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 © 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 𝐵(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 𝛾 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 [4–9]. 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 [11–18]. 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 [20–23].

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 𝐵(0, 1)𝑒1 = (1 0 00 −1 00 0 0) ,𝑒2 = ( 0 1 0−1 0 00 0 0) ,𝑒3 = (0 1 01 0 00 0 0) ,𝑒4 = (0 0 10 0 00 −1 0) ,𝑒5 = (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[𝑒1, 𝑒2] = 2𝑒3,[𝑒1, 𝑒3] = 2𝑒2,[𝑒1, 𝑒4] = [𝑒2, 𝑒5] = [𝑒3, 𝑒5] = 𝑒4,[𝑒4, 𝑒2] = [𝑒5, 𝑒1] = [𝑒3, 𝑒4] = 𝑒5,[𝑒2, 𝑒3] = 2𝑒1,[𝑒4, 𝑒4]+ = −𝑒2 − 𝑒3,[𝑒4, 𝑒5]+ = 𝑒1,[𝑒5, 𝑒5]+ = 𝑒3 − 𝑒2.(2)

To set up the generalized super-NLS-mKdV hierarchy,the spectral problem is given as follows:𝜑𝑥 = 𝑈𝜑,𝜑𝑡 = 𝑉𝜑, (3)with

𝑈 = ( 𝜆 + 𝑤 𝑢1 + 𝑢2 𝑢3𝑢1 − 𝑢2 −𝜆 − 𝑤 𝑢4𝑢4 −𝑢3 0 ) ,𝑉 = ( 𝐴 𝐵 + 𝐶 𝜎𝐵 − 𝐶 −𝐴 𝜌𝜌 −𝜎 0) ,

(4)

where 𝑤 = 𝜀((1/2)𝑢21 − (1/2)𝑢22 + 𝑢3𝑢4) with 𝜀 being anarbitrary even constant, 𝜆 is the spectral parameter, 𝑢1 and𝑢2 are even potentials, and 𝑢3 and 𝑢4 are odd potentials. Notethat spectral problem (3) with 𝜀 = 0 is reduced to super-NLS-mKdV hierarchy case [24].

Setting 𝐴 = ∑𝑚≥0

𝑎𝑚𝜆−𝑚,𝐵 = ∑𝑚≥0

𝑏𝑚𝜆−𝑚,𝐶 = ∑𝑚≥0

𝑐𝑚𝜆−𝑚,𝜎 = ∑𝑚≥0

𝜎𝑚𝜆−𝑚,𝜌 = ∑𝑚≥0

𝜌𝑚𝜆−𝑚,(5)

solving the equation 𝑉𝑥 = [𝑈,𝑉], we have𝑎𝑚𝑥 = 2𝑢2𝑏𝑚 − 2𝑢1𝑐𝑚 + 𝑢3𝜌𝑚 + 𝑢4𝜎𝑚,𝑏𝑚𝑥 = 2𝑐𝑚+1 − 2𝑢2𝑎𝑚 − 𝑢3𝜎𝑚 + 𝑢4𝜌𝑚 + 2𝑤𝑐𝑚,𝑐𝑚𝑥 = 2𝑏𝑚+1 − 2𝑢1𝑎𝑚 − 𝑢3𝜎𝑚 − 𝑢4𝜌𝑚 + 2𝑤𝑏𝑚,𝜎𝑚𝑥 = 𝜎𝑚+1 + 𝑤𝜎𝑚 + (𝑢1 + 𝑢2) 𝜌𝑚 − 𝑢3𝑎𝑚 − 𝑢4𝑏𝑚− 𝑢4𝑐𝑚,𝜌𝑚𝑥 = −𝜌𝑚+1 − 𝑤𝜌𝑚 + 𝑢1𝜎𝑚 − 𝑢2𝜎𝑚 − 𝑢3𝑏𝑚 + 𝑢3𝑐𝑚+ 𝑢4𝑎𝑚,(6)

and, from the above recursion relationship, we can get therecursion operator 𝐿 which meets the following:

( 𝑏𝑚+1−𝑐𝑚+1𝜌𝑚+1−𝜎𝑚+1)= 𝐿(𝑏𝑚−𝑐𝑚𝜌𝑚−𝜎𝑚), 𝑚 ⩾ 0, (7)

where the recursive operator 𝐿 is given as follows:𝐿 =((−𝑤 + 2𝑢1𝜕−1𝑢2 −12𝜕 + 2𝑢1𝜕−1𝑢1 12𝑢4 + 𝑢1𝜕−1𝑢3 −12𝑢3 − 𝑢1𝜕−1𝑢4−12𝜕 − 2𝑢2𝜕−1𝑢2 −𝑤 + 2𝑢2𝜕−1𝑢1 12𝑢4 − 𝑢2𝜕−1𝑢3 12𝑢3 + 𝑢2𝜕−1𝑢4−𝑢3 + 2𝑢4𝜕−1𝑢2 −𝑢3 + 2𝑢4𝜕−1𝑢1 −𝜕 − 𝑤 + 𝑢4𝜕−1𝑢3 −𝑢1 + 𝑢2 − 𝑢4𝜕−1𝑢4−𝑢4 − 2𝑢3𝜕−1𝑢2 𝑢4 − 2𝑢3𝜕−1𝑢1 𝑢1 + 𝑢2 − 𝑢3𝜕−1𝑢3 𝜕 − 𝑤 + 𝑢3𝜕−1𝑢4

