Nonlinear integrable couplings of a generalized super Ablowitz-Kaup-Newell-Segur hierarchy and its super bi-Hamiltonian structures

TL;DR

This paper introduces a new generalized matrix spectral problem related to the AKNS hierarchy, establishes its super soliton hierarchy, and derives super bi-Hamiltonian structures using super variational identities.

Contribution

It presents a novel generalized $5\times5$ matrix spectral problem linked to an enlarged Lie super algebra and constructs its super soliton hierarchy with Hamiltonian structures.

Findings

Proposes a new generalized spectral problem of AKNS type.

Establishes the super soliton hierarchy associated with the problem.

Derives super bi-Hamiltonian structures using super variational identities.

Abstract

In this paper, a new generalized $5\times5$ matrix spectral problem of Ablowitz-Kaup-Newell-Segur(AKNS) type associated with the enlarged matrix Lie super algebra is proposed and its corresponding super soliton hierarchy is established. The super variational identities is used to furnish super-Hamiltonian structures for the resulting super soliton hierarchy.

Equations

$$\displaystyle{\small\phi_{x}=M\phi,M=\left(\begin{array}[]{ccccc}\lambda&p&0&r&\alpha\\ q&-\lambda&s&0&\beta\\ 0&0&\lambda&p+r&0\\ 0&0&q+s&-\lambda&0\\ \beta&-\alpha&-\beta&\alpha&0\end{array}\right),\phi=\left(\begin{array}[]{c}\phi_{1}\\ \phi_{2}\\ \phi_{3}\\ \phi_{4}\\ \phi_{5}\end{array}\right),u=\left(\begin{array}[]{c}p\\ q\\ \alpha\\ \beta\\ r\\ s\end{array}\right)},$$

$$\displaystyle{\small\phi_{x}=U\phi,U=\left(\begin{array}[]{cccc}\lambda+\omega&p&0&r\\ q&-\lambda-\omega&s&0\\ 0&0&\lambda+\omega&p\\ 0&0&q&-\lambda-\omega\end{array}\right),\phi=\left(\begin{array}[]{c}\phi_{1}\\ \phi_{2}\\ \phi_{3}\\ \phi_{4}\end{array}\right),u=\left(\begin{array}[]{c}p\\ q\\ r\\ s\end{array}\right),}$$

$$\displaystyle E_{1}=\left(\begin{array}[]{ccc}1&0&0\\ 0&-1&0\\ 0&0&0\end{array}\right),E_{2}=\left(\begin{array}[]{ccc}0&1&0\\ 0&0&0\\ 0&0&0\end{array}\right),E_{3}=\left(\begin{array}[]{ccc}0&0&0\\ 1&0&0\\ 0&0&0\end{array}\right),$$

$$\displaystyle E_{4}=\left(\begin{array}[]{ccc}0&0&1\\ 0&0&0\\ 0&-1&0\end{array}\right),E_{5}=\left(\begin{array}[]{ccc}0&0&0\\ 0&0&1\\ 1&0&0\end{array}\right).$$

$$[E_1,E_2]=2E_2,[E_1,E_3]=-2E_3,[E_2,E_3]=E_1,$$

$$[E_1,E_4]=[E_2,E_5]=E_4,[E_1,E_5]=[E_4,E_3]=-E_5,[E_2,E_4]=[E_3,E_5]=0,$$

$$[E_4,E_4{]}_{+}=-2E_2,[E_5,E_5{]}_{+}=2E_3,[E_4,E_5{]}_{+}=[E_5,E_4{]}_{+}=E_1.$$

$$\displaystyle e_{1}=\left(\begin{array}[]{ccccc}1&0&0&0&0\\ 0&-1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&-1&0\\ 0&0&0&0&0\end{array}\right),e_{2}=\left(\begin{array}[]{ccccc}0&1&0&0&0\\ 0&0&0&0&0\\ 0&0&0&1&0\\ 0&0&0&0&0\\ 0&0&0&0&0\end{array}\right),e_{3}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\ 1&0&0&0&0\\ 0&0&0&0&0\\ 0&0&1&0&0\\ 0&0&0&0&0\end{array}\right),$$

