Negative Times of the Davey-Stewartson Integrable Hierarchy
We use the example of the Davey-Stewartson hierarchy to show that, in addition to the standard equations given by the Lax operator and evolutions of time with positive numbers, one can consider time evolutions with negative numbers and the same Lax operator. We derive corresponding Lax pairs and int...
Збережено в:
| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2021 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2021
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211436 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Negative Times of the Davey-Stewartson Integrable Hierarchy. Andrei K. Pogrebkov. SIGMA 17 (2021), 091, 12 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859789727120490496 |
|---|---|
| author | Pogrebkov, Andrei K. |
| author_facet | Pogrebkov, Andrei K. |
| citation_txt | Negative Times of the Davey-Stewartson Integrable Hierarchy. Andrei K. Pogrebkov. SIGMA 17 (2021), 091, 12 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We use the example of the Davey-Stewartson hierarchy to show that, in addition to the standard equations given by the Lax operator and evolutions of time with positive numbers, one can consider time evolutions with negative numbers and the same Lax operator. We derive corresponding Lax pairs and integrable equations.
|
| first_indexed | 2026-03-16T04:01:36Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 091, 12 pages
Negative Times of the Davey–Stewartson
Integrable Hierarchy
Andrei K. POGREBKOV ab
a) Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
E-mail: pogreb@mi-ras.ru
b) Skolkovo Institute of Science and Technology, Moscow, Russia
Received June 08, 2021, in final form October 01, 2021; Published online October 12, 2021
https://doi.org/10.3842/SIGMA.2021.091
Abstract. We use example of the Davey–Stewartson hierarchy to show that in addition to
the standard equations given by Lax operator and evolutions of times with positive num-
bers, one can consider time evolutions with negative numbers and the same Lax operator.
We derive corresponding Lax pairs and integrable equations.
Key words: commutator identities; integrable hierarchies; reductions
2020 Mathematics Subject Classification: 37K10; 70H06
To Leon A. Takhtajan with best wishes
1 Introduction
In [3], we proposed a method of derivation of (2+1)-dimensional nonlinear integrable equations
based on commutator identities on associative algebras. Taking into account the algebraic
similarity of operator commutators and derivatives, we have transformed commutator identities
into linear partial differential equations. A characteristic property of these linear equations is the
possibility to lift them up to nonlinear, integrable ones. In [4, 5], this approach was extended to
differential-difference and difference equations, where the analogy of similarity transformations
and shifts of independent variables was used. In [6], we developed this result for non-Abelian
identities of commutators.
To formulate the main aspects of this approach, we start here with the simplest examples.
Let A and B be arbitrary elements of an arbitrary associative algebra A. Then they obey the
commutator identity
4
[
A3, [A,B]
]
− 3
[
A2,
[
A2, B
]]
− [A, [A, [A, [A,B]]]] = 0. (1.1)
Being a trivial consequence of associativity, this identity easily proves that the function
B(t1, t2, t3) = et1A+t2A2+t3A3
Be−t1A−t2A2−t3A3
, (1.2)
i.e., such that Btn = [An, B], n = 1, 2, 3, obeys the linearized Kadomtsev–Petviashvili (KP)1
equation with respect to the variables tj :
4
∂2B(t)
∂t1∂t3
− 3
∂2B(t)
∂t22
− ∂4B(t)
∂t41
= 0.
This paper is a contribution to the Special Issue on Mathematics of Integrable Systems: Classical and Quan-
tum in honor of Leon Takhtajan.
The full collection is available at https://www.emis.de/journals/SIGMA/Takhtajan.html
1More precisely, KPII.
mailto:pogreb@mi-ras.ru
https://doi.org/10.3842/SIGMA.2021.091
https://www.emis.de/journals/SIGMA/Takhtajan.html
2 A.K. Pogrebkov
It was stated in [3, 7] that there are similar relations for higher commutators. In the case of KP,
they lead to higher linear equations
2n∂tn∂
n
t1B =
(
∂t2 + ∂2t1
)n
B −
(
∂t2 − ∂2t1
)n
B, n = 3, 4, . . . .
Similar results were obtained in [4, 5, 6] for difference and differential-difference equations.
In that case, we replace (1.1) with, say, the commutator identity[
A,
[
A−1, B
]]
= 2B −ABA−1 −A−1BA,
where the element A is assumed to be invertible. Thus, in addition to commutators of the
kind (1.1), we get similarity transformations here (commutators in the group sense). Therefore,
we introduce the element B depending on the number n1 and continuous variables t1 and t−1
by means of
B(t1, t−1, n1) = et1A+t−1A−1
An1BA−n1e−t1A−t−1A−1
, (1.3)
and denote the shift with respect to variable n1 as B(1) = ABA−1. Accordingly, this element B
obeys the linear differential-difference equation
Bt1,t−1 = 2B −B(1) −B(−1),
which gives a linearized version of the two-dimensional Toda system [8, 9].