)). (8)

Choosing the initial data𝑎0 = 1,𝑏0 = 𝑐0 = 𝜎0 = 𝜌0 = 0, (9)from the recursion relations in (6), we can obtain𝑎1 = 0,𝑏1 = 𝑢1,

𝑐1 = 𝑢2,𝜎1 = 𝑢3,𝜌1 = 𝑢4,𝑎2 = −12𝑢21 + 12𝑢22 − 𝑢3𝑢4,𝑏2 = 12𝑢2𝑥 − 𝑤𝑢1,

Advances in Mathematical Physics 3

𝑐2 = 12𝑢1𝑥 − 𝑤𝑢2,𝜎2 = 𝑢3𝑥 − 𝑤𝑢3,𝜌2 = −𝑢4𝑥 − 𝑤𝑢4,𝑎3 = 12 (𝑢2𝑢1𝑥 − 𝑢1𝑢2𝑥) + 𝑢3𝑢4𝑥 + 𝑢4𝑢3𝑥+ 𝑤 (𝑢21 − 𝑢22) + 2𝑢3𝑢4,𝑏3 = 14𝑢1𝑥𝑥 − 12𝑤𝑥𝑢2 + 12𝑢1𝑢22 − 12𝑢31 + 12𝑢3𝑢3𝑥− 12𝑢4𝑢4𝑥 − 𝑢1𝑢3𝑢4 − 𝑤𝑢2𝑥 + 𝑤2𝑢1,𝑐3 = 14𝑢2𝑥𝑥 − 12𝑤𝑥𝑢1 + 12𝑢32 − 12𝑢2𝑢21 + 12𝑢3𝑢3𝑥+ 12𝑢4𝑢4𝑥 − 𝑢2𝑢3𝑢4 − 𝑤𝑢1𝑥 + 𝑤2𝑢2,𝜎3 = 𝑢3𝑥𝑥 + 12𝑢3𝑢22 − 12𝑢3𝑢21 + 12𝑢4𝑢1𝑥 + 12𝑢4𝑢2𝑥+ 𝑢1𝑢4𝑥 + 𝑢2𝑢4𝑥 − 𝑤𝑥𝑢3 + 𝑤2𝑢3 − 2𝑤𝑢3𝑥,𝜌3 = 𝑢4𝑥𝑥 + 12𝑢4𝑢22 − 12𝑢4𝑢21 + 12𝑢3𝑢1𝑥 − 12𝑢3𝑢2𝑥+ 𝑢1𝑢3𝑥 − 𝑢2𝑢3𝑥 + 𝑤𝑥𝑢4 + 𝑤2𝑢4 + 2𝑤𝑢4𝑥.(10)

Then we consider the auxiliary spectral problem𝜑𝑡𝑛 = 𝑉(𝑛)𝜑, (11)with

𝑉(𝑛) = 𝑛∑𝑚=0

( 𝑎𝑚 𝑏𝑚 + 𝑐𝑚 𝜎𝑚𝑏𝑚 − 𝑐𝑚 −𝑎𝑚 𝜌𝑚𝜌𝑚 −𝜎𝑚 0 )𝜆𝑛−𝑚 + Δ 𝑛,𝑛 ≥ 0. (12)Suppose

Δ 𝑛 = ( 𝑎 𝑏 + 𝑐 𝑒𝑏 − 𝑐 −𝑎 𝑓𝑓 −𝑒 0) , (13)substituting 𝑉(𝑛) into the zero curvature equation𝑈𝑡𝑛 − 𝑉(𝑛)𝑥 + [𝑈,𝑉(𝑛)] = 0, (14)

where 𝑛 ≥ 0. Making use of (6), we have𝑤𝑡𝑛 = 𝑎𝑥,𝑏 = 𝑐 = 𝑒 = 𝑓 = 0,𝑢1𝑡𝑛 = 𝑏𝑛𝑥 − 2𝑤𝑐𝑛 + 2𝑢2𝑎𝑛 + 𝑢3𝜎𝑛 − 𝑢4𝜌𝑛 + 𝑏𝑥 − 2𝑤𝑐+ 𝑢3𝑒 − 𝑢4𝑓 + 2𝑢2𝑎 = 2𝑐𝑛+1 + 2𝑢2𝑎,𝑢2𝑡𝑛 = 𝑐𝑛𝑥 − 2𝑤𝑏𝑛 + 2𝑢1𝑎 + 𝑢3𝜎𝑛 + 𝑢4𝜌𝑛 + 𝑐𝑥 + 𝑢3𝑒+ 𝑢4𝑓 + 2𝑢1𝑎 − 2𝑤𝑏 = 2𝑏𝑛+1 + 2𝑢1𝑎,𝑢3𝑡𝑛 = 𝜎𝑛𝑥 − 𝑤𝜎𝑛 − 𝑢1𝜌𝑛 − 𝑢2𝜌𝑛 + 𝑢3𝑎𝑛 + 𝑢4𝑏𝑛 + 𝑢4𝑐𝑛+ 𝑒𝑥 − 𝑤𝑒 − 𝑢1𝑓 − 𝑢2𝑓 + 𝑢3𝑎 + 𝑢4𝑏 + 𝑢4𝑐= 𝜎𝑛+1 + 𝑢3𝑎,𝑢4𝑡𝑛 = 𝜌𝑛𝑥 + 𝑤𝜌𝑛 − 𝑢1𝜎𝑛 + 𝑢2𝜎𝑛 + 𝑢3𝑏𝑛 − 𝑢3𝑐𝑛 − 𝑢4𝑎𝑛− 𝑢4𝑎 + 𝑓𝑥 + 𝑤𝑓 − 𝑢1𝑒 + 𝑢2𝑒 + 𝑢3𝑏 − 𝑢3𝑐= −𝜌𝑛+1 − 𝑢4𝑎,

