Dressing the Dressing Chain
The dressing chain is derived by applying Darboux transformations to the spectral problem of the Korteweg-de Vries (KdV) equation. It is also an auto-Bäcklund transformation for the modified KdV equation. We show that by applying Darboux transformations to the spectral problem of the dressing chain,...
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| citation_txt | Dressing the Dressing Chain / C.A. Evripidou, P.H. van der Kamp, C. Zhang // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 32 назв. — англ. |
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| description | The dressing chain is derived by applying Darboux transformations to the spectral problem of the Korteweg-de Vries (KdV) equation. It is also an auto-Bäcklund transformation for the modified KdV equation. We show that by applying Darboux transformations to the spectral problem of the dressing chain, one obtains the lattice KdV equation as the dressing chain of the dressing chain, and that the lattice KdV equation also arises as an auto-Bäcklund transformation for a modified dressing chain. In analogy to the results obtained for the dressing chain (Veselov and Shabat proved complete integrability for odd-dimensional periodic reductions), we study the (0,n)-periodic reduction of the lattice KdV equation, which is a two-valued correspondence. We provide explicit formulas for its branches and establish complete integrability for odd n.
|
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 059, 14 pages
Dressing the Dressing Chain
Charalampos A. EVRIPIDOU †, Peter H. VAN DER KAMP † and Cheng ZHANG ‡
† Department of Mathematics and Statistics, La Trobe University,
Melbourne, Victoria 3086, Australia
E-mail: C.Evripidou@latrobe.edu.au, P.VanDerKamp@latrobe.edu.au
‡ Department of Mathematics, Shanghai University, 99 Shangda Road, Shanghai 200444, China
E-mail: ch.zhang.maths@gmail.com
Received April 18, 2018, in final form June 04, 2018; Published online June 15, 2018
https://doi.org/10.3842/SIGMA.2018.059
Abstract. The dressing chain is derived by applying Darboux transformations to the spec-
tral problem of the Korteweg–de Vries (KdV) equation. It is also an auto-Bäcklund transfor-
mation for the modified KdV equation. We show that by applying Darboux transformations
to the spectral problem of the dressing chain one obtains the lattice KdV equation as the
dressing chain of the dressing chain and, that the lattice KdV equation also arises as an auto-
Bäcklund transformation for a modified dressing chain. In analogy to the results obtained for
the dressing chain (Veselov and Shabat proved complete integrability for odd dimensional
periodic reductions), we study the (0, n)-periodic reduction of the lattice KdV equation,
which is a two-valued correspondence. We provide explicit formulas for its branches and
establish complete integrability for odd n.
Key words: discrete dressing chain; lattice KdV; Darboux transformations; Liouville inte-
grability
2010 Mathematics Subject Classification: 35Q53; 37K05; 39A14
1 Introduction
The dressing chain [23, 25, 30] appeared in the application of Darboux transformations to the
Schödinger (Sturm–Liouville) equation, which is the spectral problem for the Korteweg–de Vries
(KdV) equation. A detailed study concerning the integrability properties as well as solutions
of the model were presented in [30]. In particular, the authors proved that the dressing chain
with a periodic constraint in odd dimensions is completely integrable in the sense of Liouville–
Arnold.
The dressing chain can also be obtained as an auto-Bäcklund transformation of the modified
KdV (mKdV) equation via the celebrated Miura transformation between KdV and mKdV [19],
and a symmetry of mKdV. The two ways of deriving the dressing chain are not unrelated, as the
Miura transformation itself can be derived from factorisation of the Schrödinger equation [7].
Both the Darboux and Bäcklund approach can be seen as a discretisation process, and the two
methods have been applied to other equations. In particular, in the discrete setting, Spiridonov
and Zhedanov [26] considered a tri-diagonal discrete Schrödinger equation, for which discrete
Darboux transformations gave rise to two equivalent systems: the discrete time Toda lattice
and a system they called the discrete dressing chain [26, equations (5.30) and (5.31)]. As the
discrete Schrödinger equation considered in [26] is the spectral problem for the Toda lattice [17]
one could refer to these systems as dressing chains of the Toda lattice. Starting from the Volterra
equation and using two discrete Miura transformations, Levi and Yamilov obtained an integrable
lattice equation which they regard as a direct analogue of the dressing chain [16, equation (31)],
cf. [9]. In our context, we would refer to that equation as an auto-Bäcklund transformation for
a modified Volterra equation.
mailto:c.evripidou@latrobe.edu.au
mailto:P.VanDerKamp@latrobe.edu.au
mailto:ch.zhang.maths@gmail.com
https://doi.org/10.3842/SIGMA.2018.059
2 C.A. Evripidou, P.H. van der Kamp and C. Zhang
In general, for a given integrable equation, one can ask the following questions, see Fig. 1:
1. Is there a Miura or, more generally, a Bäcklund transformation to a modified equation
which has a symmetry? This then gives rise to an auto-Bäcklund transformation of the
modified equation, which discretises the equation.