In [3, 4, 5], we proved that any linear equation, resulting from the commutator identity, can
be lifted up to a nonlinear integrable equation using a special dressing procedure. In this paper,
our goal is to extend the class of commutator identities. For this purpose one can use arbitrary
functions f(A) with commutativity being the only condition they should obey: [f(A), g(A)] = 0.
A natural generalization of the choice of functions of the element A was suggested in [3, 7].
We assume that in algebra A there exists an element σ such that
σ2 = 1, [A, σ] = 0, {B, σ} = 0, [Aσ,B] = σ{A,B}, (1.4)
where {·, ·} denotes anticommutator. In particular, we can consider elements of A as 2 × 2
matrices, where A is proportional to the unity matrix I, B is off diagonal,
B =
(
0 B1
B2 0
)
, (1.5)
and σ = σ3 is a Pauli matrix. Thus the commutator of Aσ and B reduces to the anticommutator
of A and B, so that for any n we have the commutator identity[
A2n, B
]
= σ
[
An,
[
σAn, B
]]
.
We consider the commutators [A,B] and [Aσ,B] as generating and decompose the commu-
tators [An, B] or [Anσ,B] for n ≥ 2 in their terms. Thanks to (1.4), (1.5) it is easy to prove the
following commutator identities:
2nσn+1
[
An, B
]
= [(σ + I)A, [(σ + I)A, . . . , [(σ + I)A︸ ︷︷ ︸
n
, B], . . . ]
− [(σ − I)A, [(σ − I)A, . . . , [(σ − I)A︸ ︷︷ ︸
n
, B], . . . ], (1.6)
2nσn+1
[
σAn, B
]
= [(σ + I)A, [(σ + I)A, . . . , [(σ + I)A︸ ︷︷ ︸
n
, B], . . . ]
+ [(σ − I)A, [(σ − I)A, . . . , [(σ − I)A︸ ︷︷ ︸
n
, B], B], . . . ], (1.7)
Negative Times of the Davey–Stewartson Integrable Hierarchy 3
where n ≥ 1. These two sets of commutator identities give two sets of differential hierarchies if,
in addition to (1.2), we introduce two sets of variables, t = {t1, t2, . . . } and x = {x1, x2, . . . },
given by the equations
Btn =
[
An, B
]
, (1.8a)
Bxn
=
[
σAn, B
]
. (1.8b)
Taking n = 1 here, we get
(∂x1 ± ∂t1)
nB = [(σ ± I)A, . . . , [(σ ± I)A︸ ︷︷ ︸
n
, B], . . . ], (1.9)
so thanks to (1.6) and (1.7), we get linear differential equations for B(t, x, z),
2nσn+1∂tnB = (∂x1 + ∂t1)
nB − (∂x1 − ∂t1)
nB, (1.10)
2nσn+1∂xnB = (∂x1 + ∂t1)
nB + (∂x1 − ∂t1)
nB. (1.11)
For n = 2, these equalities are read as σBt2 = Bt1x1 and σBx2 = Bt1t1 + Bx1x1 , respectively.
In [7], these linear equations were lifted to the Davey–Stewartson equation (see [1]) and higher
equations of its hierarchy.
Here we consider “negative” version of this hierarchy, i.e., we assume negative values of n
in (1.8). In Section 2, we derive the corresponding commutator identities and the corresponding
linear differential equations. In Section 3, we introduce the realization of elements of the asso-
ciative algebra A using pseudo-differential operators. On this basis, in Section 4, we consider
the dressing procedure that enables introduction of the dressing operator and its time evolu-
tions. The Lax pair and nonlinear equations are derived in Section 5. Section 6 is devoted to
(1 + 1)-dimensional reductions of the systems under consideration. Some concluding remarks
are given in Section 7.
2 Commutator identities and linear equations
Our goal here is to construct a commutator identity based on the commutators [A,B], [σA,B],
and
[
A−1, B
]
or [A,B], [σA,B], and
[
σA−1, B
]
, where we assume existence of the inverse
element A−1. By analogy with the above, we consider commutators [A,B] and [σA,B] as
generating for the commutators
[
A−1, B
]
and
[
σA−1, B
]
. It is easy to check that we have here
the following commutator identities:[
σA,
[
σA,
[
A−1, B
]]]
−
[
A,
[
A,
[
A−1, B
]]]
+ 4[A,B] = 0, (2.1)[
σA,
[
σA,
[
σA−1, B
]]]
−
[
A,
[
A,
[
σA−1, B
]]]
− 4[σA,B] = 0. (2.2)
Taking into account that all these commutators mutually commute, we consider B as a function
of t1, x1 and t−1, or x−1, such that
Bt1 = [A,B], Bx1 = [σA,B], (2.3a)
Bt−1 =
[
A−1, B
]
, Bx−1 =
[
σA−1, B
]
. (2.3b)
Thanks to (2.3), we get from (2.1) and (2.2) linear equations of motion (cf. (1.10) and (1.11)
for n = 2)
Bx1x1t−1 −Bt1t1t−1 + 4Bt1 = 0, (2.4)
Bx1x1x−1 −Bt1t1x−1 − 4Bx1 = 0. (2.5)
4 A.K. Pogrebkov
Thus, we again have two versions of the equations: one involving ∂t−1 , and the other invol-
ving ∂x−1 . Taking into account the symmetry of these two equations with respect to the substi-
tution x−1 ↔ −t−1, we study here mainly (2.4).
