A Integrable Generalized Super-NLS-mKdV Hierarchy , Its Self-Consistent Sources , and Conservation Laws

Copyright © 2018 HanyuWei and Tiecheng Xia.This is an open access article distributed under the Creative Commons Attribution License, 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 is put into super bi-Hamiltonian form with the aid of supertrace identity. Then, the super-NLS-mKdV hierarchy with self-consistent sources is set up. Finally, the infinitely many conservation laws of integrable super-NLS-mKdV hierarchy are presented.


Introduction
The superintegrable systems have aroused strong interest in recent years; many experts and scholars do research on the field and obtain lots of results [1,2].In [3], the supertrace identity and the proof of its constant  are given by Ma et al.As an application, super-Dirac hierarchy and super-AKNS hierarchy and its super-Hamiltonian structures have been furnished.Then, like the super-C-KdV hierarchy, the super-Tu hierarchy, the multicomponent super-Yang hierarchy, and so on were proposed [4][5][6][7][8][9].In [10], the binary nonlinearization and Bargmann symmetry constraints of the super-Dirac hierarchy were given.
Soliton equations with self-consistent sources have important applications in soliton theory.They are often used to describe interactions between different solitary waves, and they can provide variety of dynamics of physical models; some important results have been got by some scholars [11][12][13][14][15][16][17][18].Conservation laws play an important role in mathematical physics.Since Miura et al. discovery of conservation laws for KdV equation in 1968 [19], lots of methods have been presented to find them [20][21][22][23].
In this work, a generalized super-NLS-mKdV hierarchy is constructed.Then, we present the super bi-Hamiltonian form for the generalized super-NLS-mKdV hierarchy with the help of the supertrace identity.In Section 3, we consider the generalized super-NLS-mKdV hierarchy with self-consistent sources based on the theory of self-consistent sources.Finally, the conservation laws of the generalized super-NLS-mKdV hierarchy are given.
solving the equation   = [, ], we have and, from the above recursion relationship, we can get the recursion operator  which meets the following: where the recursive operator  is given as follows: Choosing the initial data from the recursion relations in (6), we can obtain Then we consider the auxiliary spectral problem with Suppose substituting  () into the zero curvature equation where  ≥ 0. Making use of ( 6), we have which guarantees that the following identity holds true: Choosing  = − +1 , we arrive at the following generalized super-NLS-mKdV hierarchy: where  ≥ 0. The case of (17) with  = 0 is exactly the standard supersoliton hierarchy [24].
Through calculations, we obtain Substituting the above results into the supertrace identity [3] and balancing the coefficients of  +2 , we obtain Thus, we have Moreover, it is easy to find that where  1 is given by Advances in Mathematical Physics 5 Therefore, superintegrable hierarchy (17) possesses the following form: where and super-Hamiltonian operator  is given by ) .

Conservation Laws
In the following, we shall derive the conservation laws of supersoliton hierarchy.Introducing the variables then we obtain (37) Next, we expand  and  as series of the spectral parameter Substituting (38) into (37), we raise the recursion formulas for   and   : We write the first few terms of   and   : Note that setting  =  +  + ( 1 +  2 ) +  3 ,  =  + ( + ) + , which admitted that the conservation laws is   =   .For (19), one infers that Expanding  and  as conserved densities and currents are the coefficients   ,   , respectively.The first two conserved densities and currents are read: where  3 and  3 are given by (40).The recursion relationship for   and   is as follows: where   and   can be recursively calculated from (39).We can display the first two conservation laws of (18) as where  1 ,  1 ,  2 , and  2 are defined in (44).Then we can obtain the infinitely many conservation laws of ( 17) from (37)-(46).

Conclusions
In this work, we construct the generalized super-NLS-mKdV hierarchy with bi-Hamiltonian forms with the help of variational identity.Self-consistent sources and conservation laws are also set up.In [26][27][28][29], the nonlinearization of AKNS hierarchy and binary nonlinearization of super-AKNS hierarchy were given.Can we do the binary nonlinearization for hierarchy (17)?The question may be investigated in further work.