q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy
Using the determinant representation of gauge transformation operator, we have shown that the general form of τ function of the q-KP hierarchy is a q-deformed generalized Wronskian, which includes the q-deformed Wronskian as a special case. On the basis of these, we study the q-deformed constrained...
Збережено в:
| Дата: | 2006 |
|---|---|
| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2006
|
| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/146112 |
| 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: | q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy / Jingsong He, Yinghua Li, Yi Cheng // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 40 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-146112 |
|---|---|
| record_format |
dspace |
| spelling |
nasplib_isofts_kiev_ua-123456789-1461122025-02-23T18:15:53Z q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy He, Jingsong Li, Yinghua Cheng, Yi Using the determinant representation of gauge transformation operator, we have shown that the general form of τ function of the q-KP hierarchy is a q-deformed generalized Wronskian, which includes the q-deformed Wronskian as a special case. On the basis of these, we study the q-deformed constrained KP (q-cKP) hierarchy, i.e. l-constraints of q-KP hierarchy. Similar to the ordinary constrained KP (cKP) hierarchy, a large class of solutions of q-cKP hierarchy can be represented by q-deformed Wronskian determinant of functions satisfying a set of linear q-partial differential equations with constant coefficients. We obtained additional conditions for these functions imposed by the constraints. In particular, the effects of q-deformation (q-effects) in single q-soliton from the simplest τ function of the q-KP hierarchy and in multi-q-soliton from one-component q-cKP hierarchy, and their dependence of x and q, were also presented. Finally, we observe that q-soliton tends to the usual soliton of the KP equation when x → 0 and q → 1, simultaneously. 2006 Article q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy / Jingsong He, Yinghua Li, Yi Cheng // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 40 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37K10; 35Q51; 35Q53; 35Q55 https://nasplib.isofts.kiev.ua/handle/123456789/146112 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Using the determinant representation of gauge transformation operator, we have shown that the general form of τ function of the q-KP hierarchy is a q-deformed generalized Wronskian, which includes the q-deformed Wronskian as a special case. On the basis of these, we study the q-deformed constrained KP (q-cKP) hierarchy, i.e. l-constraints of q-KP hierarchy. Similar to the ordinary constrained KP (cKP) hierarchy, a large class of solutions of q-cKP hierarchy can be represented by q-deformed Wronskian determinant of functions satisfying a set of linear q-partial differential equations with constant coefficients. We obtained additional conditions for these functions imposed by the constraints. In particular, the effects of q-deformation (q-effects) in single q-soliton from the simplest τ function of the q-KP hierarchy and in multi-q-soliton from one-component q-cKP hierarchy, and their dependence of x and q, were also presented. Finally, we observe that q-soliton tends to the usual soliton of the KP equation when x → 0 and q → 1, simultaneously. |
| format |
Article |
| author |
He, Jingsong Li, Yinghua Cheng, Yi |
| spellingShingle |
He, Jingsong Li, Yinghua Cheng, Yi q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
He, Jingsong Li, Yinghua Cheng, Yi |
| author_sort |
He, Jingsong |
| title |
q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy |
| title_short |
q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy |
| title_full |
q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy |
| title_fullStr |
q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy |
| title_full_unstemmed |
q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy |
| title_sort |
q-deformed kp hierarchy and q-deformed constrained kp hierarchy |
| publisher |
Інститут математики НАН України |
| publishDate |
2006 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/146112 |
| citation_txt |
q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy / Jingsong He, Yinghua Li, Yi Cheng // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 40 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT hejingsong qdeformedkphierarchyandqdeformedconstrainedkphierarchy AT liyinghua qdeformedkphierarchyandqdeformedconstrainedkphierarchy AT chengyi qdeformedkphierarchyandqdeformedconstrainedkphierarchy |
| first_indexed |
2025-11-24T06:30:06Z |
| last_indexed |
2025-11-24T06:30:06Z |
| _version_ |
1849652209162125312 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications Vol. 2 (2006), Paper 060, 33 pages
q-Deformed KP Hierarchy
and q-Deformed Constrained KP Hierarchy
Jingsong HE †‡, Yinghua LI † and Yi CHENG †
† Department of Mathematics, University of Science and Technology of China, Hefei,
230026 Anhui, P.R. China
E-mail: jshe@ustc.edu.cn, chengy@ustc.edu.cn
‡ Centre for Scientific Computing, University of Warwick, Coventry CV4 7AL, United Kingdom
Received January 27, 2006, in final form April 28, 2006; Published online June 13, 2006
Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper060/
Abstract. Using the determinant representation of gauge transformation operator, we have
shown that the general form of τ function of the q-KP hierarchy is a q-deformed generalized
Wronskian, which includes the q-deformed Wronskian as a special case. On the basis of
these, we study the q-deformed constrained KP (q-cKP) hierarchy, i.e. l-constraints of q-KP
hierarchy. Similar to the ordinary constrained KP (cKP) hierarchy, a large class of solutions
of q-cKP hierarchy can be represented by q-deformed Wronskian determinant of functions
satisfying a set of linear q-partial differential equations with constant coefficients. We
obtained additional conditions for these functions imposed by the constraints. In particular,
the effects of q-deformation (q-effects) in single q-soliton from the simplest τ function of
the q-KP hierarchy and in multi-q-soliton from one-component q-cKP hierarchy, and their
dependence of x and q, were also presented. Finally, we observe that q-soliton tends to the
usual soliton of the KP equation when x→ 0 and q → 1, simultaneously.
Key words: q-deformation; τ function; Gauge transformation operator; q-KP hierarchy;
q-cKP hierarchy
2000 Mathematics Subject Classification: 37K10; 35Q51; 35Q53; 35Q55
1 Introduction
Study of the quantum calculus (or q-calculus) [1, 2] has a long history, which may go back to
the beginning of the twentieth century. F.H. Jackson was the first mathematician who studied
the q-integral and q-derivative in a systematic way starting about 1910 [3, 4]1. Since 1980’s, the
quantum calculus was re-discovered in the research of quantum group inspired by the studies
on quantum integrable model that used the quantum inverse scattering method [5] and on
noncommutative geometry [6]. In particular, S. Majid derived the q-derivative from the braided
differential calculus [7, 8].
The q-deformed integrable system (also called q-analogue or q-deformation of classical inte-
grable system) is defined by means of q-derivative ∂q instead of usual derivative ∂ with respect
to x in a classical system. It reduces to a classical integrable system as q → 1. Recently, the
q-deformation of the following three stereotypes for integrable systems attracted more attention.
The first type is q-deformed N -th KdV (q-NKdV or q-Gelfand–Dickey hierarchy) [9, 16], which
is reduced to the N -th KdV (NKdV or Gelfand–Dickey) hierarchy when q → 1. The N -th
q-KdV hierarchy becomes q-KdV hierarchy for N = 2. The q-NKdV hierarchy inherited several
integrable structures from classical N -th KdV hierarchy, such as infinite conservation laws [10],
1 Detailed notes on the initial research of q-integral, q-derivative of Jackson and wide applications of q-series
are easily available in the text.
mailto:jshe@ustc.edu.cn
mailto:chengy@ustc.edu.cn
http://www.emis.de/journals/SIGMA/2006/Paper060/
2 J.S. He, Y.H. Li and Y. Cheng
bi-Hamiltonian structure [11, 12], τ function [13, 14], Bäcklund transformation [15]. The second
type is the q-KP hierarchy [17, 22]. Its τ function, bi-Hamiltonian structure and additional
symmetries have already been reported in [20, 21, 18, 22]. The third type is the q-AKNS-D
hierarchy, and its bilinear identity and τ function were obtained in [23].
In order to get the Darboux–Bäcklund transformations, the two elementary types of gauge
transformation operators, differential-type denoted by T (or TD) and integral-type denoted
by S (or TI), for q-deformed N -th KdV hierarchy were introduced in [15]. Tu et al. obtained
not only the q-deformed Wronskian-type but also binary-type representations of τ function
of q-KdV hierarchy. On the basis of their results, He et al. [24] obtained the determinant
representation of gauge transformation operators Tn+k (n ≥ k) for q-Gelfand–Dickey hierarchy,
which is a mixed iteration of n-steps of TD and then k-steps of TI . Then, they obtained a more
general form of τ function for q-KdV hierarchy, i.e., generalized q-deformed Wronskian (q-
Wronskian) IW q
n+k [24]. It is important to note that for k = 0 IW q
n+k reduces to q-deformed
Wronskian and for k = n to binary-type determinant [15]. On the other hand, Tu introduced
the q-deformed constrained KP (q-cKP) hierarchy [22] by means of symmetry constraint of q-KP
hierarchy, which is a q-analogue of constrained KP (cKP) hierarchy [25, 31].
The purpose of this paper is to construct the τ function of q-KP and q-cKP hierarchy, and
then explore the q-effect in q-solitons. The main tool is the determinant representation of gauge
transformation operators [32, 33, 34, 35]. The paper is organized as follows: In Section 2 we
introduce some basic facts on the q-KP hierarchy, such as Lax operator, Z-S equations, the
existence of τ function. On the basis of the [15], two kinds of elementary gauge transformation
operators for q-KP hierarchy and changing rule of q-KP hierarchy under it are presented in
Section 3. In Section 4, we establish the determinant representation of gauge transformation
operator Tn+k for the q-KP hierarchy and then obtain the general form of τ function τ
(n+k)
q =
IW q
n+k. In particular, by taking n = 1, k = 0 we will show q-effect of single q-soliton solution
of q-KP hierarchy. A brief description of the sub-hierarchy of q-cKP hierarchy is presented in
Section 5, from the viewpoint of the symmetry constraint. In Section 6, we show that the q-
Wronskian is one kind of forms of τ function of q-cKP if the functions in the q-Wronskian satisfy
some restrictions. In Section 7 we consider an example which illustrates the procedure reducing
q-KP to q-cKP hierarchy. q-effects of the q-deformed multi-soliton are also discussed. The
conclusions and discussions are given in Section 8. Our presentation is similar to the relevant
papers of classical KP and cKP hierarchy [32, 34, 36, 37, 38].
At the end of this section, we shall collect some useful formulae for reader’s convenience.
The q-derivative ∂q is defined by
∂q(f(x)) =
f(qx)− f(x)
x(q − 1)
and the q-shift operator is given by
θ(f(x)) = f(qx).
Let ∂−1
q denote the formal inverse of ∂q. We should note that θ does not commute with ∂q,
(∂qθk(f)) = qkθk(∂qf), k ∈ Z.
In general, the following q-deformed Leibnitz rule holds:
∂nq ◦ f =
∑
k≥0
(
n
k
)
q
θn−k(∂kq f)∂n−kq , n ∈ Z, (1.1)
q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy 3
where the q-number and the q-binomial are defined by
(n)q =
qn − 1
q − 1
,
(
n
k
)
q
=
(n)q(n− 1)q · · · (n− k + 1)q
(1)q(2)q · · · (k)q
,
(
n
0
)
q
= 1,
and “◦” means composition of operators, defined by ∂q ◦ f = (∂q · f)+ θ(f)∂q. In the remainder
of the paper for any function f “·” is defined by ∂q · f = ∂q(f) , (∂qf). For a q-pseudo-
differential operator (q-PDO) of the form P =
n∑
i=−∞
pi∂
i
q, we decompose P into the differential
part P+ =
∑
i≥0
pi∂
i
q and the integral partP− =
∑
i≤−1
pi∂
i
q. The conjugate operation “∗” for P is
defined by P ∗ =
∑
i
(∂∗q )
ipi with ∂∗q = −∂qθ−1 = −1
q∂ 1
q
, (∂−1
q )∗ = (∂∗q )
−1 = −θ∂−1
q . We can write
out several explicit forms of (1.1) for q-derivative ∂q, as
∂q ◦ f = (∂qf) + θ(f)∂q, (1.2)
∂2
q ◦ f = (∂2
qf) + (q + 1)θ(∂qf)∂q + θ2(f)∂2
q , (1.3)
∂3
q ◦ f = (∂3
qf) + (q2 + q + 1)θ(∂2
qf)∂q + (q2 + q + 1)θ2(∂qf)∂2
q + θ3(f)∂3
q , (1.4)
and ∂−1
q
∂−1
q ◦ f = θ−1(f)∂−1
q − q−1θ−2(∂qf)∂−2
q + q−3θ−3(∂2
qf)∂−3
q − q−6θ−4(∂3
qf)∂−4
q
+
1
q10
θ−5(∂4
qf)∂−5
q + · · ·+ (−1)kq−(1+2+3+···+k)θ−k−1(∂kq f)∂−k−1
q + · · · , (1.5)
∂−2
q ◦ f = θ−2(f)∂−2
q − 1
q2
(2)qθ−3(∂qf)∂−3
q +
1
q(2+3)
(3)qθ−4(∂2
qf)∂−4
q
− 1
q(2+3+4)
(4)qθ−5(∂3
qf)∂−5
q + · · ·
+
(−1)k
q(2+3+···+k+1)
(k + 1)qθ−2−k(∂kq f)∂−2−k
q + · · · . (1.6)
More explicit expressions of ∂nq ◦ f are given in Appendix A. In particular, ∂−1
q ◦ f has different
forms,
∂−1
q ◦ f = θ−1(f)∂−1
q + ∂−1
q ◦ (∂∗qf) ◦ ∂−1
q ,
∂−1
q ◦ f ◦ ∂−1
q = (∂−1
q f)∂−1
q − ∂−1
q ◦ θ(∂−1
q f),
which will be used in the following sections. The q-exponent eq(x) is defined as follows
eq(x) =
∞∑
i=0
xn
(n)q!
, (n)q! = (n)q(n− 1)q(n− 2)q · · · (1)q.
Its equivalent expression is of the form
eq(x) = exp
( ∞∑
k=1
(1− q)k
k(1− qk)
xk
)
. (1.7)
The form (1.7) will play a crucial role in proving the existence [20] of τ function of q-KP hierarchy.