(15)

which guarantees that the following identity holds true:(12𝑢22 − 12𝑢21 + 𝑢4𝑢3)𝑡𝑛= 2𝑢2𝑏𝑛+1 − 2𝑢1𝑐𝑛+1 − 𝜌𝑛+1𝑢3 + 𝑢4𝜎𝑛+1 = 𝑎𝑛+1𝑥. (16)Choosing 𝑎 = −𝜀𝑎𝑛+1, we arrive at the following generalizedsuper-NLS-mKdV hierarchy:

𝑢𝑡𝑛 = (𝑢1𝑢2𝑢3𝑢4)𝑡𝑛 =(2𝑐𝑛+1 − 2𝜀𝑢2𝑎𝑛+12𝑏𝑛+1 − 2𝜀𝑢1𝑎𝑛+1𝜎𝑛+1 − 𝜀𝑢3𝑎𝑛+1−𝜌𝑛+1 + 𝜀𝑢4𝑎𝑛+1), (17)

where 𝑛 ≥ 0.The case of (17) with 𝜀 = 0 is exactly the standardsupersoliton hierarchy [24].

When 𝑛 = 1 in (17), the flow is trivial. Taking 𝑛 = 2,we can obtain second-order generalized super-NLS-mKdVequations𝑢1𝑡2 = 12𝑢2𝑥𝑥 + 𝑢32 − 𝑢2𝑢21 + 𝑢3𝑢3𝑥 + 𝑢4𝑢4𝑥 − 2𝑢2𝑢3𝑢4+ 𝜀 (2𝑢1𝑢2𝑢2𝑥 − 2𝑢21𝑢1𝑥 + 𝑢4𝑥𝑢3𝑢1 − 𝑢3𝑥𝑢4𝑢1− 2𝑢3𝑢4𝑢1𝑥 − 2𝑢2𝑢3𝑢4𝑥 − 2𝑢2𝑢4𝑢3𝑥 − 2𝑢2𝑢21𝑢3𝑢4+ 2𝑢32𝑢3𝑢4) + 2𝜀2𝑢2 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4)2+ 2𝜀2𝑢2 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4) (𝑢22 − 𝑢21) ,𝑢2𝑡2 = 12𝑢1𝑥𝑥 − 𝑢31 + 𝑢1𝑢22 − 2𝑢1𝑢3𝑢4 + 𝑢3𝑢3𝑥 − 𝑢4𝑢4𝑥+ 𝜀 (2𝑢22𝑢2𝑥 − 2𝑢1𝑢2𝑢1𝑥 − 𝑢3𝑥𝑢4𝑢2 + 𝑢4𝑥𝑢3𝑢2

4 Advances in Mathematical Physics− 2𝑢3𝑢4𝑢2𝑥 − 2𝑢1𝑢3𝑢4𝑥 − 2𝑢1𝑢4𝑢3𝑥)− 2𝜀2𝑢1 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4) (𝑢21 − 𝑢22)+ 2𝜀2 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4)2 𝑢1 − 2𝜀2𝑢1𝑢3 (𝑢21− 𝑢22) 𝑢4,𝑢3𝑡2 = 𝑢3𝑥𝑥 + 12𝑢3𝑢22 − 12𝑢3𝑢21 + 12𝑢4𝑢1𝑥 + 12𝑢4𝑢2𝑥+ 𝑢1𝑢4𝑥 + 𝑢2𝑢4𝑥 + 𝜀 (12𝑢3𝑢1𝑢2𝑥 − 12𝑢3𝑢2𝑢1𝑥+ 𝑢22𝑢3𝑥 − 𝑢21𝑢3𝑥 + 𝑢2𝑢2𝑥𝑢3 − 𝑢1𝑢1𝑥𝑢3− 2𝑢3𝑢4𝑢3𝑥) + 𝜀2 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4)2 𝑢3− 𝜀2𝑢3 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4) (𝑢21 − 𝑢22) ,𝑢4𝑡2 = −𝑢4𝑥𝑥 − 12𝑢4𝑢22 + 12𝑢4𝑢21 + 12𝑢3𝑢2𝑥 − 12𝑢3𝑢1𝑥− 𝑢1𝑢3𝑥 + 𝑢2𝑢3𝑥 + 𝜀 (12𝑢4𝑢2𝑢1𝑥 − 12𝑢4𝑢1𝑢2𝑥− 𝑢1𝑢1𝑥𝑢4 + 𝑢2𝑢2𝑥𝑢4 − 𝑢21𝑢4𝑥 + 𝑢22𝑢4𝑥− 2𝑢3𝑢4𝑢4𝑥) − 𝜀2 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4)2 𝑢4− 𝜀2𝑢4 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4) (𝑢21 − 𝑢22) .(18)

