Post on 18-Feb-2021
Research ArticleA Integrable Generalized Super-NLS-mKdV Hierarchy, ItsSelf-Consistent Sources, and Conservation Laws
HanyuWei 1 and Tiecheng Xia 2
1College of Mathematics and Statistics, Zhoukou Normal University, Zhoukou 466001, China2Department of Mathematics, Shanghai University, Shanghai 200444, China
Correspondence should be addressed to Hanyu Wei; weihanyu8207@163.com
Received 30 November 2017; Accepted 4 February 2018; Published 5 March 2018
Academic Editor: Zhijun Qiao
Copyright Β© 2018 HanyuWei and Tiecheng Xia.This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in anymedium, provided the originalwork is properly cited.
A generalized super-NLS-mKdV hierarchy is proposed related to Lie superalgebra π΅(0, 1); the resulting supersoliton hierarchy isput into super bi-Hamiltonian form with the aid of supertrace identity. Then, the super-NLS-mKdV hierarchy with self-consistentsources is set up. Finally, the infinitely many conservation laws of integrable super-NLS-mKdV hierarchy are presented.
1. Introduction
The superintegrable systems have aroused strong interest inrecent years; many experts and scholars do research on thefield and obtain lots of results [1, 2]. In [3], the supertraceidentity and the proof of its constant πΎ are given by Ma etal. As an application, super-Dirac hierarchy and super-AKNShierarchy and its super-Hamiltonian structures have beenfurnished. Then, like the super-C-KdV hierarchy, the super-Tu hierarchy, the multicomponent super-Yang hierarchy, andso on were proposed [4β9]. In [10], the binary nonlineariza-tion and Bargmann symmetry constraints of the super-Dirachierarchy were given.
Soliton equations with self-consistent sources haveimportant applications in soliton theory. They are often usedto describe interactions between different solitary waves, andthey can provide variety of dynamics of physical models;some important results have been got by some scholars [11β18]. Conservation laws play an important role in mathemati-cal physics. Since Miura et al. discovery of conservation lawsfor KdV equation in 1968 [19], lots of methods have beenpresented to find them [20β23].
In this work, a generalized super-NLS-mKdVhierarchy isconstructed.Then, we present the super bi-Hamiltonian formfor the generalized super-NLS-mKdVhierarchy with the helpof the supertrace identity. In Section 3, we consider thegeneralized super-NLS-mKdV hierarchy with self-consistentsources based on the theory of self-consistent sources. Finally,
the conservation laws of the generalized super-NLS-mKdVhierarchy are given.
2. The Generalized Superequation Hierarchy
Based on the Lie superalgebra π΅(0, 1)π1 = (1 0 00 β1 00 0 0) ,π2 = ( 0 1 0β1 0 00 0 0) ,π3 = (0 1 01 0 00 0 0) ,π4 = (0 0 10 0 00 β1 0) ,π5 = (0 0 00 0 11 0 0) ;
(1)
HindawiAdvances in Mathematical PhysicsVolume 2018, Article ID 1396794, 9 pageshttps://doi.org/10.1155/2018/1396794
http://orcid.org/0000-0002-2668-9550http://orcid.org/0000-0001-8235-0319https://doi.org/10.1155/2018/1396794
2 Advances in Mathematical Physics
that is, along with the communicative operation[π1, π2] = 2π3,[π1, π3] = 2π2,[π1, π4] = [π2, π5] = [π3, π5] = π4,[π4, π2] = [π5, π1] = [π3, π4] = π5,[π2, π3] = 2π1,[π4, π4]+ = βπ2 β π3,[π4, π5]+ = π1,[π5, π5]+ = π3 β π2.(2)
To set up the generalized super-NLS-mKdV hierarchy,the spectral problem is given as follows:ππ₯ = ππ,ππ‘ = ππ, (3)with
π = ( π + π€ π’1 + π’2 π’3π’1 β π’2 βπ β π€ π’4π’4 βπ’3 0 ) ,π = ( π΄ π΅ + πΆ ππ΅ β πΆ βπ΄ ππ βπ 0) ,
(4)
where π€ = π((1/2)π’21 β (1/2)π’22 + π’3π’4) with π being anarbitrary even constant, π is the spectral parameter, π’1 andπ’2 are even potentials, and π’3 and π’4 are odd potentials. Notethat spectral problem (3) with π = 0 is reduced to super-NLS-mKdV hierarchy case [24].
