Self-Consistent Sources and Conservation Laws for a Super ......2. A Super Soliton Hierarchy with...
Transcript of Self-Consistent Sources and Conservation Laws for a Super ......2. A Super Soliton Hierarchy with...
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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013, Article ID 913758, 6 pageshttp://dx.doi.org/10.1155/2013/913758
Research ArticleSelf-Consistent Sources and Conservation Laws fora Super Broer-Kaup-Kupershmidt Equation Hierarchy
Hanyu Wei1,2 and Tiecheng Xia2
1 Department of Mathematics and Information Science, Zhoukou Normal University, Zhoukou 466001, China2Department of Mathematics, Shanghai University, Shanghai 200444, China
Correspondence should be addressed to Hanyu Wei; [email protected]
Received 14 February 2013; Accepted 2 June 2013
Academic Editor: Yongkun Li
Copyright Β© 2013 H. Wei and T. Xia. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Based on the matrix Lie superalgebras and supertrace identity, the integrable super Broer-Kaup-Kupershmidt hierarchy withself-consistent sources is established. Furthermore, we establish the infinitely many conservation laws for the integrable superBroer-Kaup-Kupershmidt hierarchy. In the process of computation especially, Fermi variables also play an important role in superintegrable systems.
1. Introduction
Soliton theory has achieved great success during the lastdecades; it is being applied to mathematics, physics, biology,astrophysics, and other potential fields [1β12]. The diversityand complexity of soliton theory enable investigators to doresearch from different views, such as Hamiltonian structure,self-consistent sources, conservation laws, and various solu-tions of soliton equations.
In recent years, with the development of integrable sys-tems, super integrable systems have attractedmuch attention.Many scholars and experts do research on the topic andget lots of results. For example, in [13], Ma et al. gavethe supertrace identity based on Lie super algebras andits application to super AKNS hierarchy and super Dirachierarchy, and to get their super Hamiltonian structures, Hugave an approach to generate superextensions of integrablesystems [14]. Afterwards, super Boussinesq hierarchy [15]and super NLS-mKdV hierarchy [16] as well as their superHamiltonian structures are presented. The binary nonlin-earization of the super classical Boussinesq hierarchy [17], theBargmann symmetry constraint, and binary nonlinearizationof the super Dirac systems were given [18].
Soliton equation with self-consistent sources is an impor-tant part in soliton theory. They are usually used to describeinteractions between different solitary waves, and they are
also relevant to some problems related to hydrodynamics,solid state physics, plasma physics, and so forth. Some resultshave been obtained by some authors [19β21]. Very recently,self-consistent sources for super CKdV equation hierarchy[22] and super G-J hierarchy are presented [23].
The conservation laws play an important role in dis-cussing the integrability for soliton hierarchy. An infinitenumber of conservation laws for KdV equation were firstdiscovered by Miura et al. in 1968 [24], and then lots ofmethods have been developed to find them. This may bemainly due to the contributions of Wadati and others [25β27]. Conservation laws also play an important role in math-ematics and engineering as well. Many papers dealing withsymmetries and conservation laws were presented.The directconstruction method of multipliers for the conservation lawswas presented [28].
In this paper, starting from a Lie super algebra, isospectralproblems are designed. With the help of variational identity,Yang got super Broer-Kaup-Kupershmidt hierarchy and itsHamiltonian structure [29].Then, based on the theory of self-consistent sources, the self-consistent sources of super Broer-Kaup-Kupershmidt hierarchy are obtained by us. Further-more, we present the conservation laws for the super Broer-Kaup-Kupershmidt hierarchy. In the calculation process,extended Fermi quantities π’
1and π’
2play an important role;
namely, π’1and π’
2satisfy π’2
1= π’2
2= 0 and π’
1π’2
= βπ’2π’1
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in the whole paper. Furthermore, the operation betweenextended Fermi variables satisfies Grassmann algebra condi-tions.
2. A Super Soliton Hierarchy withSelf-Consistent Sources
Based on a Lie superalgebra πΊ,
π1= (
1 0 0
0 β1 0
0 0 0
) , π2= (
0 1 0
0 0 0
0 0 0
) ,
π3= (
0 0 0
1 0 0
0 0 0
) , π4= (
0 0 1
0 0 0
0 β1 0
) , π5= (
0 0 0
0 0 1
1 0 0
)
(1)
that is along with the communicative operation [π1, π2] =
2π2, [π1, π3] = β2π
3, [π2, π3] = π
1, [π1, π4] = [π
2, π5] =
π4, [π1, π5] = [π
4, π3] = βπ
5, [π4, π5]+= π1, [π4, π4]+= β2π
2,
and [π5, π5]+= 2π3.