By extending (1.8) to negative values of n, we arrive at a hierarchy of commutator identities
and linear equations. We can use (1.10) and (1.11), substitute n→ −n into these equations and
multiply them both by
(
∂2x1
− ∂2t1
)
. Thus, we get(
∂t−n
(
∂2x1
− ∂2t1
)n
+ 2n(σ∂x1 + ∂t1)
n − 2n(σ∂x1 − ∂t1)
n
)
B = 0, (2.6)(
σ∂x−n
(
∂2x1
− ∂2t1
)n
B − 2n(σ∂x1 + ∂t1)
n − 2n(σ∂x1 − ∂t1)
n
)
B = 0, (2.7)
where n = 1, 2, . . . and where by analogy with (2.3b)
Bt−n =
[
A−n, B
]
, Bx−n =
[
σA−n, B
]
. (2.8)
We omit here form of (2.6) and (2.7) in terns of commutator identities. It can be easely restored
with the help of (1.9). In the case of n = 1, equations (2.6) and (2.7) are reduced to (2.1)
and (2.2). Now we have to show that all these linear equations admit lift up to nonlinear
integrable ones.
3 Realization of elements of the associative algebra
To this end, we consider a special realization of the elements of the associative algebra A, see
[3, 4, 5, 6]. By analogy with the standard definition of the pseudo-differential operators, we
define an element F of A by its symbol F̃ (t, x, z). Here t and x denote (finite) subsets of
real variables t = {. . . , t−2, t−1, t1, t2 . . . }, x = {. . . , x−2, x−1, t1, t2, . . . }, and z ∈ C denotes
a complex parameter. The subsets t and x definitely include the variables t1 and x1 and at least
one of the other variables of these lists. In the following we call such subsets minimal. The
symbol of the composition of two elements of the algebra is given by means of the symbols of
cofactors in the form
F̃G(t, x, z) =
1
2π
∫
dp
∫
dy F̃ (t, x, z + ip)eip(t1−y)G̃(y, t′, x, z), (3.1)
where t′ denotes a subset t without variable t1. We see that the variable t1 plays a special
role here: the composition with respect to other variables is pointwise. In what follows we
consider elements of the algebra A such that their symbols belong to the space of tempered
distributions of their arguments. The symbol of the unity operator is 1, and we choose the
symbol of operator A as
Ã(t, x, z) = z. (3.2)
Thanks to (3.1) we have that for any F
ÃnF (t, x, z) = (z + ∂t1)
nF̃ (t, x, z), F̃An(t, x, z) = znF̃ (t, x, z),
where An is understood as n-th power of composition (3.1), where now n ∈ Z. Then, for n = 1,
we get [A,F ] = ∂t1F according to (1.8a). Further relations of these equalities give in terms of
symbols:
B̃tn(t, x, z) =
(
(z + ∂t1)
n − zn
)
B̃(t, x, z),
B̃xn
(t, x, z) = σ
(
(z + ∂t1)
n + zn
)
B̃(t, x, z).
Negative Times of the Davey–Stewartson Integrable Hierarchy 5
Because of our assumption, the symbol B̃(t, x, z) admits a Fourier transform with respect to
the variable t1, so the above relations show
B̃(t, x, z) =
∫
dp exp
(∑
n
(
(z + ip)n − zn
)
tn + σ
∑
n
(
(z + ip)n + zn
)
xn
)
f(p, z), (3.3)
where n ∈ Z and f(p, z) is an arbitrary 2× 2 off diagonal matrix function independent of all tn
and xn. Note that here we do not specify set of “times” ti and xi involved in the evolution
equation. We know that this set includes at least three times: t1, x1 and one of times tn, or xn
with n ̸= 0 and 1. It can include more times, but t1 and x1 and every third time gives an
evolution equation generated by the commutator identity. Thus, in (3.3), summation in the
exponent goes over finite number of terms, corresponding to times that are “switched on” while
other times are equal to zero.
