sigma10-018. 241KB Jun 04 2011 12:10:37 AM
Symmetry, Integrability and Geometry: Methods and Applications
SIGMA 6 (2010), 018, 10 pages
The Integrability of New Two-Component
KdV Equation⋆
Ziemowit POPOWICZ
Institute for Theoretical Physics, University of Wroclaw, Wroclaw 50204, Poland
E-mail: ziemek@ift.uni.wroc.pl
URL: http://www.ift.uni.wroc.pl/∼ziemek/
Received October 19, 2009, in final form February 04, 2010; Published online February 12, 2010
doi:10.3842/SIGMA.2010.018
Abstract. We consider the bi-Hamiltonian representation of the two-component coupled KdV equations discovered by Drinfel’d and Sokolov and rediscovered by Sakovich and
Foursov. Connection of this equation with the supersymmetric Kadomtsev–Petviashvilli–
Radul–Manin hierarchy is presented. For this new supersymmetric equation the Lax representation and odd Hamiltonian structure is given.
Key words: KdV equation; Lax representation; integrability; supersymmetry
2010 Mathematics Subject Classification: 35J05; 81Q60
1
Introduction
The scalar KdV equation admits various generalizations to the multifield case and have been
often considered in the literature [1, 2, 3, 4, 5]. However, the present classification of such
systems is not complete and depends on the assumption which we made on the very beginning.
Svinolupov [1] has introduced the class of equations
uit = uixxx + aij,k uj ukx ,
(1)
where i, j, k = 1, 2, . . . , N , ui are functions depending on the variables x and t and aij,k are
constants. The aij,k satisfy the same relations as the structural constants of Jordan algebra
anj,k ajn,r arm,s − aim,r arn,s + cyclic(j, k, m) = 0.
These equations possess infinitely many higher generalized symmetries.
G¨
urses and Karasu [2] extended the Svinolupov construction to the system of equations
ujt = bij uixxx + sij,k uj ukx ,
(2)
where bij , sij,k are constants.
In general, existence of infinitely many conserved quantities is admitted as the definition
of integrability. This implies existence of infinitely many generalized symmetries. G¨
urses and
Karasu, in order to check the integrability of the system of equations (2), assumed that the
system is integrable if it admits a recursion operator. Assuming the general form of the second
and fourth order recursion operator they found the conditions on the coefficients bij , sij,k so that
the equations in (2) are integrable.
⋆
This paper is a contribution to the Proceedings of the XVIIIth International Colloquium on Integrable Systems and Quantum Symmetries (June 18–20, 2009, Prague, Czech Republic). The full collection is available at
http://www.emis.de/journals/SIGMA/ISQS2009.html
2
Z. Popowicz
A quite different generalization of multicomponent KdV system has been found by Antonoodinger operator. Ma [4] also presented
wicz and Fordy [3] considering the energy dependent Schr¨
a multicomponent KdV system considering decomposable hereditary operators.
Several years ago Foursov [5] found the conditions on the coefficients bij , sij,k under which the
two-component system (2) possesses at least 5 generalized symmetries and conserved quantities.
He carried out computer algebra computations and found that there are five such systems which
are not symmetrical and not triangular. Three of them are known to be integrable, while two of
them are new. Foursov conjectured that these two new systems should be integrable. However,
it appeared that one of these new systems is not new and has been known for many years.
Drinfel’d and Sokolov in 1981 [6] presented the Lax pair for one of these new equation and hence
this equation is integrable.
In this paper we present the bi-Hamiltonian formulation and recursion operator for the new
equation. These results have been obtained during the study of the so called supersymmetric
Manin–Radul hierarchy. The application of the supersymmetry to the construction of new
integrable systems appeared almost in parallel to the use of this symmetry in the quantum field
theory. The quantum field theories with exact correspondences between bosonic and fermionic
helicity states are not the only basic ingredients for superstring theories, but have been utilised
both in theoretical and experimental research in particle physics. The first results, concerned the
construction of classical field theories with fermionic and bosonic fields depending on time and
one space variable, can be found in [7, 8, 9, 10]. In many cases, addition of fermion fields does
not guarantee that the final theory becomes supersymmetric invariant. Therefore this method
was named as a fermionic extension in order to distinguish it from the fully supersymmetrical
method which was developed later [11, 12, 13, 14, 15]. There are many recipes how some classical
models could be embedded in fully supersymmetric superspace. The main idea is simple: in
order to get such generalization we should construct a supermultiplet containing the classical
functions. It means that we have to add to a system of k bosonic equations kN fermions and
k(N − 1) bosons (k = 1, 2, . . . , N = 1, 2, . . . ) in such a way that they create superfields. Now
working with this supermultiplet we can step by step apply integrable Hamiltonians methods to
our considerations depending on what we would like to construct.
Manin and Radul in 1985 [10], introduced a new system of equations for an infinite set of
even and odd functions, depending on an even-odd pair of space variables and even-odd times.
This system of equations now called the Manin–Radul supersymmetric Kadomtsev–Petviashvili
hierarchy (MR-SKP). It appeared that this hierarchy contains the supersymmetric generalization
of the Korteweg–de Vries equation, the Sawada–Kotera equation and as we show in this paper
two-component coupled KdV equations discovered by Drinfel’d–Sokolov.
2
Two-component KdV systems
Let us consider a system of two equations
ut = F [u, v],
vt = G[u, v],
where F [u, v] = F (u, v, ux , vx , . . . ) denotes a differential polynomial function of u and v.
By the triangular system we understand such system which involves either an equation depending only on u or an equation depending only on v while by the symmetrical we understand
such system in which G[u, v] = F [v, u].
Definition 1. A system of t-independent evolution equations
ut = Q1 [u, v],
vt = Q2 [u, v]
The Integrability of New Two-Component KdV Equation
3
is said to be a generalized symmetry of (1) if their flows formally commute
DK (Q) − DQ (K) = 0.
Here Q = (Q1 , Q2 ), K[u, v] = (F [u, v], G[u, v]), and DK denotes the Fr´echet derivative.
The first three systems in the Foursov classification are known to be integrable equations
and are
ut = uxxx + 6uux − 12vvx ,
vt = −2vxxx − 6uvx ;
ut = uxxx + 3uux + 3vvx ,
vt = ux v + uvx ;
ut = uxxx + 2vux + uvx ,
vt = uux .
The first pair of equations is the Hirota–Satsuma system [16], second is the Ito system [17],
third is the rescaled Drinfel’d–Sokolov equation [5].
The fourth system of equations is a new one founded by Foursov
ut = uxxx + vxxx + 2vux + 2uvx ,
vt = vxxx − 9uux + 6vux + 3uvx + 2vvx .
Foursov showed that this system possesses generalized symmetries of weights 7, 9, 11, 13, 15, 17
and 19, as well as conserved densities of weights 2, 4, 6, 8, 10, 12 and 14, and conjectured that
this system is integrable and should possess infinitely many generalized symmetries.
The last system in this classification is
ut = 4uxxx + 3vxxx + 4uux + vux + 2uvx ,
vt = 3uxxx + vxxx − 4vux − 2uvx − 2vvx ,
(3)
and has been first considered many years ago by Drinfel’d and Sokolov [6] and rediscovered by
S.Yu. Sakovich [18].