2. Is there an associated spectral problem, whose factorisation yields a dressing chain?
3. Does the Bäcklund transformation (1) arise in the factorisation of the spectral problem (2)?
4. Does the auto-Bäcklund transformation for the modified equation coincide with the dress-
ing chain of the equation?
5. Can the original equation be recovered by appropriate continuum limits?
6. Is the auto-Bäcklund transformation/dressing chain integrable?
mod. equation (v)
Bäcklund
u = f(v)
symmetry
σ
equation (u)
factorisation
L = AB + µ
integrability
(5)
cont.
limit
auto-Bäcklund
ū = f(σ(v))
dressing chain
L̂ = BA + µ
(1) (2)
(4)
(3)
(6)
(6)
(5)
Figure 1. Darboux and Bäcklund transformations.
In this paper, our starting point is the dressing chain. We factorise its discrete spectral
problem, which itself is an exact discretisation, cf. [32], of the (continuous) Schrödinger equa-
tion. It turns out that the discrete dressing chain (of the dressing chain) coincides with the
(non-autonomous) lattice Korteweg–de Vries (lKdV) equation. By studying a related Lax rep-
resentation we identify a Bäcklund transformation to a modified dressing chain which admits
a symmetry. The derived auto-Bäcklund transformation is again given by the lKdV equation.
In analogy to the continuous case, cf. [30], we study the (0,n)-periodic reduction1 of the
(discrete) dressing chain of the dressing chain (a.k.a. the lKdV equation), which is a two-valued
correspondence (i.e., multi-valued map). We provide explicit formulas for its two branches, and
establish linear growth of multi-valuedness. Moreover, we prove (in odd dimensions) that the
map is Liouville integrable with respect to a quadratic Poisson structure of Lotka–Volterra type.
2 Background
We clarify Fig. 1 by succinctly providing some details for the KdV equation. We hope it also
makes clear to the reader that how the dressing chain is related to the KdV equation is completely
analogous to how the lattice KdV equation is related to the dressing chain.
The KdV equation ut = uxxx − 6uux arises as the compatibility condition, Lt = [L,M ],
for the system of linear equations Lφ = λφ, φt = Mφ where L is the Schrödinger operator
1The (n,0)-periodic reduction gives rise to the same maps, up to a minus sign.
Dressing the Dressing Chain 3
L = −D2 + u, M = 4D3 − 3(uD +Du), and λ is a spectral parameter. One can check that if v
satisfies the mKdV equation vt = vxxx− 6(v2 +α)vx, then u given by the Miura transformation
u = vx + v2 + α (2.1)
satisfies the KdV equation. As the mKdV equation is invariant under v → −v another Miura
transformation is given by
ū = −vx + v2 + α. (2.2)
Combining the two equations (2.1) and (2.2) yields an auto-Bäcklund transformation for the
mKdV equation,
(v̄ + v)x = v2 − v̄2 + α− ᾱ,
which coincides with the dressing chain [30]. A related chain, which is an auto-Bäcklund trans-
formation for the potential KdV equation, was already written down by Wahlquist and Es-
tabrook [31], who used Bianchi’s permutability theorem to show that it generates hierarchies
of solutions due to a nonlinear superposition principle. More general auto-Bäcklund transfor-
mations (and their interpretation as differential-difference equations) were given in [14, 15].
Auto-Bäcklund transformations for differential-difference equations are lattice equations, and
some examples were presented in [10, 16].
Darboux transformations for differential and difference equations (also known as dressing
transformations) are maps of the functions and the coefficients that preserve the form of the
equations [5]. They can be obtained by factorisation of operators, cf. [3, 7, 12, 21, 22], and they
provide an effective way to construct exact solutions of a wide range of integrable equations (see,
e.g., the monograph [18]).
Recall that the Schrödinger operator L can be decomposed as
L = −(D + v)(D − v) + α,
subject to the constraint (2.1). Darboux [5] showed that under the transformation
φ 7→ φ̃ = (D − v)φ, (2.3)
and L 7→ L̃, where (interchanging the two factors in the decomposition)
L̃ = −(D − v)(D + v) + α
the form of the Schödinger equation is unchanged: L̃φ̃ = λφ̃. The ˜ operation characterises
a Darboux transformation for the Schrödinger equation L, if L̃ is still a Schrödinger operator,
i.e., L̃ = −D2 + ũ with ũ = −vx + v2 + α, cf. equation (2.2). Iterated Darboux transformations
result in the dependency of the functions u and φ on shifts in the ˜ direction, and α becomes
a lattice parameter. Eliminating u, ũ in the above decompositions yields the dressing chain
(ṽ + v)x = v2 − ṽ2 + α− α̃. (2.4)
Denoting v = vi, ṽ = vi+1, etc., and adding a periodic constraint, i.e., vi+n = vi and αi+n = αi,
one gets the finite dimensional systems of ordinary differential equations
(vi+1 + vi)x = v2i − v2i+1 + αi − αi+1, 1 ≤ i ≤ n, (2.5)
which was shown to be completely integrable for odd n [30].