$$\displaystyle e_{4}=\left(\begin{array}[]{ccccc}0&0&1&0&0\\ 0&0&0&-1&0\\ 0&0&1&0&0\\ 0&0&0&-1&0\\ 0&0&0&0&0\end{array}\right),e_{5}=\left(\begin{array}[]{ccccc}0&0&0&1&0\\ 0&0&0&0&0\\ 0&0&0&1&0\\ 0&0&0&0&0\\ 0&0&0&0&0\end{array}\right)e_{6}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\ 0&0&1&0&0\\ 0&0&0&0&0\\ 0&0&1&0&0\\ 0&0&0&0&0\end{array}\right)$$

$$\displaystyle e_{7}=\left(\begin{array}[]{ccccc}0&0&0&0&1\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&-1&0&1&0\end{array}\right)e_{8}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\ 0&0&0&0&1\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 1&0&-1&0&0\end{array}\right).$$

$$[e_1,e_2]=2e_2,[e_1,e_3]=-2e_3,[e_1,e_5]=-[e_2,e_4]=[e_4,e_5]=2e_5,[e_2,e_3]=e_1,$$

$$[e_1,e_6]=-[e_3,e_4]=[e_4,e_6]=-2e_6,[e_1,e_7]=[e_2,e_8]=e_7,[e_1,e_8]=-[e_3,e_7]=-e_8,$$

$$[e_2,e_6]=-[e_3,e_5]=[e_5,e_6]=e_4,[e_1,e_4]=[e_2,e_5]=[e_2,e_7]=[e_3,e_6]=[e_3,e_8]=0,$$

$$[e_4,e_7]=[e_4,e_8]=[e_5,e_7]=[e_5,e_8]=[e_6,e_7]=[e_6,e_8]=0$$

$$[e_7,e_8{]}_{+}=e_1-e_4,[e_7,e_7{]}_{+}=2e_5-2e_2,[e_8,e_8{]}_{+}=2e_3-2e_6.$$

$$sl(4,1)=sl(4,1)\otimes [\lambda ,{\lambda }^{-1}]=e_i{\lambda }^m,e_i\in sl(4,1),i=1,2,\dots ,8;m=0,\pm 1,\pm 2,\dots .$$

$$[e_i{\lambda }^m,e_j{\lambda }^n]=[e_i,e_j]{\lambda }^{m+n},\forall e_i,e_j\in sl(4,1).$$

$$\displaystyle{\small\phi_{x}=M\phi,M=\left(\begin{array}[]{ccccc}\lambda+h&p&0&r&\alpha\\ q&-\lambda-h&s&0&\beta\\ 0&0&\lambda+h&p+r&0\\ 0&0&q+s&-\lambda-h&0\\ \beta&-\alpha&-\beta&\alpha&0\end{array}\right),\phi=\left(\begin{array}[]{c}\phi_{1}\\ \phi_{2}\\ \phi_{3}\\ \phi_{4}\\ \phi_{5}\end{array}\right),u=\left(\begin{array}[]{c}p\\ q\\ \alpha\\ \beta\\ r\\ s\end{array}\right)},$$

$$N_x=[M,N].$$

$$\displaystyle{\small N=\left(\begin{array}[]{ccccc}A&B&E&F&\rho\\ C&-A&G&-E&\delta\\ 0&0&A+E&B+F&0\\ 0&0&C+G&-A-E&0\\ \delta&-\rho&-\delta&\rho&0\end{array}\right)},$$

$$\displaystyle\left\{\begin{array}[]{l}A_{x}=pC-qB-\alpha\delta+\beta\rho,\\ B_{x}=2\lambda B-2pA-2\alpha\rho+2hB,\\ C_{x}=-2\lambda C+2qA+2\beta\delta-2hC,\\ E_{x}=(p+r)G+rC-sB-(q+s)F-\alpha\delta-\beta\rho,\\ F_{x}=2\lambda F-2(p+r)E-2rA+2\alpha\rho+2hF,\\ G_{x}=-2\lambda G+2(q+s)E+2sA-2\beta\delta-2hG,\\ \rho_{x}=\lambda\rho+p\delta-\alpha A-\beta B+h\rho,\\ \delta_{x}=-\lambda\delta+\beta A-\alpha C+q\rho-h\delta.\\ \end{array}\right.$$

