t0,00(t0,00(t0,01)2t1,02+at1,00t−1,01t1,01t0,02+at0,00(t1,01)2t−1,02t1,00t0,01t1,01t0,02)2+2at2,01(t0,00(t0,01)2t1,02+at1,00t−1,01t1,01t0,02+at0,00(t1,01)2t−1,02)t1,00t0,01(t1,01)2t0,02+12(t0,00t1,02t1,00t0,02+a2t1,00t−2,02t0,00t−1,02+a(t0,00t1,01t−1,02+t1,00t−1,01t0,02)2t0,00t1,00(t0,01)2t−1,02t0,02)2).
In a similar way we investigate the system around the point λ = 0. To this end we first change the variables, Φ¯=H¯Y where
H¯=(ξ−100001−ab2t0,11t0,02t0,00(t0,01)2−abt0,12(t0,01)200100001),(4.18)
that reduces the system to the suitable form
Y1,0=f¯(ξ)Y, Y0,1=(g¯(−1)ξ−1+g¯(0))Y,
where
f¯(ξ) is analytic at the vicinity of
ξ = 0 and the matrix
g¯(−1) has the appropriate block-diagonal structure:
g¯(−1)=(0b3t0,01t0,11t0,10t0,0200bt0,00t0,12t0,11t0,0100000000000).
Therefore one can perform the diagonalization procedure and find the local conservation laws:
- 1.
(Dm−1)(−b(t0,11)2t0,02+a(t0,01)2t0,22t0,00t1,00t0,12)=(Dn−1)(a2t0,00t0,32t0,10t0,22+t0,10t0,02t0,00t0,12+2at0,01t0,21(t0,11)2+at0,00(t0,21)2t0,12t0,10(t0,11)2t0,22+at0,10(t0,01)2t0,22t0,00(t0,11)2t0,12),
- 2.
(Dm−1)(b2t0,11t0,02t0,00t1,01−abt0,02t0,−10t1,00t0,11t0,00t1,01(t0,01)2−abt1,00t0,−11t0,12(t0,01)2t1,01)=(Dn−1)(a2t0,−10t0,22t0,00t0,12+at0,01t0,21(t0,11)2+at0,−10(t0,11)2t0,02t0,00t0,12(t0,01)2+t0,10t0,02t0,00t0,12+a(t0,01)2t0,10t0,22(t0,11)2t0,00t0,12+at0,11t0,−11(t0,01)2),
- 3.
(Dm−1)(−ab0,20(t0,01)2t0,00t0,10t1,00−b22(t0,02(t0,11)2+a(t0,01)2t0,22t0,00t1,00t0,12)2−bt1,00t0,10(t0,10(t0,11)2t0,02+at0,00t0,01t0,21t0,12+at0,10(t0,01)2t0,22t0,00t0,11t0,12)2)=(Dn−1)(a2t0,00t0,30t0,10t0,20+a3t0,00(t0,31)2t0,20(t0,21)2+at0,20t0,00(at0,00(t0,11)2t0,32+t0,01t0,21t0,10t0,22+t0,00(t0,21)2t0,12t0,10t0,11t0,21t0,22)2+2a2t0,31(at0(t0,11)2t0,32+t0,10t0,01t0,21t0,22+t0,00(t0,21)2t0,12)t0,10t0,11(t0,21)2t0,22+12(a2t0,00t0,32t0,10t0,22+t0,10t0,02t0,00t0,12+a(t0,00t0,21t0,12+t0,10t0,01t0,22)2t0,00t0,10(t0,11)2t0,12t0,22)2).