It is natural to impose on B̃(t, x, z) the conditions of convergence of the integral and the
boundedness of the limits of B̃(t, x, z) as t, or x tends to infinity. Two obvious conditions are
sufficient for this. The first one is given by the choice f(p, z) = δ(p+2zIm)g(z), where δ denotes
delta-function, so that (3.3) takes the form
B̃(t, x, z) = exp
(∑
n
(
zn − zn
)
tn + σ
∑
n
(
zn + zn
)
xn
)
g(z), (3.4)
where g(z) is an arbitrary bounded function of its argument. But in order to get B̃(t, x, z)
bounded with respect to variables xn, it is necessary to perform substitution
xn → ixn, (3.5)
where the new xn are real.
The second case is given by reduction f(p, z) = δ(zRe)h(p, zIm), where z = zRe + izIm
and h(p, zIm) is an arbitrary function of its arguments. Then (3.3) takes the form
B̃(t, x, z) =
∫
dp exp
(∑
n
in
(
(zIm + p)n − znIm
)
tn + σin
∑
n
(
(zIm + p)n + znIm
)
xn
)
× h(p, zIm)δ(zRe), (3.6)
Here we see that B̃(t, x, z) is bounded with respect to variables tn and xn with odd numbers,
and in order to make it bounded for variables with even numbers, we need to make a substitution
t2n → it2n, x2n → ix2n. (3.7)
Thus we have two types of systems defined by the choices (3.4) and (3.6).
4 Dressing procedure
Specific property of the above set of operators is the possibility of defining operation of ∂̄-
differentiation by the complex variable z, F → ∂̄F . In terms of symbols, this is defined, see [3], as
( ˜̄∂F )(t, x, z) = ∂F̃ (t, x, z)
∂z
, (4.1)
where derivative is understood in the sense of distributions. Thanks to (3.2), we get the equality
∂̄A = 0, (4.2)
which plays essential role in what follows.
6 A.K. Pogrebkov
Now we can define a dressing operator K with symbol K̃(t, x, z) by means of ∂-problem
∂K = KB, (4.3)
where the product in r.h.s. is understood in the sense of the composition law (3.1). Thanks
to (3.1) and (4.1), the equality (4.3) takes the explicit form
∂K̃(t, x, z)
∂z
= K̃(t, x, z) exp
(∑
n
(
zn − zn
)
tn + σ
∑
n
(
zn + zn
)
xn
)
g(z), (4.4)
for time evolutions given by (3.4) and the form
∂K̃(t, x, z)
∂z
= δ(zRe)
∫
dp K̃(t, x, ip)×
× exp
(∑
n
in
(
(pn − znIm)tn + σ(pn + znIm)xn
))
h(p− zIm, zIm), (4.5)
for time evolutions given by (3.6). Thus, in the case of (4.4), the equation (4.3) gives the ∂-
problem, while in the case of (4.5) we get Riemann–Hilbert problem. In both these cases, we
normalize solution K of the equation (4.3) by the asymptotic condition
K̃(t, x, z) → 1, z → ∞. (4.6)
In what follows, we assume unique solvability of the problem (4.3), (4.6). The time evolution
of the dressing operator follows from these equations. Say, due to (1.8) and (2.3) we get
∂Ktn = KtnB +K
[
An, B
]
, ∂Kxn = KxnB +K
[
σAn, B
]
. (4.7)
Accordingly,
∂Ktmtn = KtmtnB +Ktn
[
Am, B
]
+Ktm
[
An, B
]
+K
[
Am,
[
An, B
]]
thus, taking into account the commutativity of Am and An, we get ∂(Ktmtn−Ktntm) = (Ktmtn−
Ktntm)B by (4.3). Thus, the commutativity of derivatives
Ktmtn = Ktntm (4.8)
follows due to the unique solvability of the problem (4.3), (4.6). Similarly, we prove thatKxmtn =
Ktnxm and Kxmxn = Kxnxm .
In [7], the time derivatives of the dressing operator for positive times (n > 0 in (1.8)) were
calculated in terms of the asymptotic decomposition of the dressing operator K
K̃(t, x, z) = 1 + u(t, x)z−1 + v(t, x)z−2 + w(t, x)z−3 + o
(
z−3
)
, (4.9)
where u, v, and w are multiplication operators, i.e., their symbols do not depend on z. Say, using
(4.7) for n = 1 we get ∂Kt1 = Kt1B +K[A,B]. This can be written as ∂(Kt1 +KA) = (Kt1 +
KA)B, where (4.2) and (4.3) were used. Due to the condition of unique solvability of (4.3), (4.6)
we derive that there exists multiplication operator X such that Kt1 +KA = (A+X)K. Thanks
to (4.9), it is easy to see that it equals to zero, so we have
Kt1 = [A,K]. (4.10)
The situation with Kx1 is more involved, here analogous multiplication operator does not vanish
and by (4.6) we get
Kx1 = [σA,K]− [σ, u]K, (4.11)
Negative Times of the Davey–Stewartson Integrable Hierarchy 7
where the multiplication operator u is defined in (4.9). Combining (4.10) and (4.11) we get
Kx1 = σKt1 + [σ,K]A− [σ, u]K. (4.12)
Our goal here is to extend the approach of [7] to the negative numbers of times in (1.8). More
exactly, we start with the times t1 and x1 as above and we choose either t−1 or x−1 as the third
time according to (2.3b).