From (3) and (11), one infers the following:

𝑉(2) = (𝑉(2)11 𝑉(2)12 𝑉(2)13𝑉(2)21 −𝑉(2)11 𝑉(2)23𝑉(2)23 −𝑉(2)13 0 ) , (19)with 𝑉(2)11 = 𝜆2 + 12𝑢22 − 12𝑢21 − 𝑢3𝑢4− 𝜀 (12𝑢2𝑢1𝑥 − 12𝑢1𝑢2𝑥 + 𝑢4𝑢3𝑥 + 𝑢3𝑢4𝑥)− 2𝜀2 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4)2 ,𝑉(2)12 = (𝑢1 + 𝑢2) 𝜆 + 12 (𝑢1 + 𝑢2)𝑥− 𝜀 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4) (𝑢1 + 𝑢2) ,

𝑉(2)13 = 𝑢3𝜆 + 𝑢3𝑥 − 𝜀 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4) 𝑢3,𝑉(2)21 = (𝑢1 − 𝑢2) 𝜆 + 12 (𝑢2 − 𝑢1)𝑥+ 𝜀 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4) (𝑢2 − 𝑢1) ,𝑉(2)23 = 𝑢4𝜆 − 𝑢4𝑥 − 𝜀 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4) 𝑢4.(20)

In what follows, we shall set up super-Hamiltonianstructures for the generalized super-NLS-mKdV hierarchy.

Through calculations, we obtain

Str(𝑉𝜕𝑈𝜕𝜆 ) = 2𝐴,Str( 𝜕𝑈𝜕𝑢1𝑉) = 2 (𝐵 + 𝜀𝑢1𝐴) ,Str( 𝜕𝑈𝜕𝑢2𝑉) = −2 (𝐶 + 𝜀𝑢2𝐴) ,Str( 𝜕𝑈𝜕𝑢3𝑉) = 2 (𝜌 + 𝜀𝑢4𝐴) ,Str( 𝜕𝑈𝜕𝑢4𝑉) = −2 (𝜎 + 𝜀𝑢3𝐴) .

(21)

Substituting the above results into the supertrace identity[3] and balancing the coefficients of 𝜆𝑛+2, we obtain𝛿𝛿𝑢 ∫ 𝑎𝑛+2𝑑𝑥 = (𝛾 − 𝑛 − 1)(

𝑏𝑛+1 + 𝜀𝑢1𝑎𝑛+1−𝑐𝑛+1 − 𝜀𝑢2𝑎𝑛+1𝜌𝑛+1 + 𝜀𝑢4𝑎𝑛+1−𝜎𝑛+1 − 𝜀𝑢3𝑎𝑛+1),𝑛 ≥ 0.(22)

Thus, we have𝐻𝑛+1 = ∫− 𝑎𝑛+2𝑛 + 1𝑑𝑥,𝛿𝐻𝑛+1𝛿𝑢 =(

𝑏𝑛+1 + 𝜀𝑢1𝑎𝑛+1−𝑐𝑛+1 − 𝜀𝑢2𝑎𝑛+1𝜌𝑛+1 + 𝜀𝑢4𝑎𝑛+1−𝜎𝑛+1 − 𝜀𝑢3𝑎𝑛+1),𝑛 ⩾ 0.(23)

Moreover, it is easy to find that

( 𝑏𝑛+1−𝑐𝑛+1𝜌𝑛+1−𝜎𝑛+1)= 𝑅1(𝑏𝑛+1 + 𝜀𝑢1𝑎𝑛+1−𝑐𝑛+1 − 𝜀𝑢2𝑎𝑛+1𝜌𝑛+1 + 𝜀𝑢4𝑎𝑛+1−𝜎𝑛+1 − 𝜀𝑢3𝑎𝑛+1), 𝑛 ⩾ 0, (24)

where 𝑅1 is given by

Advances in Mathematical Physics 5

𝑅1 =(1 − 2𝜀𝑢1𝜕−1𝑢2 −2𝜀𝑢1𝜕−1𝑢1 𝜀𝑢1𝜕−1𝑢3 𝜀𝑢1𝜕−1𝑢42𝜀𝑢2𝜕−1𝑢2 1 + 2𝜀𝑢2𝜕−1𝑢1 𝜀𝑢2𝜕−1𝑢3 −𝜀𝑢2𝜕−1𝑢4−2𝜀𝑢4𝜕−1𝑢2 −2𝜀𝑢4𝜕−1𝑢1 1 − 𝜀𝑢4𝜕−1𝑢3 𝜀𝑢4𝜕−1𝑢42𝜀𝑢3𝜕−1𝑢2 2𝜀𝑢3𝜕−1𝑢1 𝜀𝑢3𝜕−1𝑢3 1 − 𝜀𝑢3𝜕−1𝑢4). (25)

