a integrable generalized super-nls-mkdv hierarchy, its ...hierarchy and its super-hamiltonian...
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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; [email protected]
Received 30 November 2017; Accepted 4 February 2018; Published 5 March 2018
Academic Editor: Zhijun Qiao
Copyright © 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 ðµ(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 ðŸ 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 [4â9]. 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 [11â18]. 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 [20â23].
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 ðµ(0, 1)ð1 = (1 0 00 â1 00 0 0) ,ð2 = ( 0 1 0â1 0 00 0 0) ,ð3 = (0 1 01 0 00 0 0) ,ð4 = (0 0 10 0 00 â1 0) ,ð5 = (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
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that is, along with the communicative operation[ð1, ð2] = 2ð3,[ð1, ð3] = 2ð2,[ð1, ð4] = [ð2, ð5] = [ð3, ð5] = ð4,[ð4, ð2] = [ð5, ð1] = [ð3, ð4] = ð5,[ð2, ð3] = 2ð1,[ð4, ð4]+ = âð2 â ð3,[ð4, ð5]+ = ð1,[ð5, ð5]+ = ð3 â ð2.(2)
To set up the generalized super-NLS-mKdV hierarchy,the spectral problem is given as follows:ðð¥ = ðð,ðð¡ = ðð, (3)with
ð = ( ð + ð€ ð¢1 + ð¢2 ð¢3ð¢1 â ð¢2 âð â ð€ ð¢4ð¢4 âð¢3 0 ) ,ð = ( ðŽ ðµ + ð¶ ððµ â ð¶ âðŽ ðð âð 0) ,
(4)
where ð€ = ð((1/2)ð¢21 â (1/2)ð¢22 + ð¢3ð¢4) with ð being anarbitrary even constant, ð is the spectral parameter, ð¢1 andð¢2 are even potentials, and ð¢3 and ð¢4 are odd potentials. Notethat spectral problem (3) with ð = 0 is reduced to super-NLS-mKdV hierarchy case [24].
Setting ðŽ = âðâ¥0
ðððâð,ðµ = âðâ¥0
ðððâð,ð¶ = âðâ¥0
ðððâð,ð = âðâ¥0
ðððâð,ð = âðâ¥0
ðððâð,(5)
solving the equation ðð¥ = [ð,ð], we haveððð¥ = 2ð¢2ðð â 2ð¢1ðð + ð¢3ðð + ð¢4ðð,ððð¥ = 2ðð+1 â 2ð¢2ðð â ð¢3ðð + ð¢4ðð + 2ð€ðð,ððð¥ = 2ðð+1 â 2ð¢1ðð â ð¢3ðð â ð¢4ðð + 2ð€ðð,ððð¥ = ðð+1 + ð€ðð + (ð¢1 + ð¢2) ðð â ð¢3ðð â ð¢4ððâ ð¢4ðð,ððð¥ = âðð+1 â ð€ðð + ð¢1ðð â ð¢2ðð â ð¢3ðð + ð¢3ðð+ ð¢4ðð,(6)
and, from the above recursion relationship, we can get therecursion operator ð¿ which meets the following:
( ðð+1âðð+1ðð+1âðð+1)= ð¿(ððâððððâðð), ð â©Ÿ 0, (7)
