1 Introduction

As soliton theory has been applied in almost all the natural sciences especially in the physics branches, such as fluid mechanics, plasma physics, nonlinear optics, classical field theory, and quantum field theory, the study on integrable systems has attracted the interest of many mathematicians and physicists. Conservation laws are an important feature of integrable systems in soliton theory. An infinite number of conservation laws for the KdV equation was first discovered by Miura, Gardner, and Kruscal in 1968 [1]. Then a lot of methods have been developed to find those conservation laws, mainly due to the contribution of Wadati et al. [2–4].

Super-integrable systems are an important part in integrable systems. A lot of scholars and experts have done research on the topic, and have achieved extensive results. In 1990, Hu presented the super-trace identity in his Ph.D. thesis [5]. Then the super-trace identity was mentioned and applied to establish the super-Hamiltonian structure of super-integrable systems, but he did not give its rigorous proof [6, 7]. Recently, Ma gave a systematic proof of the super-trace identity and the expression of its constant γ [8]. For application, he also obtained the super-Hamiltonian structures of the super-AKNS hierarchy and the super-Dirac hierarchy. Then many super-integrable hierarchies and their super-Hamiltonian structures are presented [9–14].

Soliton equations with self-consistent sources are also an important topic in recent research of soliton theory. Physically, the sources may result in solitary waves with a nonconstant velocity and, therefore, lead to a variety of dynamics of physical models. They are usually used to describe interactions between different solitary waves. Very recently, self-consistent sources for a super-CKdV equation hierarchy and super G-J hierarchy are presented [15–17]. In this paper, the self-consistent sources of super-Yang hierarchy is presented based on the theory of self-consistent sources. In fact, the self-consistent sources of the soliton hierarchy is just constraint flow and good supplements to related research in the field of noncommuting soliton equations [18–21]. Finally, we obtain infinitely many conservation laws for the hierarchy. In the calculation, it is worth noting that odd variables in the spectral problem are Fermi variables. The operation between extended Fermi variables constitute a Grassmann algebra.

2 A super-soliton hierarchy with self-consistent sources

In the following, we consider a set of super-Lie algebras G,

(1)

along with the communicative operation

Consider an auxiliary linear problem

(2)

where u=(u 1,…,u s )T,ϕ i =ϕ(x,t) are field variables on \(x\in R,t\in R;e_{i}=e_{i}(\lambda)\in \tilde{sl}(3)\ (i=1,2,\ldots,\allowbreak5); R_{1} \) is a pseudoregular element.

The compatibility of Eq. (2) gives rise to the following zero curvature equations:

$$ U_{nt}-V_{x}+[U_n,V_n]=0,\quad n=1,2, \ldots. $$
(3)

If an equation

$$ u_t=K(u) $$
(4)

can be worked out through Eq. (3), Eq. (3) is called a zero curvature representation of the super-evolution Eq. (4). If there is a super-Hamiltonian operator J and a functional J such that

$$ u_t=K(u)=\frac{\delta H_{n+1}}{\delta u}, $$
(5)

where

(6)

then Eq. (4) is called a super-Hamiltonian. If so, we say that Eq. (4) has a super-Hamiltonian structure.

Based on Eq. (2), we consider a new auxiliary linear problem. For N distinct λ j ,j=1,…,N, the systems of (2) become the following:

(7)

Based on the result in [22], we show that the following equation

$$ \frac{\delta H_k}{\delta u}+\sum_{j=1}^N \alpha_j\frac{\delta\lambda_j}{\delta u}=0 $$
(8)

holds true, where α j are constants, and Eq. (8) determines a finite-dimensional invariant set for the flows (6).