4 J.S. He, Y.H. Li and Y. Cheng
2 q-KP hierarchy
Similarly to the general way of describing the classical KP hierarchy [36, 37], we shall give a brief
introduction of q-KP based on [20]. Let L be one q-PDO given by
L = ∂q + u0 + u−1∂
−1
q + u−2∂
−2
q + · · · , (2.1)
which is called Lax operator of q-KP hierarchy. There exist infinite q-partial differential equa-
tions relating to dynamical variables {ui(x, t1, t2, t3, . . .), i = 0,−1,−2,−3, . . .}, and they can
be deduced from generalized Lax equation,
∂L
∂tn
= [Bn, L], n = 1, 2, 3, . . . , (2.2)
which are called q-KP hierarchy. Here Bn = (Ln)+ =
n∑
i=0
bi∂
i
q means the positive part of q-PDO,
and we will use Ln− = Ln − Ln+ to denote the negative part. By means of the formulae given
in (1.2)–(1.6) and in Appendices A and B, the first few flows in (2.2) for dynamical variables
{u0, u−1, u−2, u−3} can be written out as follows. The first flow is
∂t1u0 = θ(u−1)− u−1,
∂t1u−1 = (∂qu−1) + θ(u−2) + u0u−1 − u−2 − u−1θ
−1(u0),
∂t1u−2 = (∂qu−2) + θ(u−3) + u0u−2 − u−3 − u−2θ
−2(u0) +
1
q
u−1θ
−2(∂qu0),
∂t1u−3 = (∂qu−3) + θ(u−4) + u0u−3 − u−4 −
1
q3
u−1θ
−3(∂2
qu0)
+
1
q2
(2)qu−2θ
−3(∂qu0)− u−3θ
−3(u0).
The second flow is
∂t2u0 = θ(∂qu−1) + θ2(u−2) + θ(u0)θ(u−1) + u0θ(u−1)
−
(
(∂qu−1) + u−1u0 + u−1θ
−1(u0) + u−2
)
,
∂t2u−1 = q−1u−1θ
−2(∂qu0) + u−1(∂qu0) + (∂2
qu−1) +
(
θ(u0) + u0
)
(∂qu−1)
+ (q + 1)θ(∂qu−2) + θ(u0)θ(u−2) + u0θ(u−2) + θ(u−1)u−1 + u2
0u−1
− u−1θ
−1(u2
0)− u−1θ
−1(u−1)− u−2θ
−1(u0)− u−2θ
−2(u0) + θ3(u−3)− u−3,
∂t2u−2 = (∂2
qu−2) + (q + 1)θ(∂qu−3) + (∂qu−2)v1 + θ2(u−4) + θ(u−3)v1 + u−2v0
−
(
q−3u−1θ
−3(∂2
qv1)− q−1u−1θ
−2(∂qv0)− q−2(2)qu−2θ
−3(∂qv1)
+ u−2θ
−2(v0) + u−3θ
−3(v1) + u−4
)
,
∂t2u−3 = (∂2
qu−3) + (q + 1)θ(∂qu−4) + (∂qu−3)v1 + θ2(u−5) + θ(u−4)v1 + u−3v0
−
(
− q−6θ−4(∂3
qv1) + q−3u−1θ
−3(∂2
qv0) + q−5(3)qu−2θ
−4(∂2
qv1)
− q−2(2)qu−2θ
−3(∂qv0)− q−3(3)qu−3θ
−4(∂qv1) + u−3θ
−3(v0)
+ u−4θ
−4(v1) + u−5
)
.
The third flow is
∂t3u0 = (∂3
qu0) + (3)qθ(∂2
qu−1) + s̃2(∂2
qu0) + (3)qθ2(∂qu−2) + (2)qθ(∂qu−1)s̃2
+ (∂qu0)s̃1 + θ3(u−3) + θ2(u−2)s̃2 + θ(u−1)s̃1 + u0s̃0
−
(
− q−1θ−2(∂q s̃2)u−1 + u0s̃0 + u−1θ
−1(s̃1) + u−2θ
−2(s̃2) + u−3 + (∂q s̃0)
)
,
q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy 5
∂t3u−1 = (∂3
qu−1) + (3)qθ(∂2
qu−2) + s̃2(∂2
qu−1) + (3)qθ2(∂qu−3) + (2)q s̃2θ(∂qu−2)
+ s̃1(∂qu−1) + θ3(u−4) + s̃2θ
3(u−3) + s̃1θ(u−2) + s̃0u−1
−
(
q−3u−1θ
−3(∂2
q s̃2)− q−1u−1θ
−2(∂q s̃1)− q−2(2)qu−2θ
−3(∂q s̃2)
+ u−1θ
−1(s̃0) + u−2θ
−2(s̃1) + u−3θ
−3(s̃2) + u−4
)
,
∂t3u−2 = (∂3
qu−2) + (3)qθ(∂2
qu−3) + s̃2(∂2
qu−2) + (3)qθ2(∂qu−4) + (2)q s̃2θ(∂qu−3)
+ s̃1(∂qu−2) + θ3(u−5) + s̃2θ
2(u−4) + s̃1θ(u−3) + s̃0u−2
−
(
− q−6u−1θ
−4(∂3
q s̃2) + q−3u−1θ
−3(∂2
q s̃1) + q−5(3)qu−2θ
−4(∂2
q s̃2)
− q−1u−1θ
−2(∂q s̃0)− q−2(2)qu−2θ
−3(∂q s̃1)− q−3(3)qu−3θ
−4(∂q s̃2)
+ u−2θ
−2(s̃0) + u−3θ
−3(s̃1) + u−4θ
−4(s̃2) + u−5
)
,
∂t3u−3 = (∂2
qu−3) + (3)qθ(∂2
qu−4) + s̃2(∂2
qu−3) + (3)qθ2(∂qu−5) + (2)q s̃2θ(∂qu−4)
+ s̃1(∂qu−3) + θ3(u−6) + s̃2θ
2(u−5) + s̃1θ(u−4) + s̃0u−3
−
(
q−10u−1θ
−5(∂4
q s̃2)− q−6u−1θ
−4(∂3
q s̃1)− q−9(4)qθ−5(∂3
q s̃2)
+ q−3u−1θ
−3(∂2
q s̃0) + q−5(3)qu−2θ
−4(∂2
q s̃1) + q−7 (3)q(4)4
(2)q
u−3θ
−5(∂2
q s̃2)
− q−2(2)qu−2θ
−3(∂q s̃0)− q−3(3)qu−3θ
−4(∂q s̃1)− q−4(4)qu−4θ
−5(∂q s̃2)
+ u−3θ
−3(s̃0) + u−4θ
−4(s̃1) + u−5θ
−5(s̃2) + u−6
)
.
Obviously, ∂t1 = ∂ and equations of flows here are reduced to usual KP flows (4.10) and (4.11)
in [39] when q → 1 and u0 = 0. If we only consider the first three flows, i.e. flows of (t1, t2, t3),
then u−1 = u−1(t1, t2, t3) is a q-deformation of the solution of KP equation [39]
∂
∂t1
(
4
∂u
∂t3
− 12u
∂u
∂t1
− ∂3u
∂t31
)
− 3
∂2u
∂t22
= 0.
In other words, u−1 = u(t1, t2, t3) in the above equation when q → 1, and hence u−1 is called
a q-soliton if u(t1, t2, t3) = lim
q→1
u−1 is a soliton solution of KP equation.
On the other hand, L in (2.1) can be generated by dressing operator S = 1 +
∞∑
k=1
sk∂
−k
q in
the following way
L = S ◦ ∂q ◦ S−1. (2.3)
Further, the dressing operator S satisfies the Sato equation
∂S
∂tn
= −(Ln)−S, n = 1, 2, 3, . . . . (2.4)
The q-wave function wq(x, t) and q-adjoint wave function w∗q(x, t) for q-KP hierarchy are defined
by
wq(x, t; z) =
(
Seq(xz) exp
( ∞∑
i=1
tiz
i
))
(2.5)
and
w∗(x, t; z) =
(
(S∗)−1|x/qe1/q(−xz) exp
(
−
∞∑
i=1
tiz
i
))
, (2.6)
6 J.S. He, Y.H. Li and Y. Cheng
where t = (t1, t2, t3, . . .). Here, for a q-PDO P =
∑
i
pi(x)∂iq, the notation
P |x/t =
∑
i
pi(x/t)ti∂iq
is used in (2.6). Note that wq(x, t) and w∗q(x, t) satisfy following linear q-differential equations,
(Lwq) = zwq,
∂wq
∂tn
= (Bnwq),
(L∗|x/qw∗q) = zw∗q ,
∂w∗q
∂tn
= −((Bn|x/q)∗w∗q). (2.7)
Furthermore, wq(x, t) and w∗q(x, t) can be expressed by sole function τq(x, t) as
ωq =
τq(x; t− [z−1])
τq(x; t)
eq(xz) exp
( ∞∑
i=1
tiz
i
)
, (2.8)
ω∗q =
τq(x; t+ [z−1])
τq(x; t)
e1/q(−xz) exp
(
−
∞∑
i=1
tiz
i
)
,
where
[z] =
(
z,
z2
2
,
z3
3
, . . .
)
.
From comparison of (2.5) and (2.8), the dressing operator S has the form of
S = 1−
(
1
τq
∂
∂t1
τq
)
∂−1
q +
[
1
2τq
(
∂2
∂t21
− ∂
∂t2
)
τq
]
∂−2
q + · · · . (2.9)
Using (2.9) in (2.3), and then comparing with Lax operator in (2.1), we can show that all
dynamical variables ui (i = 0,−1,−2,−3, . . .) can be expressed by τq(x, t), and the first two are
u0 = s1 − θ(s1) = −x(q − 1)∂qs1 = x(q − 1)∂q∂t1 ln τq,
u−1 = −∂qs1 + s2 − θ(s2) + θ(s1)s1 − s21, (2.10)
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
We can see u0 = 0, and u−1 = (∂2
x log τ) as classical KP hierarchy when q → 1, where τ =
τq(x, t)|q→1. By considering u−1 depending only on (q, x, t1, t2, t3), we can regard u−1 as q-
deformation of solution of classical KP equation. We shall show the q-effect of this solution for
q-KP hierarchy after we get τq in next section. In order to guarantee that eq(x) is convergent,
we require the parameter 0 < q < 1 and parameter x to be bounded.
Beside existence of the Lax operator, q-wave function, τq for q-KP hierarchy, another impor-
tant property is the q-deformed Z-S equation and associated linear q-differential equation. In
other words, q-KP hierarchy also has an alternative expression, i.e.,
∂Bm
∂tn
− ∂Bn
∂tm
+ [Bm, Bn] = 0, m, n = 1, 2, 3, . . . . (2.11)
The “eigenfunction” φ and “adjoint eigenfunction” ψ of q-KP hierarchy associated to (2.11) are
defined by
∂φ
∂tn
= (Bnφ), (2.12)
∂ψ
∂tn
= −(B∗nψ), (2.13)
where φ = φ(λ;x, t) and ψ = ψ(µ;x, t). Here (2.13) is different from the second equation in (2.7).
φi ≡ φ(λi;x, t) and ψi ≡ ψ(µi;x, t) will be generating functions of gauge transformations.
q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy 7
3 Gauge transformations of q-KP hierarchy
The authors in [15] reported two types of elementary gauge transformation operator only for q-
Gelfand–Dickey hierarchy. We extended the elementary gauge transformations given in [15], for
the q-KP hierarchy. At the same time, we shall add some vital operator identity concerning to the
q-differential operator and its inverse. Here we shall prove two transforming rules of τ function,
“eigenfunction” and “adjoint eigenfunction” of the q-KP hierarchy under these transformations.
Majority of the proofs are similar to the classical case given by [32, 33] and [35], so we will omit
part of the proofs.
Suppose T is a pseudo-differential operator, and
L(1) = T ◦ L ◦ T−1, B(1)
n ≡
(
L(1)
)n
+
,
so that
∂
∂tn
L(1) =
[
B(1)
n , L(1)
]
still holds for the transformed Lax operator L(1); then T is called a gauge transformation operator
of the q-KP hierarchy.
Lemma 1. The operator T is a gauge transformation operator, if
(
T ◦Bn ◦ T−1
)
+
= T ◦Bn ◦ T−1 +
∂T
∂tn
◦ T−1, (3.1)
or
(
T ◦Bn ◦ T−1
)
− = − ∂T
∂tn
◦ T−1. (3.2)
If the initial Lax operator of q-KP is a “free” operator L = ∂q, then the gauge transformation
operator is also a dressing operator for new q-KP hierarchy whose Lax operator L(1) = T ◦ ∂q ◦
T−1, because of (3.2) becomes
Ttn = −
(
T ◦Bn ◦ T−1
)
− ◦ T = −
(
T ◦ ∂nq ◦ T−1
)
− ◦ T = −
(
L(1)
)n
− ◦ T, (3.3)
which is the Sato equation (2.4). In order to prove existence of two types of the gauge transfor-
mation operator, the following operator identities are necessary.
Lemma 2. Let f be a suitable function, and A be a q-deformed pseudo-differential operator,
then
(1)
(
θ(f) ◦ ∂q ◦ f−1 ◦A ◦ f ◦ ∂−1
q ◦ (θ(f))−1
)
+
= θ(f) ◦ ∂q ◦ f−1 ◦A+ ◦ f ◦ ∂−1
q ◦ (θ(f))−1
− θ(f) ◦
[
∂q
(
f−1 · (A+ · f)
)]
◦ ∂−1
q ◦ (θ(f))−1, (3.4)
(2)
(
θ−1(f−1) ◦ ∂−1
q ◦ f ◦A ◦ f−1 ◦ ∂q ◦ θ−1(f)
)
−
= θ−1(f−1) ◦ ∂−1
q ◦ f ◦A− ◦ f−1 ◦ ∂q ◦ θ−1(f)
− θ−1(f−1) ◦ ∂−1
q ◦ θ−1(f) ◦ ∂q
(
θ−1[f−1 ·
(
A∗+ · f
)
]
)
. (3.5)
Remark 1. This lemma is a q-analogue of corresponding identities of pseudo-differential ope-
rators given by [33].
8 J.S. He, Y.H. Li and Y. Cheng
Theorem 1. There exist two kinds of gauge transformation operator for the q-KP hierarchy,
namely
Type I : TD(φ1) = θ(φ1) ◦ ∂q ◦ φ−1
1 , (3.6)
Type II : TI(ψ1) = (θ−1(ψ1))−1 ◦ ∂−1
q ◦ ψ1. (3.7)
Here φ1 and ψ1 are defined by (2.12) and (2.13) that are called the generating functions of gauge
transformation.