where the recursive operator ð¿ is given as follows:ð¿ =((âð€ + 2ð¢1ðâ1ð¢2 â12ð + 2ð¢1ðâ1ð¢1 12ð¢4 + ð¢1ðâ1ð¢3 â12ð¢3 â ð¢1ðâ1ð¢4â12ð â 2ð¢2ðâ1ð¢2 âð€ + 2ð¢2ðâ1ð¢1 12ð¢4 â ð¢2ðâ1ð¢3 12ð¢3 + ð¢2ðâ1ð¢4âð¢3 + 2ð¢4ðâ1ð¢2 âð¢3 + 2ð¢4ðâ1ð¢1 âð â ð€ + ð¢4ðâ1ð¢3 âð¢1 + ð¢2 â ð¢4ðâ1ð¢4âð¢4 â 2ð¢3ðâ1ð¢2 ð¢4 â 2ð¢3ðâ1ð¢1 ð¢1 + ð¢2 â ð¢3ðâ1ð¢3 ð â ð€ + ð¢3ðâ1ð¢4
)). (8)
Choosing the initial datað0 = 1,ð0 = ð0 = ð0 = ð0 = 0, (9)from the recursion relations in (6), we can obtainð1 = 0,ð1 = ð¢1,
ð1 = ð¢2,ð1 = ð¢3,ð1 = ð¢4,ð2 = â12ð¢21 + 12ð¢22 â ð¢3ð¢4,ð2 = 12ð¢2ð¥ â ð€ð¢1,
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ð2 = 12ð¢1ð¥ â ð€ð¢2,ð2 = ð¢3ð¥ â ð€ð¢3,ð2 = âð¢4ð¥ â ð€ð¢4,ð3 = 12 (ð¢2ð¢1ð¥ â ð¢1ð¢2ð¥) + ð¢3ð¢4ð¥ + ð¢4ð¢3ð¥+ ð€ (ð¢21 â ð¢22) + 2ð¢3ð¢4,ð3 = 14ð¢1ð¥ð¥ â 12ð€ð¥ð¢2 + 12ð¢1ð¢22 â 12ð¢31 + 12ð¢3ð¢3ð¥â 12ð¢4ð¢4ð¥ â ð¢1ð¢3ð¢4 â ð€ð¢2ð¥ + ð€2ð¢1,ð3 = 14ð¢2ð¥ð¥ â 12ð€ð¥ð¢1 + 12ð¢32 â 12ð¢2ð¢21 + 12ð¢3ð¢3ð¥+ 12ð¢4ð¢4ð¥ â ð¢2ð¢3ð¢4 â ð€ð¢1ð¥ + ð€2ð¢2,ð3 = ð¢3ð¥ð¥ + 12ð¢3ð¢22 â 12ð¢3ð¢21 + 12ð¢4ð¢1ð¥ + 12ð¢4ð¢2ð¥+ ð¢1ð¢4ð¥ + ð¢2ð¢4ð¥ â ð€ð¥ð¢3 + ð€2ð¢3 â 2ð€ð¢3ð¥,ð3 = ð¢4ð¥ð¥ + 12ð¢4ð¢22 â 12ð¢4ð¢21 + 12ð¢3ð¢1ð¥ â 12ð¢3ð¢2ð¥+ ð¢1ð¢3ð¥ â ð¢2ð¢3ð¥ + ð€ð¥ð¢4 + ð€2ð¢4 + 2ð€ð¢4ð¥.(10)
Then we consider the auxiliary spectral problemðð¡ð = ð(ð)ð, (11)with
ð(ð) = ðâð=0
( ðð ðð + ðð ðððð â ðð âðð ðððð âðð 0 )ððâð + Î ð,ð ⥠0. (12)Suppose
Î ð = ( ð ð + ð ðð â ð âð ðð âð 0) , (13)substituting ð(ð) into the zero curvature equationðð¡ð â ð(ð)ð¥ + [ð,ð(ð)] = 0, (14)
where ð ⥠0. Making use of (6), we haveð€ð¡ð = ðð¥,ð = ð = ð = ð = 0,ð¢1ð¡ð = ððð¥ â 2ð€ðð + 2ð¢2ðð + ð¢3ðð â ð¢4ðð + ðð¥ â 2ð€ð+ ð¢3ð â ð¢4ð + 2ð¢2ð = 2ðð+1 + 2ð¢2ð,ð¢2ð¡ð = ððð¥ â 2ð€ðð + 2ð¢1ð + ð¢3ðð + ð¢4ðð + ðð¥ + ð¢3ð+ ð¢4ð + 2ð¢1ð â 2ð€ð = 2ðð+1 + 2ð¢1ð,ð¢3ð¡ð = ððð¥ â ð€ðð â ð¢1ðð â ð¢2ðð + ð¢3ðð + ð¢4ðð + ð¢4ðð+ ðð¥ â ð€ð â ð¢1ð â ð¢2ð + ð¢3ð + ð¢4ð + ð¢4ð= ðð+1 + ð¢3ð,ð¢4ð¡ð = ððð¥ + ð€ðð â ð¢1ðð + ð¢2ðð + ð¢3ðð â ð¢3ðð â ð¢4ððâ ð¢4ð + ðð¥ + ð€ð â ð¢1ð + ð¢2ð + ð¢3ð â ð¢3ð= âðð+1 â ð¢4ð,
(15)
which guarantees that the following identity holds true:(12ð¢22 â 12ð¢21 + ð¢4ð¢3)ð¡ð= 2ð¢2ðð+1 â 2ð¢1ðð+1 â ðð+1ð¢3 + ð¢4ðð+1 = ðð+1ð¥. (16)Choosing ð = âððð+1, we arrive at the following generalizedsuper-NLS-mKdV hierarchy:
ð¢ð¡ð = (ð¢1ð¢2ð¢3ð¢4)ð¡ð =(2ðð+1 â 2ðð¢2ðð+12ðð+1 â 2ðð¢1ðð+1ðð+1 â ðð¢3ðð+1âðð+1 + ðð¢4ðð+1), (17)
where ð ⥠0.The case of (17) with ð = 0 is exactly the standardsupersoliton hierarchy [24].
