Show Summary Details
More options …

# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo

IMPACT FACTOR 2018: 0.726
5-year IMPACT FACTOR: 0.869

CiteScore 2018: 0.90

SCImago Journal Rank (SJR) 2018: 0.323
Source Normalized Impact per Paper (SNIP) 2018: 0.821

Mathematical Citation Quotient (MCQ) 2018: 0.34

ICV 2018: 152.31

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 15, Issue 1

# Bi-integrable and tri-integrable couplings of a soliton hierarchy associated with SO(4)

Jian Zhang
• Corresponding author
• Department of Mathematics, Harbin University of Science and Technology, Rongcheng Campus, Rongcheng 264300, China
• Email
• Other articles by this author:
/ Chiping Zhang
/ Yunan Cui
Published Online: 2017-03-11 | DOI: https://doi.org/10.1515/math-2017-0017

## Abstract

In our paper, the theory of bi-integrable and tri-integrable couplings is generalized to the discrete case. First, based on the six-dimensional real special orthogonal Lie algebra SO(4), we construct bi-integrable and tri-integrable couplings associated with SO(4) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Moreover, Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by the variational identities.

MSC 2010: 37K05; 37K10; 35Q53

## 1 Introduction

An integrable coupling of a given system $ut=K(u)=K(x,t,u,ux,ut,uxx,uxt,utt,⋯)$(1)

is a triangular integrable system of the following form [1]: $ut=K(u),vt=S(u,v),$

where u is a function of variables t and x, ${u}_{x}=\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{u}_{t}=\frac{\mathrm{\partial }u}{\mathrm{\partial }t}.$ If S is nonlinear with respect to the second dependent variable v, the integrable coupling is called nonlinear.

A bi-integrable coupling of a given integrable system (1) is an enlarged triangular integrable system of the following form [2]: $ut=K(u),u1,t=S1(u,u1),u2,t=S2(u,u1,u2).$

Similarly, by a tri-integrable coupling, we mean an enlarged triangular integrable system of the following form [2]: $ut=K(u),u1,t=S1(u,u1),u2,t=S2(u,u1,u2),u3,t=S3(u,u1,u2,u3).$

Integrable couplings correspond to non-semisimple Lie algebras , and such Lie algebras can be written as semi-direct sums [3]: $g¯=g⊎gc,g−semisimple,gc−solvable.$

The notion of semi-direct sums $\overline{g}=g\uplus {g}_{c}$ means that g and gc satisfy [g, gc] gc, where [g, gc] = {[A, B] | Ag, Bgc}, with [·,·] denoting the Lie bracket of . Obviously, gc is an ideal of . The subscript c indicates a contribution to the construction of coupling systems. We also require the closure property between g and gc under the matrix multiplication: ggc, gcggc, where ggc = {AB | Ag, Bgc}.

Integrable couplings are effective tools for describing and explaining nonlinear phenomena of new evaluation equations. The study of integrable couplings generalizes the symmetry problem and other integrable properties of integrable equations. To enrich multi-component integrable equations, it has been an important task to explore more integrable properties for multi-integrable couplings. For example, one can find work on the integrable couplings [46]. It is always interesting to explore any new procedure for generating integrable couplings for different soliton hierarchies, even from existing non-semisimple Lie algebras.

The trace identity provides a powerful tool for constructing Hamiltonian structures of the hierarchies, which is proposed by Tu Guizhang [7, 8]. It is based on the Killing form on a semisimple Lie algebra. On a non-semisimple Lie algebra, the Killing form is always degenerate, and thus, the trace identity is no longer applicable. Recently, a variational identity is proposed in the theory of integrable couplings, which can be used to obtain the Hamiltonian structures in the case of non-semisimple Lie algebras [911].

Searching for integrable couplings of systems will become more and more meaningful to discuss the structures of integrable systems. To generate integrable couplings, bi-integrable couplings and tri-integrable couplings of soliton hierarchies, Ma Wenxiu propose a new way to generate integrable couplings through a few classes of matrix Lie algebras consisting of block matrices [2]. Recently, bi-integrable couplings and tri-integrable couplings for the Kdv hierarchy and the AKNS hierarchy have been studied considerably [12, 13]. Bi-integrable couplings of new soliton hierarchies associated with SO(3) and SO(4) have been constructed [14, 15].