4 C.A. Evripidou, P.H. van der Kamp and C. Zhang
3 Dressing the dressing chain
By eliminating the x-derivatives in the Schrödinger equation Lφ = λφ, using equation (2.3), one
obtains the discrete Schrödinger equation
Kφ = λφ, K = −T 2 + hT + α (3.1)
where
h = −v − ṽ, (3.2)
and T : z → z̃ represents a shift operator. The discrete Schrödinger operator K is the dual, with
respect to (2.3), to the continuous operator L, cf. [24]. We note that the compatibility condition
Kx = [N,K], with N = T + v, provides a Lax representation for the dressing chain (2.4).
The operator K can be decomposed as, cf. [32],
K = −(T + f)(T − g)− β.
Here β does not depend on the ˜ direction (as α does) but will depend on another discrete
direction introduced below. In order that such a decomposition holds, one needs
h = g̃ − f (3.3)
and
fg = α+ β. (3.4)
Eliminating f leads to (h− g̃)g + α+ β = 0. This can be solved by posing g = ψ̃ψ−1, where ψ
is a special solution of (3.1) with λ = −β. Now that f and g are well defined, we can apply
the usual tactics (interchanging the two factors in the decomposition) to generate a Darboux
transformation for (3.1). With
φ̂ = Gφ, G = T − g, (3.5)
and
K̂ = −(T − g)(T + f)− β
we have K̂φ̂ = λφ̂. Letting K̂ = −T 2 + ĥT + α imposes another constraint
ĥ = g − f̃ , (3.6)
which together with (3.3) and (3.4) yields the non-autonomous lattice KdV equation (lKdV)
f̃ − f̂ =
α+ β
f
− α̃+ β̂̂̃
f
, (3.7)
or, in terms of g,
g − ̂̃g =
α̃+ β
g̃
− α+ β̂
ĝ
. (3.8)
Shifts in the ̂ direction correspond to a second discrete direction, created by iterated Dar-
boux transformations, and the parameter β varies in this direction. The linear system of equa-
tions (3.1) and (3.5) provides a Lax representation for the lKdV equation, K̂G = G̃K.
Dressing the Dressing Chain 5
In the light of Fig. 1, equation (3.7), or (3.8), is the dressing chain of the dressing chain. In
analogy to the continuous case, we will consider a periodic reduction in the ̂ direction, i.e.,
fi+n = fi and βi+n = βi, and we take α̃ = α to be a constant. The finite dimensional system of
difference equations we will study is
f̃i − fi+1 =
α+ βi
fi
− α+ βi+1
f̃i+1
, 1 ≤ i ≤ n. (3.9)
As we make explicitly in Section 5, it gives rise to a two-valued correspondence.
It would also be justified to refer to the above system (3.7) as the discrete dressing chain,
since its continuum limit coincides with (2.4). Using equations (3.3) and (3.6), one can express
f = ̂̃w − w, g = ŵ − w̃. Substituting them into (3.4) gives the lattice potential KdV equation
( ̂̃w − w)(ŵ − w̃) = α+ β, whose continuum limit with respect to the ̂ direction is [11, 20]
(w̃ + w)x = (w̃ − w)2 + α. (3.10)
From (3.2), (3.3) and the above expressions for f and g, one obtains v = w− w̃ which relates the
potential dressing chain (3.10) to the dressing chain (2.4). We will refer to the (0, n)-reduction
of the lKdV equation (3.9) as the n-dimensional discrete dressing chain.
4 The modified dressing chain
One next wonders if the discrete dressing chain (3.7) (or (3.8)) is the auto-Bäcklund transfor-
mation of a modified dressing chain. This is indeed the case. The Lax equation Gx = N̂G−GN
gives rise to the system g(v̂−v)−gx = ṽ− v̂−g+ g̃ = 0, which together with (h =)−v− ṽ = g̃−f
yields
v =
1
2
(
f − g − gx
g
)
, ṽ =
1
2
(
f + g +
gx
g
)
− g̃ (4.1)
(as well as v̂ = 1
2
(
f − g + gx
g
)
). The system (4.1) provides a Bäcklund transformation, cf. [11,
Definition 2.1.1] between the dressing chain (2.4) and the following equation
g̃ +
α̃+ β
g̃
− g̃x
g̃
= g +
α+ β
g
+
gx
g
, (4.2)
which we will refer to as the modified dressing chain. The modified dressing chain (4.2) admits
the symmetry
σ(g, g̃, x) = (g̃, g,−x).