Let us notice that the integrable Hirota–Satsuma equation has the following Lax representation [19]
L = ∂2 + u + v ∂2 + u − v ,
3/4
∂L
= 4 L+ , L ,
∂t
while the integrable Drinfel’d–Sokolov equation possesses the following Lax representation [2, 20]
L = ∂ 3 + (u − v)∂ + (ux − vx )/2 ∂ 3 + (u + v)∂ + (ux + vx )/2 ,
3/4
∂L
= 4 L+ , L .
∂t
On the other side, the Lax operators of the Hirota–Satsuma equation and of the Drinfel’d–
Sokolov equation could be considered as special reduced Lax operators of the fourth and sixth
order respectively. Indeed, the Hirota–Satsuma Lax operator could be rewritten as
L = ∂ 4 + g2 ∂ 2 + g1 ∂ + g0 ,
(4)
where
g2 = 2u,
g1 = 2(ux − vx ),
g0 = uxx + u2 − vxx − v 2 .
4
Z. Popowicz
In this context, one can ask what kind of the equations follows from the fifth-order Lax
operator which is parametrised by two functions of same weight. Let us therefore consider the
following Lax operator
L = ∂ 5 + h 2 ∂ 3 + h3 ∂ 2 + h 4 ∂ + h 5 ,
where hi , i = 2, 3, 4, 5 are polynomials in u and v and their derivatives of the dimension i.
Computing the Lax representation for this operator
∂L
3/5
= 5 L, L+
∂t
we obtained then
L = ∂ 3 + 32 u∂ + 13 ux ∂ 2 − 13 v
(5)
produces the system of equation (3).
Let us notice that the Lax operator (4) is factorized as the product of two Lax operators.
The first one is the Lax operator of the Kaup–Kupershmidt equation while the second is the
Lax operator of the Korteweg–de Vries equation. It is exactly the same Lax operator which has
been found by Drinfel’d and Sokolov [6].
Hence we encounter the situation in which the Lax operator of the Korteweg–de Vries and
the Kaup–Kupershmidt equations can be used for construction of additional equations. This
could be schematically presented as:
˜ KdV
L
˜ KK
L
LKdV
Hirota–Satsuma
equations (3)
LKK
equations (3)
Drinfel’d–Sokolov
˜ Kdv are two different Lax operators of the Korteweg–de Vries equation while LKK
where LKdV , L
˜
and LKK are two different Lax operators of the Kaup–Kupershmidt equation.
3
The recursion operator and bi-Hamiltonian structure
From the knowledge of the Lax operator for evolution equations one can infer a lot of properties
of these equations. The generalized symmetries are obtained by computing the higher flow of
the Lax representation while the conserved charges follow from the trace formula [21] of the Lax
operator.
Using this technique we found first three conserved quantities for the equation (3)
Z
H1 = dx v 2 + 4u2 + 6uv ,
Z
H2 = dx 495u4x u − 510u2x u + 32u4 + 2v 4 + 630v4x u + 180v4x v − 210vxx vu − 210vx2 u
+ 75vx2 v − 525vx ux u + 14v 3 u + 28v 2 u2 − 105vuxx u + 56vu3 ,
Z
H3 = dx 182250u8x u + 769500u4x uxx u + 445500u2xxx u + 259200u2xx u2 + 223425uxx u2x u
− 104400u2x u3 + 1344u6 + 222750v8x u + 70875v8x v − 148500v6x vu − 160875v5x ux u
− 594000v5x vx u − 1113750v5x vxx u − 128250v5x vxx v + 54450v5x v 2 u − 825v5x vu2
2
2
v− 61875vxxx uxxx u− 217800vxxx ux u2 + 267300vxxx vx vu
u− 74250vxxx
− 742500vxxx
2 2
2 2
2
+ 70125vxx
u + 19575vxx
v + 163350vxx
2vu − 193050vxx u2x u + 297000vxx vx2 u
The Integrability of New Two-Component KdV Equation
5
+ 17550vxx vx2 v + 199650vxx vx ux u − 9900vxx v 3 u + 15400vxx vu3 − 32175vx2 uxx u
− 4400vx2 u3 + 3600vx2 v 3 − 19800vx2 v 2 u + 13200vx2 vu2 − 185625vx u5x u
+ 188100vx uxx ux u − 79200vx ux u3 + 825vx v 2 ux u + 57750vx vuxxx u + 21v 6
+ 198v 5 u + 660v 4 u2 − 7425v 3 uxx u + 440v 3 u3 + 44550v 2 u4x u − 11550v 2 u2x u
+ 2640v 2 u4 − 111375vu6x u − 4950vu2xx u − 59400vu2x u2 + 3168vu5 .
Taking into the account a simple form of the first Hamiltonian it is possible to guess the first
Hamiltonian structure
δH
δH
1
1
d
u
3∂ 3 + ∂u + u∂
0
δu
δu
=P
=
.
0
3∂ 3 − 2(∂v + v∂)
dt v
δH1
δH1
δv
δv
In order to define the second Hamiltonian structure we first found the recursion operator. We
used the technique described in [22] and we found the following tenth-order recursion operator
!
18 10
11 10
− 125
∂ + 268 terms − 375
∂ + 268 terms
R=
.
7 10
11 10
∂ + 268 terms − 375
∂ + 268 terms
− 365
Next we assumed that this operator could be factorized as R = J −1 P where J −1 is the inverse
Hamiltonian operator. Due to the diagonal form of the first Hamiltonian structure it is easy to
carry out such procedure and as a result we obtained the second Hamiltonian structure
J
−1
d
dt
u
v
=
δH4
δu
δH4
δv
,
where
H4 =
Z
dx 21u10x u + 26v10x u + 95 terms)
and the explicit form of H4 and J −1 is given in the appendix.
4
The derivation of the Lax representation
The Lax operator of equations (5) has been discovered accidentally during the investigations of
the supersymmetric Manin–Radul hierarchy. This hierarchy can be described by the supersymmetric Lax operator
L = D + f0 +
∞
X
j=1
bj ∂ −j D +
∞
X
fj ∂ −j ,
(6)
j=1
where the coefficients bj , fj are bosonic and fermionic superfield functions, respectively. We shall
∂
+ θ∂. As usual, (x,
use the following notation throughout the paper: ∂ and D = ∂θ
P θ) denotes
N = 1 superspace coordinates. For any super pseudodifferential operator A =
aj/2 Dj the
j
P
aj/2 Dj or its purely pseudo-differential
subscripts ± denote its purely differential part A+ =
j≥0
P
a−j/2 D−j respectively. For any A the super-residuum is defined as Res A = a−1/2 .
part A− =
j≥1
6
Z. Popowicz
The constrained (r, m) supersymmetric Manin–Radul hierarchy [25] is defined by the following Lax operator
L = Dr +
r−1
X
Ψj/2 Dj +
j=0
m
X
j=0
Υ m−j D−1 Ψj/2 .
2
This hierarchy for even r has been widely studied in the literature in contrast to the odd r which
is less known. Further we will consider this hierarchy for odd r = 3, 5 and m = 0, Υ = Ψ = 0.
The Lax operator for r = 3 and m = 0, Υ = Ψ = 0 has been considered recently by Tian
and Liu [24]
L = D3 + Φ,
where Φ is a superfermion function. Let us consider the following tower of equations
k/3
Lt,k = 9 L, L+ .