When ð = 1 in (17), the flow is trivial. Taking ð = 2,we can obtain second-order generalized super-NLS-mKdVequationsð¢1ð¡2 = 12ð¢2ð¥ð¥ + ð¢32 â ð¢2ð¢21 + ð¢3ð¢3ð¥ + ð¢4ð¢4ð¥ â 2ð¢2ð¢3ð¢4+ ð (2ð¢1ð¢2ð¢2ð¥ â 2ð¢21ð¢1ð¥ + ð¢4ð¥ð¢3ð¢1 â ð¢3ð¥ð¢4ð¢1â 2ð¢3ð¢4ð¢1ð¥ â 2ð¢2ð¢3ð¢4ð¥ â 2ð¢2ð¢4ð¢3ð¥ â 2ð¢2ð¢21ð¢3ð¢4+ 2ð¢32ð¢3ð¢4) + 2ð2ð¢2 (12ð¢21 â 12ð¢22 + ð¢3ð¢4)2+ 2ð2ð¢2 (12ð¢21 â 12ð¢22 + ð¢3ð¢4) (ð¢22 â ð¢21) ,ð¢2ð¡2 = 12ð¢1ð¥ð¥ â ð¢31 + ð¢1ð¢22 â 2ð¢1ð¢3ð¢4 + ð¢3ð¢3ð¥ â ð¢4ð¢4ð¥+ ð (2ð¢22ð¢2ð¥ â 2ð¢1ð¢2ð¢1ð¥ â ð¢3ð¥ð¢4ð¢2 + ð¢4ð¥ð¢3ð¢2
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4 Advances in Mathematical Physicsâ 2ð¢3ð¢4ð¢2ð¥ â 2ð¢1ð¢3ð¢4ð¥ â 2ð¢1ð¢4ð¢3ð¥)â 2ð2ð¢1 (12ð¢21 â 12ð¢22 + ð¢3ð¢4) (ð¢21 â ð¢22)+ 2ð2 (12ð¢21 â 12ð¢22 + ð¢3ð¢4)2 ð¢1 â 2ð2ð¢1ð¢3 (ð¢21â ð¢22) ð¢4,ð¢3ð¡2 = ð¢3ð¥ð¥ + 12ð¢3ð¢22 â 12ð¢3ð¢21 + 12ð¢4ð¢1ð¥ + 12ð¢4ð¢2ð¥+ ð¢1ð¢4ð¥ + ð¢2ð¢4ð¥ + ð (12ð¢3ð¢1ð¢2ð¥ â 12ð¢3ð¢2ð¢1ð¥+ ð¢22ð¢3ð¥ â ð¢21ð¢3ð¥ + ð¢2ð¢2ð¥ð¢3 â ð¢1ð¢1ð¥ð¢3â 2ð¢3ð¢4ð¢3ð¥) + ð2 (12ð¢21 â 12ð¢22 + ð¢3ð¢4)2 ð¢3â ð2ð¢3 (12ð¢21 â 12ð¢22 + ð¢3ð¢4) (ð¢21 â ð¢22) ,ð¢4ð¡2 = âð¢4ð¥ð¥ â 12ð¢4ð¢22 + 12ð¢4ð¢21 + 12ð¢3ð¢2ð¥ â 12ð¢3ð¢1ð¥â ð¢1ð¢3ð¥ + ð¢2ð¢3ð¥ + ð (12ð¢4ð¢2ð¢1ð¥ â 12ð¢4ð¢1ð¢2ð¥â ð¢1ð¢1ð¥ð¢4 + ð¢2ð¢2ð¥ð¢4 â ð¢21ð¢4ð¥ + ð¢22ð¢4ð¥â 2ð¢3ð¢4ð¢4ð¥) â ð2 (12ð¢21 â 12ð¢22 + ð¢3ð¢4)2 ð¢4â ð2ð¢4 (12ð¢21 â 12ð¢22 + ð¢3ð¢4) (ð¢21 â ð¢22) .(18)
From (3) and (11), one infers the following:
ð(2) = (ð(2)11 ð(2)12 ð(2)13ð(2)21 âð(2)11 ð(2)23ð(2)23 âð(2)13 0 ) , (19)with ð(2)11 = ð2 + 12ð¢22 â 12ð¢21 â ð¢3ð¢4â ð (12ð¢2ð¢1ð¥ â 12ð¢1ð¢2ð¥ + ð¢4ð¢3ð¥ + ð¢3ð¢4ð¥)â 2ð2 (12ð¢21 â 12ð¢22 + ð¢3ð¢4)2 ,ð(2)12 = (ð¢1 + ð¢2) ð + 12 (ð¢1 + ð¢2)ð¥â ð (12ð¢21 â 12ð¢22 + ð¢3ð¢4) (ð¢1 + ð¢2) ,
ð(2)13 = ð¢3ð + ð¢3ð¥ â ð (12ð¢21 â 12ð¢22 + ð¢3ð¢4) ð¢3,ð(2)21 = (ð¢1 â ð¢2) ð + 12 (ð¢2 â ð¢1)ð¥+ ð (12ð¢21 â 12ð¢22 + ð¢3ð¢4) (ð¢2 â ð¢1) ,ð(2)23 = ð¢4ð â ð¢4ð¥ â ð (12ð¢21 â 12ð¢22 + ð¢3ð¢4) ð¢4.(20)
In what follows, we shall set up super-Hamiltonianstructures for the generalized super-NLS-mKdV hierarchy.
Through calculations, we obtain