Proof. First of all, for the Type I case (see (3.6)),
B(1)
n ≡
(
L(1)
)n
+
=
(
TD ◦ (L)n ◦ T−1
D
)
+
= TD ◦Bn ◦ T−1
D − θ(φ1) · ∂q
(
φ−1
1 · (Bn · φ1)
)
◦ ∂−1
q ◦ (θ(φ1))−1
= TD ◦B(0)
n ◦ T−1
D −
(
θ(φ1) ◦ ∂q ◦
(φ1)tn
φ1
◦ ∂−1
q ◦ (θ(φ1))−1
− θ(φ1) ◦ θ
((φ1)tn
φ1
)
◦ ∂q ◦ ∂−1
q ◦ (θ(φ1))−1
)
= TD ◦Bn ◦ T−1
D + θ
((φ1)tn
φ1
)
− θ(φ1) ◦ ∂q ◦
(φ1)tn
φ1
◦ ∂−1
q ◦ (θ(φ1))−1.
Here the operator identity (3.4), Bn = (L)n+, (φ1)tn = (Bn · φ1) and (1.2) were used. On the
other hand,
∂TD
∂tn
◦ T−1
D =
(
θ(φ1) ◦ ∂q ◦ φ−1
1
)
tn
◦ T−1
D = θ((φ1)tn) ◦ ∂q ◦ φ−1
1 ◦ φ1 ◦ ∂−1
q ◦ (θ(φ1))−1
− θ(φ1) ◦ ∂q ◦
(φ1)tn
φ2
1
◦ φ1 ◦ ∂−1
q ◦ (θ(φ1))−1
= θ
((φ1)tn
φ1
)
− θ(φ1) ◦ ∂q ◦
(φ1)tn
φ1
◦ ∂−1
q ◦ (θ(φ1))−1.
Taking this expression back into B(1)
n , we get
B(1)
n ≡
(
L(1)
)n
+
= TD ◦Bn ◦ T−1
D +
∂TD
∂tn
◦ T−1
D ,
and that indicates that TD(φ1) is indeed a gauge transformation operator via Lemma 1. Second,
we want to prove that the equation (3.2) holds for Type II case (see (3.7)). By direct calculation
the left hand side of (3.2) is in the form of(
TI ◦Bn ◦ T−1
I
)
− =
(
(θ−1(ψ1))−1 ◦ ∂−1
q ◦ ψ1 ◦Bn ◦ ψ−1
1 ◦ ∂q ◦ θ−1(ψ1)
)
−
= (θ−1(ψ1))−1 ◦ ∂−1
q ◦ ψ1 ◦ (Bn)− ◦ ψ−1
1 ◦ ∂q ◦ θ−1(ψ1)
− (θ−1(ψ1))−1 ◦ ∂−1
q ◦ θ−1(ψ1) ◦
(
∂qθ
−1
((
B∗n · ψ1
)
ψ1
))
= (θ−1(ψ1))−1 ◦ ∂−1
q ◦ θ−1(ψ1) ◦
(
∂qθ
−1
((
ψ1
)
tn
ψ1
))
.
In the above calculation, the operator identity (3.5), (Bn)− = 0, (ψ1)tn = −(B∗n ·ψ1) were used.
Moreover, with the help of (1.2), we have
−∂TI
∂tn
◦ T−1
I = − ∂
∂tn
(
(θ−1(ψ1))−1 ◦ ∂−1
q ◦ ψ1
)
◦ ψ−1
1 ◦ ∂q ◦ θ−1(ψ1)
q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy 9
=
θ−1((ψ1)tn)
(θ−1(ψ1))2
◦ ∂−1
q ◦ ψ1 ◦ ψ−1
1 ◦ ∂q ◦ θ−1(ψ1)
− (θ−1(ψ1))−1 ◦ ∂−1
q ◦ (ψ1)tn ◦ ψ−1
1 ◦ ∂q ◦ θ−1(ψ1)
= θ−1
(
(ψ1)tn
ψ1
)
− 1
θ−1(ψ1)
◦ ∂−1
q ◦
[
∂q ◦ θ−1
(
(ψ1)tn
ψ1
)
−
(
∂q · θ−1
(
(ψ1)tn
ψ1
))]
◦ θ−1(ψ1) = θ−1
(
(ψ1)tn
ψ1
)
− θ−1
(
(ψ1)tn
ψ1
)
+
1
θ−1(ψ1)
◦ ∂−1
q ◦
(
∂q · θ−1
(
(ψ1)tn
ψ1
))
◦ θ−1(ψ1)
=
1
θ−1(ψ1)
◦ ∂−1
q ◦
(
∂q · θ−1
(
(ψ1)tn
ψ1
))
◦ θ−1(ψ1).
The two equations obtained above show that TI(ψ1) satisfies (3.2), so TI(ψ1) is also a gauge
transformation operator of the q-KP hierarchy according to Lemma 1. �
Remark 2. There are two convenient expressions for TD and TI ,
TD = ∂q − α1, T−1
D = ∂−1
q + θ−1(α1)∂−2
q + · · · , α1 =
∂qφ1
φ1
, (3.8)
TI = (∂q + β1)−1 = ∂−1
q − θ−1(β1)∂−2
q + · · · , β1 =
∂qθ
−1(ψ1)
ψ1
. (3.9)
In order to get a new solution of q-KP hierarchy from the input solution, we should know
the transformed expressions of u(1)
i , τ (1)
q , φ(1)
i , ψ(1)
i . The following two theorems are related to
this. Before we start to discuss explicit forms of them, we would like to define the generalized
q-Wronskian for a set of functions {ψk, ψk−1, . . . , ψ1;φ1, φ2, . . . , φn } as
IW q
k,n(ψk, . . . , ψ1;φ1, . . . , φn) =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
∂−1
q ψkφ1 ∂−1
q ψkφ2 · · · ∂−1
q ψkφn
...
... · · ·
...
∂−1
q ψ1φ1 ∂−1
q ψ1φ2 · · · ∂−1
q ψ1φn
φ1 φ2 · · · φn
∂qφ1 ∂qφ2 · · · ∂qφn
...
... · · ·
...
∂n−k−1
q φ1 ∂n−k−1
q φ2 · · · ∂n−k−1
q φn
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
,
which reduce to the q-Wronskian when k = 0,
W q
n(φ1, · · · , φn) =
∣∣∣∣∣∣∣∣∣
φ1 φ2 · · · φn
∂qφ1 ∂qφ2 · · · ∂qφn
...
... · · ·
...
∂n−1
q φ1 ∂n−1
q φ2 · · · ∂n−1
q φn
∣∣∣∣∣∣∣∣∣ .
Suppose φ1(λ1;x, t) is a known “eigenfunction” of q-KP with the initial function τq, which
generates gauge transformation operator TD(φ1). Then we have
Theorem 2. Under the gauge transformation L(1) = TD(φ1) ◦L ◦ (TD(φ1))−1, new “eigenfunc-
tion”, “adjoint eigenfunction” and τ function of the transformed q-KP hierarchy are
φ −→ φ(1)(λ;x, t) = (TD(φ1) · φ) =
W q
2 (φ1, φ)
φ1
,
10 J.S. He, Y.H. Li and Y. Cheng
ψ −→ ψ(1)(λ;x, t) =
((
TD(φ1)−1
)∗ · ψ) =
θ(∂−1
q φ1ψ)
θ(φ1)
,
τq −→ τ (1)
q = φ1τq.
φ
(1)
k = φ(1)(λ = λk;x, t), ψ
(1)
k = ψ(1)(λ = λk;x, t). Note φ(1)
1 = 0.
Proof. (1) By direct calculations, we have(
∂tnφ
(1)
)
= (∂tn(TD · φ)) = (∂tnTD) · φ+ (TD · ∂tnφ)
=
(
∂tnTD ◦ T−1
D
)
· (TDφ) + TD · (Bnφ) =
(
∂tnTD ◦ T−1
D + TD ◦Bn ◦ T−1
D
)
· (TDφ)
=
(
B(1)
n · φ(1)
)
,
in which (2.12) and (3.1) were used.
(2) Similarly, with the help of (B(1)
n )∗ = (T−1
D )∗ ◦ ∂tnT ∗D + (T−1
D )∗ ◦ B∗n ◦ T ∗D and (2.13), we
can obtain
∂tnψ
(1) =
(
(T−1
D )∗ · ψ
)
tn
=
(
− (T ∗D)−1 ◦ ∂tnT ∗D ◦ (T ∗D)−1
)
· ψ + (T ∗D)−1 · ∂tnψ
= −
(
(T−1
D )∗ ◦ ∂tnT ∗D + (T−1
D )∗ ◦B∗n ◦ T ∗D
)
·
(
(T−1
D )∗ · ψ
)
= −
(
B(1)
n
)∗ · ψ(1).
(3) According to the definition of TD in (3.6) and with the help of (3.8), L(1) can be expressed
as
L(1)
q = ∂q + u
(1)
0 + u
(1)
−1∂
−1
q + · · · , u
(1)
0 = x(q − 1)∂qα1 + θ(u0).
On the other hand, (φ1)t1 = ((L)+φ1) implies α1 = ∂t1 lnφ1 − u0, then u(1)
0 becomes
u
(1)
0 = x(q − 1)∂q∂t1 lnφ1 + u0 = x(q − 1)∂q∂t1 lnφ1 + x(q − 1)∂q∂t1 ln τq
= x(q − 1)∂q∂t1 lnφ1τq.
Then
τ (1)
q = φ1τq.
This completes the proof of the theorem. �
For the gauge transformation operator of Type II, there exist similar results. Let ψ1(µ1;x, t)
be a known “adjoint eigenfunction” of q-KP with the initial function τq, which generates the
gauge transformation operator TI(ψ1). Then we have
Theorem 3. Under the gauge transformation L(1) = TI(ψ1) ◦ L ◦ (TI(ψ1))−1, new “eigenfunc-
tion”, “adjoint eigenfunction” and τ function of the transformed q-KP hierarchy are
φ −→ φ(1)(λ;x, t) = (TI(ψ1) · φ) =
(∂−1
q ψ1φ)
θ−1(ψ1)
,
ψ −→ ψ(1)(λ;x, t) =
((
TI(ψ1)−1
)∗ · ψ) =
W̃ q
2 (ψ1, ψ)
ψ1
,
τq −→ τ (1)
q = θ−1(ψ1)τq.
φ
(1)
k = φ(1)(λ = λk;x, t), ψ
(1)
k = ψ(1)(λ = λk;x, t). Note ψ(1)
1 = 0. W̃ q
n is obtained from W q
n by
replacing ∂q with ∂∗q .
Proof. The proof is analogous to the proof of the previous theorem. So it is omitted. �
q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy 11
4 Successive applications of gauge transformations
We now discuss successive applications of the two types of gauge transformation operators in a
general way, which is similar to the classical case [32, 34, 35]. For example, consider the chain
of gauge transformation operators,
L
T
(1)
D
(
φ1
)
−−−−−−→ L(1)
T
(2)
D
(
φ
(1)
2
)
−−−−−−−→ L(2)
T
(3)
D
(
φ
(2)
3
)
−−−−−−−→ L(3) −→ · · · −→ L(n−1)
T
(n)
D
(
φ
(n−1)
n
)
−−−−−−−−→ L(n)
T
(n+1)
I
(
ψ1
)
−−−−−−−→ L(n+1)
T
(n+2)
I
(
ψ
(n+1)
2
)
−−−−−−−−−−→ L(n+2) −→ · · · −→ L(n+k−1)
T
(n+k)
I
(
ψ
(n+k−1)
k
)
−−−−−−−−−−−−→ L(n+k). (4.1)
Here the index “i“ in a gauge transformation operator means the i-th gauge transformation,
and φ(j)
i (or ψ(j)
i ) is transformed by j-steps gauge transformations from φi (or ψi), L(i) is trans-
formed by j-step gauge transformations from the initial Lax operator L. Successive applications
of gauge transformation operator in (4.1) can be represented by
Tn+k = T
(n+k)
I
(
ψ
(n+k−1)
k
)
· · ·T (n+2)
I
(
ψ
(n+1)
2
)
◦ T (n+1)
I
(
ψ
(n)
1
)
◦ T (n)
D
(
φ(n−1)
n
)
· · ·T (2)
D
(
φ
(1)
2
)
◦ T (1)
D (φ1).
Our goal is to obtain φ(n+k), ψ(n+k), τ (n+k)
q of L(n+k) transformed from L by the Tn+k in the
above chain. All of these are based on the determinant representation of gauge transformation
operator Tn+k. As the proof of the determinant representation of Tn+k is similar extremely to
the case of classical KP hierarchy [34], we will omit it.
Lemma 3. The gauge transformation operator Tn+k has the following determinant representa-
tion (n > k):
Tn+k =
1
IW q
k,n(ψk, . . . , ψ1;φ1, . . . , φn)
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
∂−1
q ψkφ1 · · · ∂−1
q ψkφn ∂−1
q ◦ ψk
... · · ·
...
...
∂−1
q ψ1φ1 · · · ∂−1
q ψ1φn ∂−1
q ◦ ψ1
φ1 · · · φn 1
∂qφ1 · · · ∂qφn ∂q
... · · ·
...
...
∂n−kq φ1 · · · ∂n−kq φn ∂n−kq
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
and
T−1
n+k =
∣∣∣∣∣∣∣∣∣
φ1 ◦ ∂−1
q θ(∂−1
q ψkφ1) · · · θ(∂−1
q ψ1φ1) θ(φ1) · · · θ(∂n−k−2
q φ1)
φ2 ◦ ∂−1
q θ(∂−1
q ψkφ2) · · · θ(∂−1
q ψ1φ2) θ(φ2) · · · θ(∂n−k−2
q φ2)
...
... · · ·
...
... · · ·
...
φn ◦ ∂−1
q θ(∂−1
q ψkφn) · · · θ(∂−1
q ψ1φn) θ(φn) · · · θ(∂n−k−2
q φn)
∣∣∣∣∣∣∣∣∣
× (−1)n−1
θ(IW q
k,n(ψk, . . . , ψ1;φ1, . . . , φn)
.