The first four consistent nontrivial equations are
Φt,2 = Φx ,
Φτ,7 = Φ1,xx + 21 Φ21 + 3ΦΦx x ,
Φt,10 = Φ5x + 5Φxxx Φ1 + 5Φxx Φ1,x + 5Φx Φ21 ,
Φτ,11 = Φ1,5x + 3Φ1,xxx Φ1 + 6Φ1,xx Φ1,x + 2Φ1,x Φ21
− 3Φ4x Φ − 2Φxxx Φx − 6Φxx ΦΦ1 − 6Φx ΦΦ1,x ,
where t is a usual time while τ is an odd time.
The third equation in the hierarchy in the component Φ = ξ + θw reads
ξt = ξ5x + 5wξxxx + 5wx ξxx + 5u2 ξx ,
wt = w5x + 5wwxxx + 5wx wxx + 5w2 wx − 5ξxxx ξx
and it is a supersymmetric generalization of the Sawada–Kotera equation. This equation is a biHamiltonian system with odd supersymmetric Poisson brackets [23]. The proper Hamiltonian
operator which satisfies the Jacobi identity and generates the supersymmetric N = 1 Sawada–
Kotera equation is
Φt,10 = P
where H1 =
R
δH1
,
δΦ
ΦΦx dxdθ and
P = D∂ 2 + 2∂Φ + 2Φ∂ + DΦD ∂ −1 D∂ 2 + 2∂Φ + 2Φ∂ + DΦD .
The implectic operator for this equation was defined in [23] as
JΦt =
δH3
,
δΦ
J = ∂xx + (DΦ) − ∂ −1 (DΦ)x + ∂ −1 Φx D + Φx ∂ −1 D,
where
H3 =
Z
dxdθ Φ7x Φ + 8Φxxx Φ(DΦ)xx + Φx Φ(4(DΦ)4x
+ 20(DΦ)xx (DΦ) + 10(DΦ)2x + 38 (DΦ)3 ) .
(7)
The Integrability of New Two-Component KdV Equation
7
This supersymmetric equation possesses an infinite number of conserved charges [24] which are
generated by the supertrace formula of the Lax operator. However, these charges are not reduced
to the known conserved charges in the bosonic limit. Hence we can not in general conclude that
from the supersymmetric integrability follow the integrability of the bosonic sector.
Let us now consider the Lax operator (6) for r = 5 and m = 0
L = D5 + 31 (∂U + U ∂) − 31 DV D,
where U and V are superfermionic functions U = ξ + θu, V = ψ + θv. The first nontrivial
equations in the hierarchy generated by this Lax operator is given as
6/5
Lt = L+ , L ,
Ut = 4Uxxx + 3Vxxx − 2Ux (DU + DV ) + U (6DUx + 2DVx ) − Vx DV + V (3DUx + DVx ),
Vt = 3Uxxx + Vxxx + 8Ux DU − U (8DUx − 6DVx ) + Vx (4DU + DV ) − V (4DUx + 3DVx ).
The bosonic sector of the latter system where ξ = 0, ψ = 0 gives us the system of two interacted
KdV type equations discovered by Drinfel’d–Sokolov.
Interestingly, the Lax operator equations (7) did not reduce in the bosonic sector to our Lax
operator (5), however, its second power reduces that one can easy verify. As we checked, this
system possesses the same properties as the supersymmetric Sawada–Kotera equation. Namely,
this model, due to the Lax representation, has an infinite number of conserved quantities, which
are not reduced to the usual conserved charges in the bosonic limit. For example, the first two
conserved charges are
Z
H1 = dxdθ(4Ux U + 6Vx V + Vx V ),
Z
H2 = dxdθ 75Uxxx U + 32Ux U (DU ) − 24Ux U (DV ) + 90Vxxx U + 30Vxxx V
+ 36Vx U (DU ) − 6Vx U (DV ) − 4Vx V (DV ) − 30V U (DVx ) .
(8)
We found the following odd Hamiltonian structure for our supersymmetric equation (8)
d
dt
U
V
=
1
30
1
− 10
1
10
2
− 15
!
δH
2
δu
.
δH2
δv
Unfortunately, we have been not able to found second Hamiltonian structure for our superequation.
A
Appendix
The conserved quantity H4 is
2
7312
6638 2
2032 2
3496
584 3
315 u5x uxxx u − 315 u4x u − 945 uxxx u − 315 uxxx uxx ux u − 945 uxx u
3
7
6196 4
448 2 4
8704
26
8
338
2416 2
2025 uxx u − 6075 ux u − 1215 ux u + 3189375 u + 21 v10x u + 21 v10x v − 315 v8x vu
2
1612
832
52
832
1234
208
315 v7x vx u − 63 v6x vxx u + 105 v6x v u − 35 v5x vxxx u + 315 v5x vxxx v + 63 v5x vx vu
2
416
1121 2
10127
494 2
1976
572
1575 v5x vux u − 35 v4x u + 315 v4x v + 315 v4x u4x u + 315 v4x vxx vu + 105 v4x vx u
2
2
3
2 2
6136
572
416
416 2
53 2
754 2
2835 v4x vx ux u − 4725 v4x v u − 14175 v4x v u − 945 vxxx u − 315 vxxx v + 189 vxxx vu
2
23582
416
1144
872
3952
315 vxxx u5x u+ 189 vxxx vxx ux u+ 63 vxxx vxx vx u− 945 vxxx vxx vx v− 4725 vxxx vx v u
H4 = u10x u −
+
−
+
+
+
8
Z. Popowicz
−
−
−
+
+
−
−
+
−
−
+
2
52
104 3
74 3
416 2
2288 2 3
130
567 vxxx vx vu − 945 vxxx vuxxx u + 27 vxx u − 945 vxx v − 945 vxx uxx u + 14175 vxx u
2
2
2
88 2 3
7592 2 2
26 2
2782
3484
2548
2025 vxx v − 14175 vxx v u − 135 vxx vu + 45 vxx u6x u + 135 vxx uxx u − 2025 vxx vx vu
4
3 2
2
2236
104
208
104
494 4
2835 vxx vx uxxx u + 6075 vxx v u + 42525 vxx v u + 405 vxx v uxx u − 2025 vx u
2
832 2 4
26 3
1976 2
1924 2 2
38 2 4
233 4
6075 vx v + 567 vx ux u − 4725 vx uxx u − 1575 vx ux u − 14175 vx u − 6075 vx v
3
52 2 3
208 2 2 2
286 2
3328 2
728
8164
1215 vx v u − 8505 vx v u + 567 vx vuxx u − 42525 vx vu + 45 vx u7x u − 315 vx u4x ux u
3
7
11518
12688
68
3952
10348
945 vx uxxx uxx u − 4725 vx ux u + 4725 vx vu5x u − 14175 vx vuxx ux u − 3189375 v
6
5 2
4 3
2
104
416
104 4
832
1144 3
1664 3
455625 v u − 455625 v u + 8505 v uxx u − 637875 v u − 14175 v u4x u + 42525 v uxx u
2
1664 3 4
416 3 2
338 2
208 2
104 2
14175 v ux u + 637875 v u + 945 v u6x u + 4725 v u4x u + 4725 v uxxx ux u
3
2
1664 2
3328 2 5
10816 2 2 2
247
208
104 2 2
4725 v uxx u + 14175 v uxx u + 42525 v ux u + 455625 v u − 315 vu8x u + 105 vu6x u
3
2
2
10868
14872
4576
29692
102128
945 vu5x ux u − 315 vu4x uxx u + 4725 vu4x u − 945 vuxxx u + 14175 vuxxx ux u
2
2
2
4
2 3
6
65416
123682
1664
4576
3328
14175 vuxx u + 14175 vuxx ux u + 6075 vuxx u + 6075 vux u + 455625 vu .