Str(ððððð ) = 2ðŽ,Str( ðððð¢1ð) = 2 (ðµ + ðð¢1ðŽ) ,Str( ðððð¢2ð) = â2 (ð¶ + ðð¢2ðŽ) ,Str( ðððð¢3ð) = 2 (ð + ðð¢4ðŽ) ,Str( ðððð¢4ð) = â2 (ð + ðð¢3ðŽ) .
(21)
Substituting the above results into the supertrace identity[3] and balancing the coefficients of ðð+2, we obtainð¿ð¿ð¢ â« ðð+2ðð¥ = (ðŸ â ð â 1)(
ðð+1 + ðð¢1ðð+1âðð+1 â ðð¢2ðð+1ðð+1 + ðð¢4ðð+1âðð+1 â ðð¢3ðð+1),ð ⥠0.(22)
Thus, we haveð»ð+1 = â«â ðð+2ð + 1ðð¥,ð¿ð»ð+1ð¿ð¢ =(
ðð+1 + ðð¢1ðð+1âðð+1 â ðð¢2ðð+1ðð+1 + ðð¢4ðð+1âðð+1 â ðð¢3ðð+1),ð â©Ÿ 0.(23)
Moreover, it is easy to find that
( ðð+1âðð+1ðð+1âðð+1)= ð 1(ðð+1 + ðð¢1ðð+1âðð+1 â ðð¢2ðð+1ðð+1 + ðð¢4ðð+1âðð+1 â ðð¢3ðð+1), ð â©Ÿ 0, (24)
where ð 1 is given by
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ð 1 =(1 â 2ðð¢1ðâ1ð¢2 â2ðð¢1ðâ1ð¢1 ðð¢1ðâ1ð¢3 ðð¢1ðâ1ð¢42ðð¢2ðâ1ð¢2 1 + 2ðð¢2ðâ1ð¢1 ðð¢2ðâ1ð¢3 âðð¢2ðâ1ð¢4â2ðð¢4ðâ1ð¢2 â2ðð¢4ðâ1ð¢1 1 â ðð¢4ðâ1ð¢3 ðð¢4ðâ1ð¢42ðð¢3ðâ1ð¢2 2ðð¢3ðâ1ð¢1 ðð¢3ðâ1ð¢3 1 â ðð¢3ðâ1ð¢4). (25)
Therefore, superintegrable hierarchy (17) possesses thefollowing form:
ð¢ð¡ð = ð 2( ðð+1âðð+1ðð+1âðð+1)= ð 2ð 1(ðð+1 + ðð¢1ðð+1âðð+1 â ðð¢2ðð+1ðð+1 + ðð¢4ðð+1âðð+1 â ðð¢3ðð+1)= ðœð¿ð»ð+1ð¿ð¢ , ð ⥠0,
(26)
where
ð 2 =( â4ðð¢2ðâ1ð¢2 â2 â 4ðð¢2ðâ1ð¢1 â2ðð¢2ðâ1ð¢3 2ðð¢2ðâ1ð¢42 â 4ðð¢1ðâ1ð¢2 â4ðð¢1ðâ1ð¢1 â2ðð¢1ðâ1ð¢3 2ðð¢1ðâ1ð¢4â4ðð¢3ðâ1ð¢2 â4ðð¢3ðâ1ð¢1 â2ðð¢3ðâ1ð¢3 â1 + 2ðð¢3ðâ1ð¢42ðð¢4ðâ1ð¢2 2ðð¢4ðâ1ð¢1 â1 + ðð¢4ðâ1ð¢3 âðð¢4ðâ1ð¢4 ), (27)
and super-Hamiltonian operator ðœ is given byðœ = ð 2ð 1 =( â8ðð¢2ð
â1ð¢2 â2 â 8ðð¢2ðâ1ð¢1 â4ðð¢2ðâ1ð¢3 4ðð¢2ðâ1ð¢42 â 8ðð¢1ðâ1ð¢2 â8ðð¢1ðâ1ð¢1 â4ðð¢1ðâ1ð¢3 4ðð¢1ðâ1ð¢4â6ðð¢3ðâ1ð¢2 â6ðð¢3ðâ1ð¢1 â3ðð¢3ðâ1ð¢3 â1 + 3ðð¢3ðâ1ð¢44ðð¢4ðâ1ð¢2 4ðð¢4ðâ1ð¢1 â1 + 2ðð¢4ðâ1ð¢3 â2ðð¢4ðâ1ð¢4 ). (28)In addition, generalized super-NLS-mKdV hierarchy (17)
also possesses the following super-Hamiltonian form:
ð¢ð¡ð = ð 2ð¿( ððâððððâðð)= ð 2ð¿ð 1(ðð + ðð¢1ððâðð â ðð¢2ðððð + ðð¢4ððâðð â ðð¢3ðð)=ðð¿ð»ðð¿ð¢ , ð ⥠0,
(29)
where ð = ð 2ð¿ð 1 = (ððð)4Ã4 is the second super-Hamiltonian operator.
3. Self-Consistent Sources
Consider the linear system
(ð1ðð2ðð3ð)ð¥ = ð(ð1ðð2ðð3ð),
(ð1ðð2ðð3ð)ð¡ = ð(ð1ðð2ðð3ð).