For (7), we may know that

(9)

where \(\operatorname{Str}\) denotes the super-trace of a super-matrix and

(10)

From Eq. (8) and Eq. (9), a kind of super-Hamiltonian soliton equation hierarchies with self-consistent sources is presented as follows:

$$ u_{t}=J\frac{\delta H_{n+1}}{\delta u_i}+J\sum_{j=1}^N \frac{\delta\lambda_j}{\delta u_i}. $$
(11)

3 The super-Yang hierarchy with self-consistent sources

As we know, the basis of Lie algebra A 1 can be as follows:

(12)

And corresponding loop algebra \(\tilde{A}_{1}\) is {e 1(n),e 2(n),e 3(n)|n∈ℤ} and satisfies x(n)=x⊗λ n,x∈A 1. By using the Tu scheme [23], we can obtain many evolution equations hierarchy. Based on the above basis, we consider the following spectral problem:

(13)

Take \(V^{(n)} = \sum_{m=0}^{n}(a_{m}e_{3}(n-m) + b_{m}e_{2}(n-m) + c_{m}e_{1}(n-m) + b_{n+1}e_{2}(0))\), the Yang hierarchy are obtained as follows [23]:

(14)

where

In the following, we will generalize Lie loop algebra \(\tilde{A_{1}}\) to the Lie loop super-algebra \(G_{2}= \{\sum_{i=1}^{n}\lambda _{i}e_{i},i=1,2,\ldots,5\}\)

(15)

where e 1,e 2,e 3 are even elements and e 4,e 5 are odd elements, and [⋅,⋅] and [⋅,⋅]+ denote the commutator and the anticommutator. Consider the following super isospectral problem [9]:

(16)

Denote

(17)

solving the stationary zero curvature equation

$$ V_{x}=[U,V], $$
(18)

we have

$$ \everymath{\displaystyle} \begin{cases} a_{mx} = 2c_{m+1} - 2rb_{m} + 2qc_{m} + \beta\rho_{m} + \alpha\delta _{m},\\\noalign{\vspace{3pt}} b_{mx} = -2ra_{m} + 2sc_{m} - \alpha\rho_{m} - \beta\delta_{m},\\ \noalign{\vspace{3pt}} c_{mx} = -2a_{m+1} - 2qa_{m} + 2sb_{m} - \alpha\rho_{m} + \beta \delta_{m},\\\noalign{\vspace{3pt}} \rho_{mx} = \delta_{m+1} - \alpha a_{m} - \beta b_{m} - \beta c_{m} + s\rho_{m} + q\delta_{m} + r\delta_{m},\\\noalign{\vspace{3pt}} \delta_{mx} = -\rho_{m+1} + \beta a_{m} + \alpha b_{m} - \alpha c_{m} - q\rho_{m} + r\rho_{m} - s\delta_{m},\\\noalign{\vspace{3pt}} a_{0} = c_{0} = \rho_{0} = \delta_{0} = 0, b_{0} = \mu= \mbox {constant} \neq0,\\\noalign{\vspace{3pt}} a_{1} = \mu s, b_{1} = 0, c_{1} = \mu r,\rho_{1} = \mu \alpha,\delta_{1} =\mu\beta,\\ \noalign{\vspace{3pt}} a_{2} = -\frac{1}{2}\mu r_{x} - \mu qs ,\\\noalign{\vspace{3pt}} b_{2} = \frac{1}{2}\mu r^{2} + \frac{1}{2}\mu s^{2} + \mu\alpha\beta, c_{2} = \frac{1}{2}\mu s_{x} - \mu qr, \\ \noalign{\vspace{3pt}}\rho_{2} = -\mu\beta_{x} - \mu q\alpha , \delta_{2} = \mu\alpha_{x} - \mu q\beta,\\\noalign{\vspace{3pt}} a_{3} = \mu\biggl(-\frac{1}{4} s_{xx} + \frac{1}{2} q_{x}r + qr_{x} + q^{2}s + \frac{1}{2} r^{2}s + \frac{1}{2} s^{3} + s\alpha\beta + \frac{1}{2}\alpha\beta_{x} - \frac{1}{2}\alpha_{x}\beta\biggr ),\\\noalign{\vspace{3pt}} b_{3} = \mu\biggl(\frac{1}{2}rs_{x} - \frac{1}{2}r_{x}s - qr^{2} - qs^{2} + \alpha\alpha_{x} + \beta\beta_{x} - 2q\alpha\beta\biggr),\\\noalign{\vspace{3pt}} c_{3} = \mu\biggl(-\frac{1}{4}r_{xx} - \frac{1}{2}q_{x}s - qs_{x} + \frac {1}{2} r^{3} + \frac{1}{2} rs^{2} + r\alpha\beta + q^{2}r - \frac{1}{2} \alpha\alpha_{x}+ \frac{1}{2}\beta\beta _{x}\biggr),\\\noalign{\vspace{3pt}} \rho_{3} = \mu\biggl(-\alpha_{xx} + q_{x}\beta+ 2q\beta_{x} - \frac {1}{2}r_{x}\beta+ \frac{1}{2}r^{2}\alpha + \frac{1}{2}s^{2}\alpha - \frac{1}{2}s_{x}\alpha+ q^{2}\alpha- r\beta_{x} - s\alpha _{x}\biggr) ,\\\noalign{\vspace{3pt}} \delta_{3} = \mu\biggl( -\beta_{xx} - q_{x}\alpha- 2 q\alpha_{x} - \frac {1}{2} r_{x}\alpha+ \frac{1}{2} r^{2}\beta + \frac{1}{2} s^{2}\beta+ \frac{1}{2} s_{x}\beta+ s\beta_{x} + q^{2}\beta- r\alpha_{x}\biggr). \end{cases} $$
(19)