The inverse Hamiltonian operator J −1 has the following form
!
−1
−1
J
J
1,1
1,2
J −1 =
,
−1 ∗
−1
− J1,2
J2,2
where
−1
3 7
∂ + a1,1,5 ∂ 5 + a1,1,3 ∂ 3 + a1,1,1 ∂ + b1,1 ∂ −1 + b1,1,1 ∂ −1 b1,1,2 − h.c.,
J1,1
= − 125
−1
11 7
∂ +
J1,2
= − 375
5
X
a1,2,i ∂ i + b1,2 ∂ −1 + ∂ −1 c1,2 + b1,2,1 ∂ −1 b1,2,2 ,
i=0
−1
J2,2
=
7 7
− 750
∂
+ a2,2,5 ∂ 5 + a2,2,3 ∂ 3 + a2,2,1 ∂ + b2,2 ∂ −1 + b2,2,1 ∂ −1 b2,2,2 − h.c.,
a1,1,5 = (−43u + 14v)/1125,
a1,1,3 = (225uxx − 448u2 − 690vxx − 77v 2 − 42vu)/16875,
a1,1,1 = − 3150u4x − 1800uxx u + 2397u2x − 888u3 + 720v4x − 2520vxx u + 420vxx v
+ 798vx2 − 1512vx ux + 56v 3 − 126v 2 u − 728vu2 /101250,
b1,1 = − 6750u6x − 9540u4x u − 19080uxxx ux − 14310u2xx − 5520uxx u2 − 5520u2x u − 352u4
2
− 4050v6x − 3780v4x u + 1890v4x v − 810v3x ux + 6480v3x vx + 3510vxx
+ 2160vxx uxx
− 2160vxx u2 − 540vxx v 2 − 1440vxx vu+ 720vx2 u − 810vx2 v+ 5940vx uxxx − 1080vx ux u
− 2160vx vux + 18v 4 + 96v 3 u − 1080v 2 uxx − 288v 2 u2 + 2970vu4x − 1080vuxx u
− 540vu2x − 576vu3 /759375,
b1,1,1 ∂ −1 b1,1,2 = − 990u4x − 1020uxx u − 510u2x − 128u3 − 630v4x − 420vxx u
+ 210vxx v + 210vx2 + 210vx ux − 14v 3 − 56v 2 u + 210vuxx
− 168vu2 ∂ −1 (4u + 3v)/759375,
a1,2,5 = (−38u + 19v)/1125,
a1,2,4 = (−36u + 68v)x /1125,
a1,2,3 = −45uxx − 124u2 + 515vxx − 31v 2 − 36vu /5625,
a1,2,2 = 885u3x − 656ux u + 1680v3x + 178vx u − 354vx v − 332vux )/16875,
a1,2,1 = 2520u4x − 1428uxx u − 1599u2x − 328u3 + 2610v4x − 156vxx u
− 957vxx v − 876vx2 − 1716vx ux + 41v 3 + 114v 2 u − 1371vuxx − 228vu2 /50625,
The Integrability of New Two-Component KdV Equation
9
a1,2,0 = 1575u5x + 534u3x u + 276uxx ux − 128ux u2 + 1080v5x + 198v3x u
− 534v3x v − 873vxx ux − 1206vxx vx − 1338vx uxx + 84vx u2
+ 96vx v 2 + 166vx vu + 128v 2 ux − 792vu3x + 8vux u /50625,
b1,2 = 3375u6x + 4770u4x u + 9540u3x ux + 7155u2xx + 2760uxx u2
+ 2760u2x u + 176u4 + 2025v6x + 1890v4x u − 945v4x v + 405v3x ux − 3240v3x vx
2
− 1080vxx uxx + 1080vxx u2 + 270vxx v 2 + 720vxx vu − 360vx2 u
− 1755vxx
+ 405vx2 v − 2970vx u3x + 540vx ux u + 1080vx vux − 9v 4 − 48v 3 u
+ 540v 2 uxx + 144v 2 u2 − 1485vu4x + 540vuxx u + 270vu2x + 288vu3 /759375,
c1,2 = −4050u6x − 3780u4x u − 14310u3x ux − 9045u2xx − 2160uxx u2 − 3780u2x u − 144u4
2
− 2700v6x − 2160v4x u + 1530v4x v− 4320v3x ux + 3060v3x vx + 2295vxx
− 1080vxx uxx
− 2160vxx u2 − 390vxx v 2 − 540vxx vu − 270vx2 u − 390vx2 v + 1080vx u3x − 4320vx ux u
− 540vx vux + 11v 4 + 72v 3 u − 540v 2 uxx + 144v 2 u2 + 1890vu4x − 1440vuxx u
− 360vu2x − 192vu3 /759375,
b1,2,1 ∂ −1 b1,2,2 =
−990u4x − 1020uxx u − 510u2x − 128u3 − 630v4x − 420vxx u + 210vxx v
+ 210vx2 + 210vx ux + 14v 3 − 56v 2 u + 210vuxx − 168vu2 ∂ −1 (3u + v)
− (4u + 3v)∂ −1 630u4x + 420uxx u + 525u2x + 56u3 + 360v4x − 150vxx v
− 75vx2 + 8v 3 + 42v 2 u − 210vuxx + 56vu2 /759375,
a2,2,5 = (−9u + 7v)/1125,
a2,2,3 = 540uxx − 122u2 − 30vxx − 33v 2 − 28vu /16875,
a2,2,1 = −1170u4x + 900uxx u + 1203u2x − 120u3 + 540v4x − 150vxx v
+ 162vx2 − 108vx ux + 30v 3 + 110v 2 u − 570vuxx + 120vu2 /101250,
b2,2 = 4050u6x + 3780u4x u + 14310u3x ux + 9045u2xx + 2160uxx u2 + 3780u2x u
+ 144u4 + 2700v6x + 2160v4x u − 1530v4x v + 4320v3x ux − 3060v3x vx
2
− 2295vxx
+ 1080vxx uxx + 2160vxx u2 + 390vxx v 2 + 540vxx vu + 270vx2 u + 390vx2 v
− 1080vx u3x + 4320vx ux u + 540vx vux − 11v 4 − 72v 3 u + 540v 2 uxx
− 144v 2 u2 − 1890vu4x + 1440vuxx u + 360vu2x + 192vu3 /1518750,
b2,2,1 ∂ −1 b2,2,2 = −630u4x − 420uxx u − 525u2x − 56u3 − 360v4x
+ 150vxx v + 75vx2 − 8v 3 − 42v 2 u + 210vuxx − 56vu2 ∂ −1 (3u + v)/759375.