Setting π΄ = βπβ₯0
πππβπ,π΅ = βπβ₯0
πππβπ,πΆ = βπβ₯0
πππβπ,π = βπβ₯0
πππβπ,π = βπβ₯0
πππβπ,(5)
solving the equation ππ₯ = [π,π], we haveπππ₯ = 2π’2ππ β 2π’1ππ + π’3ππ + π’4ππ,πππ₯ = 2ππ+1 β 2π’2ππ β π’3ππ + π’4ππ + 2π€ππ,πππ₯ = 2ππ+1 β 2π’1ππ β π’3ππ β π’4ππ + 2π€ππ,πππ₯ = ππ+1 + π€ππ + (π’1 + π’2) ππ β π’3ππ β π’4ππβ π’4ππ,πππ₯ = βππ+1 β π€ππ + π’1ππ β π’2ππ β π’3ππ + π’3ππ+ π’4ππ,(6)
and, from the above recursion relationship, we can get therecursion operator πΏ which meets the following:
( ππ+1βππ+1ππ+1βππ+1)= πΏ(ππβππππβππ), π β©Ύ 0, (7)
where the recursive operator πΏ is given as follows:πΏ =((βπ€ + 2π’1πβ1π’2 β12π + 2π’1πβ1π’1 12π’4 + π’1πβ1π’3 β12π’3 β π’1πβ1π’4β12π β 2π’2πβ1π’2 βπ€ + 2π’2πβ1π’1 12π’4 β π’2πβ1π’3 12π’3 + π’2πβ1π’4βπ’3 + 2π’4πβ1π’2 βπ’3 + 2π’4πβ1π’1 βπ β π€ + π’4πβ1π’3 βπ’1 + π’2 β π’4πβ1π’4βπ’4 β 2π’3πβ1π’2 π’4 β 2π’3πβ1π’1 π’1 + π’2 β π’3πβ1π’3 π β π€ + π’3πβ1π’4
)). (8)
Choosing the initial dataπ0 = 1,π0 = π0 = π0 = π0 = 0, (9)from the recursion relations in (6), we can obtainπ1 = 0,π1 = π’1,
π1 = π’2,π1 = π’3,π1 = π’4,π2 = β12π’21 + 12π’22 β π’3π’4,π2 = 12π’2π₯ β π€π’1,
Advances in Mathematical Physics 3
π2 = 12π’1π₯ β π€π’2,π2 = π’3π₯ β π€π’3,π2 = βπ’4π₯ β π€π’4,π3 = 12 (π’2π’1π₯ β π’1π’2π₯) + π’3π’4π₯ + π’4π’3π₯+ π€ (π’21 β π’22) + 2π’3π’4,π3 = 14π’1π₯π₯ β 12π€π₯π’2 + 12π’1π’22 β 12π’31 + 12π’3π’3π₯β 12π’4π’4π₯ β π’1π’3π’4 β π€π’2π₯ + π€2π’1,π3 = 14π’2π₯π₯ β 12π€π₯π’1 + 12π’32 β 12π’2π’21 + 12π’3π’3π₯+ 12π’4π’4π₯ β π’2π’3π’4 β π€π’1π₯ + π€2π’2,π3 = π’3π₯π₯ + 12π’3π’22 β 12π’3π’21 + 12π’4π’1π₯ + 12π’4π’2π₯+ π’1π’4π₯ + π’2π’4π₯ β π€π₯π’3 + π€2π’3 β 2π€π’3π₯,π3 = π’4π₯π₯ + 12π’4π’22 β 12π’4π’21 + 12π’3π’1π₯ β 12π’3π’2π₯+ π’1π’3π₯ β π’2π’3π₯ + π€π₯π’4 + π€2π’4 + 2π€π’4π₯.(10)
Then we consider the auxiliary spectral problemππ‘π = π(π)π, (11)with
π(π) = πβπ=0
( ππ ππ + ππ ππππ β ππ βππ ππππ βππ 0 )ππβπ + Ξ π,π β₯ 0. (12)Suppose
Ξ π = ( π π + π ππ β π βπ ππ βπ 0) , (13)substituting π(π) into the zero curvature equationππ‘π β π(π)π₯ + [π,π(π)] = 0, (14)
where π β₯ 0. Making use of (6), we haveπ€π‘π = ππ₯,π = π = π = π = 0,π’1π‘π = πππ₯ β 2π€ππ + 2π’2ππ + π’3ππ β π’4ππ + ππ₯ β 2π€π+ π’3π β π’4π + 2π’2π = 2ππ+1 + 2π’2π,π’2π‘π = πππ₯ β 2π€ππ + 2π’1π + π’3ππ + π’4ππ + ππ₯ + π’3π+ π’4π + 2π’1π β 2π€π = 2ππ+1 + 2π’1π,π’3π‘π = πππ₯ β π€ππ β π’1ππ β π’2ππ + π’3ππ + π’4ππ + π’4ππ+ ππ₯ β π€π β π’1π β π’2π + π’3π + π’4π + π’4π= ππ+1 + π’3π,π’4π‘π = πππ₯ + π€ππ β π’1ππ + π’2ππ + π’3ππ β π’3ππ β π’4ππβ π’4π + ππ₯ + π€π β π’1π + π’2π + π’3π β π’3π= βππ+1 β π’4π,
(15)
which guarantees that the following identity holds true:(12π’22 β 12π’21 + π’4π’3)π‘π= 2π’2ππ+1 β 2π’1ππ+1 β ππ+1π’3 + π’4ππ+1 = ππ+1π₯. (16)Choosing π = βπππ+1, we arrive at the following generalizedsuper-NLS-mKdV hierarchy:
π’π‘π = (π’1π’2π’3π’4)π‘π =(2ππ+1 β 2ππ’2ππ+12ππ+1 β 2ππ’1ππ+1ππ+1 β ππ’3ππ+1βππ+1 + ππ’4ππ+1), (17)
where π β₯ 0.The case of (17) with π = 0 is exactly the standardsupersoliton hierarchy [24].