In this paper, we will construct bi-integrable and tri-integrable couplings associated with SO(4) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Our work is essentially motivated by [1517].

## 2.1 Bi-integrable couplings associated with SO(4)

We take the class of 3 × 3 block matrices in the form of $M2(A,A1,A2)=AA1A20A+αA1A1+αA200A+αA1,$(2)

where α is an arbitrary nonzero constant and A, A1 and A2 are square matrices of the same order. In the following, we define the corresponding non-semisimple Lie algebra by a semi-direct sum $g¯(λ)=g⊎gC,$

with $g={M2(A,0,0)|A∈SO(4)~},gc={M2(0,A1,A2)|A1,A2∈SO(4)~},$

where the loop algebra $\stackrel{~}{SO\left(4\right)}$ is defined by $SO(4~)={A(λ)∈SO(4)|entriesofA(λ)−Laurent series inλ}.$

The corresponding matrix product reads $[M2(A,A1,A2),M2(B,B1,B2)]=M2(C,C1,C2),$

with C, C1 and C2 being defined by $C=[A,B],C1=[A,B1]+[A1,B]+α[A1,B1],C2=[A,B2]+[A2,B]+[A1,B1]+α[A1,B2]+α[A2,B1].$

We consider the Lie algebra G = {e1, e2, e3, e4, e5, e6} in SO(4) [18], where $e1=000−1000000001000,e2=000000−1001000000,e3=0−100100000000000,e4=00000000000−10010,e5=00100000−10000000,e6=0000000−100000100.$

The soliton hierarchy introduced in [16] has a spectral problem $ϕx=Uϕ=U(u,λ)ϕ,u=(u1,u2,u3,u4)T,$

with the spectral matrix U given as $U=U(u,λ)=λe2+u1e3+u2e4+u3e5+u4e6=0−u1u30u10−λ−u4−u3λ0−u20u4u20.$(3)

Based on this special non-semisimple Lie algebra (λ)we choose the corresponding enlarged spectral matrix Ū1 = M2(U, U1, U2) and the following supplementary spectral matrices: $U1=U1(u1,λ)=0−u1′u3′0u1′00−u4′−u3′00−u2′0u4′u2′0,u1=u1′u2′u3′u4′,$(4) $U2=U2(u2,λ)=0−u1″u3″0u1″00−u4″−u3″00−u2″0u4″u2″0,u2=u1″u2″u3″u4″.$(5)

In order to solve the enlarged stationary zero curvature equation ${\overline{V}}_{1x}=\left[{\overline{U}}_{1},{\overline{V}}_{1}\right],$ we take ${\overline{V}}_{1}={M}_{2}\left(V,{V}_{1},{V}_{2}\right),$ where V is defined as in [16] $V=V(u,λ)=0−cf−ac0−b−g−fb0−dagd0=∑i≥00−cifi−aici0−bi−gi−fibi0−diaigidi0λ−i,$(6)

V1 and V2 are defined similarly $V1=V1(u,u1,λ)=0−c′f′−a′c′0−b′−g′−f′b′0−d′a′g′d′0=∑i≥00−ci′fi′−ai′ci′0−bi′−gi′−fi′bi′0−di′ai′gi′di′0λ−i,$(7) $V2=V2(u,u1,u2,λ)=0−c″f″−a″c″0−b″−g″−f″b″0−d″a″g″d″0=∑i≥00−ci″fi″−ai″ci″0−bi″−gi″−fi″bi″0−di″ai″gi″di″0λ−i.$(8)

It now follows from the enlarged stationary zero curvature equation ${\overline{V}}_{1x}=\left[{\overline{U}}_{1},{\overline{V}}_{1}\right]$ that $Vx=[U,V],V1x=[U,V1]+[U1,V]+α[U1,V1],V2x=[U,V2]+[U2,V]+[U1,V1]+α[U1,V2]+α[U2,V1].$(9)