References
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407–408.
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335–338.
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262, solv-int/9901005.
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Phys. 8 (2001), 311–312, nlin.SI/0104072.
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SIGMA 6 (2010), 018, 10 pages
The Integrability of New Two-Component
KdV Equation⋆
Ziemowit POPOWICZ
Institute for Theoretical Physics, University of Wroclaw, Wroclaw 50204, Poland
E-mail: ziemek@ift.uni.wroc.pl
URL: http://www.ift.uni.wroc.pl/∼ziemek/
Received October 19, 2009, in final form February 04, 2010; Published online February 12, 2010
doi:10.3842/SIGMA.2010.018
Abstract. We consider the bi-Hamiltonian representation of the two-component coupled KdV equations discovered by Drinfel’d and Sokolov and rediscovered by Sakovich and
Foursov. Connection of this equation with the supersymmetric Kadomtsev–Petviashvilli–
Radul–Manin hierarchy is presented. For this new supersymmetric equation the Lax representation and odd Hamiltonian structure is given.
Key words: KdV equation; Lax representation; integrability; supersymmetry
2010 Mathematics Subject Classification: 35J05; 81Q60
1
Introduction
The scalar KdV equation admits various generalizations to the multifield case and have been
often considered in the literature [1, 2, 3, 4, 5]. However, the present classification of such
systems is not complete and depends on the assumption which we made on the very beginning.
Svinolupov [1] has introduced the class of equations
uit = uixxx + aij,k uj ukx ,
(1)
where i, j, k = 1, 2, . . . , N , ui are functions depending on the variables x and t and aij,k are
constants. The aij,k satisfy the same relations as the structural constants of Jordan algebra
anj,k ajn,r arm,s − aim,r arn,s + cyclic(j, k, m) = 0.
These equations possess infinitely many higher generalized symmetries.
G¨
urses and Karasu [2] extended the Svinolupov construction to the system of equations
ujt = bij uixxx + sij,k uj ukx ,
(2)
where bij , sij,k are constants.
In general, existence of infinitely many conserved quantities is admitted as the definition
of integrability. This implies existence of infinitely many generalized symmetries. G¨
urses and
Karasu, in order to check the integrability of the system of equations (2), assumed that the
system is integrable if it admits a recursion operator. Assuming the general form of the second
and fourth order recursion operator they found the conditions on the coefficients bij , sij,k so that
the equations in (2) are integrable.
⋆
This paper is a contribution to the Proceedings of the XVIIIth International Colloquium on Integrable Systems and Quantum Symmetries (June 18–20, 2009, Prague, Czech Republic). The full collection is available at
http://www.emis.de/journals/SIGMA/ISQS2009.html
2
Z. Popowicz
A quite different generalization of multicomponent KdV system has been found by Antonoodinger operator. Ma [4] also presented
wicz and Fordy [3] considering the energy dependent Schr¨
a multicomponent KdV system considering decomposable hereditary operators.
Several years ago Foursov [5] found the conditions on the coefficients bij , sij,k under which the
two-component system (2) possesses at least 5 generalized symmetries and conserved quantities.
He carried out computer algebra computations and found that there are five such systems which
are not symmetrical and not triangular. Three of them are known to be integrable, while two of
them are new. Foursov conjectured that these two new systems should be integrable. However,
it appeared that one of these new systems is not new and has been known for many years.
Drinfel’d and Sokolov in 1981 [6] presented the Lax pair for one of these new equation and hence
this equation is integrable.
In this paper we present the bi-Hamiltonian formulation and recursion operator for the new
equation. These results have been obtained during the study of the so called supersymmetric
Manin–Radul hierarchy. The application of the supersymmetry to the construction of new
integrable systems appeared almost in parallel to the use of this symmetry in the quantum field
theory. The quantum field theories with exact correspondences between bosonic and fermionic
helicity states are not the only basic ingredients for superstring theories, but have been utilised
both in theoretical and experimental research in particle physics. The first results, concerned the
construction of classical field theories with fermionic and bosonic fields depending on time and
one space variable, can be found in [7, 8, 9, 10]. In many cases, addition of fermion fields does
not guarantee that the final theory becomes supersymmetric invariant. Therefore this method
was named as a fermionic extension in order to distinguish it from the fully supersymmetrical
method which was developed later [11, 12, 13, 14, 15]. There are many recipes how some classical
models could be embedded in fully supersymmetric superspace. The main idea is simple: in
order to get such generalization we should construct a supermultiplet containing the classical
functions. It means that we have to add to a system of k bosonic equations kN fermions and
k(N − 1) bosons (k = 1, 2, . . . , N = 1, 2, . . . ) in such a way that they create superfields. Now
working with this supermultiplet we can step by step apply integrable Hamiltonians methods to
our considerations depending on what we would like to construct.
Manin and Radul in 1985 [10], introduced a new system of equations for an infinite set of
even and odd functions, depending on an even-odd pair of space variables and even-odd times.
This system of equations now called the Manin–Radul supersymmetric Kadomtsev–Petviashvili
hierarchy (MR-SKP). It appeared that this hierarchy contains the supersymmetric generalization
of the Korteweg–de Vries equation, the Sawada–Kotera equation and as we show in this paper
two-component coupled KdV equations discovered by Drinfel’d–Sokolov.
2
Two-component KdV systems
Let us consider a system of two equations
ut = F [u, v],
vt = G[u, v],
where F [u, v] = F (u, v, ux , vx , . . . ) denotes a differential polynomial function of u and v.
By the triangular system we understand such system which involves either an equation depending only on u or an equation depending only on v while by the symmetrical we understand
such system in which G[u, v] = F [v, u].
Definition 1. A system of t-independent evolution equations
ut = Q1 [u, v],
vt = Q2 [u, v]
The Integrability of New Two-Component KdV Equation
3
is said to be a generalized symmetry of (1) if their flows formally commute
DK (Q) − DQ (K) = 0.
Here Q = (Q1 , Q2 ), K[u, v] = (F [u, v], G[u, v]), and DK denotes the Fr´echet derivative.
The first three systems in the Foursov classification are known to be integrable equations
and are
ut = uxxx + 6uux − 12vvx ,
vt = −2vxxx − 6uvx ;
ut = uxxx + 3uux + 3vvx ,
vt = ux v + uvx ;
ut = uxxx + 2vux + uvx ,
vt = uux .
The first pair of equations is the Hirota–Satsuma system [16], second is the Ito system [17],
third is the rescaled Drinfel’d–Sokolov equation [5].
The fourth system of equations is a new one founded by Foursov
ut = uxxx + vxxx + 2vux + 2uvx ,
vt = vxxx − 9uux + 6vux + 3uvx + 2vvx .
Foursov showed that this system possesses generalized symmetries of weights 7, 9, 11, 13, 15, 17
and 19, as well as conserved densities of weights 2, 4, 6, 8, 10, 12 and 14, and conjectured that
this system is integrable and should possess infinitely many generalized symmetries.
The last system in this classification is
ut = 4uxxx + 3vxxx + 4uux + vux + 2uvx ,
vt = 3uxxx + vxxx − 4vux − 2uvx − 2vvx ,
(3)
and has been first considered many years ago by Drinfel’d and Sokolov [6] and rediscovered by
S.Yu. Sakovich [18].