Applying this symmetry to the right hand sides of (4.1) and transforming the left hand sides by
(v, ṽ)→ (˜̄v, v̄), we obtain another Bäcklund transformation
˜̄v =
1
2
(
f̃ − g̃ +
g̃x
g̃
)
, v̄ =
1
2
(
f̃ + g̃ − g̃x
g̃
)
− g.
Combining the two Bäcklund transformations
(
v̄ + ˜̄v = v + ṽ
)
we obtain f̃ − g = f̄ − ¯̃g, which
shows that the lKdV equation is an auto-Bäcklund transformation for the modified dressing
chain (4.2).
6 C.A. Evripidou, P.H. van der Kamp and C. Zhang
5 Explicit formulas for the n-dimensional discrete dressing
chain, and linear growth of multivaluedness
In this section we consider the (0, n)-reduction of the lattice KdV equation, which is a two-
valued correspondence. We give explicit formulas for both branches (M,N), and prove that
MNM = N . The latter implies that the l-th iteration of the correspondence is 2l-valued, cf.
[29, Section 6.2].
In the finite reduction (3.9), without loss of generality, we set α = 0 since it can be absorbed
into the parameters βj . Having fixed n ∈ N and taking i ∈ I = {1, 2, . . . , n} subject to the
periodic boundary conditions fn+i = fi, βn+i = βi for all i ∈ I, the system of equations (3.9)
reads
E1 : f̃1 +
β2
f̃2
= f2 +
β1
f1
,
E2 : f̃2 +
β3
f̃3
= f3 +
β2
f2
,
...
En : f̃n +
β1
f̃1
= f1 +
βn
fn
.
(5.1)
These equations define a two-valued correspondence on Rn. One solution of the system (5.1)
is given by
f̃i =
βi
fi
, i ∈ I (5.2)
(which is f̃i = gi, cf. (3.4)). This defines a map
N : (f1, f2, . . . , fn) 7→
(
β1
f1
,
β2
f2
, . . . ,
βn
fn
)
,
which is an involution. The other solution of the system (5.1) gives rise to a more intriguing
map on Rn, which will be denoted by M . We next provide explicit formulas for M and for its
inverse.
Remark 5.1. Consider the finite version of system (3.9) defined by choosing n ∈ N and re-
stricting i ∈ I subject to the open boundary condition fn+1 = βn+1 = 0. The resulting system
then takes the form
f̃i +
βi+1
f̃i+1
= fi+1 +
βi
fi
, for i = 1, 2, . . . , n− 1, and f̃n =
βn
fn
,
whose unique solution is given by the involution (5.2).
In order to describe the nontrivial solution of (5.1) we introduce some notation. With
1 < n ∈ N and k ∈ I we consider Rk with coordinates f1, f2, . . . , fk. We fix the parameters
β1, β2, . . . , βn, and define functions Fk : Rk → R by F1(f1) = 1 and
Fk = Fk(f1, f2, . . . , fk) = f1f
2
2 f
2
3 · · · f2k−1fk, k > 1. (5.3)
We also define a function G : R2n → R by
G = G(f1, f2, . . . , fn, β1, β2, . . . , βn)
=
n−1∑
i=0
i∏
j=1
βj · Fn−i(fi+1, fi+2, . . . , fn)
= Fn(f1, f2, . . . , fn) + β1Fn−1(f2, f3, . . . , fn) + · · ·+ β1β2 · · ·βn−1.
(5.4)
Dressing the Dressing Chain 7
For n = 3 the function G reads
G = f1f
2
2 f3 + β1f2f3 + β1β2.
We will make use of the following cyclic permutation τ : I → I
τi =
{
i+ 1, if i < n,
1, if i = n,
and of the involution σ : I → I defined by
σi = n+ 1− i.
Simply stated, the permutation τ is a shift modulo n, and σ is to reverse the elements of I.
By some abuse of notation we will write, for any function H depending on the variables f1,
f2, . . . , fn and the parameters β1, β2, . . . , βn,
τH(f1, f2, . . . , fn, β1, β2, . . . , βn) = H(fτ1 , fτ2 , . . . , fτn , βτ1 , βτ2 , . . . , βτn)
= H(f2, f3, . . . , f1, β2, β3, . . . , β1),
and similarly
σH(f1, f2, . . . , fn, β1, β2, . . . , βn) = H(fσ1 , fσ2 , . . . , fσn , βσ1 , βσ2 , . . . , βσn)
= H(fn, fn−1, . . . , f1, βn, βn−1, . . . , β1).