When π = 1 in (17), the flow is trivial. Taking π = 2,we can obtain second-order generalized super-NLS-mKdVequationsπ’1π‘2 = 12π’2π₯π₯ + π’32 β π’2π’21 + π’3π’3π₯ + π’4π’4π₯ β 2π’2π’3π’4+ π (2π’1π’2π’2π₯ β 2π’21π’1π₯ + π’4π₯π’3π’1 β π’3π₯π’4π’1β 2π’3π’4π’1π₯ β 2π’2π’3π’4π₯ β 2π’2π’4π’3π₯ β 2π’2π’21π’3π’4+ 2π’32π’3π’4) + 2π2π’2 (12π’21 β 12π’22 + π’3π’4)2+ 2π2π’2 (12π’21 β 12π’22 + π’3π’4) (π’22 β π’21) ,π’2π‘2 = 12π’1π₯π₯ β π’31 + π’1π’22 β 2π’1π’3π’4 + π’3π’3π₯ β π’4π’4π₯+ π (2π’22π’2π₯ β 2π’1π’2π’1π₯ β π’3π₯π’4π’2 + π’4π₯π’3π’2
4 Advances in Mathematical Physicsβ 2π’3π’4π’2π₯ β 2π’1π’3π’4π₯ β 2π’1π’4π’3π₯)β 2π2π’1 (12π’21 β 12π’22 + π’3π’4) (π’21 β π’22)+ 2π2 (12π’21 β 12π’22 + π’3π’4)2 π’1 β 2π2π’1π’3 (π’21β π’22) π’4,π’3π‘2 = π’3π₯π₯ + 12π’3π’22 β 12π’3π’21 + 12π’4π’1π₯ + 12π’4π’2π₯+ π’1π’4π₯ + π’2π’4π₯ + π (12π’3π’1π’2π₯ β 12π’3π’2π’1π₯+ π’22π’3π₯ β π’21π’3π₯ + π’2π’2π₯π’3 β π’1π’1π₯π’3β 2π’3π’4π’3π₯) + π2 (12π’21 β 12π’22 + π’3π’4)2 π’3β π2π’3 (12π’21 β 12π’22 + π’3π’4) (π’21 β π’22) ,π’4π‘2 = βπ’4π₯π₯ β 12π’4π’22 + 12π’4π’21 + 12π’3π’2π₯ β 12π’3π’1π₯β π’1π’3π₯ + π’2π’3π₯ + π (12π’4π’2π’1π₯ β 12π’4π’1π’2π₯β π’1π’1π₯π’4 + π’2π’2π₯π’4 β π’21π’4π₯ + π’22π’4π₯β 2π’3π’4π’4π₯) β π2 (12π’21 β 12π’22 + π’3π’4)2 π’4β π2π’4 (12π’21 β 12π’22 + π’3π’4) (π’21 β π’22) .(18)
From (3) and (11), one infers the following:
π(2) = (π(2)11 π(2)12 π(2)13π(2)21 βπ(2)11 π(2)23π(2)23 βπ(2)13 0 ) , (19)with π(2)11 = π2 + 12π’22 β 12π’21 β π’3π’4β π (12π’2π’1π₯ β 12π’1π’2π₯ + π’4π’3π₯ + π’3π’4π₯)β 2π2 (12π’21 β 12π’22 + π’3π’4)2 ,π(2)12 = (π’1 + π’2) π + 12 (π’1 + π’2)π₯β π (12π’21 β 12π’22 + π’3π’4) (π’1 + π’2) ,
π(2)13 = π’3π + π’3π₯ β π (12π’21 β 12π’22 + π’3π’4) π’3,π(2)21 = (π’1 β π’2) π + 12 (π’2 β π’1)π₯+ π (12π’21 β 12π’22 + π’3π’4) (π’2 β π’1) ,π(2)23 = π’4π β π’4π₯ β π (12π’21 β 12π’22 + π’3π’4) π’4.(20)
In what follows, we shall set up super-Hamiltonianstructures for the generalized super-NLS-mKdV hierarchy.
Through calculations, we obtain
Str(πππππ ) = 2π΄,Str( ππππ’1π) = 2 (π΅ + ππ’1π΄) ,Str( ππππ’2π) = β2 (πΆ + ππ’2π΄) ,Str( ππππ’3π) = 2 (π + ππ’4π΄) ,Str( ππππ’4π) = β2 (π + ππ’3π΄) .