To determine the evolutions with respect to t−1 or x−1 for the dressing operator we differen-
tiate (4.3) and use (2.3b):
∂Kt−1 = Kt−1B +K
[
A−1, B
]
, ∂Kx−1 = Kx−1B +K
[
σA−1, B
]
, (4.13)
so for the first equality, we have ∂Kt−1 = Kt−1B +KA−1B −KBA−1, i.e., thanks to (4.2)
∂(Kt−1A+K) =
(
Kt−1A+K
)
A−1BA. (4.14)
We see that situation here is more complicated than in the case of positive numbers of times.
There we were able to reduce the equations to the form ∂(Ktn +KAn) = (Ktn +KAn)B due
to (4.2). While for negative n, this equality gives an additional delta-term. Therefore, to use
the relation (4.14), we must find replacement for A−1BA.
This can be done by introducing a discrete variable, cf. [4] and (1.3) here. We assume that
the symbols of B, K, etc. depend on an intermediate variable n ∈ Z. Denote B̃(1)(t, x, n, z) =
B̃(t, x, n+ 1, z), K̃(1)(t, x, n, z) = K̃(t, x, n+ 1, z) and set
B(1) = ABA−1, B(−1) = A−1BA, . . . . (4.15)
It is easy to see that these shifts commute with times t and x:
(
B(1)
)
tj
= (Btj )
(1),
(
B(1)
)
xj
=
(Bxj )
(1) and we extend definition of composition law (3.1) to symbols that depend on n pointwise
with respect to this variable. Now ∂K(1) = K(1)ABA−1 because of (4.3), so that due to the
unique solvability of the problem (4.3), (4.6) there exists a multiplication operator ψ such that
K(1)A = (A+ ψ)K, (4.16)
and thanks to (4.9) we get
ψ = u(1) − u, (4.17)
where u(1)(t, x, n) = u(t, x, n + 1). Let us shift n → n + 1 of (4.14) that due to (4.15) gives
∂
(
K
(1)
t−1
A + K(1)
)
=
(
K
(1)
t−1
A + K(1)
)
B so that, because of (4.6), there exists multiplication
operator Z such that K
(1)
t−1
A+K(1) = ZK. Thanks to (4.9), we get that Z = 1 + u
(1)
t−1
.
It looks like we have constructed a (3+1)-dimensional integrable system with the independent
variables t1, x1, t−1, and n. But in fact, we have two different systems here: t1, x1, n (see (4.16))
and t1, x1, t−1, because the dependence on n can be excluded. Indeed, substituting K(1) for K
by means of (4.16) and using ψ as new dependent variable in (4.17) instead of u(1), we get
Kt1t−1 +Kt1A
−1 +Kt−1A+ ψ
(
Kt−1 +KA−1
)
− ut1K = 0, (4.18)
Kt1x−1 +Kt1σA
−1 +Kx−1A+Kσ + ψ
(
Kx−1 +KσA−1
)
− (σ + ux−1)K = 0. (4.19)
Here the equation (4.19) is derived by analogy using the second equality in (4.13). The compa-
tibility of any of these equations with (4.12) can be proved like in (4.8).
Compatible evolutions (4.12) and (4.18) or (4.12) and (4.19) admit higher (in fact, lower)
versions that involve the times t−n and x−n, n > 1, see (2.8). By analogy with (4.13), we get
for this case by (2.8)
∂Kt−n = Kt−nB +K
[
A−n, B
]
. (4.20)
8 A.K. Pogrebkov
Multiplying this equality by An from the right, we use n-multiple application of (4.15): B[−n] =
A−nBAn. Thus (4.20) takes the form
∂
(
Kt−nA
n +K
)
=
(
Kt−nA
n +K
)
B[−n],
cf. (4.14). Again thanks to the assumed unique solvability of the Inverse problem (4.3), (4.6) we
get that there exist multiplication operators α0, . . . , αn−1 such that
K
[n]
t−n
An +K [n] =
n−1∑
j=0
αjA
jK, (4.21)
where we applied n-fold shift operation. The operators αj are defined in terms of operators u,
v, w, etc. in (4.9). We omit these calculations here.