The above equation system equivalently leads to $ax=u4c−u2f−u1g+u3d,bx=u3c−u2g−u1f+u4d,cx=λf−u4a−u3b,dx=λg−u3a−u4b,fx=−λc+u1b+u2a,gx=−λd+u1a+u2b,$(10) $ax′=u4c′−u2f′−u1g′+u3d′+u4′c−u2′f−u1′g+u3′d+α(u4′c′−u2′f′−u1′g′+u3′d′),bx′=u3c′−u2g′−u1f′+u4d′+u3′c−u2′g−u1′f+u4′d+α(u3′c′−u2′g′−u1′f′+u4′d′),cx′=λf′−u4a′−u3b′−u4′a−u3′b−α(u4′a′+u3′b′),dx′=λg′−u3a′−u4b′−u3′a−u4′b−α(u3′a′+u4′b′),fx′=−λc′+u1b′+u2a′+u1′b+u2′a+α(u1′b′+u2′a′),gx′=−λd′+u1a′+u2b′+u1′a+u2′b+α(u2′b′+u1′a′),$(11)

and $ax″=u4c″−u2f″−u1g″+u3d″+u4″c−u2″f−u1″g+u3″d+α(u4′c″−u2′f″−u1′g″+u3′d″)+u4′c′−u2′f′−u1′g′+u3′d′+α(u4″c′−u2″f′−u1″g′+u3″d′),bx″=u3c″−u2g″−u1f″+u4d″+u3″c−u2″g−u1″f+u4″d+α(u3′c″−u2′g″−u1′f″+u4′d″)+u3′c′−u2′g′−u1′f′+u4′d′+α(u3″c′−u2″g′−u1″f′+u4″d′),cx″=λf″−u4a″−u3b″−u4″a−u3″b−α(u4′a″+u3′b″)−u4′a′−u3′b′−α(u4″a′+u3″b′),dx″=λg″−u3a″−u4b″−u3″a−u4″b−α(u3′a″+u4′b″)−u3′a′−u4′b′−α(u3″a′+u4″b′),fx″=−λc″+u1b″+u2a″+u1″b+u2″a+α(u1′b″+u2′a″)+u1′b′−u2′a′+α(u1″b′+u2″a′),gx″=−λd″+u1a″+u2b″+u1″a+u2″b+α(u1′a″+u2′b″)+u1′a′+u2′b′+α(u1″a′+u2″b′).$(12)

Now, we define the enlarged Lax matrices ${\overline{V}}_{1}^{\left[m\right]}=\left({\lambda }^{m}{\overline{V}}_{1}{\right)}_{+}={M}_{2}\left({V}^{\left[m\right]},{V}_{1}^{\left[m\right]},{V}_{2}^{\left[m\right]}\right),m\ge 0,$ where V[m]is defined as ${V}^{\left[m\right]}=\left({\lambda }^{m}V{\right)}_{+},\text{\hspace{0.17em}and}\phantom{\rule{thinmathspace}{0ex}}{V}_{i}^{\left[m\right]}=\left({\lambda }^{m}{V}_{i}{\right)}_{+},\phantom{\rule{thinmathspace}{0ex}}i=1,2.$

Solving the enlarged zero curvature equations ${\overline{U}}_{1{t}_{m}}-{\overline{V}}_{1x}^{\left[m\right]}+\left[{\overline{U}}_{1},{\overline{V}}_{1}^{\left[m\right]}\right]=0,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}m\ge 0,$ we get bi-integrable couplings of the soliton hierarchy in [16]: $u¯tm=utmu1tmu2tm=J1000J1000J1P1mP2mP3m=J1000J1000J1P1,m+1,$(13)

where $utm=u1u2u3u4tm=fm+1gm+1−cm+1−dm+1,u1tm=u1′u2′u3′u4′=fm+1′gm+1′−cm+1′−dm+1′,u2tm=u1″u2″u3″u4″tm=fm+1″gm+1″−cm+1″−dm+1″,J1=00−120000−121200001200,$

and $P1m=−2cm+1−2dm+1−2fm+1−2gm+1,P2m=−2cm+1′−2dm+1′−2fm+1′−2gm+1′,P3m=−2cm+1″−2dm+1″−2fm+1″−2gm+1″.$

## 2.2 Hamiltonian structures

In this section, in order to generate the Hamiltonian structure of the hierarchy (13), we use the corresponding variational identity [19] $δδu¯∫{V¯,U¯λ}dx=λ−γ∂∂λλγ{V¯,U¯u},$(14)

where {·,·} is a required bilinear form, which is symmetric, non-degenerate, and invariant under the Lie bracket.