$$A=\sum _{m\ge 0}^n{a}_m{\lambda }^{-m},B=\sum _{m\ge 0}^n{b}_m{\lambda }^{-m},C=\sum _{m\ge 0}^n{c}_m{\lambda }^{-m},E=\sum _{m\ge 0}^n{e}_m{\lambda }^{-m},$$

$$F=\sum _{m\ge 0}^n{f}_m{\lambda }^{-m},G=\sum _{m\ge 0}^n{g}_m{\lambda }^{-m},\rho =\sum _{m\ge 0}^n{\rho }_m{\lambda }^{-m},\delta =\sum _{m\ge 0}^n{\delta }_m{\lambda }^{-m},$$

$$\displaystyle\left\{\begin{array}[]{l}a_{m,x}=pc_{m}-qb_{m}+\alpha\delta_{m}+\beta\rho_{m},\\ e_{m,x}=rc_{m}-sb_{m}+(p+r)g_{m}-(q+s)f_{m}-\alpha\delta_{m}-\beta\rho_{m},\\ b_{m+1}=\frac{1}{2}b_{m,x}+pa_{m}+\alpha\rho_{m}-hb_{m},\\ c_{m+1}=-\frac{1}{2}c_{m,x}+qa_{m}+\beta\delta_{m}-hc_{m},\\ f_{m+1}=\frac{1}{2}f_{m,x}+ra_{m}+(p+r)e_{m}-\alpha\rho_{m}-hf_{m},\\ g_{m+1}=-\frac{1}{2}g_{m,x}+sa_{m}+(q+s)e_{m}-\beta\delta_{m}-hg_{m},\\ \rho_{m+1}=\rho_{m,x}+\alpha a_{m}+\beta b_{m}-p\delta_{m}-h\rho_{m},\\ \delta_{m+1}=-\delta_{m,x}+\beta a_{m}-\alpha c_{m}+q\rho_{m}-h\delta_{m},\\ \end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{l}(c_{m+1},b_{m+1},\delta_{m+1},\rho_{m+1},g_{m+1},f_{m+1})^{T}=L(c_{m},b_{m},\delta_{m},\rho_{m},g_{m},f_{m})^{T},\\ a_{m}=\partial^{-1}(rc_{m}-qb_{m}+\alpha\delta_{m}+\beta\rho_{m}),\\ e_{m}=\partial^{-1}(rc_{m}-sb_{m}-\alpha\delta_{m}-\beta\rho_{m}+(p+r)g_{m}-(q+s)f_{m}),\end{array}\right.$$

$$L=\left(\begin{array}[]{ccc}L_{1}&L_{2}&0\\ L_{3}&L_{4}&0\\ L_{5}&-L_{2}&L_{6}\end{array}\right).$$

$$\displaystyle L_{1}=\left(\begin{array}[]{cc}q\partial^{-1}p-\frac{1}{2}\partial-h&-q\partial^{-1}q\\ p\partial^{-1}p&-p\partial^{-1}q+\frac{1}{2}\partial-h\end{array}\right),L_{2}=\left(\begin{array}[]{cc}q\partial^{-1}\alpha+\beta&q\partial^{-1}\beta\\ p\partial^{-1}\alpha&p\partial^{-1}\beta+\alpha\end{array}\right),$$

$$\displaystyle L_{3}=\left(\begin{array}[]{cc}\beta\partial^{-1}p-\alpha&-\beta\partial^{-1}q\\ \alpha\partial^{-1}p&-\alpha\partial^{-1}q+\beta\end{array}\right),L_{4}=\left(\begin{array}[]{cc}\beta\partial^{-1}\alpha-\partial-h&\beta\partial^{-1}\beta+q\\ \alpha\partial^{-1}\alpha-p&\alpha\partial^{-1}\beta+\partial-h\end{array}\right),$$

$$\displaystyle L_{5}=\left(\begin{array}[]{cc}s\partial^{-1}p+(q+s)\partial^{-1}r&-s\partial^{-1}q-(q+s)\partial^{-1}s\\ r\partial^{-1}p+(p+r)\partial^{-1}r&-r\partial^{-1}q-(p+r)\partial^{-1}s\end{array}\right),$$