Next, we execute an (n− 1)-fold shift of a discrete variable in equation (4.16), which gives
K [n]An =
(
A+ ψ[n−1]
)(
A+ ψ[n−2]
)
· · ·
(
A+ ψ
)
K, (4.22)
where the multiplication operator ψ was defined in (4.17). The final expression follows as a result
of inserting of K [n] from (4.22) to (4.21), which again cancels dependence on the auxiliary
variable n. The consideration of dependence on x−n is similar.
5 Lax pair and nonlinear equations
In (4.8), we proved that the commutativity of evolutions (4.10), (4.11), (4.13), and (4.16) is
a direct consequence of commutativity of evolutions (2.3) and (4.15) and the consequence of the
unique solvability of the problem (4.3), (4.6). These conditions lead to the compatibility of the
equation (4.12) with (4.18), or (4.19), which give nonlinear equations of motion. To simplify
these equations, it is reasonable to rewrite them in terms of the Jost solutions defined by means
of the symbol of dressing operator:
φ(t, x, z) = K̃(t, x, z)ezt1+σzx1+z−1t−1+σz−1x−1 . (5.1)
Here we omit dependence on the discrete variable n, since it was excluded from (4.18) and (4.19).
Thanks to this substitution coefficients of the equations (4.12), (4.18), and (4.19) become
independent on z:
φx1 − σφt1 + [σ, u]φ = 0, (5.2)
φt1t−1 + ψφt−1 − (1 + ut−1)φ = 0, (5.3)
φt1x−1 + ψφx−1 − (σ + ux−1)φ = 0, (5.4)
where the first equation is the famous two-dimensional linear Zakharov–Shabat problem.
One can also rewrite (4.3) in terms of the Jost solutions. Say, by means of (3.4) we get
∂φ(t, x, z)
∂z
= φ(t, x, z)g(z), (5.5)
and by means of (3.6)
∂φ(t, x, z)
∂z
= δ(zRe)
∫
dpφ(t, x, ip)h(p− zIm, zIm). (5.6)
Negative Times of the Davey–Stewartson Integrable Hierarchy 9
We see that the equations on the Jost solutions are independent on all “time” variables t and x.
The dependence on them, as well as on z in (5.2)–(5.4) is given by (4.6), which, thanks to (5.1),
takes the form
lim
z→∞
φ(t, x, z)e−zt1−σzx1−z−1t−1−σz−1x−1 = 1. (5.7)
Note that (5.5) is a standard ∂-problem with the normalization condition (5.7), where we must
perform substitution mentioned in (3.5). At the same time, (5.6) shows that the Jost solution
in this case is analytic in the left and right half planes of z with discontinuity on the imaginary
axis. Thus here inverse problem is given in terms of the Riemann–Hilbert problem, i.e., we
define the boundary values of the Jost solution as φ±(t, x, izIm) = lim
zRe→±0
φ(t, x, z) and set
φ+(t, x, izIm)− φ−(t, x, izIm) =
∫
dpφ−(t, x, ip)h(p− zIm, zIm),
under the condition (5.7) and substitution given in (3.7). The difference of these two formulations
of the inverse problem results from the condition of boundedness of the symbol of operator B
in (3.4) and (3.6). In the case of (5.5) tn are real and xn are pure imaginary, while in the case
of (5.6) tn and xn with odd n are real and are pure imaginary for even n.
The compatibility of (5.2) with (5.3) and (5.4) follows from (4.8) and (5.1). Thus, we get the
following theorem.
Theorem 5.1.
(a) Let the minimal subset of independent variables includes t1, x1, and t−1. Then compati-
bility of (5.2) and (5.3) gives
ut1t−1σ − ux1t−1 − [σ, ψ(1 + ut−1)] + [ut−1 , [σ, u]] = 0, (5.8)
ψx1 − σψt1 − [σ, ut1 ] + [σ, ψ]ψ + [[σ, u], ψ] = 0. (5.9)
(b) Let the minimal subset of independent variables includes t1, x1, and x−1. Then compati-
bility of (5.2) and (5.4) gives
ut1x−1σ − ux1x−1 − [σ, ψ(σ + ux−1)] + [σ + ux−1 , [σ, u]] = 0, (5.10)
where the equation for ψ coincides with (5.9).
It is natural to decompose both matrices u and ψ into diagonal and anti-diagonal parts:
u = ud + ua, ψ = ψd + ψa,
so
[
σ, ud
]
= 0, [σ, ua] = 2σua thanks to (1.4). Then anti-diagonal parts of the equations (5.8)
and (5.10) give
uat−1t1 + σuat−1x1
+ 2ψa
(
1 + udt−1
)
+ 2ψduat−1
+ 2
[
ua, udt−1
]
= 0, (5.11)
uax−1t1 + σuax−1x1
+ 2ψa
(
σ + udx−1
)
+ 2ψduax−1
− 4σua + 2
[
ua, udx−1
]
= 0, (5.12)
while their diagonal parts reduces to the derivative of (5.8) with respect to t−1 and of (5.10)
with respect to x−1 of one and the same equation
udt1 − σudx1
− 2
(
ua
)2
= 0 (5.13)
10 A.K. Pogrebkov
that we have integrated here with respect to t−1 (or, correspondingly, to x−1) under the as-
sumption of the rapid decay of u as (t1, x1) → ∞.