$\mathrm{\forall }a=\left({a}_{1},\cdots ,{a}_{6},{a}_{{1}^{},}^{\prime }\cdots ,{a}_{6}^{\prime },{a}_{1}^{″},\cdots ,{a}_{6}^{″}\right)\in {\mathrm{R}}^{18},b=\left({b}_{1},\cdots ,{b}_{6},{b}_{1}^{\prime },\cdots ,{b}_{6}^{\prime },{b}_{1}^{″},\cdots ,{b}_{6}^{″}\right)\in {\mathrm{R}}^{18},$ we define the Lie bracket {·,·} on R18 as follows: $[a,b]=aTR¯1(b),R¯1(b)=R(b)R1(b)R2(b)0R(b)+αR1(b)R1(b)+αR2(b)00R(b)+αR1(b),$

where $Rb=00b6b5−b4−b300b5b6−b3−b4−b6−b500b2b1−b5−b600b1b2b4b3−b2−b100b3b4−b1−b200,$(15) $R1b=00b6′b5′−b4′−b3′00b5′b6′−b3′−b4′−b6′−b5′00b2′b1′−b5′−b6′00b1′b2′b4′b3′−b2′−b1′00b3′b4′−b1′−b2′00,$(16)

and $R2b=00b6″b5″−b4″−b3″00b5″b6″−b3″−b4″−b6″−b5″00b2″b1″−b5″−b6″00b1″b2″b4″b3″−b2″−b1″00b3″b4″−b1″−b2″00.$(17)

Following the properties of the matrix ${F}_{1}:{F}_{1}\left({\overline{R}}_{1}\left(b\right){\right)}^{T}=-{\overline{R}}_{1}\left(b\right){F}_{1}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{F}_{1}={F}_{1}^{T},$ we have $F1=η1η22η3η2αη2+2η32αη32η32αη30⊗r1r20000r2r1000000r1r20000r2r1000000r1r20000r2r1,$

where η1, η2, η3, r1, r2 are arbitrary constants. We choose r1 ±r2, then we easily have $det(F1)=4096(α2η1−αη2+2η3)6η312(r12−r22)9≠0.$

In order to get the Hamiltonian structure of the Lax integrable system, we define a bilinear form {a, b} on R18 as follows: ${a,b}=aTF1b.$(18)

Now we can compute that ${V¯1,U¯1,λ}=(r2a+r1b)η1+(r2a′+r1b′)η2+2(r2a″+r1b″)η3,{V¯1,U¯1,u1}=(r1c+r2d)η1+(r1c′+r2d′)η2+2(r1c″+r2d″)η3,{V¯1,U¯1,u2}=(r2c+r1d)η1+(r2c′+r1d′)η2+2(r2c″+r1d″)η3,{V¯1,U¯1,u3}=(r1f+r2g)η1+(r1f′+r2g′)η2+2(r1f″+r2g″)η3,{V¯1,U¯1,u4}=(r2f+r1g)η1+(r2f′+r1g′)η2+(2r2f″+2r1g″)η3,{V¯1,U¯1,u1′}=[r1c+r2d+α(r1c′+r2d′)]η2+2[r1c′+r2d′+α(r1c″+r2d″)]η3,{V¯1,U¯1,u2′}=[r2c+r1d+α(r2c′+r1d′)]η2+2[r2c′+r1d′+α(r2c″+r1d″)]η3,{V¯1,U¯1,u3′}=[r1f+r2g+α(r1f′+r2g′)]η2+2[r1f′+r2g′+α(r1f″+r2g″)]η3,{V¯1,U¯1,u4′}=[r2f+r1g+α(r2f′+r1g′)]η2+2[r2f′+r1g′+α(r2f″+r1g″)]η3,{V¯1,U¯1,u1″}=2[r1c+r2d+α(r1c′+r2d′)]η3,{V¯1,U¯1,u2″}=2[r2c+r1d+α(r2c′+r1d′)])η3,{V¯1,U¯1,u3″}=2[r1f+r2g+α(r1f′+r2g′)]η3,{V¯1,U¯1,u4″}=2[r2f+r1g+α(r2f′+r1g′)]η3,$