(21)
Substituting the above results into the supertrace identity[3] and balancing the coefficients of ππ+2, we obtainπΏπΏπ’ β« ππ+2ππ₯ = (πΎ β π β 1)(
ππ+1 + ππ’1ππ+1βππ+1 β ππ’2ππ+1ππ+1 + ππ’4ππ+1βππ+1 β ππ’3ππ+1),π β₯ 0.(22)
Thus, we haveπ»π+1 = β«β ππ+2π + 1ππ₯,πΏπ»π+1πΏπ’ =(
ππ+1 + ππ’1ππ+1βππ+1 β ππ’2ππ+1ππ+1 + ππ’4ππ+1βππ+1 β ππ’3ππ+1),π β©Ύ 0.(23)
Moreover, it is easy to find that
( ππ+1βππ+1ππ+1βππ+1)= π 1(ππ+1 + ππ’1ππ+1βππ+1 β ππ’2ππ+1ππ+1 + ππ’4ππ+1βππ+1 β ππ’3ππ+1), π β©Ύ 0, (24)
where π 1 is given by
Advances in Mathematical Physics 5
π 1 =(1 β 2ππ’1πβ1π’2 β2ππ’1πβ1π’1 ππ’1πβ1π’3 ππ’1πβ1π’42ππ’2πβ1π’2 1 + 2ππ’2πβ1π’1 ππ’2πβ1π’3 βππ’2πβ1π’4β2ππ’4πβ1π’2 β2ππ’4πβ1π’1 1 β ππ’4πβ1π’3 ππ’4πβ1π’42ππ’3πβ1π’2 2ππ’3πβ1π’1 ππ’3πβ1π’3 1 β ππ’3πβ1π’4). (25)
Therefore, superintegrable hierarchy (17) possesses thefollowing form:
π’π‘π = π 2( ππ+1βππ+1ππ+1βππ+1)= π 2π 1(ππ+1 + ππ’1ππ+1βππ+1 β ππ’2ππ+1ππ+1 + ππ’4ππ+1βππ+1 β ππ’3ππ+1)= π½πΏπ»π+1πΏπ’ , π β₯ 0,
(26)
where
π 2 =( β4ππ’2πβ1π’2 β2 β 4ππ’2πβ1π’1 β2ππ’2πβ1π’3 2ππ’2πβ1π’42 β 4ππ’1πβ1π’2 β4ππ’1πβ1π’1 β2ππ’1πβ1π’3 2ππ’1πβ1π’4β4ππ’3πβ1π’2 β4ππ’3πβ1π’1 β2ππ’3πβ1π’3 β1 + 2ππ’3πβ1π’42ππ’4πβ1π’2 2ππ’4πβ1π’1 β1 + ππ’4πβ1π’3 βππ’4πβ1π’4 ), (27)
and super-Hamiltonian operator π½ is given byπ½ = π 2π 1 =( β8ππ’2π
β1π’2 β2 β 8ππ’2πβ1π’1 β4ππ’2πβ1π’3 4ππ’2πβ1π’42 β 8ππ’1πβ1π’2 β8ππ’1πβ1π’1 β4ππ’1πβ1π’3 4ππ’1πβ1π’4β6ππ’3πβ1π’2 β6ππ’3πβ1π’1 β3ππ’3πβ1π’3 β1 + 3ππ’3πβ1π’44ππ’4πβ1π’2 4ππ’4πβ1π’1 β1 + 2ππ’4πβ1π’3 β2ππ’4πβ1π’4 ). (28)In addition, generalized super-NLS-mKdV hierarchy (17)
also possesses the following super-Hamiltonian form:
π’π‘π = π 2πΏ( ππβππππβππ)= π 2πΏπ 1(ππ + ππ’1ππβππ β ππ’2ππππ + ππ’4ππβππ β ππ’3ππ)=ππΏπ»ππΏπ’ , π β₯ 0,
(29)
where π = π 2πΏπ 1 = (πππ)4Γ4 is the second super-Hamiltonian operator.
3. Self-Consistent Sources
Consider the linear system
(π1ππ2ππ3π)π₯ = π(π1ππ2ππ3π),
(π1ππ2ππ3π)π‘ = π(π1ππ2ππ3π).