Similarly, we derive that diagonal and anti-diagonal parts of (5.9) obey
ψd
t1 − σψd
x1
− 2σ
(
ψa
)2
+ 2
{
ua, ψa
}
= 0, (5.14)
ψa
t1 − σψa
x1
+ 2uat1 − 2ψaψd + 2
[
ψd, ua
]
= 0, (5.15)
where {·, ·} denotes anticommutator.
Corollary 5.2. Each of these systems of four equations, (5.11), (5.13), (5.14), (5.15),
and (5.12), (5.13), (5.14), (5.15), has only one evolution equation specific to one or another
system. The other three equations play an auxiliary role and coincide.
6 Dimensional reductions
Here we introduce (1 + 1)-dimensional reductions of (2 + 1)-dimensional nonlinear integrable
equations constructed above. Such reductions follow due to time evolutions (3.4), (3.6), which,
due to (4.3) and (4.6), lead to the same reductions of the dressing operator K, and then to
reductions of all coefficients of the series (4.9). The reduction of time dependence of the ope-
rator B, in turn, is the result of conditions on the supports of the functions g(z) and h(p, zIm)
in (3.4) and (3.6), which reduce the number of independent time variables. For example, for the
operator B̃(t, x, z) in (3.4), depending on times t1, x1, and t−1, we can cancel dependence on x1
by imposing condition
g(z) = δ(zRe)G(zIm).
Thanks to (3.4), this gives
B̃(t, x, z) = exp
(
−2i
[
zImt1 −
t−1
zIm
])
δ(zRe)G(zIm). (6.1)
It is clear that this dependence on two variables is preserved in evolution and that thanks to
the ∂-problem (4.3) and (4.6) and the composition law (3.1) (or due to (4.4)) we get the symbol
of the operator K also independent of x1. Moreover, this operator is now analytic function for
zRe ̸= 0. Taking into account the independence of the operator K from x1, we must change the
definition of the Jost solution, cf. (5.1),
φ(t1, t−1, z) = K̃(t, x, z)ezt1+z−1t−1 ,
so thanks to (5.2) and (5.3), the Jost solution obeys a Lax pair, where the first equation reads as
σφt1 − [σ, u]φ = zφσ, (6.2)
(cf. (5.2)) and the second equation coincides with (5.3).
In the same way, we derive compatibility conditions for these equations from (5.8) and (5.9):
ut1t−1σ − [σ, ψ(1 + ut−1)] + [ut−1 , [σ, u]] = 0,
σψt1 + [σ, ut1 ]− [σ, ψ]ψ − [[σ, u], ψ] = 0.
We see that the ∂-problem in this case is the Riemann–Hilbert problem for a function analytic
in the right and left half planes on the complex z-plane with discontinuity given by (6.1) on
Negative Times of the Davey–Stewartson Integrable Hierarchy 11
the imaginary axis. The function K is normalized by the condition (4.6) at z → ∞. Summari-
zing, (6.2) is nothing but Zakharov–Shabat linear problem [10] that has been extensively studied
in the literature, e.g., [2].
This is not the only reduction applicable to (3.4). Setting there
g(z) = δ(|z| − 1)g(zIm),
we get the scattering data, i.e., symbol of operator B, depending on two variables t1−t−1 and x1:
B̃(t, x, z) = δ(|z| − 1) exp
(
−2izIm(t1 − t−1) + 2σzRex1
)
g(zIm). (6.3)
Thus after shifting t1 → t1 + t−1, we exclude the dependence on t−1 from B, and then from K.
Now, because of the delta-function in (6.3), we reduce the inverse problem (4.3) to the Riemann–
Hilbert problem on the circle |z| = 1 and the normalization condition (4.6). Now we define the
Jost solution by means of the relation
φ(t1, x1, z) = K̃(t1 + t−1, t−1, x1, z)e
zt1+σzx1 ,
where the r.h.s. does not depend on t−1. The integrable equation follows from (5.8):
ut1t1σ − ux1t1 − [σ, ψ(1 + ut1)] + [ut1 , [σ, u]] = 0,
where the second equation (5.9) is left unchanged.
By analogy, we can consider the reductions of the symbol of operator B in (4.5), i.e.,
when t1, x1, and x−1 are chosen as independent variables.