Therefore, superintegrable hierarchy (17) possesses thefollowing form:

𝑢𝑡𝑛 = 𝑅2( 𝑏𝑛+1−𝑐𝑛+1𝜌𝑛+1−𝜎𝑛+1)= 𝑅2𝑅1(𝑏𝑛+1 + 𝜀𝑢1𝑎𝑛+1−𝑐𝑛+1 − 𝜀𝑢2𝑎𝑛+1𝜌𝑛+1 + 𝜀𝑢4𝑎𝑛+1−𝜎𝑛+1 − 𝜀𝑢3𝑎𝑛+1)= 𝐽𝛿𝐻𝑛+1𝛿𝑢 , 𝑛 ≥ 0,

(26)

where

𝑅2 =( −4𝜀𝑢2𝜕−1𝑢2 −2 − 4𝜀𝑢2𝜕−1𝑢1 −2𝜀𝑢2𝜕−1𝑢3 2𝜀𝑢2𝜕−1𝑢42 − 4𝜀𝑢1𝜕−1𝑢2 −4𝜀𝑢1𝜕−1𝑢1 −2𝜀𝑢1𝜕−1𝑢3 2𝜀𝑢1𝜕−1𝑢4−4𝜀𝑢3𝜕−1𝑢2 −4𝜀𝑢3𝜕−1𝑢1 −2𝜀𝑢3𝜕−1𝑢3 −1 + 2𝜀𝑢3𝜕−1𝑢42𝜀𝑢4𝜕−1𝑢2 2𝜀𝑢4𝜕−1𝑢1 −1 + 𝜀𝑢4𝜕−1𝑢3 −𝜀𝑢4𝜕−1𝑢4 ), (27)

and super-Hamiltonian operator 𝐽 is given by𝐽 = 𝑅2𝑅1 =( −8𝜀𝑢2𝜕

−1𝑢2 −2 − 8𝜀𝑢2𝜕−1𝑢1 −4𝜀𝑢2𝜕−1𝑢3 4𝜀𝑢2𝜕−1𝑢42 − 8𝜀𝑢1𝜕−1𝑢2 −8𝜀𝑢1𝜕−1𝑢1 −4𝜀𝑢1𝜕−1𝑢3 4𝜀𝑢1𝜕−1𝑢4−6𝜀𝑢3𝜕−1𝑢2 −6𝜀𝑢3𝜕−1𝑢1 −3𝜀𝑢3𝜕−1𝑢3 −1 + 3𝜀𝑢3𝜕−1𝑢44𝜀𝑢4𝜕−1𝑢2 4𝜀𝑢4𝜕−1𝑢1 −1 + 2𝜀𝑢4𝜕−1𝑢3 −2𝜀𝑢4𝜕−1𝑢4 ). (28)In addition, generalized super-NLS-mKdV hierarchy (17)

also possesses the following super-Hamiltonian form:

𝑢𝑡𝑛 = 𝑅2𝐿( 𝑏𝑛−𝑐𝑛𝜌𝑛−𝜎𝑛)= 𝑅2𝐿𝑅1(𝑏𝑛 + 𝜀𝑢1𝑎𝑛−𝑐𝑛 − 𝜀𝑢2𝑎𝑛𝜌𝑛 + 𝜀𝑢4𝑎𝑛−𝜎𝑛 − 𝜀𝑢3𝑎𝑛)=𝑀𝛿𝐻𝑛𝛿𝑢 , 𝑛 ≥ 0,

(29)

where 𝑀 = 𝑅2𝐿𝑅1 = (𝑀𝑖𝑗)4×4 is the second super-Hamiltonian operator.

3. Self-Consistent Sources

Consider the linear system

(𝜑1𝑗𝜑2𝑗𝜑3𝑗)𝑥 = 𝑈(𝜑1𝑗𝜑2𝑗𝜑3𝑗),

(𝜑1𝑗𝜑2𝑗𝜑3𝑗)𝑡 = 𝑉(𝜑1𝑗𝜑2𝑗𝜑3𝑗).

(30)

From the result in [25], we set𝛿𝜆𝑗𝛿𝑢𝑖 = 13Str(Ψ𝑗 𝜕𝑈 (𝑢, 𝜆𝑗)𝛿𝑢𝑖 ) , 𝑖 = 1, 2, . . . , 4, (31)

6 Advances in Mathematical Physics

with Str on behalf of the supertrace, and

Ψ𝑗 = (𝜑1𝑗𝜑2𝑗 −𝜑21𝑗 𝜑1𝑗𝜑3𝑗𝜑22𝑗 −𝜑1𝑗𝜑2𝑗 𝜑2𝑗𝜑3𝑗𝜑2𝑗𝜑3𝑗 −𝜑1𝑗𝜑3𝑗 0 ) , 𝑗 = 1, 2, . . . , 𝑁. (32)From system (30), we get 𝛿𝜆𝑗/𝛿𝑢 as follows:𝑁∑𝑗=1