and furthermore, we use the formula [19]: $γ=−λ2ddλln⁡|{V¯,V¯}|,$(19)

to obtain that γ = 0. Applying the corresponding variational identity, we obtain the following Hamiltonian structure for the hierarchy of bi-integrable coupling (13): $u¯tm=K¯1,m(u¯)=J¯1δH¯1,mδu¯;m≥0,$

with the Hamiltonian operator $J¯1=η1η22η3η2αη2+2η32αη32η32αη30−1⊗00−120000−121200001200,$

and the Hamiltonian functionals $H¯1,m=−∫(r2am+2+r1bm+2)η1+(r2am+2′+r1bm+2′)η2+(2r2am+2″+2r1bm+2″)η3m+1dx,m≥0.$

Based on (10), (11), (12), a direct computation yields a recursion relation $P1,m+1=L¯1P1,m,$

where $L¯1=M2T(L1,L11,L21)=L100L11L1+αL110L21L11+αL21L1+αL11,$

with ${L}^{1},{L}_{1}^{1}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{L}_{2}^{1}$ being defined by $L1=l11l12l13l14l21l22l23l24l31l32l33l34l41l42l43l44,L11=l11′l12′l13′l14′l21′l22′l23′l24′l31′l32′l33′l34′l41′l42′l43′l44′,$(20) $L21=l11″l12″l13″l14″l21″l22″l23″l24″l31″l32″l33″l34″l41″l42″l43″l44″,$(21)

and $l11=l22=u1∂−1u3+u2∂−1u4,l12=l21=u1∂−1u4+u2∂−1u3,l13=l24=−∂−u1∂−1u1−u2∂−1u2,l14=l23=−u1∂−1u2−u2∂−1u1,l31=l42=∂+u4∂−1u4+u3∂−1u3,l32=l41=u4∂−1u3+u3∂−1u4,l33=l44=−u4∂−1u2−u3∂−1u1,l34=l43=−u4∂−1u1−u3∂−1u2,l11′=l22′=u1∂−1u3′+u2∂−1u4′+u1′∂−1u3+u2′∂−1u4+αu1′∂−1u3′+αu2′∂−1u4′,l12′=l21′=u1∂−1u4′+u2∂−1u3′+u1′∂−1u4+u2′∂−1u3+αu1′∂−1u4′+αu2′∂−1u3′,l13′=l24′=−u1∂−1u1′−u2∂−1u2′−u1′∂−1u1−u2′∂−1u2−αu1′∂−1u1′−αu2′∂−1u2′,l14′=l23′=−u1∂−1u2′−u2∂−1u1′−u1′∂−1u2−u2′∂−1u1−αu1′∂−1u2′−αu2′∂−1u1′,l31′=l42′=u4∂−1u4′+u3∂−1u3′+u4′∂−1u4+u3′∂−1u3+αu3′∂−1u3′+αu4′∂−1u4′,l32′=l41′=u4∂−1u3′+u3∂−1u4′+u4′∂−1u3+u3′∂−1u4+αu3′∂−1u4′+αu4′∂−1u3′,l33′=l44′=−u4∂−1u2′−u3∂−1u1′−u4′∂−1u2−u3′∂−1u1−αu3′∂−1u1′−αu4′∂−1u2′,l34′=l43′=−u4∂−1u1′−u3∂−1u2′−u4′∂−1u1−u3′∂−1u2−αu3′∂−1u2′−αu4′∂−1u1′,l11″=l22″=u1∂−1u3″+u2∂−1u4″+u1″∂−1u3+u2″∂−1u4+u1′∂−1u3′+u2′∂−1u4′+α(u1′∂−1u3″+u2′∂−1u4″+u1″∂−1u3′+u2″∂−1u4′),l12″=l21″=u1∂−1u4″+u2∂−1u3″+u1″∂−1u4+u2″∂−1u3+u1′∂−1u4′+u2′∂−1u3′+α(u1′∂−1u4″+u2′∂−1u3″+u1″∂−1u4′+u2″∂−1u3′),l13″=l24″=−u1∂−1u1″−u2∂−1u2″−u1″∂−1u1−u2″∂−1u2−u1′∂−1u1′−u2′∂−1u2′+α(−u1′∂−1u1″−u2′∂−1u2″−u1″∂−1u1′−u2″∂−1u2′),l14″=l23″=−u1∂−1u2″−u2∂−1u1″−u1″∂−1u2−u2″∂−1u1−u1′∂−1u2′−u2′∂−1u1′+α(−u1′∂−1u2″−u2′∂−1u1″−u1″∂−1u2′−u2″∂−1u1′),l31″=l42″=u4∂−1u4″+u∂−1u3″+u4″∂−1u4+u3″∂−1u+u4′∂−1u4′+u3′∂−1u3′+α(u4′∂−1u4″+u3′∂−1u3″+u4″∂−1u4′+u3″∂−1u3′),l32″=l41″=u4∂−1u″3+u3∂−1u″4+u′4′−1∂u3+u3″∂−1u4+u4′∂−1u3′+u3′∂−1u4′+α(u4′∂−1u3″+u3′∂−1u4″+u4″∂−1u3′+u3″∂−1u4′),l33″=l44″=−u4∂−1u2″−u3∂−1u1″−u4″∂−1u2−u3″∂−1u1−u4′∂−1u2′−u3′∂−1u1′+α(−u4′∂−1u2″−u3′∂−1u1″−u4″∂−1u2′−u3″∂−1u1′),l34″=l43″=−u4∂−1u1″−u3∂−1u2″−u4″∂−1u1−u3″∂−1u2−u4′∂−1u1′−u3′∂−1u2′+α(−u4′∂−1u1″−u3′∂−1u2″−u4″∂−1u1′−u3″∂−1u2′),$