We consider an auxiliary linear problem
(
π1
π2
π3
)
π₯
= π (π’, π)(
π1
π2
π3
) , π (π’, π) = π 1+
5
β
π=1
π’πππ(π) ,
(
π1
π2
π3
)
π‘π
= ππ(π’, π)(
π1
π2
π3
) ,
(2)
where π’ = (π’1, . . . , π’
π )π, ππ= π 1+π’1π1+β β β +π’
5π5, π’π(π, π‘) =
π’π(π = 1, 2, . . . , 5), π
π= π(π₯, π‘) are field variables defining
π₯ β π , π‘ β π ; ππ= ππ(π) β π π(3) and π
1is a pseudoregular
element.The compatibility of (2) gives rise to the well-known zero
curvature equation as follows:
πππ‘
β πππ₯
+ [ππ, ππ] = 0, π = 1, 2, . . . . (3)
If an equation
π’π‘= πΎ (π’) (4)
can be worked out through (3), we call (4) a super evolutionequation. If there is a super Hamiltonian operator π½ and afunctionπ»
πsuch that
π’π‘= πΎ (π’) = π½
πΏπ»π+1
πΏπ’
, (5)
where
πΏπ»π
πΏπ’
= πΏ
πΏπ»πβ1
πΏπ’
= β β β = πΏπ πΏπ»0
πΏπ’
,
π = 1, 2, . . . ,
πΏ
πΏπ’
= (
πΏ
πΏπ’1
, . . . ,
πΏ
πΏπ’5
)
π
,
(6)
then (4) possesses a superHamiltonian equation. If so, we cansay that (4) has a super Hamiltonian structure.
According to (2), now we consider a new auxiliary linearproblem. Forπ distinct π
π, π = 1, 2, . . . , π, the systems of (2)
become as follows:
(
π1π
π2π
π3π
)
π₯
= π (π’, ππ)(
π1π
π2π
π3π
) =
5
β
π=1
π’πππ(π)(
π1π
π2π
π3π
) ,
(
π1π
π2π
π3π
)
π‘π
= ππ(π’, ππ)(
π1
π2
π3
)
= [
π
β
π=0
ππ(π’) ππβπ
π+ Ξπ(π’, ππ)](
π1
π2
π3
) .
(7)
Based on the result in [30], we can show that the followingequation:
πΏπ»π
πΏπ’
+
π
β
π=1
πΌπ
πΏππ
πΏπ’
= 0 (8)
holds true, where πΌπare constants. Equation (8) determines a
finite dimensional invariant set for the flows in (6).From (7), we may know that
πΏππ
πΏπ’π
=
1
3
π tr(Ξ¨π
ππ (π’, ππ)
ππ’π
)
=
1
3
π tr (Ξ¨πππππ) , π = 1, 2, . . . 5,
(9)
where π tr denotes the trace of a matrix and
Ξ¨π= (
π1ππ2π
βπ2
1ππ1ππ3π
π2
2πβπ1ππ2π
π2ππ3π
π2ππ3π
βπ1ππ3π
0
) . (10)
From (8) and (9), a kind of super Hamiltonian solitonequation hierarchy with self-consistent sources is presentedas follows:
π’ππ‘
= π½
πΏπ»π+1
πΏπ’π
+ π½
π
β
π=1
πΌπ
πΏππ
πΏπ’
= π½πΏπ πΏπ»1
πΏπ’π
+ π½
π
β
π=1
πΌπ
πΏππ
πΏπ’
, π = 1, 2, . . . .