Lemma 4. Under the case of n = k, Tn+k is given by
Tn+n =
1
IW q
n,n(ψn, . . . , ψ1;φ1, . . . , φn)
∣∣∣∣∣∣∣∣∣
∂−1
q ψnφ1 · · · ∂−1
q ψnφn ∂−1
q ◦ ψn
... · · ·
...
...
∂−1
q ψ1φ1 · · · ∂−1
q ψ1φn ∂−1
q ◦ ψ1
φ1 · · · φn 1
∣∣∣∣∣∣∣∣∣
12 J.S. He, Y.H. Li and Y. Cheng
but T−1
n+n becomes
T−1
n+n =
∣∣∣∣∣∣∣∣∣
−1 ψn · · · ψ1
φ1 ◦ ∂−1
q θ(∂−1
q ψnφ1) · · · θ(∂−1
q ψ1φ1)
...
... · · ·
...
φn ◦ ∂−1
q θ(∂−1
q ψnφn) · · · θ(∂−1
q ψ1φn)
∣∣∣∣∣∣∣∣∣
(−1)
θ(IW q
n,n(ψn, . . . , ψ1;φ1, . . . , φn)
.
In the above lemmas, Tn+k are expanded with respect to the last column collecting all sub-
determinants on the left of the symbols ∂iq (i = −1, 0, 1, 2, . . . , n − k); T−1
n+k are expanded with
respect to the first column by means of collection of all minors on the right of φi∂−1
q . Basing on
the determinant representation, first of all, we would like to consider the case of k = 0 in (4.1),
i.e.
L
T
(1)
D
(
φ1
)
−−−−−−→ L(1)
T
(2)
D
(
φ
(1)
2
)
−−−−−−−→ L(2)
T
(3)
D
(
φ
(2)
3
)
−−−−−−−→ L(3) −→ · · · −→ L(n−1)
T
(n)
D
(
φ
(n−1)
n
)
−−−−−−−−→ L(n),
whose corresponding equivalent gauge transformation operator is
Tn = T
(n)
D
(
φ(n−1)
n
)
· · ·T (2)
D
(
φ
(1)
2
)
◦ T (1)
D (φ1). (4.2)
Theorem 4. Under the gauge transformation Tn (n ≥ 1),
φ(n)(λ;x, t) = (Tn · φ) =
W q
n+1(φ1, . . . , φn, φ)
W q
n(φ1, . . . , φn)
, (4.3)
ψ(n)(µ;x, t) =
((
T−1
n
)∗ · ψ) = (−1)nθ
(
IW q
1,n(ψ;φ1, . . . , φn)
W q
n(φ1, . . . , φn)
)
, (4.4)
τ (n)
q = W q
n(φ1, . . . , φn) · τq.
Furthermore, φ(n)
i = φ(n)(λ = λi;x, t), ψ
(n)
i = ψ(n)(µ = µi;x, t). Note φ
(n)
i = 0 if i ∈
{1, 2, . . . , n}.
Proof. (1) Successive application of Theorem 2 implies
φ(n) = T
(n)
D
(
φ(n−1)
n
)
φ(n−1) = T
(n)
D
(
φ(n−1)
n
)
T
(n−1)
D
(
φ
(n−2)
n−1
)
φ(n−2) = · · ·
= T
(n)
D
(
φ(n−1)
n
)
· · ·T (2)
D
(
φ
(1)
2
)
◦ T (1)
D (φ1)φ = (Tn · φ).
Using the determinant representation of Tn in it leads to φ(n). Here T (1)
D (φ1) = TD(φ1).
(2) Similarly, according to Theorem 2 we have
ψ(n) =
(
T
(n)
D
−1)∗
ψ(n−1) =
(
T
(n)
D
−1)∗(
T
(n−1)
D
−1)∗
ψ(n−2) = · · ·
=
((
T
(n)
D
−1)∗(
T
(n−1)
D
−1)∗ · · · (T (3)
D
−1)∗(
T
(2)
D
−1)∗(
T−1
D
)∗) · ψ =
((
T−1
n
)∗ · ψ).
Then ψ(n) can be deduced by using the determinant representation of T−1
n in the Lemma 3 with
k = 0. Here we omit the generating functions in T (i)
D (i = 1, 2, . . . , n), which are the same as (1).
(3) Meanwhile, we can get τ (n) by repeated iteration according to the rule in Theorem 2,
τ (n)
q = φ(n−1)
n τ (n−1)
q = φ(n−1)
n φ
(n−2)
n−1 τ (n−2)
q = φ(n−1)
n φ
(n−2)
n−1 φ
(n−3)
n−2 τ (n−3)
q = · · ·
= φ(n−1)
n φ
(n−2)
n−1 φ
(n−3)
n−2 · · ·φ(3)
4 φ
(2)
3 φ
(1)
2 φ1τq
=
W q
n(φ1, φ2, φ3, . . . , φn)
W q
n−1(φ1, φ2, φ3, . . . , φn−1)
W q
n−1(φ1, φ2, φ3, . . . , φn−1)
W q
n−2(φ1, φ2, φ3, . . . , φn−2)
W q
n−2(φ1, φ2, φ3, . . . , φn−2)
W q
n−3(φ1, φ2, φ3, . . . , φn−3)
· · ·W
q
4 (φ1, φ2, φ3, φ4)
W q
3 (φ1, φ2, φ3)
W q
3 (φ1, φ2, φ3)
W q
2 (φ1, φ2)
W q
2 (φ1, φ2)
W q
1 (φ1)
φ1τq = W q
n(φ1, φ2, . . . , φn)τq.
with the help of the determinant representation of Lemma 3 with k = 0. Here W q
1 (φ1) = φ1. �
q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy 13
It should be noted that there is a θ action in (4.4), which is the main difference between
the q-KP and classical KP beside different elements in determinant. Furthermore, for more
complicated chain of gauge transformation operators in (4.1), φ(n+k), ψ(n+k), τ (n+k)
q of L(n+k)
can be expressed by the generalized q-Wronskian.
Theorem 5. Under the gauge transformation Tn+k (n > k > 0),
φ(n+k)(λ;x, t) = (Tn+k · φ) =
IW q
k,n+1(ψk, . . . , ψ1;φ1, . . . , φn, φ)
IW q
k,n(ψk, . . . , ψ1;φ1, . . . , φn)
,
ψ(n+k)(µ;x, t) =
((
T−1
n+k
)∗ · ψ) = (−1)n
IW q
k+1,n(ψ,ψk, ψk−1, . . . , ψ1;φ1, . . . , φn)
IW q
k,n(ψk, . . . , ψ1;φ1, . . . , φn)
,
τ (n+k)
q = IW q
k,n(ψk, . . . , ψ1;φ1, · · · , φn) · τq.
Furthermore, φ(n+k)
i = φ(n+k)(λ = λi;x, t); ψ
(n+k)
i = ψ(n+k)(µ = µi;x, t). Note φ(n+k)
i = 0 if
i ∈ {1, 2, . . . , n}, ψ(n+k)
i = 0 if i ∈ {1, 2, . . . , k}.
Proof. (1) The repeated iteration of Theorems 2 and 3 according to the ordering of TI and TD
deduces
φ(n+k) = T
(n+k)
I
(
ψ
(n+k−1)
n+k
)
· φ(n+k−1)
= T
(n+k)
I
(
ψ
(n+k−1)
n+k
)
T
(n+k−1)
I
(
ψ
(n+k−2)
n+k−1
)
· φ(n+k−2) = · · ·
= T
(n+k)
I
(
ψ
(n+k−1)
n+k
)
T
(n+k−1)
I
(
ψ
(n+k−2)
n+k−1
)
· · ·T (n+2)
I
(
ψ
(n+1)
n+2
)
T
(n+1)
I
(
ψ
(n)
n+1
)
· φ(n).
Then taking in it φ(n) = (Tn · φ) from (4.3) , we get
φ(n+k) =
(
T
(n+k)
I
(
ψ
(n+k−1)
n+k
)
T
(n+k−1)
I
(
ψ
(n+k−2)
n+k−1
)
· · ·T (n+2)
I
(
ψ
(n+1)
n+2
)
T
(n+1)
I
(
ψ
(n)
n+1
)
Tn
)
· φ
= (Tn+k · φ).
Therefore the determinant form of φ(n+k) is given by Lemma 3.
(2) Using Theorems 2 and 3 iteratively according to the chain in (4.1), similarly to the
step (1), we can get
ψ(n+k) =
(
T
(n+k)
I
−1)∗ · ψ(n+k−1) =
(
T
(n+k)
I
−1)∗(
T
(n+k−1)
I
−1)∗ · ψ(n+k−2) = · · ·
=
(
T
(n+k)
I
−1)∗(
T
(n+k−1)
I
−1)∗ · · · (T (n+2)
I
−1)∗(
T
(n+1)
I
−1)∗ · ψ(n).
Noting that ψ(n) is given by (4.4), we get ψ(n+k) = ((T−1
n+k)
∗ · ψ). The explicit form of ψ(n+k) is
given from the determinant representation of T−1
n+k.
(3) According to the changing rule under gauge transformation in Theorems 2 and 3, the new
τ function of q-KP hierarchy τ (n+k)
q produced by chain of gauge transformations in (4.1) is
τ (n+k)
q = θ−1
(
ψ
(n+k−1)
k
)
τ (n+k−1)
q = θ−1
(
ψ
(n+k−1)
k
)
θ−1
(
ψ
(n+k−2)
k−1
)
τ (n+k−2)
q
= θ−1
(
ψ
(n+k−1)
k
)
θ−1
(
ψ
(n+k−2)
k−1
)
· · · θ−1
(
ψ
(n+1)
2
)
θ−1
(
ψ
(n)
1
)
τ (n)
q .
So the explicit form of ψ(n+i−1)
i (i = 1, 2, . . . , k) and τ (n)
q implies
τ (n+k)
q = (−1)n
IW q
k,n(ψk, ψk−1, . . . , ψ1;φ1, φ2, . . . , φn)
IW q
k−1,n(ψk−1, . . . , ψ1;φ1, φ2, . . . , φn)
14 J.S. He, Y.H. Li and Y. Cheng
× (−1)n
IW q
k−1,n(ψk−1, ψk−1, . . . , ψ1;φ1, φ2, . . . , φn)
IW q
k−2,n(ψk−2, . . . , ψ1;φ1, φ2, . . . , φn)
· · · (−1)n
IW q
2,n(ψ2, ψ1;φ1, φ2, . . . , φn)
IW q
1,n(ψ1;φ1, φ2, . . . , φn)
× (−1)n
IW q
1,n(ψ1;φ1, φ2, . . . , φn)
W q
n(φ1, φ2, . . . , φn)
W q
n(φ1, φ2, . . . , φn)τq
≈ IW q
k,n(ψk, ψk−1, . . . , ψ1;φ1, φ2, . . . , φn)τq.
We omitted the trivial factor (−1)n in the last step, because it will not affect ui in the q-KP
hierarchy. �
Remark 3. There exists another complicated chain of gauge transformation operators for q-KP
hierarchy (that may be regarded as motivated by the classical KP hierarchy)
L
T
(1)
I
(
ψ1
)
−−−−−−→ L(1)
T
(2)
I
(
ψ
(1)
2
)
−−−−−−−→ L(2)
T
(3)
I
(
ψ
(2)
3
)
−−−−−−−→ L(3) −→ · · · −→ L(n−1)
T
(n)
I
(
ψ
(n−1)
n
)
−−−−−−−−−→ L(n)
T
(n+1)
D
(
φ
(n)
1
)
−−−−−−−−→ L(n+1)
T
(n+2)
D
(
φ
(n+1)
2
)
−−−−−−−−−−→ L(n+2) −→ · · · −→ L(n+k−1)
T
(n+k)
D
(
φ
(n+k−1)
k
)
−−−−−−−−−−−−→ L(n+k),
that can lead to another form of τ (n+k)
q . This is parallel to the classical case of [32].
If the initial q-KP is a “free” operator, then L = ∂q that means the initial τ function is 1. We
can write down the explicit form of q-KP hierarchy generated by Tn+k. Under this situation,
(2.12) and (2.13) become
∂φ
∂tn
= (∂nq φ),
∂ψ
∂tn
= −(∂n∗q ψ), (4.5)
that possess set of solution {φi, ψi} as follows
φi(x; t) = eq(λi1x)e
∞∑
j=1
tjλ
j
i1 + aieq(µi1x)e
∞∑
j=1
tjµ
j
i1
, (4.6)
ψi(x; t) = e1/q(−λi2qx)e
−
∞∑
j=1
tjλ
j
i2 + bie1/q(−µi2qx)e
−
∞∑
j=1
tjµ
j
i2
. (4.7)
After the (n+k)-th step gauge transformation Tn+k, the final form of τq can be given in following
corollary, which can be deduced directly from Theorems 4 and 5.
Corollary 1. The gauge transformation can generate the following two forms of τ function of
the q-KP hierarchy,
τ (n+k)
q = IW q
k,n(ψk, . . . , ψ1;φ1, . . . , φn) =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
∂−1
q ψkφ1 ∂−1
q ψkφ2 · · · ∂−1
q ψkφn
...
... · · ·
...
∂−1
q ψ1φ1 ∂−1
q ψ1φ2 · · · ∂−1
q ψ1φn
φ1 φ2 · · · φn
∂qφ1 ∂qφ2 · · · ∂qφn
...
... · · ·
...
∂n−k−1
q φ1 ∂n−k−1
q φ2 · · · ∂n−k−1
q φn
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
,
τ (n)
q = W q
n(φ1, . . . , φn) =
∣∣∣∣∣∣∣∣∣
φ1 φ2 · · · φn
∂qφ1 ∂qφ2 · · · ∂qφn
...
... · · ·
...
∂n−1
q φ1 ∂n−1
q φ2 · · · ∂n−1
q φn
∣∣∣∣∣∣∣∣∣ . (4.8)
Here {φi, ψi} are def ined by (4.6) and (4.7).
q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy 15
On the other hand, we know from (3.3) that Tn defined by (4.2) is a dressing operator if its
generating functions are given by (4.6). Therefore we can define one q-wave function
ωq = Tn∂
−n
q eq(xz)e
∞∑
i=1
ziti
=
1
W q
n
∣∣∣∣∣∣∣∣∣
φ1 · · · φn z−n
∂qφ1 · · · ∂qφn z−n+1
... · · ·
...
...