where $\mathrm{\partial }=\frac{d}{dx}\phantom{\rule{0.056em}{0ex}}\text{\hspace{0.17em}and}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{0.056em}{0ex}}{\mathrm{\partial }}^{-1}=\int \frac{d}{dx}dx.$

## 3.1 Tri-integrable couplings associated with SO(4)

We take the class of 4 × 4 block matrices in the form of $M3(A,A1,A2,A3)=AA1A2A30A+βA1βA2A1+βA300A+βA1+μA2vA2000A+βA1,$(22)

where β, μ, v are arbitrary nonzero constants and A, A1, A2 and A3 are square matrices of the same order. In the following, we define the corresponding non-semisimple Lie algebra ḡ(λ) by a semi-direct sum $g¯(λ)=g⊎gC,$

with $g={M3(A,0,0,0)|A∈SO(4~)},gc={M3(0,A1,A2,A3)|A1,A2,A3∈SO(4~)},$

where the loop algebra $\stackrel{~}{SO\left(4\right)}$ is defined by $SO(4)~={A(λ)∈SO(4)|entries ofA(λ)−Laurent series inλ}.$

The corresponding matrix product reads $[M3(A,A1,A2,A3),M3(B,B1,B2,B3)]=M3(C,C1,C2,C3),$

with C, C1,C2 and C3 being defined by $C=[A,B],C1=[A,B1]+[A1,B]+β[A1,B1],C2=[A,B2]+[A2,B]+μ[A2,B2]+β[A1,B2]+β[A2,B1],C3=[A,B3]+[A3,B]+β[A3,B1]+β[A1,B3]+[A1,B1]+v[A2,B2].$

We will adopt the following enlarged spectral matrix to construct tri-integrable couplings for SO(4) hierarchy $U¯2=U¯2(u¯,λ)=M3(U,U1,U2,U3)∈g¯(λ),$

with U = U (u, λ) being defined as in (3), U1, U2 are defined by (4) and (5), also the supplementary spectral matrix U3 reads $U3=U3(u3,λ)=0−u1‴u3‴0u1″00−u4‴−u3‴00−u2‴0u4‴u2‴0,u3=u1‴u2‴u3‴u4‴.$