(11)
3. The Super Broer-Kaup-KupershmidtHierarchy with Self-Consistent Sources
The super Broer-Kaup-Kupershmidt spectral problem asso-ciated with the Lie super algebra is given in [29]:
ππ₯= ππ, π
π‘= ππ, (12)
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where
π = (
π + π π π’1
1 βπ β π π’2
π’2
βπ’1
0
) , π = (
π΄ π΅ π
πΆ βπ΄ π
π βπ 0
) , (13)
and π΄ = βπβ₯0
π΄ππβπ
, π΅ = βπβ₯0
π΅ππβπ
, πΆ =
βπβ₯0
πΆππβπ
, π = βπβ₯0
πππβπ, and π = β
πβ₯0πππβπ. As
π’1and π’
2are Fermi variables, they constitute Grassmann
algebra. So, we have π’1π’2= βπ’2π’1, π’2
1= π’2
2= 0.
Starting from the stationary zero curvature equationππ₯= [π,π] , (14)
we haveπ΄ππ₯
= π πΆπ+ π’1ππβ π΅π+ π’2ππ,
π΅ππ₯
= 2π΅π+1
+ 2ππ΅πβ 2π π΄
πβ 2π’1ππ,
πΆππ₯
= β2πΆπ+1
β 2ππΆπ+ 2π΄π+ 2π’2ππ,
πππ₯
= ππ+1
+ πππ+ π ππβ π’1π΄πβ π’2π΅π,
πππ₯
= βππ+1
β πππ+ ππβ π’1πΆπ+ π’2π΄π,
π΅0= πΆ0= π0= π0= 0,
π΄0= 1, π΅
1= π , πΆ
1= π,
π1= π’1, π
1= π’2, π΄
1= 0, . . . .
(15)
Then we consider the auxiliary spectral problem
ππ‘π
= π(π)
π = (πππ)+π, (16)
where
π(π)
=
π
β
π=0
(
π΄π
π΅π
ππ
πΆπ
βπ΄π
ππ
ππ
βππ
0
)ππβπ
, (17)
considering
π(π)
= π(π)
++ Ξπ, Ξπ= βπΆπ+1
π1. (18)
Substituting (18) into the zero curvature equation
ππ‘π
β π(π)
π₯+ [π,π
(π)] = 0, (19)
we get the super Broer-Kaup-Kupershmidt hierarchy
π’π‘π
= (
π
π
π’1
π’2
)
π‘
=(
(
(
0 π 0 0
π 0 π’1
βπ’2
0 π’1
0 β
1
2
0 βπ’2
β
1
2
0
)
)
)
(
β2π΄π+1
βπΆπ+1
2ππ+1
β2ππ+1
)
= π½(
β2π΄π+1
βπΆπ+1
2ππ+1
β2ππ+1
) = π½ππ+1
,
(20)
where
ππ+1= πΏππ,
πΏ=((
(
1
2π β πβ1
ππ βπ β πβ1
π π πβ1
π’1π +1
2π’1πβ1
π’2π β1
2π’2
1
2β1
2π β π
1
2π’2
0
βπ’2
2π’1
βπ β π β1
π’1β π’2π 2π π’
2π +1
2π’1π’2
π β π
))
)
.
(21)
According to super trace identity on Lie super algebras, adirect calculation reads as
πΏπ»π
πΏπ’
= (β2π΄π+1
, βπΆπ+1
, 2ππ+1
, β2ππ+1
)π
,
π»π= β«
2π΄π+2
π + 1
ππ₯, π β₯ 0.
(22)
When we take π = 2, the hierarchy (20) can be reduced tosuper nonlinear integrable couplings equations
ππ‘2
= β
1
2
ππ₯π₯
+
1
2
π π₯β 2πππ₯+ (π’1π’2)π₯+ (π’1π’2π₯)π₯,
π π‘2
=
1
2
π π₯π₯
β 2(ππ )π₯+ 2π’1π’1π₯
+ 2π π’2π’2π₯,
π’1π‘2
= π’1π₯π₯
β
3
2
ππ₯π’1+
1
2
π π₯π’2β 2ππ’
1π₯
+ (π + π’1π’2) π’2π₯,
π’2π‘2
= βπ’2π₯π₯
β
1
2
ππ₯π’2β 2ππ’
2π₯β π’1π₯.
(23)
Next, we will construct the super Broer-Kaup-Kuper-shmidt hierarchy with self-consistent sources. Consider thelinear system
(
π1π
π2π
π3π
)
π₯
= π(
π1π
π2π
π3π
) ,
(
π1π
π2π
π3π
)
π‘
= π(
π1π
π2π
π3π
) .