(30)
From the result in [25], we setð¿ððð¿ð¢ð = 13Str(Κð ðð (ð¢, ðð)ð¿ð¢ð ) , ð = 1, 2, . . . , 4, (31)
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with Str on behalf of the supertrace, and
Κð = (ð1ðð2ð âð21ð ð1ðð3ðð22ð âð1ðð2ð ð2ðð3ðð2ðð3ð âð1ðð3ð 0 ) , ð = 1, 2, . . . , ð. (32)From system (30), we get ð¿ðð/ð¿ð¢ as follows:ðâð=1
ð¿ððð¿ð¢ð = ðâð=1((((((
Str(Κð ð¿ðð¿ð¢1)Str(Κð ð¿ðð¿ð¢2)Str(Κð ð¿ðð¿ð¢3)Str(Κð ð¿ðð¿ð¢4)
))))))
=( 2ðð¢1 âšÎŠ1, Ί2â© â âšÎŠ1, Ί1â© + âšÎŠ2, Ί2â©â2ðð¢2 âšÎŠ1, Ί2â© + âšÎŠ1, Ί1â© + âšÎŠ2, Ί2â©2ðð¢4 âšÎŠ1, Ί2â© â 2 âšÎŠ2, Ί3â©2ðð¢3 âšÎŠ1, Ί2â© â 2 âšÎŠ1, Ί3â© ) ,(33)
where Ίð = (ðð1, . . . , ððð)ð, ð = 1, 2, 3.So, we obtain the self-consistent sources of generalized
super-NLS-mKdV hierarchy (17):
ð¢ð¡ð =(ð¢1ð¢2ð¢3ð¢4)ð¡ð = ðœð¿ð»ð+1ð¿ð¢ð + ðœ ðâð=1ð¿ððð¿ð¢ð
= ðœ( ðð+1 + ðð¢1ðð+1âðð+1 â ðð¢2ðð+1ðð+1 + ðð¢4ðð+1âðð+1 â ðð¢3ðð+1)+ ðœ( 2ðð¢1 âšÎŠ1, Ί2â© â âšÎŠ1, Ί1â© + âšÎŠ2, Ί2â©â2ðð¢2 âšÎŠ1, Ί2â© + âšÎŠ1, Ί1â© + âšÎŠ2, Ί2â©2ðð¢4 âšÎŠ1, Ί2â© â 2 âšÎŠ2, Ί3â©2ðð¢3 âšÎŠ1, Ί2â© â 2 âšÎŠ1, Ί3â© ) .
(34)
For ð = 2, we get supersoliton equation with self-consistent sources as follows:ð¢1ð¡2 = 12ð¢2ð¥ð¥ + ð¢32 â ð¢2ð¢21 + ð¢3ð¢3ð¥ + ð¢4ð¢4ð¥ â 2ð¢2ð¢3ð¢4+ ð (2ð¢1ð¢2ð¢2ð¥ â 2ð¢21ð¢1ð¥ + ð¢4ð¥ð¢3ð¢1 â ð¢3ð¥ð¢4ð¢1â 2ð¢3ð¢4ð¢1ð¥ â 2ð¢2ð¢3ð¢4ð¥ â 2ð¢2ð¢4ð¢3ð¥ â 2ð¢2ð¢21ð¢3ð¢4+ 2ð¢32ð¢3ð¢4) + 2ð2ð¢2 (12ð¢21 â 12ð¢22 + ð¢3ð¢4)2