For example, for n = 3 we have
τG = f1f2f
2
3 + β2f1f3 + β2β3, σG = f1f
2
2 f3 + β3f1f2 + β2β3.
A useful property of the above-defined functions is that, for any n, the expression
f1fnG− β1τG = f21 f
2
2 · · · f2n − β1β2 · · ·βn (5.5)
is invariant under τ and σ. The following formula, which can be easily proved, is also useful. For
any function H depending on f1, f2, . . . , fn, β1, β2, . . . , βn, we have τστH = σH, which implies,
for all i,
στ iH = τn−iσH. (5.6)
We now define functions
Mi = fi−1
τ i−1G
τ iG
, (5.7)
where the indices are considered modulo n and in the set I, in particular
M1 = fn
G
τG
.
Lemma 5.2. The map M : (f1, f2, . . . , fn) 7→ (M1,M2, . . . ,Mn), where Mi is defined by (5.7),
is a solution of the system (5.1).
8 C.A. Evripidou, P.H. van der Kamp and C. Zhang
Proof. By definition (5.7), we have Mi = τMi−1 for all i and a similar property holds for the
equations in (5.1) (each equation is obtained by applying τ to the previous one). Hence, it is
enough to show that the functions Mi satisfy the equation E1. Taking f̃1 = M1 and f̃2 = M2,
we have to verify
M1 +
β2
M2
= f2 +
β1
f1
⇐⇒ fn
G
τG
+
β2
f1
τ2G
τG
= f2 +
β1
f1
⇐⇒ f1fnG+ β2τ
2G = (f1f2 + β1)τG
⇐⇒ f1fnG− β1τG = τ(f1fnG− β1τG)
or equivalently that f1fnG− β1τG is fixed under τ , which holds due to (5.5). �
The maps M and N satisfy the following relation.
Lemma 5.3. The map N is a reversing symmetry of M .
Proof. The statement entails
MN = NM−1.
Applying the involution σ to all indices in system (5.1), and interchanging fi ↔ f̃i one observes
that
Ej
(
fi, f̃i
)
= En−j
(
f̃σi , fσi
)
,
for 1 ≤ j < n and En
(
fi, f̃i
)
= En
(
f̃σi , fσi
)
. If we write the nontrivial solution as f̃j = Mj(fi),
then we also have fσj = Mj
(
f̃σi
)
, and applying σ to the j index (which just enumerates the
functions), one has
fj = Mσj
(
f̃σi
)
= M−1j
(
f̃i
)
.
Thus the inverse map is
M−1 = σMσ,
which as a function of the fi has components
M−1j = σ
(
fσj−1
τσj−1G
τσjG
)
= fj+1
στn−jG
στn−j+1G
= fj+1
τ jσG
τ j−1σG
.
The latter formula is obtained using (5.6). It follows immediately that M−1j+1 = τM−1j . Therefore
it is enough to show that (MN)1 =
(
NM−1
)
1
, that is
βnG
(β1
f1
, . . . , βnfn
)
fnτG
(β1
f1
, . . . , βnfn
) =
β1σG(f1, . . . , fn)
f2τσG(f1, . . . , fn)
. (5.8)
Using the formulas (5.3) and (5.4), one has
G
(
β1
f1
, . . . ,
βn
fn
)
=
n−1∑
i=0
i∏
j=1
βj
Fn−i
(
βi+1
fi+1
, . . . ,
βn
fn
)
=
n−1∑
i=0
i∏
j=1
βj
Fn−i(βi+1, . . . , βn)
Fn−i(fi+1, . . . , fn)
=
n−1∏
j=1
βj
Fn(f1, . . . , fn)
n−1∑
i=0
n∏
j=i+2
βj
Fi+1(f1, . . . , fi+1),
Dressing the Dressing Chain 9
and similarly
τG
(
β1
f1
, . . . ,
βn
fn
)
=
n−1∏
j=1
βj+1
Fn(f2, . . . , fn, f1)
n−1∑
i=0
n∏
j=i+2
βj+1
Fi+1(f2, . . . , fi+2),
τσG(f1, . . . , fn) =
n−1∑
i=0
i∏
j=1
βn+2−j
Fn−i(fi+2, fi+1, . . . , f2),
and
σG(f1, . . . , fn) = τστG(f1, . . . , fn) =
n−1∑
i=0
i∏
j=1
βn+1−j
Fn−i(fi+1, fi, . . . , f1).
