A Integrable Generalized Super-NLS-mKdV Hierarchy, Its Self-Consistent Sources, and Conservation Laws
A generalized super-NLS-mKdV hierarchy is proposed related to Lie superalgebra ; 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.
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 , 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 [4–9]. In , 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 [11–18]. Conservation laws play an important role in mathematical physics. Since Miura et al. discovery of conservation laws for KdV equation in 1968 , lots of methods have been presented to find them [20–23].
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.
2. The Generalized Superequation Hierarchy
Based on the Lie superalgebra that is, along with the communicative operation
To set up the generalized super-NLS-mKdV hierarchy, the spectral problem is given as follows:withwhere with being an arbitrary even constant, is the spectral parameter, and are even potentials, and and are odd potentials. Note that spectral problem (3) with is reduced to super-NLS-mKdV hierarchy case .
Settingsolving the equation , we haveand, 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 datafrom the recursion relations in (6), we can obtain
Then we consider the auxiliary spectral problemwith
Supposesubstituting into the zero curvature equationwhere . Making use of (6), we havewhich guarantees that the following identity holds true:Choosing , we arrive at the following generalized super-NLS-mKdV hierarchy:where . The case of (17) with is exactly the standard supersoliton hierarchy .
When in (17), the flow is trivial. Taking , we can obtain second-order generalized super-NLS-mKdV equations
In what follows, we shall set up super-Hamiltonian structures for the generalized super-NLS-mKdV hierarchy.
Through calculations, we obtain
Substituting the above results into the supertrace identity  and balancing the coefficients of , we obtainThus, we haveMoreover, it is easy to find thatwhere is given by
Therefore, superintegrable hierarchy (17) possesses the following form:whereand super-Hamiltonian operator is given by
In addition, generalized super-NLS-mKdV hierarchy (17) also possesses the following super-Hamiltonian form:where is the second super-Hamiltonian operator.
3. Self-Consistent Sources
Consider the linear system
From the result in , we setwith on behalf of the supertrace, and
From system (30), we get as follows:where
So, we obtain the self-consistent sources of generalized super-NLS-mKdV hierarchy (17):
For , we get supersoliton equation with self-consistent sources as follows:
4. Conservation Laws
In the following, we shall derive the conservation laws of supersoliton hierarchy. Introducing the variablesthen we obtain
Note thatsetting , , which admitted that the conservation laws is . For (19), one infers that
Expanding and asconserved densities and currents are the coefficients , respectively. The first two conserved densities and currents are read:where and 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) aswhere , and are defined in (44). Then we can obtain the infinitely many conservation laws of (17) from (37)–(46).
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 [26–29], 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.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The project is supported by the National Natural Science Foundation of China (Grant nos. 11547175, 11271008, and 11601055) and the Aid Project for the Mainstay Young Teachers in Henan Provincial Institutions of Higher Education of China (Grant no. 2017GGJS145).