Let us notice that the integrable Hirota–Satsuma equation has the following Lax representation [19]
L = ∂2 + u + v ∂2 + u − v ,
3/4
∂L
= 4 L+ , L ,
∂t
while the integrable Drinfel’d–Sokolov equation possesses the following Lax representation [2, 20]
L = ∂ 3 + (u − v)∂ + (ux − vx )/2 ∂ 3 + (u + v)∂ + (ux + vx )/2 ,
3/4
∂L
= 4 L+ , L .
∂t
On the other side, the Lax operators of the Hirota–Satsuma equation and of the Drinfel’d–
Sokolov equation could be considered as special reduced Lax operators of the fourth and sixth
order respectively. Indeed, the Hirota–Satsuma Lax operator could be rewritten as
L = ∂ 4 + g2 ∂ 2 + g1 ∂ + g0 ,
(4)
where
g2 = 2u,
g1 = 2(ux − vx ),
g0 = uxx + u2 − vxx − v 2 .
4
Z. Popowicz
In this context, one can ask what kind of the equations follows from the fifth-order Lax
operator which is parametrised by two functions of same weight. Let us therefore consider the
following Lax operator
L = ∂ 5 + h 2 ∂ 3 + h3 ∂ 2 + h 4 ∂ + h 5 ,
where hi , i = 2, 3, 4, 5 are polynomials in u and v and their derivatives of the dimension i.
Computing the Lax representation for this operator
∂L
3/5
= 5 L, L+
∂t
we obtained then
L = ∂ 3 + 32 u∂ + 13 ux ∂ 2 − 13 v
(5)
produces the system of equation (3).
Let us notice that the Lax operator (4) is factorized as the product of two Lax operators.
The first one is the Lax operator of the Kaup–Kupershmidt equation while the second is the
Lax operator of the Korteweg–de Vries equation. It is exactly the same Lax operator which has
been found by Drinfel’d and Sokolov [6].
Hence we encounter the situation in which the Lax operator of the Korteweg–de Vries and
the Kaup–Kupershmidt equations can be used for construction of additional equations. This
could be schematically presented as:
˜ KdV
L
˜ KK
L
LKdV
Hirota–Satsuma
equations (3)
LKK
equations (3)
Drinfel’d–Sokolov
˜ Kdv are two different Lax operators of the Korteweg–de Vries equation while LKK
where LKdV , L
˜
and LKK are two different Lax operators of the Kaup–Kupershmidt equation.
3
The recursion operator and bi-Hamiltonian structure
From the knowledge of the Lax operator for evolution equations one can infer a lot of properties
of these equations. The generalized symmetries are obtained by computing the higher flow of
the Lax representation while the conserved charges follow from the trace formula [21] of the Lax
operator.
Using this technique we found first three conserved quantities for the equation (3)
Z
H1 = dx v 2 + 4u2 + 6uv ,
Z
H2 = dx 495u4x u − 510u2x u + 32u4 + 2v 4 + 630v4x u + 180v4x v − 210vxx vu − 210vx2 u
+ 75vx2 v − 525vx ux u + 14v 3 u + 28v 2 u2 − 105vuxx u + 56vu3 ,
Z
H3 = dx 182250u8x u + 769500u4x uxx u + 445500u2xxx u + 259200u2xx u2 + 223425uxx u2x u
− 104400u2x u3 + 1344u6 + 222750v8x u + 70875v8x v − 148500v6x vu − 160875v5x ux u
− 594000v5x vx u − 1113750v5x vxx u − 128250v5x vxx v + 54450v5x v 2 u − 825v5x vu2
2
2
v− 61875vxxx uxxx u− 217800vxxx ux u2 + 267300vxxx vx vu
u− 74250vxxx
− 742500vxxx
2 2
2 2
2
+ 70125vxx
u + 19575vxx
v + 163350vxx
2vu − 193050vxx u2x u + 297000vxx vx2 u
The Integrability of New Two-Component KdV Equation
5
+ 17550vxx vx2 v + 199650vxx vx ux u − 9900vxx v 3 u + 15400vxx vu3 − 32175vx2 uxx u
− 4400vx2 u3 + 3600vx2 v 3 − 19800vx2 v 2 u + 13200vx2 vu2 − 185625vx u5x u
+ 188100vx uxx ux u − 79200vx ux u3 + 825vx v 2 ux u + 57750vx vuxxx u + 21v 6
+ 198v 5 u + 660v 4 u2 − 7425v 3 uxx u + 440v 3 u3 + 44550v 2 u4x u − 11550v 2 u2x u
+ 2640v 2 u4 − 111375vu6x u − 4950vu2xx u − 59400vu2x u2 + 3168vu5 .
Taking into the account a simple form of the first Hamiltonian it is possible to guess the first
Hamiltonian structure
δH
δH
1
1
d
u
3∂ 3 + ∂u + u∂
0
δu
δu
=P
=
.
0
3∂ 3 − 2(∂v + v∂)
dt v
δH1
δH1
δv
δv
In order to define the second Hamiltonian structure we first found the recursion operator. We
used the technique described in [22] and we found the following tenth-order recursion operator
!
18 10
11 10
− 125
∂ + 268 terms − 375
∂ + 268 terms
R=
.
7 10
11 10
∂ + 268 terms − 375
∂ + 268 terms
− 365
Next we assumed that this operator could be factorized as R = J −1 P where J −1 is the inverse
Hamiltonian operator. Due to the diagonal form of the first Hamiltonian structure it is easy to
carry out such procedure and as a result we obtained the second Hamiltonian structure
J
−1
d
dt
u
v
=
δH4
δu
δH4
δv
,
where
H4 =
Z
dx 21u10x u + 26v10x u + 95 terms)
and the explicit form of H4 and J −1 is given in the appendix.
4
The derivation of the Lax representation
The Lax operator of equations (5) has been discovered accidentally during the investigations of
the supersymmetric Manin–Radul hierarchy. This hierarchy can be described by the supersymmetric Lax operator
L = D + f0 +
∞
X
j=1
bj ∂ −j D +
∞
X
fj ∂ −j ,
(6)
j=1
where the coefficients bj , fj are bosonic and fermionic superfield functions, respectively. We shall
∂
+ θ∂. As usual, (x,
use the following notation throughout the paper: ∂ and D = ∂θ
P θ) denotes
N = 1 superspace coordinates. For any super pseudodifferential operator A =
aj/2 Dj the
j
P
aj/2 Dj or its purely pseudo-differential
subscripts ± denote its purely differential part A+ =
j≥0
P
a−j/2 D−j respectively. For any A the super-residuum is defined as Res A = a−1/2 .
part A− =
j≥1
6
Z. Popowicz
The constrained (r, m) supersymmetric Manin–Radul hierarchy [25] is defined by the following Lax operator
L = Dr +
r−1
X
Ψj/2 Dj +
j=0
m
X
j=0
Υ m−j D−1 Ψj/2 .
2
This hierarchy for even r has been widely studied in the literature in contrast to the odd r which
is less known. Further we will consider this hierarchy for odd r = 3, 5 and m = 0, Υ = Ψ = 0.
The Lax operator for r = 3 and m = 0, Υ = Ψ = 0 has been considered recently by Tian
and Liu [24]
L = D3 + Φ,
where Φ is a superfermion function. Let us consider the following tower of equations
k/3
Lt,k = 9 L, L+ .