$$\displaystyle L_{6}=\left(\begin{array}[]{cc}(q+s)\partial^{-1}(p+r)-\frac{1}{2}\partial-h&-(q+s)\partial^{-1}(q+s)\\ (p+r)\partial^{-1}(p+r)&-(p+r)\partial^{-1}(q+s)+\frac{1}{2}\partial-h\end{array}\right).$$

$$a_1=e_1=0,b_1=p,c_1=q,f_1=p+r,g_1=q+s,{\rho }_1=\alpha ,{\delta }_1=\beta ,$$

$$a_2=-\frac{1}{2}(pq+2\alpha \beta ),b_2=\frac{1}{2}{p}_x-hp,c_2=-\frac{1}{2}{q}_x-hq,$$

$$e_2=-(ps+qr+rs+\frac{1}{2}pq-\alpha \beta ),f_2=\frac{1}{2}{p}_x+\frac{1}{2}{r}_x-hp-hr,$$

$$g_2=-\frac{1}{2}{q}_x-\frac{1}{2}{s}_x-hq-hs,{\rho }_2={\alpha }_x-h\alpha ,{\delta }_2=-{\beta }_x-h\beta ,$$

$$a_3=\frac{1}{4}(p{q}_x-p_{x}q)+\alpha {\beta }_x-{\alpha }_x\beta +h(pq+2\alpha \beta ),$$

$$b_3=\frac{1}{4}{p}_{xx}-\frac{1}{2}{h}_{x}p-h{p}_x-\frac{1}{2}(pq+2\alpha \beta )p+\alpha {\alpha }_x+h^{2}p,$$

$$c_3=\frac{1}{4}{q}_{xx}+\frac{1}{2}{h}_{x}q+h{q}_x-\frac{1}{2}(pq+2\alpha \beta )q-\beta {\beta }_x+h^{2}q,$$

$$e_3=\frac{1}{2}(p{s}_x-p_{x}s+q_{x}r-q{r}_x+r{s}_x-r_{x}s+\frac{1}{2}p{q}_x-\frac{1}{2}{p}_{x}q)$$

$$-(\alpha {\beta }_x-{\alpha }_x\beta )+2h(ps+qr+rs+\frac{1}{2}pq-\alpha \beta ),$$

$$f_3=\frac{1}{4}{p}_{xx}+\frac{1}{2}{r}_{xx}-\frac{1}{2}{h}_x(p+2r)-h{p}_x-2h{r}_x-\frac{1}{2}(pq+2\alpha \beta )r$$

$$-\alpha {\alpha }_x+h^2(p+2r)-(p+r)(\frac{1}{2}pq+ps+qr+rs-\alpha \beta ),$$

$$g_3=\frac{1}{4}{q}_{xx}+\frac{1}{2}{s}_{xx}+\frac{1}{2}{h}_x(q+2s)+h{q}_x+2h{s}_x-\frac{1}{2}(pq+2\alpha \beta )s$$

Full text

PACS: 02.30.Ik, 05.45.Yv

Keywords: Lie superalgebras; superintegrable couplings; generalized super AKNS hirearchy; super bi-Hamiltonian structures

1 Introduction

It is known that super integrable systems provide interesting and important models in the supersymmetry theory Supersymmetry is originated in 1970s when physicists have proposed simple models with supersymmetric colors in string models and mathematical physics respectively. After that, Wess and Zumino [1] applied supersymmetry to the four-dimensional spacetime. Unfortunately, the supersymmetry partners of any particle have not been found so far, and it is generally believed that this symmetry is spontaneous rupture. In order to unify two kinds of particles with different spin and statistical properties-Boson and Fermion, theoretical physicists proposed the concept of hyperspace in the study of unified field theory and quantum field theory. Inspired by this, mathematicians developed the super analysis, the hypergeometric and the super algebra.