Combining all these leads to (5.8). �
As a corollary of Lemma 5.3, using the fact that N is an involution, it follows that MN
and NM are involutions, and hence M can be written as a composition of two involutions,
M = (MN)N = N(NM). Furthermore, Lemma 5.3 implies the relations: MNM = NNN and
MNN = NNM . These relations are also satisfied by the branches of the quotient-difference
(n, 0)-correspondence. In [29, Section 6.2] it is proved that the l-th iteration of such a corre-
spondence is 2l-valued.
6 Complete integrability of the odd-dimensional discrete
dressing chain
In this section we show that the correspondence (M,N) is Liouville integrable with respect to
a quadratic Poisson structure which is of Lotka–Volterra type. Our main result is the following
theorem.
Theorem 6.1. For odd n, the correspondence defined by (5.1) is Liouville integrable.
Before proving the theorem, we introduce the relevant Poisson structures and provide some
of their basic properties.
6.1 Lotka–Volterra Poisson structures
Lotka–Volterra Poisson structures are homogeneous quadratic, and they are defined on Rn by
the formulas
{xi, xj}q = Ai,jxixj , i, j ∈ I, (6.1)
where A is a constant skew-symmetric matrix.
The rank of this Poisson structure is equal, at a generic point, to the rank of the constant
matrix A. Each null-vector of the matrix A is associated to a Casimir of the corresponding
Poisson bracket. If v = (v1, v2, . . . , vn) is such that vA = 0, then the function
n∏
i=1
xvii is a Casimir
of the Poisson bracket. Two linearly independent null-vectors correspond to two functionally
independent Casimirs (for a proof see [13, Example 8.14]).
In what follows we consider the Lotka–Volterra structures (6.1) where A is the n × n skew-
symmetric matrix with its upper triangular part defined by
Ai,j = (−1)j−i+1, 1 ≤ i < j ≤ n. (6.2)
10 C.A. Evripidou, P.H. van der Kamp and C. Zhang
The rank of the matrix A is n when n is even and n − 1 when n is odd with the null-vector
v = (1, 1, . . . , 1); a Casimir of the corresponding Poisson structure is the function x1x2 · · ·xn.
With H = x1 +x2 + · · ·+xn, the Hamiltonian vector field {·, H}q defines a system of differential
equations which, up to a simple change of variables, is isomorphic to the Bogoyavlenskij lattice
[1, 2, 4, 28].
A simpler Poisson structure is the constant Poisson structure defined by the brackets
{xi, xj}c = Bi,j , (6.3)
where B is a constant n × n skew-symmetric matrix. After some tedious but straightforward
calculations (see [6, Proposition 3]), one proves that the Poisson structures (6.1) and (6.3) are
compatible if and only if{
Bi,j = 0 for all |j − i| > 1, when n is even,
Bi,j = 0 for all 1 < |j − i| < n− 1, when n is odd.
(6.4)
For n odd, let {·, ·}b = {·, ·}q + {·, ·}c denote the sum of the brackets defined by (6.1), (6.2)
and (6.3), (6.4) where the subscript b is the vector b = (b1,2, b2,3, . . . , bn−1,n, b1,n), i.e., the non-
zero elements of the matrix B. With H = x1+x2+ · · ·+xn, the Hamiltonian vector field {·, H}b
defines a system of differential equations which can be transformed to the dressing chain (2.5).
The Poisson structure {·, ·}b is of rank n − 1 but with a more complicated Casimir than the
product x1x2 · · ·xn (see [6, 8] for an explicit construction of this Casimir).
6.2 Complete integrability
We start by showing that the maps M and N , defined in Section 5, preserve the Poisson structure
{·, ·}b = {·, ·}q + {·, ·}c with b = (−β2,−β3, . . . ,−βn, β1). To this end we define two additional
maps φ, ψ : Rn → Rn,
φ(f1, f2, . . . , fn) = (f1 + g2, f2 + g3, . . . , fn + g1),
ψ(f1, f2, . . . , fn) = (f2 + g1, f3 + g2, . . . , f1 + gn).
Lemma 6.2. For n odd, the maps M , N , φ and ψ are Poisson maps as follows
1) φ : (Rn, {·, ·}q)→ (Rn, {·, ·}b);
2) ψ : (Rn, {·, ·}q)→ (Rn, {·, ·}b);
3) M : (Rn, {·, ·}q)→ (Rn, {·, ·}q);
4) N : (Rn, {·, ·}q)→ (Rn, {·, ·}q).