∂nq φ1 · · · ∂nq φn 1
∣∣∣∣∣∣∣∣∣ eq(xz)e
∞∑
i=1
ziti
. (4.9)
Corollary 2. The relationship in (2.8) between the q-wave function and τq is satisf ied by τ (n)
q
in (4.8) and q-wave function in (4.9), i.e.,
ωq =
τ
(n)
q (x; t− [z−1])
τ
(n)
q (x; t)
eq(xz) exp
( ∞∑
i=1
tiz
i
)
. (4.10)
Proof. We follow the Dickey’s method on page 100 of [37] to prove the corollary. By direct
computations,
φk(x; t− [z−1]) = eq(λkx)e
∞∑
i=1
ziti−
(
λk
z
+
λ2
k
2z2 +···
)
+ akeq(µkx)e
∞∑
i=1
ziti−
(
µk
z
+
µ2
k
2z2 +···
)
= φk −
1
z
∂qφk
whence
τ
(n)
q (x; t− [z−1])
τ
(n)
q (x; t)
=
1
Wq
∣∣∣∣∣∣∣∣∣
φ1 − 1
z∂qφ1 φ2 − 1
z∂qφ2 · · · φn − 1
z∂qφn
∂qφ1 − 1
z∂
2
qφ1 ∂qφ2 − 1
z∂
2
qφ2 · · · ∂qφn − 1
z∂
2
qφn
...
... · · ·
...
∂n−1
q φ1 − 1
z∂
n
q φ1 ∂n−1
q φ2 − 1
z∂
n
q φ2 · · · ∂nq φn − 1
z∂
n
q φn
∣∣∣∣∣∣∣∣∣ .
Comparing the fraction above of the determinant term with (4.9), we can see that they are
similar, although the form of the determinant in the numerator is different. The determinant
in the numerator of (4.9) can be reduced to the same form of (4.10) if the second row, divided
by z, is subtracted from the first one, the third from the second etc. �
At the end of this section, we would like to discuss q-effects in the solution of q-KP hierarchy.
By direct calculation, we get that the first flow of q-KP is
∂t1u0 = x(q − 1)(∂qu1),
∂t1u−1 = (∂qu−1) + u0u−1 + θ(u−2)− u−2 − u−1θ
−1(u0)),
∂t1u−2 = (∂qu−2) + u0u−2 + θ(u−3) +
[
− u−3 + q−1u−1θ
−2(∂qu0)− u−2θ
−2(u0)
]
,
∂t1u−3 = (∂qu−3) + u0u−3 + θ(u−4) +
[
− u−4 − q−3u−1θ
−3(∂2
qu0)
+ (q−1 + q−2)u−2θ
−3(∂qu0)− u−3θ
−3(u0)
]
,
∂t1u−i = (∂qu−i) + u0u−i + θ(u−i−1) +
[
− u−i−1 + (· · · )− u−iθ
−i(u0)
]
,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
in which (· · · ) =
i−1∑
k=1
a−ku−kθ
−i(∂i−kq u0) (i = 2, 3, . . .), and a−k depends on q only. We can see
that
∂t1u0 = 0, ∂t1u−i = ∂qu−i = ∂xu−i, i ≥ 1.
16 J.S. He, Y.H. Li and Y. Cheng
when q → 1. This result shows that the variable t1 in q-KP hierarchy is corresponding to the
variable x in KP hierarchy. So we have two global parameters in q-KP hierarchy, namely x
and q. In order to show q-effect, we will write out the concrete form of single q-soliton of q-KP
equation, namely, we let u−1 depend on three variable (t1, t2, t3) beside two parameters (x, q).
We consider L(1) generated by one step of TD(φ1) from L = ∂q, and the generating function is
given by
φ1 = eq(λ1x)eξ1 +B1eq(λ2x)eξ2 (4.11)
from (4.6), then the Corollary 1 shows that the τ function of L(1) is τ (1)
q = φ1 in (4.11). Here
ξk = λkt1+λ2
kt2+λ
3
kt3 (k = 1, 2), B1 is real constant. Taking this τ (1)
q back into (2.9), then (2.10),
we get q-soliton of q-KP as
u−1 =
[
1 + x(q − 1)
(
λ1eq(λ1x)eξ1 + λ2B1eq(λ2x)eξ2
eq(λ1x)eξ1 +B1eq(λ2x)eξ2
)]
×
{
(λ2
1eq(λ1x)eξ1 +B1λ
2
2eq(λ2x)eξ2)(eq(λ1qx)eξ1 +B1eq(λ2qx)eξ2)
(eq(λ1qx)eξ1 +B1eq(λ2qx)eξ2)(eq(λ1x)eξ1 +B1eq(λ2x)eξ2)
− (λ1eq(λ1x)eξ1 +B1λ2eq(λ2x)eξ2)(λ1eq(λ1qx)eξ1 +B1λ2eq(λ2qx)eξ2)
(eq(λ1qx)eξ1 +B1eq(λ2qx)eξ2)(eq(λ1x)eξ1 +B1eq(λ2x)eξ2)
}
.
In particular, if q → 1, we have
u−1 =
B1(λ1 − λ2)2
eξ̂1−ξ̂2 +B2
1e
ξ̂2−ξ̂1 + 2B1
,
which is a single soliton of the classical KP when x→ 0. Here ξ̂k = λkx+ ξk (k = 1, 2). In order
to plot a figure for u−1, we fix λ1 = 2, λ2 = −1.5 and B1 = 1, so u−1 = u−1(x, t1, t2, t3, q).
The single q-soliton u−1(0.001, t1, t2, t3, 0.999) is plotted in Fig. 1, which is close to classical
soliton of KP equation as we analysed above. From Figs. 2–52 we can see the varying trends
of Mu−1 = u−1(0.5, t1, t2, 0, 0.999) − u−1(0.5, t1, t2, 0, q) , u−1(q = 0.999) − u−1(q) for certain
values of q, where q = 0.7, 0.5, 0.3, 0.1 respectively. Furthermore, in order to see the q-effects
more clearly, we further fixed t2 = −5 in Mu−1, which are plotted in Figs. 6–9. Dependence
of Mu−1 = u−1(x, t1,−5, 0, 0.999) − u−1(x, t1,−5, 0, 0.1) , u−1(x, q = 0.999) − u−1(x, q = 0.1)
on x is shown in Figs. 10–14, and x = 0.3, 0.4, 0.52, 0.54, 0.55 respectively. It is obvious from
figures that Mu−1 goes to zero when q → 1 and x→ 0, q-soliton (u−1) of q-KP goes to a usual
soliton of KP, which reproduces the process of q-deformation. On the other hand, Figs. 10–143
show parameter x amplifies q-effects. In other word, for a given Mq, Mu−1 will increase along x.
However, x is bounded so that eq(λkx) and eq(λkqx) (k = 1, 2) are convergent. This is the
reason for plotting u−1 with x ≤ 0.55. Obviously, the convergent interval depends on q and λk.
We would like to emphasize that from Figs. 6–144 the q-deformation does not destroy the profile
of soliton; it just similar to an “impulse” to soliton.
5 Symmetry constraint of q-KP: q-cKP hierarchy
We know that there exists a constrained version of KP hierarchy, i.e. the constrained KP hi-
erarchy (cKP) [25, 31], introduced by means of the symmetry constraint from KP hierarchy.
2For Figs. 2–5, q-effect Du−1 ≡ 4u−1 , u−1(q = 0.999) − u−1(q = i) with x = 0.5 and t3 = 0, where
i = 0.7, 0.5, 0.3, 0.1. Figs. 6–9, are projection of Figs. 2–5, by fixing t2 = −5.
3For Figs. 10–14, the variable x, varies as follows: 0.3, 0.4, 0.52, 0.54, 0.55, while q = i = 0.1 in Du−1 is fixed.
4For Figs. 6–14, Du−1, u−1(q = 0.999), are represented by continuous line and dashed line (long), respectively,
while dashed line (short) represent u−1(q = i), i = 0.7, 0.5, 0.3, 0.1 for Figs. 6–9.
q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy 17
-5
-2.5
0
2.5
5
t1 -4
-2
0
2
4
t2
0
1
2
3
u-1
-5
-2.5
0
2.5
5
t1
Figure 1. u−1(x = 0.001, q = 0.999) with t3 = 0.
-5
-2.5
0
2.5
5
t1 -4
-2
0
2
4
t2
-0.05
0
0.05
Du-1
-5
-2.5
0
2.5
5
t1
Figure 2.
-5
-2.5
0
2.5
5
t1 -4
-2
0
2
4
t2
-0.2
-0.1
0
0.1
0.2
Du-1
-5
-2.5
0
2.5
5
t1
Figure 3.
With inspiration from it, the symmetry of q-KP was established in [22]. In the same article the
authors defined one kind of constrained q-KP (q-cKP) hierarchy by using the linear combination
of generators of additional symmetry. In this section, we shall briefly introduce the symmetry
and q-cKP hierarchy [22].
The linearization of (2.2) is given by
∂tm(δL) = [δBm, L] + [Bm, δL], (5.1)
where
δBm =
(
m∑
r=1
Lm−rδLLr−1
)
+
.
We call δL = δu0 + δu1∂
−1
q + · · · the symmetry of the q-KP hierarchy, if it satisfies (5.1). Let L
be a “dressed” operator from ∂q, we find
δL = δS∂qS
−1 − S∂qS
−1δSS−1 = [δSS−1, L] = [K,L], (5.2)
where δS = δs1∂
−1
q + δs2∂
−2
q + · · · , and K = δSS−1. Therefore
δBm = [K,Lm]+ = [K,Bm]+,
the last identity is resulted by K = K− and [K,Lm− ]+ = 0. Then the linearized equation (5.1)
is equivalent to
∂tmK = [Bm,K]−, δS = KS. (5.3)
18 J.S. He, Y.H. Li and Y. Cheng
-5
-2.5
0
2.5
5
t1 -4
-2
0
2
4
t2
-0.5
-0.25
0
0.25
0.5
Du-1
-5
-2.5
0
2.5
5
t1
Figure 4.
-5
-2.5
0
2.5
5
t1 -4
-2
0
2
4
t2
-1
0
1
Du-1
-5
-2.5
0
2.5
5
t1
Figure 5.
-6 -4 -2 2 4 6
t1
0.5
1
1.5
2
2.5
3
Du-1, u-1
Figure 6.
-6 -4 -2 2 4 6
t1
0.5
1
1.5
2
2.5
3
Du-1, u-1
Figure 7.
Let Kn = −(Ln)− (n = 1, 2, . . .), then it can easily be checked that Kn satisfies (5.3). For
each Kn, δL is given by δL = −[(Ln)−, L] = [Bn, L] from (5.2). So the q-KP hierarchy admits
a reduction defined by (Ln)− = 0, which is called q-deformed n-th KdV hierarchy. For example,
n = 2, it leads to q-KdV hierarchy, whose q-Lax operator is
LqKdV = L2 = L2
+ = ∂2
q + x(q − 1)u∂q + u.
There is also another symmetry called additional symmetry, which is K = (MmLl)− [22], and
it also satisfies (5.3). Here the operator M is defined by
∂tkM = [L+
k ,M ], M = SΓqS−1,
and Γq is defined as
Γq =
∞∑
i=1
[
iti +
(1− q)i
1− qi
xi
]
∂i−1
q .
The more general generators of additional symmetry are in form of
Yq(µ, λ) =
∞∑
m=0
(µ− λ)m
m!
∞∑
l=−∞
λ−m−l−1
(
MmLm+l
)
−,
which are constructed by combination of K = (MmLl)−. The operator Yq(µ, λ) can be expressed
as
Yq(µ, λ) = ωq(x, t;µ) ◦ ∂−1
q ◦ θ(ω∗q (x, t;λ)).
q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy 19
-6 -4 -2 2 4 6
t1
-0.5
0.5
1
1.5
2
2.5
3
Du-1, u-1
Figure 8.
-6 -4 -2 2 4 6
t1
-1
1
2
3
Du-1, u-1
Figure 9.
-6 -4 -2 2 4 6
t1
0.5
1
1.5
2
2.5
3
Du-1, u-1
Figure 10.
-6 -4 -2 2 4 6
t1
-0.5
0.5
1
1.5
2
2.5
3
Du-1, u-1
Figure 11.
In order to define the q-analogue of the constrained KP hierarchy, we need to establish one
special generator of symmetry Y (t) = φ(t) ◦ ∂−1
q ◦ ψ(t) based on Yq(µ, λ), where
φ(t) =
∫
ρ(µ)ωq(x, t;µ)dµ, ψ(t) =
∫
χ(λ)θ(ω∗q (x, t;λ))dλ,
further φ(t) and ψ(t) satisfy (2.12) and (2.13). In other words, we get a new symmetry of q-KP
hierarchy,
K = φ(λ;x, t) ◦ ∂−1
q ◦ ψ(µ;x, t), (5.4)
where φ(λ;x, t) and ψ(µ;x, t) is an “eigenfunction” and an “adjoint eigenfunction”, respectively.
We can regard from the process above that K = φ(λ;x, t) ◦ ∂−1
q ◦ ψ(µ;x, t) is a special linear
combination of the additional symmetry generator (MmLl)−. It is obvious that generator K
in (5.4) satisfies (5.3), because of the following two operator identities,
(A ◦ a ◦ ∂−1
q ◦ b)− = (A · a) ◦ ∂−1
q ◦ b, (a ◦ ∂−1
q ◦ b ◦A)− = a ◦ ∂−1
q ◦ (A∗ · b). (5.5)
Here A is a q-PDO, and a and b are two functions. Naturally, q-KP hierarchy also has a multi-
component symmetry, i.e.
K =
n∑
i
φi ◦ ∂−1
q ◦ ψi.
It is well known that the integrable KP hierarchy is compatible with generalized l-constraints
of this type (Ll)− =
∑
i
qi ◦ ∂−1
x ◦ ri. Similarly, the l-constraints of q-KP hierarchy
(Ll)− = K =
m∑
i=1
φi ◦ ∂−1
q ◦ ψi
20 J.S. He, Y.H. Li and Y. Cheng
-6 -4 -2 2 4 6
t1
-1
1
2
3
Du-1, u-1
Figure 12.
-6 -4 -2 2 4 6
t1
-2
-1
1
2
3
Du-1, u-1
Figure 13.