+ 2ð2ð¢2 (12ð¢21 â 12ð¢22 + ð¢3ð¢4) (ð¢22 â ð¢21)+ 2ðð¢1 ðâð=1
ð1ðð2ð â ðâð=1
(ð21ð â ð22ð) ,ð¢2ð¡2 = 12ð¢1ð¥ð¥ â ð¢31 + ð¢1ð¢22 â 2ð¢1ð¢3ð¢4 + ð¢3ð¢3ð¥ â ð¢4ð¢4ð¥+ ð (2ð¢22ð¢2ð¥ â 2ð¢1ð¢2ð¢1ð¥ â ð¢3ð¥ð¢4ð¢2 + ð¢4ð¥ð¢3ð¢2â 2ð¢3ð¢4ð¢2ð¥ â 2ð¢1ð¢3ð¢4ð¥ â 2ð¢1ð¢4ð¢3ð¥)â 2ð2ð¢1 (12ð¢21 â 12ð¢22 + ð¢3ð¢4) (ð¢21 â ð¢22)+ 2ð2 (12ð¢21 â 12ð¢22 + ð¢3ð¢4)2 ð¢1 â 2ð2ð¢1ð¢3 (ð¢21â ð¢22) ð¢4 â 2ðð¢2 ðâ
ð=1
ð1ðð2ð + ðâð=1
(ð21ð + ð22ð) ,ð¢3ð¡2 = ð¢3ð¥ð¥ + 12ð¢3ð¢22 â 12ð¢3ð¢21 + 12ð¢4ð¢1ð¥ + 12ð¢4ð¢2ð¥+ ð¢1ð¢4ð¥ + ð¢2ð¢4ð¥ + ð (12ð¢3ð¢1ð¢2ð¥ â 12ð¢3ð¢2ð¢1ð¥+ ð¢22ð¢3ð¥ â ð¢21ð¢3ð¥ + ð¢2ð¢2ð¥ð¢3 â ð¢1ð¢1ð¥ð¢3â 2ð¢3ð¢4ð¢3ð¥) + ð2 (12ð¢21 â 12ð¢22 + ð¢3ð¢4)2 ð¢3â ð2ð¢3 (12ð¢21 â 12ð¢22 + ð¢3ð¢4) (ð¢21 â ð¢22)+ 2ðð¢4 ðâ
ð=1
ð1ðð2ð â ðâð=1
ð2ðð3ð,ð¢4ð¡2 = âð¢4ð¥ð¥ â 12ð¢4ð¢22 + 12ð¢4ð¢21 + 12ð¢3ð¢2ð¥ â 12ð¢3ð¢1ð¥â ð¢1ð¢3ð¥ + ð¢2ð¢3ð¥ + ð (12ð¢4ð¢2ð¢1ð¥ â 12ð¢4ð¢1ð¢2ð¥â ð¢1ð¢1ð¥ð¢4 + ð¢2ð¢2ð¥ð¢4 â ð¢21ð¢4ð¥ + ð¢22ð¢4ð¥â 2ð¢3ð¢4ð¢4ð¥) â ð2 (12ð¢21 â 12ð¢22 + ð¢3ð¢4)2 ð¢4â ð2ð¢4 (12ð¢21 â 12ð¢22 + ð¢3ð¢4) (ð¢21 â ð¢22)+ 2ðð¢3 ðâð=1
ð1ðð2ð â 2 ðâð=1
ð1ðð3ð,ð1ðð¥ = (ð + ð€) ð1ð + (ð¢1 + ð¢2) ð2ð + ð¢3ð3ð,ð2ðð¥ = (ð¢1 â ð¢2) ð1ð â (ð + ð€) ð2ð + ð¢4ð3ð,ð3ðð¥ = ð¢4ð1ð â ð¢3ð2ð, ð = 1, . . . , ð.(35)
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Advances in Mathematical Physics 7
4. Conservation Laws