The first four consistent nontrivial equations are
Φt,2 = Φx ,
Φτ,7 = Φ1,xx + 21 Φ21 + 3ΦΦx x ,
Φt,10 = Φ5x + 5Φxxx Φ1 + 5Φxx Φ1,x + 5Φx Φ21 ,
Φτ,11 = Φ1,5x + 3Φ1,xxx Φ1 + 6Φ1,xx Φ1,x + 2Φ1,x Φ21
− 3Φ4x Φ − 2Φxxx Φx − 6Φxx ΦΦ1 − 6Φx ΦΦ1,x ,
where t is a usual time while τ is an odd time.
The third equation in the hierarchy in the component Φ = ξ + θw reads
ξt = ξ5x + 5wξxxx + 5wx ξxx + 5u2 ξx ,
wt = w5x + 5wwxxx + 5wx wxx + 5w2 wx − 5ξxxx ξx
and it is a supersymmetric generalization of the Sawada–Kotera equation. This equation is a biHamiltonian system with odd supersymmetric Poisson brackets [23]. The proper Hamiltonian
operator which satisfies the Jacobi identity and generates the supersymmetric N = 1 Sawada–
Kotera equation is
Φt,10 = P
where H1 =
R
δH1
,
δΦ
ΦΦx dxdθ and
P = D∂ 2 + 2∂Φ + 2Φ∂ + DΦD ∂ −1 D∂ 2 + 2∂Φ + 2Φ∂ + DΦD .
The implectic operator for this equation was defined in [23] as
JΦt =
δH3
,
δΦ
J = ∂xx + (DΦ) − ∂ −1 (DΦ)x + ∂ −1 Φx D + Φx ∂ −1 D,
where
H3 =
Z
dxdθ Φ7x Φ + 8Φxxx Φ(DΦ)xx + Φx Φ(4(DΦ)4x
+ 20(DΦ)xx (DΦ) + 10(DΦ)2x + 38 (DΦ)3 ) .
(7)
The Integrability of New Two-Component KdV Equation
7
This supersymmetric equation possesses an infinite number of conserved charges [24] which are
generated by the supertrace formula of the Lax operator. However, these charges are not reduced
to the known conserved charges in the bosonic limit. Hence we can not in general conclude that
from the supersymmetric integrability follow the integrability of the bosonic sector.
Let us now consider the Lax operator (6) for r = 5 and m = 0
L = D5 + 31 (∂U + U ∂) − 31 DV D,
where U and V are superfermionic functions U = ξ + θu, V = ψ + θv. The first nontrivial
equations in the hierarchy generated by this Lax operator is given as
6/5
Lt = L+ , L ,
Ut = 4Uxxx + 3Vxxx − 2Ux (DU + DV ) + U (6DUx + 2DVx ) − Vx DV + V (3DUx + DVx ),
Vt = 3Uxxx + Vxxx + 8Ux DU − U (8DUx − 6DVx ) + Vx (4DU + DV ) − V (4DUx + 3DVx ).
The bosonic sector of the latter system where ξ = 0, ψ = 0 gives us the system of two interacted
KdV type equations discovered by Drinfel’d–Sokolov.
Interestingly, the Lax operator equations (7) did not reduce in the bosonic sector to our Lax
operator (5), however, its second power reduces that one can easy verify. As we checked, this
system possesses the same properties as the supersymmetric Sawada–Kotera equation. Namely,
this model, due to the Lax representation, has an infinite number of conserved quantities, which
are not reduced to the usual conserved charges in the bosonic limit. For example, the first two
conserved charges are
Z
H1 = dxdθ(4Ux U + 6Vx V + Vx V ),
Z
H2 = dxdθ 75Uxxx U + 32Ux U (DU ) − 24Ux U (DV ) + 90Vxxx U + 30Vxxx V
+ 36Vx U (DU ) − 6Vx U (DV ) − 4Vx V (DV ) − 30V U (DVx ) .
(8)
We found the following odd Hamiltonian structure for our supersymmetric equation (8)
d
dt
U
V
=
1
30
1
− 10
1
10
2
− 15
!
δH
2
δu
.
δH2
δv
Unfortunately, we have been not able to found second Hamiltonian structure for our superequation.
A
Appendix
The conserved quantity H4 is
2
7312
6638 2
2032 2
3496
584 3
315 u5x uxxx u − 315 u4x u − 945 uxxx u − 315 uxxx uxx ux u − 945 uxx u
3
7
6196 4
448 2 4
8704
26
8
338
2416 2
2025 uxx u − 6075 ux u − 1215 ux u + 3189375 u + 21 v10x u + 21 v10x v − 315 v8x vu
2
1612
832
52
832
1234
208
315 v7x vx u − 63 v6x vxx u + 105 v6x v u − 35 v5x vxxx u + 315 v5x vxxx v + 63 v5x vx vu
2
416
1121 2
10127
494 2
1976
572
1575 v5x vux u − 35 v4x u + 315 v4x v + 315 v4x u4x u + 315 v4x vxx vu + 105 v4x vx u
2
2
3
2 2
6136
572
416
416 2
53 2
754 2
2835 v4x vx ux u − 4725 v4x v u − 14175 v4x v u − 945 vxxx u − 315 vxxx v + 189 vxxx vu
2
23582
416
1144
872
3952
315 vxxx u5x u+ 189 vxxx vxx ux u+ 63 vxxx vxx vx u− 945 vxxx vxx vx v− 4725 vxxx vx v u
H4 = u10x u −
+
−
+
+
+
8
Z. Popowicz
−
−
−
+
+
−
−
+
−
−
+
2
52
104 3
74 3
416 2
2288 2 3
130
567 vxxx vx vu − 945 vxxx vuxxx u + 27 vxx u − 945 vxx v − 945 vxx uxx u + 14175 vxx u
2
2
2
88 2 3
7592 2 2
26 2
2782
3484
2548
2025 vxx v − 14175 vxx v u − 135 vxx vu + 45 vxx u6x u + 135 vxx uxx u − 2025 vxx vx vu
4
3 2
2
2236
104
208
104
494 4
2835 vxx vx uxxx u + 6075 vxx v u + 42525 vxx v u + 405 vxx v uxx u − 2025 vx u
2
832 2 4
26 3
1976 2
1924 2 2
38 2 4
233 4
6075 vx v + 567 vx ux u − 4725 vx uxx u − 1575 vx ux u − 14175 vx u − 6075 vx v
3
52 2 3
208 2 2 2
286 2
3328 2
728
8164
1215 vx v u − 8505 vx v u + 567 vx vuxx u − 42525 vx vu + 45 vx u7x u − 315 vx u4x ux u
3
7
11518
12688
68
3952
10348
945 vx uxxx uxx u − 4725 vx ux u + 4725 vx vu5x u − 14175 vx vuxx ux u − 3189375 v
6
5 2
4 3
2
104
416
104 4
832
1144 3
1664 3
455625 v u − 455625 v u + 8505 v uxx u − 637875 v u − 14175 v u4x u + 42525 v uxx u
2
1664 3 4
416 3 2
338 2
208 2
104 2
14175 v ux u + 637875 v u + 945 v u6x u + 4725 v u4x u + 4725 v uxxx ux u
3
2
1664 2
3328 2 5
10816 2 2 2
247
208
104 2 2
4725 v uxx u + 14175 v uxx u + 42525 v ux u + 455625 v u − 315 vu8x u + 105 vu6x u
3
2
2
10868
14872
4576
29692
102128
945 vu5x ux u − 315 vu4x uxx u + 4725 vu4x u − 945 vuxxx u + 14175 vuxxx ux u
2
2
2
4
2 3
6
65416
123682
1664
4576
3328
14175 vuxx u + 14175 vuxx ux u + 6075 vuxx u + 6075 vux u + 455625 vu .