(30)
From the result in [25], we setπΏπππΏπ’π = 13Str(Ξ¨π ππ (π’, ππ)πΏπ’π ) , π = 1, 2, . . . , 4, (31)
6 Advances in Mathematical Physics
with Str on behalf of the supertrace, and
Ξ¨π = (π1ππ2π βπ21π π1ππ3ππ22π βπ1ππ2π π2ππ3ππ2ππ3π βπ1ππ3π 0 ) , π = 1, 2, . . . , π. (32)From system (30), we get πΏππ/πΏπ’ as follows:πβπ=1
πΏπππΏπ’π = πβπ=1((((((
Str(Ξ¨π πΏππΏπ’1)Str(Ξ¨π πΏππΏπ’2)Str(Ξ¨π πΏππΏπ’3)Str(Ξ¨π πΏππΏπ’4)
))))))
=( 2ππ’1 β¨Ξ¦1, Ξ¦2β© β β¨Ξ¦1, Ξ¦1β© + β¨Ξ¦2, Ξ¦2β©β2ππ’2 β¨Ξ¦1, Ξ¦2β© + β¨Ξ¦1, Ξ¦1β© + β¨Ξ¦2, Ξ¦2β©2ππ’4 β¨Ξ¦1, Ξ¦2β© β 2 β¨Ξ¦2, Ξ¦3β©2ππ’3 β¨Ξ¦1, Ξ¦2β© β 2 β¨Ξ¦1, Ξ¦3β© ) ,(33)
where Ξ¦π = (ππ1, . . . , πππ)π, π = 1, 2, 3.So, we obtain the self-consistent sources of generalized
super-NLS-mKdV hierarchy (17):
π’π‘π =(π’1π’2π’3π’4)π‘π = π½πΏπ»π+1πΏπ’π + π½ πβπ=1πΏπππΏπ’π
= π½( ππ+1 + ππ’1ππ+1βππ+1 β ππ’2ππ+1ππ+1 + ππ’4ππ+1βππ+1 β ππ’3ππ+1)+ π½( 2ππ’1 β¨Ξ¦1, Ξ¦2β© β β¨Ξ¦1, Ξ¦1β© + β¨Ξ¦2, Ξ¦2β©β2ππ’2 β¨Ξ¦1, Ξ¦2β© + β¨Ξ¦1, Ξ¦1β© + β¨Ξ¦2, Ξ¦2β©2ππ’4 β¨Ξ¦1, Ξ¦2β© β 2 β¨Ξ¦2, Ξ¦3β©2ππ’3 β¨Ξ¦1, Ξ¦2β© β 2 β¨Ξ¦1, Ξ¦3β© ) .
(34)
For π = 2, we get supersoliton equation with self-consistent sources as follows:π’1π‘2 = 12π’2π₯π₯ + π’32 β π’2π’21 + π’3π’3π₯ + π’4π’4π₯ β 2π’2π’3π’4+ π (2π’1π’2π’2π₯ β 2π’21π’1π₯ + π’4π₯π’3π’1 β π’3π₯π’4π’1β 2π’3π’4π’1π₯ β 2π’2π’3π’4π₯ β 2π’2π’4π’3π₯ β 2π’2π’21π’3π’4+ 2π’32π’3π’4) + 2π2π’2 (12π’21 β 12π’22 + π’3π’4)2
+ 2π2π’2 (12π’21 β 12π’22 + π’3π’4) (π’22 β π’21)+ 2ππ’1 πβπ=1
π1ππ2π β πβπ=1
(π21π β π22π) ,π’2π‘2 = 12π’1π₯π₯ β π’31 + π’1π’22 β 2π’1π’3π’4 + π’3π’3π₯ β π’4π’4π₯+ π (2π’22π’2π₯ β 2π’1π’2π’1π₯ β π’3π₯π’4π’2 + π’4π₯π’3π’2β 2π’3π’4π’2π₯ β 2π’1π’3π’4π₯ β 2π’1π’4π’3π₯)β 2π2π’1 (12π’21 β 12π’22 + π’3π’4) (π’21 β π’22)+ 2π2 (12π’21 β 12π’22 + π’3π’4)2 π’1 β 2π2π’1π’3 (π’21β π’22) π’4 β 2ππ’2 πβ
π=1
π1ππ2π + πβπ=1
(π21π + π22π) ,π’3π‘2 = π’3π₯π₯ + 12π’3π’22 β 12π’3π’21 + 12π’4π’1π₯ + 12π’4π’2π₯+ π’1π’4π₯ + π’2π’4π₯ + π (12π’3π’1π’2π₯ β 12π’3π’2π’1π₯+ π’22π’3π₯ β π’21π’3π₯ + π’2π’2π₯π’3 β π’1π’1π₯π’3β 2π’3π’4π’3π₯) + π2 (12π’21 β 12π’22 + π’3π’4)2 π’3β π2π’3 (12π’21 β 12π’22 + π’3π’4) (π’21 β π’22)+ 2ππ’4 πβ
π=1
π1ππ2π β πβπ=1
π2ππ3π,π’4π‘2 = βπ’4π₯π₯ β 12π’4π’22 + 12π’4π’21 + 12π’3π’2π₯ β 12π’3π’1π₯β π’1π’3π₯ + π’2π’3π₯ + π (12π’4π’2π’1π₯ β 12π’4π’1π’2π₯β π’1π’1π₯π’4 + π’2π’2π₯π’4 β π’21π’4π₯ + π’22π’4π₯β 2π’3π’4π’4π₯) β π2 (12π’21 β 12π’22 + π’3π’4)2 π’4β π2π’4 (12π’21 β 12π’22 + π’3π’4) (π’21 β π’22)+ 2ππ’3 πβπ=1
π1ππ2π β 2 πβπ=1
π1ππ3π,π1ππ₯ = (π + π€) π1π + (π’1 + π’2) π2π + π’3π3π,π2ππ₯ = (π’1 β π’2) π1π β (π + π€) π2π + π’4π3π,π3ππ₯ = π’4π1π β π’3π2π, π = 1, . . . , π.(35)
Advances in Mathematical Physics 7
4. Conservation Laws
In the following, we shall derive the conservation laws ofsupersoliton hierarchy. Introducing the variablesπΎ = π2π1 ,πΊ = π3π1 , (36)then we obtainπΎπ₯ = π’1 β π’2 β 2 (π + π€)πΎ + π’4πΊ β (π’1 + π’2)πΎ2β π’3πΎπΊ,πΊπ₯ = π’4 β π’3πΎ β (π + π€)πΊ β (π’1 + π’2) πΊπΎ β π’3πΊ2. (37)