From Eqs. (19), we can obtain the following recursions for

(20)

with

(21)

Denoting

a direct calculation reads,

(22)

Considering

$$ \Delta_{n} = b_{n+1}e_{2}(0), V^{(n)} = V^{(n)}_{+} + \Delta_{n}. $$
(23)

Substituting Eq. (23) into a zero curvature equation \(U_{t} - V^{(n)}_{x} + [U, V^{(n)}]=0\), we have the following super-integrable system:

(24)

where P n+1 meets as follows: P n+1=LP n . If taking α=β=0 in system (24), it can be reduced to the Yang hierarchy (14). Therefore, we call system (24) a super-Yang hierarchy.

Based on Lie super-algebra G 2 in (15) and the associated corresponding loop super-algebra \(\widetilde{G_{2}}\), a direct calculation gives

(25)

where \(a = a_{1}e_{1} + a_{2}e_{2} + a_{3}e_{3} + a_{4}e_{4} + a_{5}e_{5}\in\tilde{B}(0,1)\), \(\mathrm{ad}_{a}b=[a,b],a,b\in \tilde{B}(0,1)\), and the bracket [⋅,⋅] is the Lie super-bracket of \(\widetilde{G_{2}}\). If we define the compact super-trace as follows:

(26)

where c=(c ij )3×3,P=(p ij )5×5 and ab is matrix product of a and b, then we have

$$ \mathrm{str}(\mathrm{ad}_{a}\mathrm{ad}_{b}) = 3 \mathrm{str}(ab). $$
(27)

By using the super-trace identity in [8], we have

(28)

Comparing the coefficient of λ −n−1 yields

(29)

Since str(ad V ad V )=6μ 2=constant≠0, we can obtain γ=0 from the computation formula in [8]. Therefore, we conclude that

$$ P_{n+1} = \frac{\delta\mathcal{H}_{n}}{\delta u},\quad\mathcal{H}_{n} = \int \frac{2b_{n+2}}{n+1}\,\mathrm{d} x. $$
(30)

Hence, the super-Yang hierarchy (24) has the following super-Hamiltonian structure:

$$ u_{t} = JP_{n+1} = J\frac{\delta\mathcal{H}_{n}}{\delta u},\quad n\geq0. $$
(31)

For the integrable system (14) and super-integrable system (24), n=2 μ=1, we can get the nonlinear evolution equations and super-integrable couplings (also see [9]).

Next, we will construct the super-Yang hierarchy with self-consistent sources. Consider the linear system

(32)

From Eqs. (8)–(9), we have the following \(\frac{\delta\lambda_{j}}{\delta u}\):

where Ψ 1=(ϕ i1,…,ϕ N1)T, (i=1,2,3). And 〈⋅,⋅〉 stand for standard inner product.