The inverse Hamiltonian operator J −1 has the following form
!
−1
−1
J
J
1,1
1,2
J −1 =
,
−1 ∗
−1
− J1,2
J2,2
where
−1
3 7
∂ + a1,1,5 ∂ 5 + a1,1,3 ∂ 3 + a1,1,1 ∂ + b1,1 ∂ −1 + b1,1,1 ∂ −1 b1,1,2 − h.c.,
J1,1
= − 125
−1
11 7
∂ +
J1,2
= − 375
5
X
a1,2,i ∂ i + b1,2 ∂ −1 + ∂ −1 c1,2 + b1,2,1 ∂ −1 b1,2,2 ,
i=0
−1
J2,2
=
7 7
− 750
∂
+ a2,2,5 ∂ 5 + a2,2,3 ∂ 3 + a2,2,1 ∂ + b2,2 ∂ −1 + b2,2,1 ∂ −1 b2,2,2 − h.c.,
a1,1,5 = (−43u + 14v)/1125,
a1,1,3 = (225uxx − 448u2 − 690vxx − 77v 2 − 42vu)/16875,
a1,1,1 = − 3150u4x − 1800uxx u + 2397u2x − 888u3 + 720v4x − 2520vxx u + 420vxx v
+ 798vx2 − 1512vx ux + 56v 3 − 126v 2 u − 728vu2 /101250,
b1,1 = − 6750u6x − 9540u4x u − 19080uxxx ux − 14310u2xx − 5520uxx u2 − 5520u2x u − 352u4
2
− 4050v6x − 3780v4x u + 1890v4x v − 810v3x ux + 6480v3x vx + 3510vxx
+ 2160vxx uxx
− 2160vxx u2 − 540vxx v 2 − 1440vxx vu+ 720vx2 u − 810vx2 v+ 5940vx uxxx − 1080vx ux u
− 2160vx vux + 18v 4 + 96v 3 u − 1080v 2 uxx − 288v 2 u2 + 2970vu4x − 1080vuxx u
− 540vu2x − 576vu3 /759375,
b1,1,1 ∂ −1 b1,1,2 = − 990u4x − 1020uxx u − 510u2x − 128u3 − 630v4x − 420vxx u
+ 210vxx v + 210vx2 + 210vx ux − 14v 3 − 56v 2 u + 210vuxx
− 168vu2 ∂ −1 (4u + 3v)/759375,
a1,2,5 = (−38u + 19v)/1125,
a1,2,4 = (−36u + 68v)x /1125,
a1,2,3 = −45uxx − 124u2 + 515vxx − 31v 2 − 36vu /5625,
a1,2,2 = 885u3x − 656ux u + 1680v3x + 178vx u − 354vx v − 332vux )/16875,
a1,2,1 = 2520u4x − 1428uxx u − 1599u2x − 328u3 + 2610v4x − 156vxx u
− 957vxx v − 876vx2 − 1716vx ux + 41v 3 + 114v 2 u − 1371vuxx − 228vu2 /50625,
The Integrability of New Two-Component KdV Equation
9
a1,2,0 = 1575u5x + 534u3x u + 276uxx ux − 128ux u2 + 1080v5x + 198v3x u
− 534v3x v − 873vxx ux − 1206vxx vx − 1338vx uxx + 84vx u2
+ 96vx v 2 + 166vx vu + 128v 2 ux − 792vu3x + 8vux u /50625,
b1,2 = 3375u6x + 4770u4x u + 9540u3x ux + 7155u2xx + 2760uxx u2
+ 2760u2x u + 176u4 + 2025v6x + 1890v4x u − 945v4x v + 405v3x ux − 3240v3x vx
2
− 1080vxx uxx + 1080vxx u2 + 270vxx v 2 + 720vxx vu − 360vx2 u
− 1755vxx
+ 405vx2 v − 2970vx u3x + 540vx ux u + 1080vx vux − 9v 4 − 48v 3 u
+ 540v 2 uxx + 144v 2 u2 − 1485vu4x + 540vuxx u + 270vu2x + 288vu3 /759375,
c1,2 = −4050u6x − 3780u4x u − 14310u3x ux − 9045u2xx − 2160uxx u2 − 3780u2x u − 144u4
2
− 2700v6x − 2160v4x u + 1530v4x v− 4320v3x ux + 3060v3x vx + 2295vxx
− 1080vxx uxx
− 2160vxx u2 − 390vxx v 2 − 540vxx vu − 270vx2 u − 390vx2 v + 1080vx u3x − 4320vx ux u
− 540vx vux + 11v 4 + 72v 3 u − 540v 2 uxx + 144v 2 u2 + 1890vu4x − 1440vuxx u
− 360vu2x − 192vu3 /759375,
b1,2,1 ∂ −1 b1,2,2 =
−990u4x − 1020uxx u − 510u2x − 128u3 − 630v4x − 420vxx u + 210vxx v
+ 210vx2 + 210vx ux + 14v 3 − 56v 2 u + 210vuxx − 168vu2 ∂ −1 (3u + v)
− (4u + 3v)∂ −1 630u4x + 420uxx u + 525u2x + 56u3 + 360v4x − 150vxx v
− 75vx2 + 8v 3 + 42v 2 u − 210vuxx + 56vu2 /759375,
a2,2,5 = (−9u + 7v)/1125,
a2,2,3 = 540uxx − 122u2 − 30vxx − 33v 2 − 28vu /16875,
a2,2,1 = −1170u4x + 900uxx u + 1203u2x − 120u3 + 540v4x − 150vxx v
+ 162vx2 − 108vx ux + 30v 3 + 110v 2 u − 570vuxx + 120vu2 /101250,
b2,2 = 4050u6x + 3780u4x u + 14310u3x ux + 9045u2xx + 2160uxx u2 + 3780u2x u
+ 144u4 + 2700v6x + 2160v4x u − 1530v4x v + 4320v3x ux − 3060v3x vx
2
− 2295vxx
+ 1080vxx uxx + 2160vxx u2 + 390vxx v 2 + 540vxx vu + 270vx2 u + 390vx2 v
− 1080vx u3x + 4320vx ux u + 540vx vux − 11v 4 − 72v 3 u + 540v 2 uxx
− 144v 2 u2 − 1890vu4x + 1440vuxx u + 360vu2x + 192vu3 /1518750,
b2,2,1 ∂ −1 b2,2,2 = −630u4x − 420uxx u − 525u2x − 56u3 − 360v4x
+ 150vxx v + 75vx2 − 8v 3 − 42v 2 u + 210vuxx − 56vu2 ∂ −1 (3u + v)/759375.
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