-6 -4 -2 2 4 6
t1
-2
-1
1
2
3
Du-1, u-1
Figure 14.
also lead to q-cKP hierarchy. The flow equations of this q-cKP hierarchy
∂tkL
l = [Lk+, L
l], Ll = (Ll)+ +
m∑
i=1
φi ◦ ∂−1
q ◦ ψi (5.6)
are compatible with
(φi)tk = ((Lk)+φi), (ψi)tk = −((L∗k)+ψi).
It can be obtained directly by using the operator identities in (5.5). An important fact is that
there exist two m-th order q-differential operators
A = ∂mq + am−1∂
m−1
q + · · ·+ a0, B = ∂mq + bm−1∂
m−1
q + · · ·+ b0,
such that ALl and LlB are differential operators. From (ALl)− = 0 and (LlB)− = 0, we get
that A and B annihilate the functions φi and ψi, i.e., A(φ1) = · · · = A(φm) = 0, B∗(ψ1) =
· · · = B∗(ψm) = 0, that implies φi ∈ Ker (A). It should be noted that Ker (A) has dimension m.
We will use this fact to reduce the number of components of the q-cKP hierarchy in the next
section.
6 q-Wronskian solutions of q-cKP hierarchy
We know from Corollary 1 that q-Wronskian
τ (N)
q = W q
N (φ1, . . . , φN ) =
∣∣∣∣∣∣∣∣∣
φ1 φ2 · · · φN
∂qφ1 ∂qφ2 · · · ∂qφN
...
... · · ·
...
∂N−1
q φ1 ∂N−1
q φ2 · · · ∂N−1
q φN
∣∣∣∣∣∣∣∣∣ , (6.1)
q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy 21
is a τ function of q-KP hierarchy. Here φi (i = 1, 2, . . . , N) satisfy linear q-partial differential
equations,
∂φi
∂tn
= (∂nq φi), n = 1, 2, 3, . . . . (6.2)
In this section, we will reduce τ (N)
q in (6.1) to a τ function of q-cKP hierarchy. To this end, we
will find the additional conditions satisfied by φi except the linear q-differential equation (6.2).
Corollary 1 also shows that the q-KP hierarchy with Lax operator L(N) = TN ◦ ∂q ◦ T−1
N is
generated from the “free” Lax operator L = ∂q, which has the τ function τ (N)
q in (6.1). In order
to get the explicit form of such Lax operator L(N), the following lemma is necessary.
Lemma 5.
TN =
1
W q
N (φ1, . . . , φN )
∣∣∣∣∣∣∣∣∣
φ1 · · · φN 1
∂qφ1 · · · ∂qφN ∂q
... · · ·
...
...
∂Nq φ1 · · · ∂Nq φN ∂Nq
∣∣∣∣∣∣∣∣∣
and
T−1
N =
∣∣∣∣∣∣∣∣∣
φ1 ◦ ∂−1
q θ(φ1) · · · θ(∂N−2
q φ1)
φ2 ◦ ∂−1
q θ(φ2) · · · θ(∂N−2
q φ2)
...
... · · ·
...
φN ◦ ∂−1
q θ(φN ) · · · θ(∂N−2
q φN )
∣∣∣∣∣∣∣∣∣ ·
(−1)N−1
θ(W q
N (φ1, . . . , φN ))
=
N∑
i=1
φi ◦ ∂−1
q ◦ gi
with
gi = (−1)N−iθ
( W q
N (φ1, . . . , φi−1, î, φi+1, . . . , φN )
W q
N (φ1, . . . , φi−1, φi, φi+1, . . . , φN )
)
. (6.3)
Here î means that the column containing φi is deleted from W q
N (φ1, . . . , φi−1, φi, φi+1, . . . , φN ),
and the last row is also deleted.
Proof. The proof is a direct consequence of Lemma 3 and Theorem 4 from the initial “free”
Lax operator L = ∂q. The generating functions {φi, i = 1, 2, . . . , N} of TN satisfies equa-
tions (6.2), which is obtained from definition of “eigenfunction” (2.12) of the KP hierarchy
under Bn = ∂nq . �
In particular, (TN · φ1) = (TN · φ2) = · · · = (TN · φN ) = 0.
Now we can give one theorem reducing the q-Wronskian τ function τ
(N)
q in (6.1) of q-KP
hierarchy to the q-cKP hierarchy defined by (5.6).
Theorem 6. τ (N)
q is also a τ function of the q-cKP hierarchy whose Lax operator Ll = (Ll)+ +
M∑
i=1
qi ◦∂−1
q ◦ri with some suitable functions {qi, i = 1, 2, . . . ,M} and {ri, i = 1, 2, . . . ,M} if and
only if
W q
N+M+1(φ1, . . . , φN , ∂
l
qφi1 , . . . , ∂
l
qφiM+1) = 0 (6.4)
for any choice of (M + 1)-indices (i1, i2, . . . , iM+1) 1 6 i1 < · · · < iM+1 ≤ N , which can be
expressed equivalently as
W q
M+1
(
W q
N+1(φ1, . . . , φN , ∂
l
qφi1)
W q
N (φ1, . . . , φN )
,
W q
N+1(φ1, . . . , φN , ∂
l
qφi2)
W q
N (φ1, . . . , φN )
, . . . ,
22 J.S. He, Y.H. Li and Y. Cheng
W q
N+1(φ1, . . . , φN , ∂
l
qφiM+1)
W q
N (φ1, . . . , φN )
)
= 0 (6.5)
for all indices. Here {φi, i = 1, 2, . . . , N} satisfy (6.2).
Remark 4. This theorem is a q-analogue of the classical theorem on cKP hierarchy given
by [38].
Proof. The q-Wronskian identity proven in Appendix C
W q
M+1
(
W q
N+1(φ1, . . . , φN , f1)
W q
N (φ1, . . . , φN )
, . . . ,
W q
N+1(φ1, . . . , φN , fM+1)
W q
N (φ1, . . . , φN )
)
=
W q
N+M+1(φ1, . . . , φN , f1, . . . , fM+1)
W q
N (φ1, . . . , φN )
implies equivalence between (6.4) and (6.5). Using TN and T−1
N in Lemma 5 and the operator
identity in (5.5) we have
(Ll)− = (TN ◦ ∂lq ◦ T−1
N )− =
N∑
i=1
(TN (∂lqφi)) ◦ ∂−1
q ◦ gi, (6.6)
where gi is given by (6.3) and TN acting on (∂lqφi) is TN (∂lqφi) =
W q
N+1(φ1, φ2, . . . , φN , ∂
l
qφi)
W q
N (φ1, φ2, . . . , φN )
.
So τ (N)
q is automatically a tau function of N -component q-cKP hierarchy with the form (6.6).
Next, we can reduce the N -component to the M -component (M < N) by a suitable constraint
of φi.
Suppose that the M -component (M < N) q-cKP hierarchy is obtained by constraint of qKP
hierarchy generated by TN , i.e., there exist suitable functions {qi, ri} such that
(Ll)− =
M∑
i=1
qi ◦ ∂−1
q ◦ ri =
N∑
i=1
(TN (∂lqφi)) ◦ ∂−1
q ◦ gi.
As we pointed out in previous section, for a Lax operator whose negative part is in the form of
(Ll)− =
M∑
i=1
qi ◦ ∂−1
q ◦ ri, there exists an M -th order q-differential operator A such that ALl is
a q-differential operator, then we have
0 = ALl(TN (φi)) = ATN∂
l
q(φi) = A(TN (∂lqφi))
from TN (φi) = 0that implies TN (∂lqφi) ∈ Ker (A). Therefore, at most M of these functions
TN (∂lqφi) can be linearly independent because the Kernel of A has dimension M . So (6.5) is
deduced.
Conversely, suppose (6.5) is true, we will show that there exists one M-component q-ckP
(M < N) constrained from (6.6). The equation (6.5) implies that at most M of functions
TN (∂lqφi) (i = 1, 2, . . . , N) are linearly independent. Then we can find suitable M functions
{q1, q2, . . . , qM}, which are linearly independent, to express functions TN (∂lqφi) as
TN (∂lqφi) =
W q
N+1(φ1, φ2 · · · , φN , ∂lqφi)
W q
N (φ1, φ2, . . . , φN )
=
M∑
j=1
cijqj , i = 1, · · · , N
q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy 23
with some constants cij . Taking this back into (6.6), it becomes
(Ll)− =
N∑
i=1
M∑
j=1
cijqj
◦ ∂−1
q ◦ gi =
M∑
j=1
qj ◦ ∂−1
q ◦
(
N∑
i=1
cijgi
)
=
M∑
j=1
qj ◦ ∂−1
q ◦ rj ,
which is an M -component q-cKP hierarchy as we expected. �
7 Example reducing q-KP to q-cKP hierarchy
To illustrate the method in Theorem 6 reducing the q-KP to multi-component a q-cKP hierarchy,
we discuss the q-KP generated by TN |N=2. In order to obtain the concrete solution, we only
consider the three variables (t1, t2, t3) in t. Furthermore, the q1, r1 and u−1 are constructed in
this section.
According to Theorem 6, the q-KP hierarchy generated by TN |N=2 possesses a tau function
τ (2)
q = W q
2 (φ1, φ2) = φ1(∂qφ2)− φ2(∂qφ1)
= (λ2 − λ1)eq(λ1x)eq(λ2x)eξ1+ξ2 + (λ3 − λ1)eq(λ1x)eq(λ3x)eξ1+ξ3
+ (λ3 − µ)eq(µx)eq(λ3x)eξ+ξ3 + (λ2 − µ)eq(µx)eq(λ2x)eξ+ξ2 (7.1)
with
φ1 = eq(λ1x)eξ1 + eq(µx)eξ, φ2 = eq(λ2x)eξ2 + eq(λ3x)eξ3 .
Here ξi = ci+λit1 +λ2
i t2 +λ3
i t3 (i = 1, 2, 3), and ξ = d+µt1 +µ2t2 +µ3t3, ci and d are arbitrary
constants. These functions satisfy the linear equations
∂φi
∂tn
= ∂nq φi, n = 1, 2, 3, i = 1, 2,
as a special case of (6.2). From (6.6), the q-KP hierarchy generated by TN |N=2 is in the form of
Ll = (Ll)+ + (T2(∂lqφ1)) ◦ ∂−1
q ◦ g1 + (T2(∂lqφ2)) ◦ ∂−1
q ◦ g2, (7.2)
constraint===== (Ll)+ + q1 ◦ ∂q ◦ r1. (7.3)
Here q1 and r1 are undetermined, which can be expressed by φ1 and φ2 as follows.
According to (6.4), the restriction for φ1 and φ2 to reduce (7.2) to (7.3) is given by
0 = W q
2 (φ1, φ2, ∂
l
qφ1, ∂
l
qφ2) = (µl − λl1)(λ
l
2 − λl3)V (λ1, λ2, λ3, µ)ec1+c2+c3+de(λ1+λ2+λ3+µ)t1
× e(λ
2
1+λ2
2+λ2
3+µ2)t2e(λ
3
1+λ3
2+λ3
3+µ3)t3eq(λ1x)eq(λ2x)eq(λ3x)eq(µx) (7.4)
with
V (λ1, λ2, λ3, µ) =
∣∣∣∣∣∣∣∣
1 λ1 λ2
1 λ3
1
1 λ2 λ2
2 λ3
2
1 λ3 λ2
3 λ3
3
1 µ µ2 µ3
∣∣∣∣∣∣∣∣ .
Obviously, we can let µ = λ2 and d = c2 such that (7.4) holds for φ1 and φ2. Then the τ function
of a single component q-cKP defined by (7.3) is
τqcKP = (λ2 − λ1)eq(λ1x)eq(λ2x)eξ1+ξ2 + (λ3 − λ1)eq(λ1x)eq(λ3x)eξ1+ξ3
24 J.S. He, Y.H. Li and Y. Cheng
+ (λ3 − λ2)eq(λ2x)eq(λ3x)eξ2+ξ3 ,
which is deduced from (7.1). That means we indeed reduce the τ function τ
(2)
q in (7.1) of
the q-KP hierarchy generated by TN |N=2 to the τ function τqcKP of the one-component q-cKP
hierarchy. Furthermore, we would like to get the explicit expression of (q1, r1) of q-cKP in (7.3).
Using the determinant representation of TN |N=2 and T−1
N |N=2, we have
f1 , (T2(∂lqφ1)) =
(λl1 − λl2)(λ3 − λ2)(λ2 − λ1)(λ3 − λ1)eq(λ1x)eq(λ2x)eq(λ3x)eξ1+ξ2+ξ3
τqcKP
,
f2 , (T2(∂lqφ2) =
(λl3 − λl2)(λ3 − λ2)(λ2 − λ1)(λ3 − λ1)eq(λ1x)eq(λ2x)eq(λ3x)eξ1+ξ2+ξ3
τqcKP
,
g1 = −θ
(
φ2
τqcKP
)
, g2 = θ
(
φ1
τqcKP
)
,
under the restriction µ = λ2 and d = c2. One can find that f1 and f2 are linearly dependent,
and (λl3 − λl2)f1 = (λl1 − λl2)f2. So (7.2) and (7.3) reduce to
Ll− = f1 ◦ ∂−1
q ◦ g1 + f2 ◦ ∂−1
q ◦ g2
= (λl3 − λl2)f1 ◦ ∂−1
q ◦ g1
(λl3 − λl2)
+ (λl1 − λl2)f2 ◦ ∂−1
q ◦ g2
(λl1 − λl2)
= q1 ◦ ∂−1
q ◦ r1,
in which
q1 , (λl3 − λl2)f1 = (λl1 − λl2)f2
=
(λl1 − λl2)λ
l
3 − λl2)(λ3 − λ2)(λ2 − λ1)(λ3 − λ1)eq(λ1x)eq(λ2x)eq(λ3x)eξ1+ξ2+ξ3
τqcKP
,
r1 ,
(
g1
(λl3 − λl2)
+
g2
(λl1 − λl2)
)
=
1
(λl1 − λl2)(λ
l
3 − λl2)
× θ
(
e−(ξ1+ξ2+ξ3)
(
(λl3 − λl2)eq(λ1x)eξ1 + (λl3 − λl1)eq(λ2x)eξ2 + (λl2 − λl1)eq(λ3x)eξ3
)
τqcKP
)
.