Proof. The proof of items (1), (2) and (4) follows from straightforward computations. For
example, for any i = 1, 2, . . . , n
{fi, fi+1}b ◦ φ = (fi + gi+1)(fi+1 + gi+2)− βi+1 = fifi+1 + figi+2 + gi+1gi+2
= {fi + gi+1, fi+1 + gi+2}q,
{fi, fi+1}b ◦ ψ = (fi+1 + gi)(fi+2 + gi+1)− βi+1 = fi+1fi+2 + gifi+2 + gigi+1
= {fi+1 + gi, fi+2 + gi+1}q ,
where the indices are considered modulo n and in the set I. Item (3) follows from items (1)
and (2) combined with
φ ◦M = ψ and φ ◦N = ψ,
which are a consequence of (5.1). �
Dressing the Dressing Chain 11
Remark 6.3. The involution N preserves any Poisson structure of Lotka–Volterra form. It
follows that item (4), of the previous proposition, is (trivially) true for all n ∈ N. However, this
is not the case for items (1)− (3) as for even n the equations in the previous proof do not hold
for i = n. Note, for even n the bracket {·, ·}b is not Poisson, see condition (6.4).
To derive sufficiently many independent invariants for the map (5.7), we employ a matrix
version of the Lax representation. Let Φ̃i = ViΦi, Φ̂i = Φi+1 = GiΦi, where
Vi =
(
0 1
−λ g̃i − fi
)
, Gi =
(
−gi 1
−λ −fi
)
and Φ =
(
φ, φ̃
)T
. The compatibility condition is now written (modulo the periodic condition
n+ j = j) as Vi+1Gi = G̃iVi. Then one has
G̃nG̃n−1 · · · G̃1 = V1GnGn−1 · · · G1V−11 ,
which implies that the (monodromy) matrix L = GnGn−1 · · · G1 is isospectral. The eigenvalues
of L (and therefore its trace) are invariants of our map (5.7).
Using a decomposition of the Lax matrix Gi similar to the one used in [6, 30] (explained in
a more general manner in [27]), we are able to calculate the trace of L and provide a succinct
formula for the invariants. Futhermore, using the results of [30] we show that these invariants are
in involution with respect to the Poisson bracket (6.1) with matrix (6.2) and therefore complete
the proof of Theorem 6.1.
Proposition 6.4. Let r = n−1
2 ∈ N. There exist functions I0, I1, . . . , Ir, which are polynomials
of the variables
x1 = −(g2 + f1), x2 = −(g3 + f2), . . . , xn = −(g1 + fn)
of degree deg(Ii) = 2i+ 1, such that tr(L) =
r∑
i=0
Iiλ
i. With
Di =
∂2
∂xi∂xi+1
, D =
∑
i∈I
Di,
where xn+1 = x1 is understood, we have
I0 =
∏
i∈I
(1− βiDi)
∏
j∈I
xj , Ij =
(−1)j
j!
DjI0.
Furthermore, these functions are in involution with respect to the Poisson bracket {·, ·}q.
Proof. We decompose the matrix Gi as follows
Gi = Aii − (λ+ βi)K,
where
K =
(
0 0
1 0
)
, Aij =
(
−1
fi
)
·
(
gj −1
)
.
This structure simplifies the computations. For example we can immediately verify the following
properties
K2 = 0, AijAkl = −(gj + fk)Ail, AijKAkl = Ail, KAijK = K.
12 C.A. Evripidou, P.H. van der Kamp and C. Zhang
The monodromy matrix is, with yi = −(λ+ βi), in the form
L =
(
Ann + ynK
)(
An−1n−1 + yn−1K
)
· · ·
(
A1
1 + y1K
)
= AnnAn−1n−1 · · · A
1
1 +
∑
i∈I
yiAnnAn−1n−1 · · · A
i+1
i+1KA
i−1
i−1 · · · A
1
1 +
∑
i,j∈I
· · ·
= xnxn−1 · · ·x2An1 +
∑
i∈I
yixnxn−1 · · ·xi+2xi−1 · · ·x1An1 +
∑
i,j∈I
· · · ,
and its trace
tr(L) =
∏
i∈I
xi +
∑
i∈I
yi
xi+1xi
∏
j∈I
xj +
∑
i>j+1∈I
yi
xi+1xi
yj
xj+1xj
∏
k∈I
xj + · · ·
=
∏
i∈I
(1 + yiDi)
∏
j∈I
xi =
k∑
i=0
Iiλ
i.
Substituting λ = 0 yields the expression for I0, i.e.,
I0 = (1− β1D1)(1− β2D2) · · · (1− βnDn)
∏
i∈I
xi, (6.5)
while
I1 =
∑
j∈I
(1− β1D1)(1− β2D2) · · · (−Dj) · · · (1− βnDn)
∏
i∈I
xi,
I2 =
∑
j<k∈I
(1− β1D1)(1− β2D2) · · · (−Dj) · · · (−Dk) · · · (1− βnDn)
∏
i∈I
xi,
...