Next, we expand πΎ and πΊ as series of the spectralparameter π
πΎ = ββπ=1
ππππ,πΊ = ββπ=1
ππππ. (38)Substituting (38) into (37), we raise the recursion formulas forππ and ππ:ππ+1 = β12πππ₯ β π€ππ + 12π’4ππ β 12 (π’1 + π’2) πβ1β
π=1
ππππβπβ 12π’3πβ1β
π=1
ππππβπ,ππ+1 = βπππ₯ β π’3ππ β π€ππ β (π’1 + π’2) πβ1β
π=1
ππππβπβ π’3πβ1βπ=1
ππππβπ, π β₯ 2.(39)
We write the first few terms of ππ and ππ:π1 = 12 (π’1 β π’2) ,π1 = π’4,π2 = β14 (π’1 β π’2) π₯β 12π (π’1 β π’2) (12π’21 β 12π’22 + π’3π’4) ,π2 = βπ’4π₯ β 12 (π’1 β π’2) (π’3 + ππ’1 + ππ’2) ,
π3 = 18 (π’1 β π’2)π₯π₯ β 18 (π’1 + π’2) (π’1 β π’2)2+ 14π€π₯ (π’1 β π’2) + 12 (π’1 β π’2)π₯ π€+ 12 (π’1 β π’2) π€2 β 12π’4π’4π₯,π3 = π’4π₯π₯ + 12 (π’1 β π’2) π’3π₯ β 12 (π’21 β π’22) π’4+ 34 (π’1 β π’2)π₯ π’3 + π€π₯π’4 + 2π€π’4π₯+ π€ (π’1 β π’2) π’3 + π€2π’4, . . . .(40)
Note that πππ‘ [π + π€ + (π’1 + π’2)πΎ + π’3πΊ]= πππ₯ [π΄ + (π΅ + πΆ)πΎ + ππΊ] , (41)setting πΏ = π+π€+ (π’1 +π’2)πΎ+π’3πΊ, π = π΄+ (π΅+πΆ)πΎ+ππΊ,which admitted that the conservation laws is πΏπ‘ = ππ₯. For (19),one infers thatπ΄ = π2 + 12π’22 β 12π’21 β π’3π’4,π΅ = π’1π + 12π’2π₯ β ππ’1 (12π’21 β 12π’22 + π’3π’4) ,πΆ = π’2π + 12π’1π₯ β ππ’2 (12π’21 β 12π’22 + π’3π’4) ,π = π’3π + π’3π₯ β 12ππ’3 (π’21 β π’22) .
(42)
Expanding πΏ and π asπΏ = π + π€ + ββ
π=1
πΏππβπ,π = π2 + 12π’22 β 12π’21 β π’3π’4 + ββ
π=1
πππβπ, (43)conserved densities and currents are the coefficients πΏπ, ππ,respectively. The first two conserved densities and currentsare read:πΏ1 = 12 (π’21 β π’22) + π’3π’4,π1 = β14 (π’1 + π’2) (π’1 β π’2) π₯ + 14 (π’1 β π’2) (π’1 + π’2)β π₯ β 14π (π’21 β π’22) (12π’21 β 12π’22 + π’3π’4) β 12 (π’21
8 Advances in Mathematical Physics
β π’22) (12π’21 β 12π’22 + π’3π’4) β 12π (π’21 β π’22) π’3 β 12β π (π’21 β π’22) π’3π’4 β π’3π’4π₯ + π’3π₯π’4,πΏ2 = β14 (π’1 + π’2) (π’2 β π’1)π₯ β 12π (π’21 β π’22) (12π’21β 12π’22 + π’3 + π’3π’4) β π’3π’4π₯,π2 = (π’1 + π’2) π3 β 14 [(π’1 + π’2)π₯β π (π’1 + π’2) (12π’21 β 12π’22 + π’3π’4)]β [12 (π’1 β π’2) π₯ + (π’1 β π’2)Γ (12π’21 β 12π’22 + π’3π’4)] + π’3π3 β [π’3π₯β 12π (π’21 β π’22) π’3] [π’4π₯+ 12 (π’1 β π’2) (π’3 + ππ’1 + ππ’2)] ,(44)
where π3 and π3 are given by (40).The recursion relationshipfor πΏπ and ππ is as follows:πΏπ = (π’1 + π’2) ππ + π’3ππ,ππ = (π’1 + π’2) ππ+1 + 12 [(π’1 + π’2)π₯β π (π’1 + π’2) (12π’21 β 12π’22 + π’3π’4)] ππ + π’3ππ+1+ [π’3π₯ β 12π (π’21 β π’22) π’3] ππ,
(45)
where ππ and ππ can be recursively calculated from (39). Wecan display the first two conservation laws of (18) asπΏ1π‘ = π1π₯,πΏ2π‘ = π2π₯, (46)where πΏ1, π1, πΏ2, and π2 are defined in (44). Then we canobtain the infinitely many conservation laws of (17) from(37)β(46).