𝛿𝜆𝑗𝛿𝑢𝑖 = 𝑁∑𝑗=1((((((

Str(Ψ𝑗 𝛿𝑈𝛿𝑢1)Str(Ψ𝑗 𝛿𝑈𝛿𝑢2)Str(Ψ𝑗 𝛿𝑈𝛿𝑢3)Str(Ψ𝑗 𝛿𝑈𝛿𝑢4)

))))))

=( 2𝜀𝑢1 ⟨Φ1, Φ2⟩ − ⟨Φ1, Φ1⟩ + ⟨Φ2, Φ2⟩−2𝜀𝑢2 ⟨Φ1, Φ2⟩ + ⟨Φ1, Φ1⟩ + ⟨Φ2, Φ2⟩2𝜀𝑢4 ⟨Φ1, Φ2⟩ − 2 ⟨Φ2, Φ3⟩2𝜀𝑢3 ⟨Φ1, Φ2⟩ − 2 ⟨Φ1, Φ3⟩ ) ,(33)

where Φ𝑖 = (𝜑𝑖1, . . . , 𝜑𝑖𝑁)𝑇, 𝑖 = 1, 2, 3.So, we obtain the self-consistent sources of generalized

super-NLS-mKdV hierarchy (17):

𝑢𝑡𝑛 =(𝑢1𝑢2𝑢3𝑢4)𝑡𝑛 = 𝐽𝛿𝐻𝑛+1𝛿𝑢𝑖 + 𝐽 𝑁∑𝑗=1𝛿𝜆𝑗𝛿𝑢𝑖

= 𝐽( 𝑏𝑛+1 + 𝜀𝑢1𝑎𝑛+1−𝑐𝑛+1 − 𝜀𝑢2𝑎𝑛+1𝜌𝑛+1 + 𝜀𝑢4𝑎𝑛+1−𝜎𝑛+1 − 𝜀𝑢3𝑎𝑛+1)+ 𝐽( 2𝜀𝑢1 ⟨Φ1, Φ2⟩ − ⟨Φ1, Φ1⟩ + ⟨Φ2, Φ2⟩−2𝜀𝑢2 ⟨Φ1, Φ2⟩ + ⟨Φ1, Φ1⟩ + ⟨Φ2, Φ2⟩2𝜀𝑢4 ⟨Φ1, Φ2⟩ − 2 ⟨Φ2, Φ3⟩2𝜀𝑢3 ⟨Φ1, Φ2⟩ − 2 ⟨Φ1, Φ3⟩ ) .

(34)

For 𝑛 = 2, we get supersoliton equation with self-consistent sources as follows:𝑢1𝑡2 = 12𝑢2𝑥𝑥 + 𝑢32 − 𝑢2𝑢21 + 𝑢3𝑢3𝑥 + 𝑢4𝑢4𝑥 − 2𝑢2𝑢3𝑢4+ 𝜀 (2𝑢1𝑢2𝑢2𝑥 − 2𝑢21𝑢1𝑥 + 𝑢4𝑥𝑢3𝑢1 − 𝑢3𝑥𝑢4𝑢1− 2𝑢3𝑢4𝑢1𝑥 − 2𝑢2𝑢3𝑢4𝑥 − 2𝑢2𝑢4𝑢3𝑥 − 2𝑢2𝑢21𝑢3𝑢4+ 2𝑢32𝑢3𝑢4) + 2𝜀2𝑢2 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4)2

+ 2𝜀2𝑢2 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4) (𝑢22 − 𝑢21)+ 2𝜀𝑢1 𝑁∑𝑗=1

𝜑1𝑗𝜑2𝑗 − 𝑁∑𝑗=1

(𝜑21𝑗 − 𝜑22𝑗) ,𝑢2𝑡2 = 12𝑢1𝑥𝑥 − 𝑢31 + 𝑢1𝑢22 − 2𝑢1𝑢3𝑢4 + 𝑢3𝑢3𝑥 − 𝑢4𝑢4𝑥+ 𝜀 (2𝑢22𝑢2𝑥 − 2𝑢1𝑢2𝑢1𝑥 − 𝑢3𝑥𝑢4𝑢2 + 𝑢4𝑥𝑢3𝑢2− 2𝑢3𝑢4𝑢2𝑥 − 2𝑢1𝑢3𝑢4𝑥 − 2𝑢1𝑢4𝑢3𝑥)− 2𝜀2𝑢1 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4) (𝑢21 − 𝑢22)+ 2𝜀2 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4)2 𝑢1 − 2𝜀2𝑢1𝑢3 (𝑢21− 𝑢22) 𝑢4 − 2𝜀𝑢2 𝑁∑

𝑗=1

𝜑1𝑗𝜑2𝑗 + 𝑁∑𝑗=1

(𝜑21𝑗 + 𝜑22𝑗) ,𝑢3𝑡2 = 𝑢3𝑥𝑥 + 12𝑢3𝑢22 − 12𝑢3𝑢21 + 12𝑢4𝑢1𝑥 + 12𝑢4𝑢2𝑥+ 𝑢1𝑢4𝑥 + 𝑢2𝑢4𝑥 + 𝜀 (12𝑢3𝑢1𝑢2𝑥 − 12𝑢3𝑢2𝑢1𝑥+ 𝑢22𝑢3𝑥 − 𝑢21𝑢3𝑥 + 𝑢2𝑢2𝑥𝑢3 − 𝑢1𝑢1𝑥𝑢3− 2𝑢3𝑢4𝑢3𝑥) + 𝜀2 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4)2 𝑢3− 𝜀2𝑢3 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4) (𝑢21 − 𝑢22)+ 2𝜀𝑢4 𝑁∑