7 Concluding remarks
In the above derivation of nonlinear integrable equations we needed some essential assumptions,
the main was the condition of unique solvability of the ∂-problem (4.3), (4.6). But when non-
linear equation is derived, these assumptions are not necessary: the nonlinear equation is given
as a compatibility condition of a Lax pair. On the other hand, the existence of linear equations
given by commutator identities always leads to nonlinear integrable equations, as was shown
above.
Acknowledgements
This research is supported by a grant from the Russian Science Foundation (Project No. 19-11-
00062).
References
[1] Davey A., Stewartson K., On three-dimensional packets of surface waves, Proc. Roy. Soc. London Ser. A
338 (1974), 101–110.
[2] Faddeev L.D., Takhtajan L.A., Hamiltonian methods in the theory of solitons, Springer Series in Soviet
Mathematics, Springer-Verlag, Berlin, 1987.
[3] Pogrebkov A.K., Commutator identities on associative algebras and the integrability of nonlinear evolution
equations, Theoret. and Math. Phys. 154 (2008), 405–417, arXiv:nlin.SI/0703018.
[4] Pogrebkov A.K., 2D Toda chain and associated commutator identity, in Geometry, Topology, and Math-
ematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 224, Amer. Math. Soc., Providence, RI, 2008,
261–269, arXiv:0711.0969.
[5] Pogrebkov A.K., Hirota difference equation and a commutator identity on an associative algebra, St. Pe-
tersburg Math. J. 22 (2010), 473–483.
https://doi.org/10.1098/rspa.1974.0076
https://doi.org/10.1007/978-3-540-69969-9
https://doi.org/10.1007/s11232-008-0035-6
https://arxiv.org/abs/nlin.SI/0703018
https://doi.org/10.1090/trans2/224/13
https://arxiv.org/abs/0711.0969
https://doi.org/10.1090/S1061-0022-2011-01153-7
https://doi.org/10.1090/S1061-0022-2011-01153-7
12 A.K. Pogrebkov
[6] Pogrebkov A.K., Commutator identities on associative algebras, the non-Abelian Hirota difference equation
and its reductions, Theoret. and Math. Phys. 187 (2016), 823–834.
[7] Pogrebkov A.K., Commutator identities and integrable hierarchies, Theoret. and Math. Phys. 205 (2020),
1585–1592.
[8] Toda M., Vibration of a chain with nonlinear interaction, J. Phys. Soc. Japan 22 (1967), 431–436.
[9] Toda M., Wave propagation in anharmonic lattices, J. Phys. Soc. Japan 23 (1967), 501–506.
[10] Zakharov V.E., Shabat A.B., Exact theory of two-dimensional self-focusing and one-dimensional self-
modulation of waves in nonlinear media, Soviet Phys. JETP 61 (1971), 62–69.
https://doi.org/10.1134/S0040577916060039
https://doi.org/10.1134/S004057792012003X
https://doi.org/10.1143/JPSJ.22.431
https://doi.org/10.1143/JPSJ.23.501
1 Introduction
2 Commutator identities and linear equations
3 Realization of elements of the associative algebra
4 Dressing procedure
5 Lax pair and nonlinear equations
6 Dimensional reductions
7 Concluding remarks
References
|
| id | nasplib_isofts_kiev_ua-123456789-211436 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-16T04:01:36Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Pogrebkov, Andrei K. 2026-01-02T08:32:59Z 2021 Negative Times of the Davey-Stewartson Integrable Hierarchy. Andrei K. Pogrebkov. SIGMA 17 (2021), 091, 12 pages 1815-0659 2020 Mathematics Subject Classification: 37K10; 70H06 arXiv:2106.03835 https://nasplib.isofts.kiev.ua/handle/123456789/211436 https://doi.org/10.3842/SIGMA.2021.091 We use the example of the Davey-Stewartson hierarchy to show that, in addition to the standard equations given by the Lax operator and evolutions of time with positive numbers, one can consider time evolutions with negative numbers and the same Lax operator. We derive corresponding Lax pairs and integrable equations. This research is supported by a grant from the Russian Science Foundation (Project No. 19-1100062). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Negative Times of the Davey-Stewartson Integrable Hierarchy Article published earlier |
| spellingShingle | Negative Times of the Davey-Stewartson Integrable Hierarchy Pogrebkov, Andrei K. |
| title | Negative Times of the Davey-Stewartson Integrable Hierarchy |
| title_full | Negative Times of the Davey-Stewartson Integrable Hierarchy |
| title_fullStr | Negative Times of the Davey-Stewartson Integrable Hierarchy |
| title_full_unstemmed | Negative Times of the Davey-Stewartson Integrable Hierarchy |
| title_short | Negative Times of the Davey-Stewartson Integrable Hierarchy |
| title_sort | negative times of the davey-stewartson integrable hierarchy |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211436 |
| work_keys_str_mv | AT pogrebkovandreik negativetimesofthedaveystewartsonintegrablehierarchy |