5. Conclusions
In this work, we construct the generalized super-NLS-mKdV hierarchy with bi-Hamiltonian forms with the help ofvariational identity. Self-consistent sources and conservationlaws are also set up. In [26β29], the nonlinearization ofAKNS hierarchy and binary nonlinearization of super-AKNShierarchy were given. Can we do the binary nonlinearizationfor hierarchy (17)? The question may be investigated infurther work.
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper.
Acknowledgments
The project is supported by the National Natural ScienceFoundation of China (Grant nos. 11547175, 11271008, and11601055) and the Aid Project for the Mainstay Young Teach-ers in Henan Provincial Institutions of Higher Education ofChina (Grant no. 2017GGJS145).
References
[1] A. Roy Chowdhury and S. Roy, βOn the Backlund transforma-tion and Hamiltonian properties of superevaluation equations,βJournal of Mathematical Physics, vol. 27, no. 10, pp. 2464β2468,1986.
[2] X.-B. Hu, βAn approach to generate superextensions of inte-grable systems,β Journal of Physics A:Mathematical andGeneral,vol. 30, no. 2, pp. 619β632, 1997.
[3] W.-X. Ma, J.-S. He, and Z.-Y. Qin, βA supertrace identityand its applications to superintegrable systems,β Journal ofMathematical Physics, vol. 49, no. 3, Article ID 033511, 2008.
[4] Y. S. Li and L. N. Zhang, βHamiltonian structure of the superevolution equation,β Journal ofMathematical Physics, vol. 31, no.2, pp. 470β475, 1990.
[5] S.-X. Tao and T.-C. Xia, βLie algebra and lie super algebrafor integrable couplings of C-KdV hierarchy,β Chinese PhysicsLetters, vol. 27, no. 4, Article ID 040202, 2010.
[6] X.-Z. Wang and X.-K. Liu, βTwo types of Lie super-algebra forthe super-integrable Tu-hierarchy and its super-Hamiltonianstructure,βCommunications in Nonlinear Science andNumericalSimulation, vol. 15, no. 8, pp. 2044β2049, 2010.
[7] F. Yu and L. Li, βA new matrix Lie algebra, the multicompo-nent Yang hierarchy and its super-integrable coupling system,βApplied Mathematics and Computation, vol. 207, no. 2, pp. 380β387, 2009.
[8] H.-X. Yang and Y.-P. Sun, βHamiltonian and super-hamiltonianextensions related to Broer-Kaup-Kupershmidt system,β Inter-national Journal of Theoretical Physics, vol. 49, no. 2, pp. 349β364, 2010.
[9] S. Tao and T. Xia, βTwo super-integrable hierarchies and theirsuper-Hamiltonian structures,β Communications in NonlinearScience and Numerical Simulation, vol. 16, no. 1, pp. 127β132,2011.
[10] J. Yu, J. He, W. Ma, and Y. Cheng, βThe Bargmann symmetryconstraint and binary nonlinearization of the super Diracsystems,β Chinese Annals of Mathematics, Series B, vol. 31, no.3, pp. 361β372, 2010.
[11] V. K. Melβnikov, βIntegration of the nonlinear Schrodingerequation with a source,β Inverse Problems, vol. 8, no. 1, articleno. 009, pp. 133β147, 1992.
[12] V. S. Shchesnovich and E. V. Doktorov, βModified Manakovsystem with self-consistent source,β Physics Letters A, vol. 213,no. 1-2, pp. 23β31, 1996.
[13] T. C. Xia, βTwo new integrable couplings of the soliton hierar-chies with self-consistent sources,βChinese Physics B, vol. 19, no.10, Article ID 100303, 2010.
Advances in Mathematical Physics 9
[14] H.-Y. Wei and T.-C. Xia, βA new six-component super solitonhierarchy and its self-consistent sources and conservation laws,βChinese Physics B, vol. 25, no. 1, Article ID 010201, 2015.
[15] Y. Fa-Jun, βThemulti-function jaulent-miodek equation hierar-chy with self-consistent sources,βChinese Physics Letters, vol. 27,no. 5, Article ID 050201, 2010.
[16] F.-J. Yu, βConservation laws and self-consistent sources for asuper-classical- boussinesq hierarchy,β Chinese Physics Letters,vol. 28, no. 12, Article ID 120201, 2011.
[17] H. Wang and T. Xia, βSuper Jaulent-Miodek hierarchy and itssuper Hamiltonian structure, conservation laws and its self-consistent sources,β Frontiers of Mathematics in China, vol. 9,no. 6, pp. 1367β1379, 2014.