(24)
From (8), for the system (12), we set
πΏπ»π
πΏπ’
=
π
β
π=1
πΏππ
πΏπ’
(25)
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and obtain the following πΏππ/πΏπ’:
π
β
π=1
πΏππ
πΏπ’
=
π
β
π=1
(
(
(
(
(
(
(
(
(
(
π tr(Ξ¨π
πΏπ
πΏπ
)
π tr(Ξ¨π
πΏπ
πΏπ
)
π tr(Ξ¨π
πΏπ
πΏπΌ
)
π tr(Ξ¨π
πΏπ
πΏπ½
)
)
)
)
)
)
)
)
)
)
)
=
(
(
(
(
2β¨Ξ¦1, Ξ¦2β©
β¨Ξ¦2, Ξ¦2β©
β2 β¨Ξ¦2, Ξ¦3β©
2 β¨Ξ¦1, Ξ¦3β©
)
)
)
)
,
(26)
where Ξ¦π= (ππ1, . . . , π
ππ)π, π = 1, 2, 3.
According to (11), the integrable super Broer-Kaup-Kupershmidt hierarchy with self-consistent sources is pro-posed as follows:
π’π‘π
= (
π
π
π’1π‘
π’2π‘
)
π‘π
= π½(
β2π΄π+1
βπΆπ+1
2ππ+1
β2ππ+1
) + π½
(
(
(
(
2β¨Ξ¦1, Ξ¦2β©
β¨Ξ¦2, Ξ¦2β©
β2 β¨Ξ¦2, Ξ¦3β©
2 β¨Ξ¦1, Ξ¦3β©
)
)
)
)
,
(27)
where Ξ¦π= (ππ1, . . . , π
ππ)π, π = 1, 2, 3, satisfy
π1ππ₯
= (π + π) π1π
+ π π2π
+ π’1π3π,
π2ππ₯
= π1π
β (π + π) π2π
+ π’2π3π,
π3ππ₯
= π’2π1π
β π’1π2π,
π = 1, . . . , π.
(28)
For π = 2, we obtain the super Broer-Kaup-Kupershmidtequation with self-consistent sources as follows:
ππ‘2
= β
1
2
ππ₯π₯
+
1
2
π π₯β 2πππ₯+ (π’1π’2)π₯+ (π’1π’2π₯)π₯+ π
π
β
π=1
π2
2π,
π π‘2
=
1
2
π π₯π₯
β 2(ππ )π₯+ 2π’1π’1π₯
+ 2π π’2π’2π₯
+ 2π
π
β
π=1
π1ππ2π
β 2π’1
π
β
π=1
π2ππ3π
β 2π’2
π
β
π=1
π1ππ3π,
π’1π‘2
= π’1π₯π₯
β
3
2
ππ₯π’1+
1
2
π π₯π’2β 2ππ’
1π₯
+ (π + π’1π’2) π’2π₯
+ π’1
π
β
π=1
π2
2πβ
π
β
π=1
π1ππ3π,
π’2π‘2
= βπ’2π₯π₯
β
1
2
ππ₯π’2β 2ππ’
2π₯β π’1π₯
β π’2
π
β
π=1
π2
2π+
π
β
π=1
π2ππ3π,
(29)
where Ξ¦π= (ππ1, . . . , π
ππ)π, π = 1, 2, 3, satisfy
π1ππ₯
= (π + π) π1π
+ π π2π
+ π’1π3π,
π2ππ₯
= π1π
β (π + π) π2π
+ π’2π3π,
π3ππ₯
= π’2π1π
β π’1π2π,
π = 1, . . . , π.
(30)
4. Conservation Laws for the SuperBroer-Kaup-Kupershmidt Hierarchy
In the following, we will construct conservation laws of thesuper Broer-Kaup-Kupershmidt hierarchy. We introduce thevariables
πΈ =
π2
π1
, πΎ =
π3
π1
. (31)
From (7) and (12), we have
πΈπ₯= 1 β 2ππΈ β 2ππΈ + π’
2πΎ β π πΈ
2β π’1πΈπΎ,
πΎπ₯= π’2β ππΎ β π’
1πΈ β ππΎ β π πΎπΈ β π’
1πΎ2.