Due to the importance of supersymmetry in physics(especially in the exploration of the relationship between supersymmetric conformal field and chord theory), which has attracted great attention for the study of super integrable systems associated with Lie super algebra, many classical solition equations have been extended to be the super completely integrable system. For examples, the super Ablowitz-Kaup-Newell-Segur(AKNS) hierarchy [2-10], the super Dirac hierarchy [4,11-14], the super Kaup-Newell(KN) hierarchy [14-18], and others [19-28]. Among those, Hu [29] and Ma [30] has made a great contribution. Hu [29] proposed the super-trace identity, which is an effective tool to constructing super Hamiltonian structures of super integrable equations. In 2008, Ma given the proof of the super-trace identity and the super Hamiltonian structure of many super integrable equations is established by the super-trace identity [30, 31]. Meanwhile, constructing nonlinear superintegrable couplings with enlarging matrix Lie super algebra is one of pretty interesting topics in supermodel theory [32-38]. There are much richer mathematical structures behind nonlinear super integrable couplings than scalar superintegrable equations. Moreover, the study of superintegrable couplings generalizes the classical integrable couplings theory and provides clues toward complete classification of superintegrable equations [39-45].

In Ref.[34], You considered an enlarged super AKNS matrix spectral problem is given by

[TABLE]

where $\lambda$ is the spectral parameter, $p,q,r$ and $s$ are even potentials, but $\alpha$ and $\beta$ are odd ones. Take $\alpha=\beta=0$, the hierarchy (1.17) reduces to a nonlinear integrable couplings of the classical AKNS hierarchy [39]. Whose super Hamiltonian structure is furnished by super trace identity. Recently, Shen et al [46]. considered a generalized spatial spectral problem of AKNS integrable coupling as follows

[TABLE]

where $\omega=\epsilon(ps+qr)$ and $\lambda$ is the spectral parameter, $p,q,r$ and $s$ are commuting variables. Obviously, when $\epsilon=0$, this generalized spatial spectral problem (1.30) is reduced to a new case of AKNS integrable couplings [47]. Whose bi-Hamiltonian structures were constructed by using the component-trace identity in [46]. Inspired by those spatial spectral problem, in this paper, we would like to construct nonlinear super integrable couplings of a generalized super AKNS hirearchy.

The rest of this paper is organized as follows. In Section 2, we will enlarge the Lie superalgebra $sl(2,1)$ to the Lie superalgebra $sl(4,1)$. In Section 3, we will construct a generalization of the super AKNS integrable coupling hierarchy from zero curvature equations, based on the above-mentioned generalized spatial spectral problem (1.17). In Section 4, the super bi-Hamiltonian form will be presented for the obtained super integrable couplings of the generalized super AKNS hierarchy by making use of the super trace identity. For the sake of convenience, we will use the mathematical software Maple to deal with some complicated symbolic computations. And the last section is devoted to conclusions and discussions.

2 Enlargement of a Lie Superalgebra

In this section, we consider the Lie superalgebra $sl(2,1)$. Its basis is

[TABLE]

where $E_{1},E_{2},E_{3}$ are even elements and $E_{4},E_{5}$ are odd ones, $[.,.]$ and $[.,.]_{+}$ denote the commutator and the anticommutator, satisfy the following operational relations:

[TABLE]

Let us enlarge the Lie superalgebra $sl(2,1)$ to the Lie superalgebra $sl(4,1)$ with a basis

[TABLE]

where $e_{1},e_{2},e_{3},e_{4},e_{5},e_{6}$ are even and $e_{7},e_{8}$ are odd, $[.,.]$ and $[.,.]_{+}$ denote the commutator and the anticommutator. The generators of the Lie superalgebra $sl(4,1),e_{i},0\leq i\leq 8$, satisfy the following operational relations:

[TABLE]

Define a loop super algebra corresponding to the Lie super algebra $sl(4,1)$, and denote by

[TABLE]

The corresponding (anti)commutative relations are given as

[TABLE]

3 Nonlinear generalized super integrable couplings of the super AKNS hierarchy

In this section, we shall construct nonlinear integrable couplings of a generalized super AKNS hierarchy from an enlarging matrix Lie super algebra. Consider the following spatial spectral problem

[TABLE]

where $h=\mu(ps+qr+rs-2\alpha\beta)$ with $\mu$ being an arbitrary even constant, $\lambda$ is the spectral parameter, $p,q,r$ and $s$ are even potentials, and $\alpha$ and $\beta$ are odd potentials. Obviously, the spatial spectral problem (3.17) with $\mu=0$ reduces to the standard nonlinear integrable couplings of super AKNS hierarchy case [34].