According to (11), the integrable super-Yang hierarchy with self-consistent sources is proposed

(33)

For n=1, μ=1, we obtain the super-Yang equation with self-consistent sources

$$\everymath{\displaystyle} \begin{cases} q_t=rr_x+ss_x+4\alpha_x\beta+4\alpha\beta_x+2sqr-2qs\\ \noalign{\vspace{3pt}} \qquad{}-\partial\sum_{j=1}^N \bigl(\phi_{1j}^2+\phi_{2j}^2\bigr) -2s\sum_{j=1}^N \phi_{1j}\phi_{2j}\\ \noalign{\vspace{3pt}} \qquad{} +2r\sum_{j=1}^N \phi_{1j}\phi_{2j}-\beta\sum_{j=1}^N \phi_{2j}\phi_{3j}\\ \noalign{\vspace{3pt}} \qquad{}-\alpha\sum_{j=1}^N \phi_{1j}\phi_{3j},\\ \noalign{\vspace{3pt}} r_t=-sr^2-s^3+r_x+2qs-s\sum_{j=1}^N \bigl(\phi_{1j}^2+\phi _{2j}^2\bigr)\\ \noalign{\vspace{3pt}} \qquad{}-2\sum_{j=1}^N \phi_{1j}\phi_{2j},\\ \noalign{\vspace{3pt}} s_t=-r^3-rs^2+s^2-2qr-r\sum_{j=1}^N \bigl(\phi_{1j}^2+\phi _{2j}^2\bigr)\\ \qquad{}+2\sum_{j=1}^N \phi_{1j}\phi_{2j},\\ \noalign{\vspace{3pt}} \alpha_t=\frac{1}{2}\bigl(r^2+s^2\bigr)+\alpha_x-q\beta-\frac {1}{2}\beta\sum_{j=1}^N\bigl(\phi_{1j}^2+\phi_{2j}^2\bigr)\\ \qquad{}-\sum_{j=1}^N\phi_{2j}\phi_{3j},\\ \noalign{\vspace{3pt}} \beta_t=-\frac{1}{2}\alpha\bigl(r^2+s^2\bigr)+\beta_x+q\alpha\\ \noalign{\vspace{3pt}} \qquad{}-\frac{1}{2}\alpha \sum_{j=1}^N\bigl(\phi_{1j}^2+\phi_{2j}^2\bigr)\sum_{j=1}^N\phi _{1j}\phi_{2j}. \end{cases} $$

4 Conservation laws for the super-Yang hierarchy

In what follows, we will construct conservation laws of the super-Yang hierarchy. We introduce the variables

$$ K=\frac{\psi_{2}}{\psi_{1}},\qquad G=\frac{\psi_{3}}{\psi_{1}}. $$
(34)

From Eq. (7) and Eq. (12), we have

(35)

Expand K,G in the power of λ −1

$$ K=\sum_{j=1}^\infty k_j \lambda^{-j},\qquad G=\sum_{j=1}^\infty g_j\lambda^{-j}. $$
(36)

Substituting Eq. (30) into Eq. (29) and comparing the coefficients of the same power of λ, we obtain

(37)

and a recursion formula for k n and g n

(38)

Because of

$$ \frac{\partial}{\partial t} \bigl[s+(\lambda+q+r)K+2G \bigr]=\frac{\partial}{\partial t}(a+bK+\rho G) $$
(39)

where

Assume that σ=s+(λ+q+r)K+2G,θ=a+bK+ρG. Then Eq. (33) can be written as σ t =θ x , which is the right form of conservation laws. We expand σ and θ as series in powers of λ with the coefficients, which are called conserved densities and currents, respectively,

$$\sigma=s+\sum_{j=1}^\infty \sigma_j\lambda^{-j},\qquad\theta=m_0\lambda+\sum _{j=1}^\infty\theta_j \lambda^{-j} $$

where m 0,m 1 are constants of integration. The first two conserved densities and currents read

The recursion relation for σ n and θ n are

(40)

where f n and g n can be calculated from Eq. (40). The infinitely many conservation laws of Eq. (24) can be easily obtained from Eq. (29)–(34), respectively.

5 Remarks and conclusions

As there is little research on the topic of the integrable couplings of the self-consistent sources on super-integrable hierarchy, the super-Yang hierarchy with self-consistent sources enriched the content of self-consistent sources. Meanwhile, we also get the conservation laws of hierarchy. We must point out that the super-integrable hierarchies involve Fermi variables. In detail, the potential function in the coupling terms are Fermi variables, which satisfy the Grassmann algebra, namely α 2=β 2=0, αβ+βα=0 is used in the operations of the paper. We can also use this method to get more super-hierarchies with self-consistent sources.