In the following, we shall derive the conservation laws ofsupersoliton hierarchy. Introducing the variablesðŸ = ð2ð1 ,ðº = ð3ð1 , (36)then we obtainðŸð¥ = ð¢1 â ð¢2 â 2 (ð + ð€)ðŸ + ð¢4ðº â (ð¢1 + ð¢2)ðŸ2â ð¢3ðŸðº,ðºð¥ = ð¢4 â ð¢3ðŸ â (ð + ð€)ðº â (ð¢1 + ð¢2) ðºðŸ â ð¢3ðº2. (37)
Next, we expand ðŸ and ðº as series of the spectralparameter ð
ðŸ = ââð=1
ðððð,ðº = ââð=1
ðððð. (38)Substituting (38) into (37), we raise the recursion formulas forðð and ðð:ðð+1 = â12ððð¥ â ð€ðð + 12ð¢4ðð â 12 (ð¢1 + ð¢2) ðâ1â
ð=1
ððððâðâ 12ð¢3ðâ1â
ð=1
ððððâð,ðð+1 = âððð¥ â ð¢3ðð â ð€ðð â (ð¢1 + ð¢2) ðâ1â
ð=1
ððððâðâ ð¢3ðâ1âð=1
ððððâð, ð ⥠2.(39)
We write the first few terms of ðð and ðð:ð1 = 12 (ð¢1 â ð¢2) ,ð1 = ð¢4,ð2 = â14 (ð¢1 â ð¢2) ð¥â 12ð (ð¢1 â ð¢2) (12ð¢21 â 12ð¢22 + ð¢3ð¢4) ,ð2 = âð¢4ð¥ â 12 (ð¢1 â ð¢2) (ð¢3 + ðð¢1 + ðð¢2) ,
ð3 = 18 (ð¢1 â ð¢2)ð¥ð¥ â 18 (ð¢1 + ð¢2) (ð¢1 â ð¢2)2+ 14ð€ð¥ (ð¢1 â ð¢2) + 12 (ð¢1 â ð¢2)ð¥ ð€+ 12 (ð¢1 â ð¢2) ð€2 â 12ð¢4ð¢4ð¥,ð3 = ð¢4ð¥ð¥ + 12 (ð¢1 â ð¢2) ð¢3ð¥ â 12 (ð¢21 â ð¢22) ð¢4+ 34 (ð¢1 â ð¢2)ð¥ ð¢3 + ð€ð¥ð¢4 + 2ð€ð¢4ð¥+ ð€ (ð¢1 â ð¢2) ð¢3 + ð€2ð¢4, . . . .(40)
Note that ððð¡ [ð + ð€ + (ð¢1 + ð¢2)ðŸ + ð¢3ðº]= ððð¥ [ðŽ + (ðµ + ð¶)ðŸ + ððº] , (41)setting ð¿ = ð+ð€+ (ð¢1 +ð¢2)ðŸ+ð¢3ðº, ð = ðŽ+ (ðµ+ð¶)ðŸ+ððº,which admitted that the conservation laws is ð¿ð¡ = ðð¥. For (19),one infers thatðŽ = ð2 + 12ð¢22 â 12ð¢21 â ð¢3ð¢4,ðµ = ð¢1ð + 12ð¢2ð¥ â ðð¢1 (12ð¢21 â 12ð¢22 + ð¢3ð¢4) ,ð¶ = ð¢2ð + 12ð¢1ð¥ â ðð¢2 (12ð¢21 â 12ð¢22 + ð¢3ð¢4) ,ð = ð¢3ð + ð¢3ð¥ â 12ðð¢3 (ð¢21 â ð¢22) .