(32)
Expand πΈ, πΎ in the power of π as follows:
πΈ =
β
β
π=1
πππβπ, πΎ =
β
β
π=1
πππβπ. (33)
Substituting (33) into (32) and comparing the coefficients ofthe same power of π, we obtain
π1=
1
2
, π1= π’2, π
2= β
1
2
π,
π2= βπ’2π₯
β
1
2
π’1β ππ’2,
π3=
1
4
ππ₯β
1
2
π’2π’2π₯
+
1
2
π2β
1
2
π’1π’2β
1
8
π ,
π3= π’2π₯π₯
+ ππ₯π’2+ 2ππ’
2π₯+
1
2
π’1π₯
+ ππ’1+ π2π’2β
1
2
π π’2, . . .
(34)
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Journal of Applied Mathematics 5
and a recursion formula for ππand ππ
ππ+1
= β
1
2
ππ,π₯
β πππ+
1
2
π’2ππβ
1
2
π
πβ1
β
π=1
ππππβπ
β
1
2
π’1
πβ1
β
π=1
ππππβπ
,
ππ+1
= βππ,π₯
β π’1ππβ πππβ π
πβ1
β
π=1
ππππβπ
β π’1
πβ1
β
π=1
ππππβπ
,
(35)
because of
π
ππ‘
[π + π + π πΈ + π’1πΎ] =
π
ππ₯
[π΄ + π΅πΈ + ππΎ] , (36)
where
π΄ = π0π2+ π1π +
1
2
π0π β π0π’1π’2,
π΅ = π0π π +
1
2
π0π π₯β π0ππ + π
1π ,
π = π0π’1π + π
0π’1π₯
β π0ππ’1+ π1π’1.
(37)
Assume that πΏ = π + π + π πΈ + π’1πΎ, π = π΄ + π΅πΈ + ππΎ.
Then (36) can be written as πΏπ‘= ππ₯, which is the right form of
conservation laws. We expand πΏ and π as series in powers ofπ with the coefficients, which are called conserved densitiesand currents, respectively,
πΏ = π + π +
β
β
π=1
πΏππβπ,
π = π0π2+ π1π + π
0π +
β
β
π=1
πππβπ,
(38)
where π0, π1are constants of integration. The first two
conserved densities and currents are read as follows:
πΏ1=
1
2
π + π’1π’2,
π1= π0(
1
4
π π₯β π π β π’
1π’2π₯
+ π’2π’1π₯
β 2ππ’1π’2)
+ π1(
1
2
π + π’1π’2) ,
πΏ2= β
1
2
π π β π’1π’2π₯
β ππ’1π’2,
π2= π0(
1
4
π ππ₯β
1
2
π π’2π’2π₯
+ π π2β π π’1π’2β
1
8
π 2
β
1
4
ππ π₯+ π’1π’2π₯π₯
+ ππ₯π’1π’2+ 3ππ’
1π’2π₯
+2π2π’1π’2β π’1π₯π’2π₯
β ππ’1π₯π’2)
β π1(
1
2
π π + π’1π’2π₯
+ ππ’1π’2) .
(39)
The recursion relation for πΏπand ππare
πΏπ= π ππ+ π’1ππ,
ππ= π0(π π+1
+
1
2
π π₯ππβ ππ ππ+ π’1ππ+1
+ π’1π₯ππβ ππ’1ππ)
+ π1(π ππ+ π’1ππ) ,
(40)
where ππand π
πcan be calculated from (35). The infinitely
many conservation laws of (20) can be easily obtained from(32)β(40), respectively.
5. Conclusions
Starting from Lie super algebras, we may get super equa-tion hierarchy. With the help of variational identity, theHamiltonian structure can also be presented. Based on Liesuper algebra, the self-consistent sources of super Broer-Kaup-Kupershmidt hierarchy can be obtained. It enriched thecontent of self-consistent sources of super soliton hierarchy.Finally, we also get the conservation laws of the superBroer-Kaup-Kupershmidt hierarchy. It is worth to note thatthe coupling terms of super integrable hierarchies involvefermi variables; they satisfy the Grassmann algebra which isdifferent from the ordinary one.
Acknowledgments
This project is supported by the National Natural ScienceFoundation of China (Grant nos. 11271008, 61072147, and11071159), the First-Class Discipline of Universities in Shang-hai, and the Shanghai University Leading Academic Disci-pline Project (Grant no. A13-0101-12-004).
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