In order to derive super integrable couplings of a generalized super integrable hierarchy associated with the spatial spectral problem (3.17), we solve the stationary zero curvature equation

[TABLE]

where

[TABLE]

in which the corresponding $A,B,C,E,F,G$ are even elements and $\rho,\delta$, are odd elements.

Substituting $M$ in (3.17) and $N$ in (3.24) into Eq.(3.18) yields

[TABLE]

Choosing

[TABLE]

and comparing the coefficients of the same powers of $\lambda$ in Eq.(3.33), we have

[TABLE]

which results in the recurrence relations

[TABLE]

where the recursion operator $L$ has the following form

[TABLE]

with

[TABLE]

Upon choosing the initial conditions $a_{0}=e_{0}=1,b_{0}=c_{0}=e_{0}=f_{0}=g_{0}=\rho_{0}=\delta_{0}=0$, all other a_{j},b_{j},c_{j},\rho_{j},\delta_{j}$$(j\geq 1) can be worked out uniquely by the recurrence relations (3.34) and by using of symbolic computation software(Maple). We list the first three sets as follows:

[TABLE]

Let us consider the spectral problem (3.17) with the following auxiliary spectral problem:

[TABLE]

where

[TABLE]

with $\Delta_{n}$ being the modification term. We setting

[TABLE]

and substitute Eq.(3.17) and Eq.(3.65) into the following zero curvature equation

[TABLE]

where $n\geq 0$. Making use of Eq.(3.33) yields

[TABLE]

which guarantees the following identity:

[TABLE]

Choosing $a=-2\mu e_{n+1}$, we can obtain the following hierarchy:

[TABLE]

When $n=2$ in Eq.(3.76), we obtain the first non-trivial flow as follows

[TABLE]

whose Lax pair consists of $M$ and $N^{(2)}$. $M$ is defined by (3.17) and $N^{(2)}$

[TABLE]

with

[TABLE]

4 Super bi-Hamiltonian structures

In what follows we shall find super bi-Hamiltonian structures of the nonlinear super integrable couplings of a generalized super AKNS hirearchy (3.76). To this end, we shall apply the super variational identities, which was discussed in [41]

[TABLE]

where $Str$ denotes the super trace. It is not difficult to find that

[TABLE]

Substituting Eq.(4.2) into Eq.(4.1), and comparing the coefficient of $\lambda^{-n-2}$ of both sides of Eq.(4.1) yields

[TABLE]

The identity with $n=0$ tells $\gamma=0$. Thus, we have

[TABLE]

where $\tilde{H}_{n+1}=-2\int\frac{2a_{n+2}+e_{n+2}}{n+1}dx$. Moreover, a direct calculation yields to the following recursive relationship

[TABLE]

where $R$ is given by

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Thus, the hierarchy (3.76) possesses the following super-Hamiltonian structure

[TABLE]

where

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

It could be proved that $J$ is a super Hamiltonian operator.

Specially, by making use of the recursive relationship (3.47), the hierarchy (3.76) possesses the following super-bi-Hamiltonian structure

[TABLE]

where the second compatible super-Hamiltonian operator $P=QLR=(P_{ij})_{6\times 6},i,j=1,2,\ldots,6,$ is given by

[TABLE]

with

[TABLE]

5 Conclusion and discussions

In this paper, we presented an approach for constructing nonlinear super integrable couplings of super soliton equations through enlarging matrix Lie super algebras. We took the Lie algebra $sl(2,1)$ as an example to illustrate the introduced idea to extend Lie super algebras. Based on the enlarged Lie super algebra $sl(4,1)$, we worked out nonlinear integrable couplings for a generalized super AKNS soliton hierarchy. The presented method in this paper can be applied to other generalized super integrable hierarchies, which will be our future problems to construct super integrable couplings.

**Acknowledgements

**

The work was supported by NSFC under the grants 11601055, 11371326, 11301331, and 11371086, NSF under the grant DMS-1664561, the 111 project of China (B16002), and the Distinguished Professorships by Shanghai University of Electric Power and Shanghai Second Polytechnic University.