𝑗=1

𝜑1𝑗𝜑2𝑗 − 𝑁∑𝑗=1

𝜑2𝑗𝜑3𝑗,𝑢4𝑡2 = −𝑢4𝑥𝑥 − 12𝑢4𝑢22 + 12𝑢4𝑢21 + 12𝑢3𝑢2𝑥 − 12𝑢3𝑢1𝑥− 𝑢1𝑢3𝑥 + 𝑢2𝑢3𝑥 + 𝜀 (12𝑢4𝑢2𝑢1𝑥 − 12𝑢4𝑢1𝑢2𝑥− 𝑢1𝑢1𝑥𝑢4 + 𝑢2𝑢2𝑥𝑢4 − 𝑢21𝑢4𝑥 + 𝑢22𝑢4𝑥− 2𝑢3𝑢4𝑢4𝑥) − 𝜀2 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4)2 𝑢4− 𝜀2𝑢4 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4) (𝑢21 − 𝑢22)+ 2𝜀𝑢3 𝑁∑𝑗=1

𝜑1𝑗𝜑2𝑗 − 2 𝑁∑𝑗=1

𝜑1𝑗𝜑3𝑗,𝜑1𝑗𝑥 = (𝜆 + 𝑤) 𝜑1𝑗 + (𝑢1 + 𝑢2) 𝜑2𝑗 + 𝑢3𝜑3𝑗,𝜑2𝑗𝑥 = (𝑢1 − 𝑢2) 𝜑1𝑗 − (𝜆 + 𝑤) 𝜑2𝑗 + 𝑢4𝜑3𝑗,𝜑3𝑗𝑥 = 𝑢4𝜑1𝑗 − 𝑢3𝜑2𝑗, 𝑗 = 1, . . . , 𝑁.(35)

Advances in Mathematical Physics 7

4. Conservation Laws

In the following, we shall derive the conservation laws ofsupersoliton hierarchy. Introducing the variables𝐾 = 𝜑2𝜑1 ,𝐺 = 𝜑3𝜑1 , (36)then we obtain𝐾𝑥 = 𝑢1 − 𝑢2 − 2 (𝜆 + 𝑤)𝐾 + 𝑢4𝐺 − (𝑢1 + 𝑢2)𝐾2− 𝑢3𝐾𝐺,𝐺𝑥 = 𝑢4 − 𝑢3𝐾 − (𝜆 + 𝑤)𝐺 − (𝑢1 + 𝑢2) 𝐺𝐾 − 𝑢3𝐺2. (37)

Next, we expand 𝐾 and 𝐺 as series of the spectralparameter 𝜆

𝐾 = ∞∑𝑗=1

𝑘𝑗𝜆𝑗,𝐺 = ∞∑𝑗=1

𝑔𝑗𝜆𝑗. (38)Substituting (38) into (37), we raise the recursion formulas for𝑘𝑗 and 𝑔𝑗:𝑘𝑗+1 = −12𝑘𝑗𝑥 − 𝑤𝑘𝑗 + 12𝑢4𝑔𝑗 − 12 (𝑢1 + 𝑢2) 𝑗−1∑

𝑙=1

𝑘𝑙𝑘𝑗−𝑙− 12𝑢3𝑗−1∑

𝑙=1

𝑘𝑙𝑔𝑗−𝑙,𝑔𝑗+1 = −𝑔𝑗𝑥 − 𝑢3𝑘𝑗 − 𝑤𝑔𝑗 − (𝑢1 + 𝑢2) 𝑗−1∑

𝑙=1

𝑔𝑙𝑘𝑗−𝑙− 𝑢3𝑗−1∑𝑙=1

𝑔𝑙𝑔𝑗−𝑙, 𝑗 ≥ 2.(39)

We write the first few terms of 𝑘𝑗 and 𝑔𝑗:𝑘1 = 12 (𝑢1 − 𝑢2) ,𝑔1 = 𝑢4,𝑘2 = −14 (𝑢1 − 𝑢2) 𝑥− 12𝜀 (𝑢1 − 𝑢2) (12𝑢21 − 12𝑢22 + 𝑢3𝑢4) ,𝑔2 = −𝑢4𝑥 − 12 (𝑢1 − 𝑢2) (𝑢3 + 𝜀𝑢1 + 𝜀𝑢2) ,

𝑘3 = 18 (𝑢1 − 𝑢2)𝑥𝑥 − 18 (𝑢1 + 𝑢2) (𝑢1 − 𝑢2)2+ 14𝑤𝑥 (𝑢1 − 𝑢2) + 12 (𝑢1 − 𝑢2)𝑥 𝑤+ 12 (𝑢1 − 𝑢2) 𝑤2 − 12𝑢4𝑢4𝑥,𝑔3 = 𝑢4𝑥𝑥 + 12 (𝑢1 − 𝑢2) 𝑢3𝑥 − 12 (𝑢21 − 𝑢22) 𝑢4+ 34 (𝑢1 − 𝑢2)𝑥 𝑢3 + 𝑤𝑥𝑢4 + 2𝑤𝑢4𝑥+ 𝑤 (𝑢1 − 𝑢2) 𝑢3 + 𝑤2𝑢4, . . . .(40)