In particular, we can let λ1 = λ, λ2 = 0, λ3 = −λ, c1 = c, c2 = −0, c3 = −c, then
q1 =
(−1)lλ2l+2eq(λx)eq(−λx)
eq(λx)eq(−λx) + eq(λx)eη+eq(−λx)e−η
2 e−λ2t2
and
r1 =
− 1
λl+1
θ
[
e−λ
2t2 + eq(λx)eη+eq(−λx)e−η
2
eλ2t2eq(λx)eq(−λx) + eq(λx)eη+eq(−λx)e−η
2
]
if l is odd,
− 1
λl+1
θ
[
eq(λx)eη−eq(−λx)e−η
2
eλ2t2eq(λx)eq(−λx) + eq(λx)eη+eq(−λx)e−η
2
]
if l is even,
where η = c+ λt1 + λ3t3.
In general, the l-constrained one-component q-KP hierarchy has the Lax operator L = ∂q +
u0 + q1 ◦ ∂−1
q ◦ r1 when l = 1. On the other hand, its Lax operator can also be expressed as
L = ∂q + u0 + u−1∂
−1
q + u−2∂
−2
q + · · · . So all of the dynamical variables {u−i, i = 1, 2, 3, . . .} of
q-KP hierarchy are given by
u−i−1 = (−1)iq−i(i+1)/2q1θ
−i−1(∂iqr1), i ≥ 0.
q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy 25
-10
-5
0
5
10
t1
-2
-1
0
1
2
3
t2
0
1
2
3
u-1
-10
-5
0
5t1
Figure 15. u−1(x = 0.001, q = 0.999) from q-cKP and t3 = 0.
For the present situation, u−1 = u−1(t1, t2, t3) = q1θ
−1(r1) represents the q-deformed solution
of the classical KP eqution, which is constructed from the components of q-cKP hierarchy, and
is of the form
q1 =
−λ4eq(λx)eq(−λx)
eq(λx)eq(−λx) + eq(λx)eη+eq(−λx)e−η
2 e−λ2t2
, (7.5)
r1 = − 1
λ2
θ
[
e−λ
2t2 + eq(λx)eη+eq(−λx)e−η
2
eλ2t2eq(λx)eq(−λx) + eq(λx)eη+eq(−λx)e−η
2
]
, (7.6)
u−1 =
λ2eq(λx)eq(−λx)
eq(λx)eq(−λx) + eq(λx)eη+eq(−λx)e−η
2 e−λ2t2
×
e−λ
2t2 + eq(λx)eη+eq(−λx)e−η
2
eλ2t2eq(λx)eq(−λx) + eq(λx)eη+eq(−λx)e−η
2
. (7.7)
Obviously, they will approach to the classical results on the cKP hierarchy in [38] when
x → 0 and q → 1. We will fix λ = 2, t3 = 0 and c = 0 to plot their figures, then get
q1 = q1(x, t1, t2, q), r1 = r1(x, t1, t2, q) and u−1 = u−1(x, t1, t2, q) from (7.5)–(7.7). To save space,
we plot the figures for u−1 and q1 in (t1, t2, t3) dimension spaces. It can be seen that Fig. 15 of
u−1(0.001, t1, t2, 0.999) and Fig. 20 of q1(0.001, t1, t2, 0.999) match with the profile of u1 and q
in [38] with the same parameters. So we define q-effects quantity 4u−1 = u−1(0.5, t1, t2, 0.999)−
u−1(0.5, t1, t2, q) = u−1(q = 0.999)−u−1(q), 4q1 = q1(0.5, t1, t2, 0.999)−q1(.5, , t1, t2, q) = q1(q =
0.999) − q1(q), to show their dependence on q. Figs. 16–195 and Figs. 21–246 are plotted for
4u−1 and 4q1, respectively, where q = 0.7, 0.5, 0.3, 0.1. Obviously, they are decreasing to
almost zero when q goes from 0.1 to 1 with fixed x = 0.5. Furthermore, Figs. 25–297 show that
the dependence of the q-effects 4u−1 = u−1(x, t1, t2, 0.999) − u−1(x, t1, t2, 0.1) = u−1(x, q =
0.999) − u−1(x, q = 0.1) on x,where x = 0.2, 0.4, 0.51, 0.53, 0.55 in order. These figures give
us again an opportunity to observe the process of q-deformation in q-soliton solution of q-KP
equation. They also demonstrate that q-deformation keep the profile of the soliton, although
there exists deformation in some degree. On the other hand, in fact, (q1, r1) can be regarded as
5For Figs. 16–19, q-effect Du−1 ≡ 4u−1 , u−1(q = 0.999) − u−1(q = i), where i = 0.7, 0.5, 0.3, 0.1, from
q-cKP with x = 0.5 and t3 = 0.
6For Figs. 21–24, q-effect Dq1 ≡ 4q1 , q1(q = 0.999)− q1(q = i), where i = 0.7, 0.5, 0.3, 0.1, from q-cKP with
x = 0.5 and t3 = 0.
7For Figs. 25–29, q-effect Du−1 ≡ 4u−1 , u−1(x = i, q = 0.999)−u−1(x = i, q = 0.1) from q-cKP with t3 = 0,
where i = 0.2, 0.4, 0.51, 0.53, 0.55.
26 J.S. He, Y.H. Li and Y. Cheng
-5
0
5t1 -2
0
2
4
t2
0
0.05Du-1
-5
0
5t1
Figure 16.
-5
0
5t1 -2
0
2
4
t2
-0.1
0
0.1Du-1
-5
0
5t1
Figure 17.
-5
0
5t1 -2
0
2
4
t2
-0.5
-0.25
0
0.25
0.5
Du-1
-5
0
5t1
Figure 18.
-5
0
5t1 -2
0
2
4
t2
-1
0
1
Du-1
-5
0
5t1
Figure 19.
a q-deformation of dynamical variables (q, r) of AKNS hierarchy, because cKP possessing Lax
operator L = ∂ + q ◦ ∂−1 ◦ r is equivalent to the AKNS hierarchy.
8 Conclusions and discussions
In this paper, we have shown in Theorem 1 that there exist two types of elementary gauge
transformation operators for the q-KP hierarchy. The changing rules of q-KP under the gauge
transformation are given in Theorems 2 and 3. We mention that these two types of elementary
gauge transformation operators are introduced first by Tu et al. [15] for q-NKdV hierarchy.
Considering successive application of gauge transformation, we established the determinant rep-
resentation of the gauge transformation operator of the q-KP hierarchy in Lemma 3 and the
corresponding results on the transformed new q-KP are given in Theorem 5. For the q-KP
hierarchy generated by Tn+k from the “free” Lax operator L = ∂q (i.e. the Lax operator is
L(n+k) = Tn+k ◦ ∂q ◦ T−1
n+k), Corollary 1 shows that the generalized q-Wronskian IW q
k,n of func-
tions {φi, ψj} (i = 1, 2, . . . , n; j = 1, 2, . . . , k) is a general τ function of it, and q-Wronskian W q
n
of functions φi(i = 1, 2, . . . , n) is also a special one. Here {φi} and {ψj} satisfy special linear
q-partial differential equations (4.5).
The symmetry and symmetry constraint of q-KP (q-cKP) hierarchy are discussed in Section 5.
On the basis of the representation of TN in Lemma 5, the q-KP hierarchy whose Lax operator
Ll = TN ◦ ∂lq ◦ T−1
N is generated from the “free” Lax operator L = ∂q. The explicit form of its
negative part Ll− is given in (6.6), which is called l-constraint of the q-KP hierarchy. Further we
found necessary and sufficient conditions that are given in Theorem 6, reducing a q-Wronskian
solution in (6.1) of the q-KP hierarchy to solutions of the multi-component q-cKP hierarchy.
One example is given in Section 7 to illustrate the method, i.e., the q-KP generated by TN |N=2
q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy 27
-20
0
20
t1
0
5
10
t2
0
5
10
15
-q1
-20
0
20
t1
Figure 20. q1(x = 0.001, q = 0.999) and t3 = 0.
-5
0
5t1 -2
0
2
t2
-0.4
-0.3
-0.2
-0.1
0
-Dq1
-5
0
5t1
Figure 21.
-5
0
5t1 -2
0
2
t2
-0.75
-0.5
-0.25
0
-Dq1
-5
0
5t1
Figure 22.
is reduced to one-component q-cKP hierarchy. By taking finite variables (t1, t2, t3) in t, the
component q1 and r1 are written out. Our results can be reduced to the classical results in [38].
As we pointed out in Section 2, u−1 is the q-analogue of the solution of classical KP equation
if we only consider three variables (t1, t2, t3) in t. Therefore, the solution u−1 is called q-soliton
of the q-KP equation, although we do not write out the q-KP equation on u−1. One can find
that the equations of dynamical variables {u0,u−i} in q-KP hierarchy are coupled with each
other and can not get one q-partial differential equation associated only with one dynamical
variable, like classical KP equation has one dynamical variable u−1. The reason is that the
q-Leibnitz rule contains q-differential operation and θ operation, however, the Leibnitz rule of
the standard calculus only contains one differential operation. We get a single q-soliton u−1 by
means of the simplest τ function τq = W q
1 (φ1) = φ1 in Section 4. Meanwhile, the multi-q-soliton
u−1 is obtained from one-component q-cKP hierarchy in Section 7. Figures of q-effect 4u−1
show that q-soliton u−1 indeed goes to classical soliton of KP equation when x→ 0 and q → 1
and q-deformation does not destroy the rough profile of the q-soliton. In other worlds, the figure
of q-soliton is similar to the classical soliton of KP equation. We also show the trends of the
q-effect 4u−1 depends on x and q; x plays a role of the amplifier of q-effects. In conclusion,
the figures of q-effects 4u−1 let us know the process of q-deformation in integrable systems for
the first time. Of course, it is a long way to explore the physical meaning of q from the soliton
theory.
In comparison with the research of classical soliton theory [40], in particular, the KP hi-
erarchy [36, 37], the cKP [25, 31] hierarchy and the AKNS [40] hierarchy, there exist at least
several topics needed to be discussed in order to research the integrability property of nonlinear
q-partial differential equations. For instance, the Hamiltonian structure the q-cKP hierarchy
and its q-W-algebra; the gauge transformation of the q-cKP hierarchy; the q-Hirota equation as-
28 J.S. He, Y.H. Li and Y. Cheng
-5
0
5t1 -2
0
2
t2
-2
-1.5
-1
-0.5
0
-Dq1
-5
0
5t1
Figure 23.
-5
0
5t1 -2
0
2
t2
-4
-2
0
-Dq1
-5
0
5t1
Figure 24.
-5
0
5t1 -2
0
2
4
t2
-0.05
0
0.05
Du-1
-5
0
5t1
Figure 25.
-5
0
5t1 -2
0
2
4
t2
-0.5
-0.25
0
0.25
0.5
Du-1
-5
0
5t1
Figure 26.
sociated with the bilinear identity of the q-KP hierarchy; the symmetry analysis of q-differential
equation and q-partial differential equations; the interaction of q-solitons; the q-AKNS hierarchy
and its properties. Since the KP hierarchy has B-type and C-type sub-hierarchies, what are
q-analogues of them? In particular, we showed in the previous sections that convergence of eq(x)
affects the q-soliton, so the analytic property of eq(x) is a basis for research the interaction of
q-solitons. We will try to investigate these questions in the future.
Acknowledgements
This work is supported partly by the 973 project of China for “Nonlinear Science”, the National
Natural Science Foundation of China (10301030) and SRFDP of China. The author (Jingsong
He) would like to thank the Centre for Scientific Computing and University of Warwick for
supporting him to visit there. Special thanks go to Dr. Rudolf A. Römer at Warwick for the
numerous helpful discussions, Dr. P. Iliev for explaining few parts of his work [20] and Professor
J. Mas for answering questions on his paper [18]. Jingsong He is also grateful to Professors
F. Calogero, A. Degasperis, D. Levi and P.M. Santini of University of Rome “La Sapienza” for
their hospitality during his visit to Rome. We thank anonymous referees very much for valuable
suggestions and corrections.
A More explicit expressions of ∂n
q ◦ f
For n ≥ 1, we have
∂4
q ◦ f = (∂4
qf) + (4)qθ(∂3f)∂q +
(4)q(3)q
(2)q
θ2(∂2
qf)∂2
q + (4)qθ3(∂qf)∂3
q + θ4(f)∂4
q ,
q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy 29
-5
0
5t1 -2
0
2
4
t2
-1
0
1
Du-1
-5
0
5t1
Figure 27.
-5
0
5t1 -2
0
2
4
t2
-2
0
2
Du-1
-5
0
5t1
Figure 28.
-5
0
5t1 -2
0
2
4
t2
-2
0
2
Du-1
-5
0
5t1
Figure 29.
∂5
q ◦ f = (∂5
qf) + (5)qθ(∂4
qf)∂q +
(5)q(4)q
(2)q
θ2(∂3
qf)∂2
q +
(5)q(4)q
(2)q
θ3(∂2
qf)∂3
q
+ (5)qθ4(∂qf)∂4
q + θ5(f)∂5
q .
On the other hand, several examples of an explicit expression for ∂−nq ◦ f (n ≥ 1) are
∂−3
q ◦ f = θ−3(f)∂−3
q − (3)q
q3
θ−4(∂qf)∂−4
q +
(3)q(4)q
(2)qq3+4
θ−5(∂2
qf)∂−5
q
− (4)q(5)q
q3+4+5(2)q
θ−6(∂3
qf)∂−6
q +
(5)q(6)q
q3+4+5+6(2)q
θ−7(∂4
qf)∂−7
q + · · ·
+
(−1)k(k + 1)q(k + 2)q
q3+4+5+···+(k+1)+(k+2)(2)q
θ−3−k(∂kq f)∂−3−k
q + · · · ,
∂−4
q ◦ f = θ−4(f)∂−4
q − (4)q
q4
θ−5(∂qf)∂−5
q +
(4)q(5)q
q4+5(2)q
θ−6(∂2
qf)∂−6
q
− (4)q(5)q(6)q
q4+5+6(2)q(3)q
θ−7(∂3
qf)∂−7
q +
(5)q(6)q(7)q
q4+5+6+7(2)q(3)q
θ−8(∂4
qf)∂−8
q + · · ·
+
(−1)k(k + 1)q(k + 2)q(k + 3)q
q4+5+6+···+(k+2)+(k+3)(2)q(3)q
θ−4−k(∂kq f)∂−4−k
q + · · · ,
∂−5
q ◦ f = θ−5(f)∂−5
q − (5)q
q5
θ−6(∂qf)∂−6
q +
(5)q(6)q
q5+6(2)q
θ−7(∂2
qf)∂−7
q
− (5)q(6)q(7)q
q5+6+7(2)q(3)q
θ−8(∂3
qf)∂−8
q +
(5)q(6)q(7)q(8)q
q5+6+7+8(2)q(3)q(4)q
θ−9(∂4
qf)∂−9
q + · · ·
+
(−1)k(k + 1)q(k + 2)q(k + 3)q(k + 4)q
q5+6+7+···+(k+3)+(k+4)(2)q(3)q(4)q
θ−5−k(∂kq f)∂−5−k
q + · · · .