[18] J. Yu, W.-X. Ma, J. Han, and S. Chen, βAn integrable generaliza-tion of the super AKNS hierarchy and its bi-Hamiltonian for-mulation,βCommunications in Nonlinear Science andNumericalSimulation, vol. 43, pp. 151β157, 2017.
[19] R. M. Miura, C. S. Gardner, and M. . Kruskal, βKorteweg-deVries equation and generalizations. II. Existence of conserva-tion laws and constants of motion,β Journal of MathematicalPhysics, vol. 9, pp. 1204β1209, 1968.
[20] M. Wadati, H. Sanuki, and K. Konno, βRelationships amonginverse method, BaΜcklund transformation and an infinite num-ber of conservation laws,β Progress of Theoretical and Experi-mental Physics, vol. 53, no. 2, pp. 419β436, 1975.
[21] H.Wei, T. Xia, and G. He, βA New Soliton Hierarchy Associatedwith so (3, R) and Its Conservation Laws,β MathematicalProblems in Engineering, vol. 2016, Article ID 3682579, 2016.
[22] N. H. Ibragimov and T. Kolsrud, βLagrangian approach to evo-lution equations: Symmetries and conservation laws,βNonlinearDynamics, vol. 36, no. 1, pp. 29β40, 2004.
[23] A. H. Kara and F.M.Mahomed, βRelationship between symme-tries and conservation laws,β International Journal ofTheoreticalPhysics, vol. 39, no. 1, pp. 23β40, 2000.
[24] Q.-L. Zhao, Y.-X. Li, X.-Y. Li, and Y.-P. Sun, βThe finite-dimensional super integrable system of a super NLS-mKdVequation,β Communications in Nonlinear Science and NumericalSimulation, vol. 17, no. 11, pp. 4044β4052, 2012.
[25] G. Z. Tu, βAn extension of a theorem on gradients of conserveddensities of integrable systems,β Northeastern MathematicalJournal, vol. 6, no. 1, pp. 28β32, 1990.
[26] C. W. Cao, βNonlinearization of the Lax system for AKNShierarchy,β Science China Mathematics, vol. 33, no. 5, pp. 528β536, 1990.
[27] C. Cao, Y.Wu, and X. Geng, βRelation between the Kadomtsev-Petviashvili equation and the confocal involutive system,β Jour-nal of Mathematical Physics, vol. 40, no. 8, pp. 3948β3970, 1999.
[28] Z. J. Qiao, βA Bargmann system and the involutive represen-tation of solutions of the Levi hierarchy,β Journal of Physics A:Mathematical and General, vol. 26, no. 17, pp. 4407β4417, 1993.
[29] J. He, J. Yu, Y. Cheng, and R. Zhou, βBinary nonlinearization ofthe super AKNS system,β Modern Physics Letters B, vol. 22, no.4, pp. 275β288, 2008.
Hindawiwww.hindawi.com Volume 2018
MathematicsJournal of
Hindawiwww.hindawi.com Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwww.hindawi.com Volume 2018
Probability and StatisticsHindawiwww.hindawi.com Volume 2018
Journal of
Hindawiwww.hindawi.com Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwww.hindawi.com Volume 2018
OptimizationJournal of
Hindawiwww.hindawi.com Volume 2018
Hindawiwww.hindawi.com Volume 2018
Engineering Mathematics
International Journal of
Hindawiwww.hindawi.com Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwww.hindawi.com Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwww.hindawi.com Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwww.hindawi.com Volume 2018
Hindawi Publishing Corporation http://www.hindawi.com Volume 2013Hindawiwww.hindawi.com
The Scientific World Journal
Volume 2018
Hindawiwww.hindawi.com Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwww.hindawi.com Volume 2018
Hindawiwww.hindawi.com
DiοΏ½erential EquationsInternational Journal of
Volume 2018
Hindawiwww.hindawi.com Volume 2018
Decision SciencesAdvances in
Hindawiwww.hindawi.com Volume 2018
AnalysisInternational Journal of
Hindawiwww.hindawi.com Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwww.hindawi.com
https://www.hindawi.com/journals/jmath/https://www.hindawi.com/journals/mpe/https://www.hindawi.com/journals/jam/https://www.hindawi.com/journals/jps/https://www.hindawi.com/journals/amp/https://www.hindawi.com/journals/jca/https://www.hindawi.com/journals/jopti/https://www.hindawi.com/journals/ijem/https://www.hindawi.com/journals/aor/https://www.hindawi.com/journals/jfs/https://www.hindawi.com/journals/aaa/https://www.hindawi.com/journals/ijmms/https://www.hindawi.com/journals/tswj/https://www.hindawi.com/journals/ana/https://www.hindawi.com/journals/ddns/https://www.hindawi.com/journals/ijde/https://www.hindawi.com/journals/ads/https://www.hindawi.com/journals/ijanal/https://www.hindawi.com/journals/ijsa/https://www.hindawi.com/https://www.hindawi.com/