Note that 𝜕𝜕𝑡 [𝜆 + 𝑤 + (𝑢1 + 𝑢2)𝐾 + 𝑢3𝐺]= 𝜕𝜕𝑥 [𝐴 + (𝐵 + 𝐶)𝐾 + 𝜎𝐺] , (41)setting 𝛿 = 𝜆+𝑤+ (𝑢1 +𝑢2)𝐾+𝑢3𝐺, 𝜃 = 𝐴+ (𝐵+𝐶)𝐾+𝜎𝐺,which admitted that the conservation laws is 𝛿𝑡 = 𝜃𝑥. For (19),one infers that𝐴 = 𝜆2 + 12𝑢22 − 12𝑢21 − 𝑢3𝑢4,𝐵 = 𝑢1𝜆 + 12𝑢2𝑥 − 𝜀𝑢1 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4) ,𝐶 = 𝑢2𝜆 + 12𝑢1𝑥 − 𝜀𝑢2 (12𝑢21 − 12𝑢22 + 𝑢3𝑢4) ,𝜎 = 𝑢3𝜆 + 𝑢3𝑥 − 12𝜀𝑢3 (𝑢21 − 𝑢22) .

(42)

Expanding 𝛿 and 𝜃 as𝛿 = 𝜆 + 𝑤 + ∞∑

𝑗=1

𝛿𝑗𝜆−𝑗,𝜃 = 𝜆2 + 12𝑢22 − 12𝑢21 − 𝑢3𝑢4 + ∞∑

𝑗=1

𝜃𝑗𝜆−𝑗, (43)conserved densities and currents are the coefficients 𝛿𝑗, 𝜃𝑗,respectively. The first two conserved densities and currentsare read:𝛿1 = 12 (𝑢21 − 𝑢22) + 𝑢3𝑢4,𝜃1 = −14 (𝑢1 + 𝑢2) (𝑢1 − 𝑢2) 𝑥 + 14 (𝑢1 − 𝑢2) (𝑢1 + 𝑢2)⋅ 𝑥 − 14𝜀 (𝑢21 − 𝑢22) (12𝑢21 − 12𝑢22 + 𝑢3𝑢4) − 12 (𝑢21

8 Advances in Mathematical Physics

− 𝑢22) (12𝑢21 − 12𝑢22 + 𝑢3𝑢4) − 12𝜀 (𝑢21 − 𝑢22) 𝑢3 − 12⋅ 𝜀 (𝑢21 − 𝑢22) 𝑢3𝑢4 − 𝑢3𝑢4𝑥 + 𝑢3𝑥𝑢4,𝛿2 = −14 (𝑢1 + 𝑢2) (𝑢2 − 𝑢1)𝑥 − 12𝜀 (𝑢21 − 𝑢22) (12𝑢21− 12𝑢22 + 𝑢3 + 𝑢3𝑢4) − 𝑢3𝑢4𝑥,𝜃2 = (𝑢1 + 𝑢2) 𝑘3 − 14 [(𝑢1 + 𝑢2)𝑥− 𝜀 (𝑢1 + 𝑢2) (12𝑢21 − 12𝑢22 + 𝑢3𝑢4)]⋅ [12 (𝑢1 − 𝑢2) 𝑥 + (𝑢1 − 𝑢2)× (12𝑢21 − 12𝑢22 + 𝑢3𝑢4)] + 𝑢3𝑔3 − [𝑢3𝑥− 12𝜀 (𝑢21 − 𝑢22) 𝑢3] [𝑢4𝑥+ 12 (𝑢1 − 𝑢2) (𝑢3 + 𝜀𝑢1 + 𝜀𝑢2)] ,(44)

where 𝑘3 and 𝑔3 are given by (40).The recursion relationshipfor 𝛿𝑗 and 𝜃𝑗 is as follows:𝛿𝑗 = (𝑢1 + 𝑢2) 𝑘𝑗 + 𝑢3𝑔𝑗,𝜃𝑗 = (𝑢1 + 𝑢2) 𝑘𝑗+1 + 12 [(𝑢1 + 𝑢2)𝑥− 𝜀 (𝑢1 + 𝑢2) (12𝑢21 − 12𝑢22 + 𝑢3𝑢4)] 𝑘𝑗 + 𝑢3𝑔𝑗+1+ [𝑢3𝑥 − 12𝜀 (𝑢21 − 𝑢22) 𝑢3] 𝑔𝑗,

(45)

where 𝑘𝑗 and 𝑔𝑗 can be recursively calculated from (39). Wecan display the first two conservation laws of (18) as𝛿1𝑡 = 𝜃1𝑥,𝛿2𝑡 = 𝜃2𝑥, (46)where 𝛿1, 𝜃1, 𝛿2, and 𝜃2 are defined in (44). Then we canobtain the infinitely many conservation laws of (17) from(37)–(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 [26–29], 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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