Applying D to I0 gives a sum of similar products where the j-th term (1 − βjDj) is replaced
by Dj , which is equal to −I1. Similarly, applying this operator k times yields (−1)kIk k! times,
namely once for each permutation of k indices. This proves the first part of the proposition.
To prove the second part of the proposition, we first note that the invariants Ii (as functions
of xi), as obtained from the trace of L, coincide with the invariants that Veselov and Shabat
provided for the continuous dressing chain [30]. In that paper, using the Lenard–Magri scheme,
they proved that the invariants Ii are in involution with respect to the Poisson bracket {·, ·}b =
{·, ·}q + {·, ·}c. Therefore, in order to prove that the invariants Ii (considered as functions of fi)
are in involution with respect to {·, ·}q, it suffices to show that the map
Rn → Rn, (f1, f2, . . . , fn) 7→ (x1, x2, . . . , xn)
is a Poisson map between (Rn, {·, ·}q) and (Rn, {·, ·}b). This is precisely item (1) of Lemma 6.2.
�
Remark 6.5. The expression for I0 (6.5) in terms of fi, gi, and βi is quite cumbersome, e.g.,
for n = 3,
I0 = −f1f2f3 − f1f2g1 − f1f3g3 − f1g1g3 − f2f3g2 − f2g1g2 − f3g2g3
− g1g2g3 + β1f2 + β3f1 + β1g3 + β2f3 + β2g1 + β3g2.
Dressing the Dressing Chain 13
However, when imposing the relation figi = βi, the expression simplifies drastically, and we have
I0 = −(f1f2f3 + g1g2g3). The fact that a similar expression can be obtained for any n can be
seen from
L |λ=0=
n∏
i=1
(
−gi 1
0 −fi
)
= (−1)n
n∏
i=1
gi −
n∑
i=1
i−1∏
j=1
fi
n∏
k=i+1
gi
0
n∏
i=1
fi
.
Remark 6.6. For n even the map M is anti-volume preserving and for odd n it is volume
preserving. The map N is measure preserving when n is even and anti-measure preserving
when n is odd. The density of the measure is
n∏
i=1
1
fi
.
Acknowledgements
This work was supported by the Australian Research Council, by the China Strategy Imple-
mentation Grant Program of La Trobe University, by the NSFC (No. 11601312) and by the
Shanghai Young Eastern Scholar program (2016-2019).
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1 Introduction
2 Background
3 Dressing the dressing chain
4 The modified dressing chain
5 Explicit formulas for the n-dimensional discrete dressing chain, and linear growth of multivaluedness
6 Complete integrability of the odd-dimensional discrete dressing chain
6.1 Lotka–Volterra Poisson structures
6.2 Complete integrability
References
|
| id | nasplib_isofts_kiev_ua-123456789-209513 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T14:34:35Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Evripidou, C.A. van der Kamp, P.H. Zhang, C. 2025-11-24T10:05:24Z 2018 Dressing the Dressing Chain / C.A. Evripidou, P.H. van der Kamp, C. Zhang // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 32 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35Q53; 37K05; 39A14 arXiv: 1804.02564 https://nasplib.isofts.kiev.ua/handle/123456789/209513 https://doi.org/10.3842/SIGMA.2018.059 The dressing chain is derived by applying Darboux transformations to the spectral problem of the Korteweg-de Vries (KdV) equation. It is also an auto-Bäcklund transformation for the modified KdV equation. We show that by applying Darboux transformations to the spectral problem of the dressing chain, one obtains the lattice KdV equation as the dressing chain of the dressing chain, and that the lattice KdV equation also arises as an auto-Bäcklund transformation for a modified dressing chain. In analogy to the results obtained for the dressing chain (Veselov and Shabat proved complete integrability for odd-dimensional periodic reductions), we study the (0,n)-periodic reduction of the lattice KdV equation, which is a two-valued correspondence. We provide explicit formulas for its branches and establish complete integrability for odd n. This work was supported by the Australian Research Council, by the China Strategy Implementation Grant Program of La Trobe University, by the NSFC (No. 11601312), and by the Shanghai Young Eastern Scholar program (2016-2019). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Dressing the Dressing Chain Article published earlier |
| spellingShingle | Dressing the Dressing Chain Evripidou, C.A. van der Kamp, P.H. Zhang, C. |
| title | Dressing the Dressing Chain |
| title_full | Dressing the Dressing Chain |
| title_fullStr | Dressing the Dressing Chain |
| title_full_unstemmed | Dressing the Dressing Chain |
| title_short | Dressing the Dressing Chain |
| title_sort | dressing the dressing chain |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209513 |
| work_keys_str_mv | AT evripidouca dressingthedressingchain AT vanderkampph dressingthedressingchain AT zhangc dressingthedressingchain |