30 J.S. He, Y.H. Li and Y. Cheng
B Positive part of Ln (n = 1, 2, 3)
The first few of Bn are in the form of
B1 = ∂q + u0, B2 = ∂2
q + v1∂q + v0, B3 = ∂3
q + s̃2∂
2
q + s̃1∂q + s̃0,
where
v1 = θ(u0) + u0, v0 = (∂qu0) + θ(u−1) + u2
0 + u−1,
v−1 = (∂qu−1) + θ(u−2) + u0u−1 + u−1θ
−1(u0) + u−2,
and
s̃2 = θ(v1) + u0, s̃1 = (∂qv1) + θ(v0) + u0v1 + u−1,
s̃0 = (∂qv0) + θ(v−1) + u0v0 + u−1θ
−1(v1) + u−2.
Note that v−1 comes from L2 = B2 + v−1∂
−1
q + v−2∂
−2
q + · · · .
C Proof of the q-Wronskian identity
1) The first N steps. Consider the gauge transformation generated by the order of {φi, i =
1, 2, . . . , N}
T
(1)
D (φ1) −→ T
(2)
D
(
φ
(1)
2
)
−→ · · · −→ T
(i)
D
(
φ
(i−1)
i
)
−→ · · · −→ T
(N)
D
(
φ
(N−1)
N
)
.
Assume there are l functions {φ(N)
N+j , j = 1, 2, . . . , l} expressed by
φ
(N)
N+j = (TN · φN+j) =
W q
N+1(φ1, φ2, . . . , φN , φN+j)
W q
N (φ1, φ2, . . . , φN )
,
which are generated by TN from {φj}. Here φi (i = 1, 2, . . . , N + l) are arbitrary functions such
that the gauge transformations can be constructed.
2) The last l − 1 steps. Let yj = φ
(N)
N+j (j = 1, 2, . . . , l). Using yj (j = 1, 2, . . . , l − 1) as
the generating functions in order of TD, we can construct (l − 1) steps of gauge transformation
operators as
T
(1)
D (y1) −→ T
(2)
D
(
y
(1)
2
)
−→ T
(3)
D
(
y
(2)
3
)
−→ · · · −→ T
(l−1)
D
(
y
(l−2)
l−1
)
.
According to the determinant of TN |N=j (j = 1, 2, . . . , l − 1), we have
y
(j)
i = (Tj · yi)
0 if j ≥ i,
W q
j+1(y1, y2, . . . , yj , yi)
W q
j (y1, y2, . . . , yj)
if j < i,
then
y1 · y(1)
2 · y(2)
3 · · · y(l−2)
l−1 y
(l−1)
l = y1
W q
2 (y1, y2)
W q
1 (y1)
W q
3 (y1, y2, y3)
W q
2 (y1, y2)
· · ·
W q
l−1(y1, y2, . . . , yl−2, yl−1)
W q
l−2(y1, y2, . . . , yl−2)
W q
l (y1, y2, . . . , yl−2, yl−1, yl)
W q
l−1(y1, y2, . . . , yl−2, yl−1)
= W q
l (y1, y2, . . . , yl) = W q
l
(
φ
(N)
N+1, φ
(N)
N+2, . . . , φ
(N)
N+l
)
. (C.1)
q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy 31
3) Combine two chains of gauge transformations above. In fact, we can combine two chains
into one,
T
(1)
D (φ1) −→ T
(2)
D
(
φ
(1)
2
)
−→ · · · −→ T
(i)
D
(
φ
(i−1)
i
)
−→ · · · −→ T
(N)
D
(
φ
(N−1)
N
)
,
T
(N+1)
D
(
φ
(N)
N+1
)
−→ T
(N+2)
D
(
φ
(N+1)
N+2
)
−→ T
(N+3)
D
(
φ
(N+2)
N+3
)
−→ · · · −→ T
(N+l−1)
D
(
φ
(N+l−2)
N+l−1
)
.
The determinant representation of TN |N+j implies (1 < i, j < l):
φ
(N+j)
N+i = (TN+j · φN+i) =
0 if j ≥ i,
W q
N+j+1(φ1, φ2 · · · , φN , φN+1, · · · , φN+j , φN+i)
W q
N+j(φ1, φ2 · · · , φN , φN+1, · · · , φN+j)
if j < i.
So
φ
(N)
N+1 · φ
(N+1)
N+2 · φ(N+2)
N+3 · · ·φ(N+l−2)
N+l−1 φ
(N+l−1)
N+l =
W q
N+1(φ1, φ2, . . . , φN , φN+1)
W q
N (φ1, φ2, . . . , φN )
×
×
W q
N+2(φ1, φ2, . . . , φN+1, φN+2)
W q
N+1(φ1, φ2, . . . , φN+1)
W q
N+3(φ1, φ2, . . . , φN+2, φN+3)
W q
N+2(φ1, φ2, . . . , φN+2)
· · ·
×
W q
N+l−1(φ1, φ2, . . . , φN+l−2, φN+l−1)
W q
N+l−2(φ1, φ2, . . . , φN+l−2)
W q
N+l(φ1, φ2, . . . , φN+l−1, φN+l)
W q
N+l−1(φ1, φ2, . . . , φN+l−1)
· · ·
=
W q
N+l(φ1, φ2, . . . , φN+l−1, φN+l)
W q
N (φ1, φ2, . . . , φN )
. (C.2)
The left hand side of (C.1) equals the left hand side of (C.2), which is followed by
W q
N+l(φ1, φ2, . . . , φN , φN+1, . . . , φN+l−1, φN+l)
W q
N (φ1, φ2, . . . , φN )
= W q
l
(
φ
(N)
N+1, φ
(N)
N+2, . . . , φ
(N)
N+l
)
.
It should be noted that the proof above is independent of the form of φk, so we can replace
φN+j with (∂lqφN+j). This completes the proof of the q-Wronskian identity.
[1] Klimyk A., Schmüdgen K., q-calculus, in Quantum Groups and Their Represntaions, Berlin, Springer, 1997,
Chapter 2, 37–52.
[2] Kac V., Cheung P., Quantum calculus, New York, Springer-Verlag, 2002.
[3] Exton H., q-hypergeometric functions and applications, Chichester, Ellis Horwood Ltd., 1983.
[4] Andrews G.E., q-series: their development and application in analysis, number theory, combinatorics,
physics, and computer algebra, Providence, American Mathematical Society, 1986.
[5] Jimbo M., Yang–Baxter equation in integrable systems, Advanced Series in Mathematical Physics, Vol. 10,
Singapore, World Scientific, 1990.
[6] Connes A., Noncommutative geometry, San Diego – London, Academic Press, 1994.
[7] Majid S., Free braided differential calculus, braided binomial theorem, and the braided exponential map,
J. Math. Phys., 1993, V.34, 4843–4856, hep-th/9302076.
[8] Majid S., Foundations of quantum group theory, Cambridge, Cambridge University Press, 1995, § 10.4.
[9] Zhang D.H., Quantum deformation of KdV hierarchies and their infinitely many conservation laws,
J. Phys. A: Math. Gen., 1993, V.26, 2389–2407.
[10] Wu Z.Y., Zhang D.H., Zheng Q.R., Quantum deformation of KdV hierarchies and their exact solutions:
q-deformed solitons, J. Phys. A: Math. Gen., 1994, V.27, 5307–5312.
[11] Frenkel E., Reshetikhin N., Quantum affine algebras and deformations of the Virasoro and W -algebras,
Comm. Math. Phys., 1996, V.178, 237–264, q-alg/9505025.
[12] Frenkel E., Deformations of the KdV hierarchy and related soliton equations, Int. Math. Res. Not., 1996,
V.2, 55–76, q-alg/9511003.
32 J.S. He, Y.H. Li and Y. Cheng
[13] Haine L., Iliev P., The bispectral property of a q-deformation of the Schur polynomials and the q-KdV
hierarchy, J. Phys. A: Math. Gen., 1997, V.30, 7217–7227.
[14] Adler M., Horozov E., van Moerbeke P., The solution to the q-KdV equation, Phys. Lett. A, 1998, V.242,
139–151, solv-int/9712015.
[15] Tu M.H., Shaw J.C., Lee C.R., On Darboux–Bäcklund transformations for the q-deformed Korteweg–de
Vries hierarchy, Lett. Math. Phys., 1999, V.49, 33–45, solv-int/9811004.
[16] Tu M.H., Shaw J.C., Lee C.R., On the q-deformed modified Korteweg–de Vries hierarchy, Phys. Lett. A,
2000, V.266, 155–159.
[17] Khesin B., Lyubashenko V., Roger C., Extensions and contractions of the Lie algebra of q-pseudodifferential
symbols on the circle, J. Funct. Anal., 1997, V.143, 55–97, hep-th/9403189.
[18] Mas J., Seco M., The algebra of q-pseudodifferential symbols and the q-W
(n)
KP algebra, J. Math. Phys., 1996,
V.37, 6510–6529, q-alg/9512025.
[19] Iliev P., Solutions to Frenkel’s deformation of the KP hierarchy, J. Phys. A: Math. Gen., 1998, V.31, L241–
L244.
[20] Iliev P., Tau function solutions to a q-deformation of the KP hierarchy, Lett. Math. Phys., 1998, V.44,
187–200.
[21] Iliev P., q-KP hierarchy, bispectrality and Calogero–Moser systems, J. Geom. Phys., 2000, V.35, 157–182.
[22] Tu M.H., q-deformed KP hierarchy: its additional symmetries and infinitesimal Bäcklund transformations,
Lett. Math. Phys., 1999, V.49, 95–103, solv-int/9811010.
[23] Wang S.K., Wu K., Wu X.N., Wu D.L., The q-deformation of AKNS-D hierarchy, J. Phys. A: Math. Gen.,
2001, V.34, 9641–9651.
[24] He J.S., Li Y.H., Cheng Y., q-deformed Gelfand–Dickey hierarchy and the determinant representation of its
gauge transformation, Chinese Ann. Math. Ser. A, 2004, V.25, 373–382 (in Chinese).
[25] Konopelchenko B.G., Sidorenko J., Strampp W., (1 + 1)-dimensional integrable systems as symmetry con-
straints of (2 + 1)-dimensional systems, Phys. Lett. A, 1991, V.157, 17–21.
[26] Cheng Y., Li Y.S., The constraint of the Kadomtsev–Petviashvili equation and its special solutions, Phys.
Lett. A, 1991, V.157, 22–26.
[27] Oevel W., Strampp W., Constrained KP hierarchy and bi-Hamiltonian structures, Comm. Math. Phys.,
1993, V.157, 51–81.
[28] Cheng Y., Constraints of the Kadomtsev–Petviashvili hierarchy, J. Math. Phys., 1992, V.33, 3774–3782.
[29] Cheng Y., Modifying the KP, the nth constrained KP hierarchies and their Hamiltonian structures, Comm.
Math. Phys., 1995, V.171, 661–682.
[30] Aratyn H., Ferreira L.A., Gomes J.F., Zimerman A.H., Constrained KP models as integrable matrix hierar-
chies, J. Math. Phys., 1997, V.38, 1559–1576, hep-th/9509096.
[31] Aratyn H., On Grassmannian description of the constrained KP hierarchy, J. Geom. Phys., 1999, V.30,
295–312, solv-int/9805006.
[32] Chau L.L., Shaw J.C., Yen H.C., Solving the KP hierarchy by gauge transformations, Comm. Math. Phys.,
1992, V.149, 263–278.
[33] Oevel W., Rogers C., Gauge transformations and reciprocal links in 2 + 1 dimensions, Rev. Math. Phys.,
1993, V.5, 299–330.
[34] He J.S., Li Y.S., Cheng Y., The determinant representation of the gauge transformation operators, Chinese
Ann. Math. Ser. B, 2002, V.23, 475–486.
[35] He J.S., Li Y.S., Cheng Y., Two choices of the gauge transformation for the AKNS hierarchy through the
constrained KP hierarchy, J. Math. Phys., 2003, V.44, 3928–3960.
[36] Date E., Kashiwara M., Jimbo M., Miwa T., Transformation group for soliton equations, in Bosonization,
Editor M. Stone, Singapore, World Scientific, 1994, 427–507.
[37] Dickey L.A., Soliton equations and Hamiltonian systems, Singapore, World Scientific, 1991.
[38] Oevel W., Strampp W., Wronskian solutions of the constrained Kadomtsev–Petviashvili hierarchy, J. Math.
Phys., 1996, V.37, 6213–6219.
[39] Ohta Y., Satsuma J., Takahashi D., Tokihiro T., An elementary introduction to Sato theory, Progr. Theoret.
Phys. Suppl., 1988, N 94, 210–241.
[40] Ablowitz M.J., Clarkson P.A., Solitons, nonlinear evolution equations and inverse scattering, Cambridge,
Cambridge University Press, 1991.
1 Introduction
2 q-KP hierarchy
3 Gauge transformations of q-KP hierarchy
4 Successive applications of gauge transformations
5 Symmetry constraint of q-KP: q-cKP hierarchy
6 q-Wronskian solutions of the q-cKP hierarchy
7 Example reducing q-KP to q-cKP hierarchy
8 Conclusions and discussions
A More explicit expressions of qnf
B Positive part of Ln (n=1,2,3)
C Proof of the q-Wronskian identity
|