(42)
Expanding ð¿ and ð asð¿ = ð + ð€ + ââ
ð=1
ð¿ððâð,ð = ð2 + 12ð¢22 â 12ð¢21 â ð¢3ð¢4 + ââ
ð=1
ðððâð, (43)conserved densities and currents are the coefficients ð¿ð, ðð,respectively. The first two conserved densities and currentsare read:ð¿1 = 12 (ð¢21 â ð¢22) + ð¢3ð¢4,ð1 = â14 (ð¢1 + ð¢2) (ð¢1 â ð¢2) ð¥ + 14 (ð¢1 â ð¢2) (ð¢1 + ð¢2)â ð¥ â 14ð (ð¢21 â ð¢22) (12ð¢21 â 12ð¢22 + ð¢3ð¢4) â 12 (ð¢21
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8 Advances in Mathematical Physics
â ð¢22) (12ð¢21 â 12ð¢22 + ð¢3ð¢4) â 12ð (ð¢21 â ð¢22) ð¢3 â 12â ð (ð¢21 â ð¢22) ð¢3ð¢4 â ð¢3ð¢4ð¥ + ð¢3ð¥ð¢4,ð¿2 = â14 (ð¢1 + ð¢2) (ð¢2 â ð¢1)ð¥ â 12ð (ð¢21 â ð¢22) (12ð¢21â 12ð¢22 + ð¢3 + ð¢3ð¢4) â ð¢3ð¢4ð¥,ð2 = (ð¢1 + ð¢2) ð3 â 14 [(ð¢1 + ð¢2)ð¥â ð (ð¢1 + ð¢2) (12ð¢21 â 12ð¢22 + ð¢3ð¢4)]â [12 (ð¢1 â ð¢2) ð¥ + (ð¢1 â ð¢2)à (12ð¢21 â 12ð¢22 + ð¢3ð¢4)] + ð¢3ð3 â [ð¢3ð¥â 12ð (ð¢21 â ð¢22) ð¢3] [ð¢4ð¥+ 12 (ð¢1 â ð¢2) (ð¢3 + ðð¢1 + ðð¢2)] ,(44)
where ð3 and ð3 are given by (40).The recursion relationshipfor ð¿ð and ðð is as follows:ð¿ð = (ð¢1 + ð¢2) ðð + ð¢3ðð,ðð = (ð¢1 + ð¢2) ðð+1 + 12 [(ð¢1 + ð¢2)ð¥â ð (ð¢1 + ð¢2) (12ð¢21 â 12ð¢22 + ð¢3ð¢4)] ðð + ð¢3ðð+1+ [ð¢3ð¥ â 12ð (ð¢21 â ð¢22) ð¢3] ðð,
(45)
where ðð and ðð can be recursively calculated from (39). Wecan display the first two conservation laws of (18) asð¿1ð¡ = ð1ð¥,ð¿2ð¡ = ð2ð¥, (46)where ð¿1, ð1, ð¿2, and ð2 are defined in (44). Then we canobtain the infinitely many conservation laws of (17) from(37)â(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 [26â29], 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).
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