We take $V¯2=V¯2(u¯,λ)=M3(V,V1,V2,V3)=VV1V2V30V+βV1βV2V1+βV300V+βV1+μV2vV2000V+βV1,$

where V, V1, V2 are defined by (6), (7) and (8), also V3 reads $V3=V3(u,u1,u2,u3,λ)=0−c‴f‴−a‴c‴0−b‴−g‴−f‴b‴0−d‴a‴g‴d‴0,$(23)

and $a‴=∑i≥0ai‴λ−i,b″=∑i≥0bi‴λ−i,c‴=∑i≥0ci‴λ−i,f‴=∑i≥0fi‴λ−i,g‴=∑i≥0gi‴λ−i.$

It now follows from the enlarged stationary zero curvature equation ${\overline{V}}_{2x}=\left[{\overline{U}}_{2},{\overline{V}}_{2}\right]$that $Vx=[U,V],V1x=[U,V1]+[U1,V]+β[U1,V1],V2x=[U,V2]+[U2,V]+μ[U2,V2]+β[U1,V2]+β[U2,V1],V3x=[U,V3]+[U3,V]+β[U3,V1]+[U1,V1]+β[U1,V3]+v[U2,V2].$

The above equation system equivalently leads to $ax=u4c−u2f−u1g+u3d,bx=u3c−u2g−u1f+u4d,cx=λf−u4a−u3b,dx=λg−u3a−u4b,fx=−λc+u1b+u2a,gx=−λd+u1a+u2b,$(24) $ax′=u4c′−u2f′−u1g′+u3d′+u4′c−u2′f−u1′g+u3′d+β(u4′c′−u2′f′−u1′g′+u3′d′),bx′=u3c′−u2g′−u1f′+u4d′+u3′c−u2′g−u1′f+u4′d+β(u3′c′−u2′g′−u1′f′+u4′d′),cx′=λf′−u4a′−u3b′−u4′a−u3′b−β(u4′a′+u3′b′),dx′=λg′−u3a′−u4b′−u3′a−u4′b−β(u3′a′+u4′b′),fx′=−λc′+u1b′+u2a′+u1′b+u2′a+β(u1′b′+u2′a′),gx′=−λd′+u1a′+u2b′+u1′a+u2′b+β(u2′b′+u1′a′),$(25) $ax″=u4c″−u2f″−u1g″+u3d″+u4″c−u2″f−u1″g+u3″d+μu4″c″−u2″f″−u1″g″+u3″d″+βu4″c′−u2″f′−u1″g′+u3″d′,+βu4′c″−u2′f″−u1′g″+u3′d″bx″=u3c″−u2g″−u1f″+u4d″+u3″c−u2″g−u1″f+u4″d+μu3′c′−u2′g′−u1′f′+u4′d′+βu3″c′−u2′g′−u1″f′+u4″d′,+βu3′c″−u2′g″−u1′f″+u4′d″cx″=λf″−u4a″−u3b″−u4″a−u3″b−βu4′a″+u3′b″−μu4″a″+u3″b″−βu4″a′+u3″b′,dx″=λg″−u3a″−u4b″−u3″a−u4″b−βu3′a″+u4′b″−μu3″a″+u4″b″−βu3″a′+u4″b′,fx″=−λc″+u1b″+u2a″+u1″b+u2″a+βu1′b″+u2′a″+μu1″b″+u2″a″+βu1″b′+u2″a′,gx″=−λd″+u1a″+u2b″+u1″a+u2″b+βu1′a″+u2′b″+μu1″a″+u2″b″+βu1″a′+u2″b′,$(26) $ax‴=u4c‴−u2f‴−u1g‴+u3d‴+u4‴c−u2‴f−u1‴g+u3‴d+βu4‴c′−u2‴f′−u1‴g′+u3‴d′+βu4′c‴−u2′f‴−u1′g‴+u3′d‴+u4′c′−u2′f′−u1′g′+u3′d′+vu4″c″−u2″f″−u1″g″+u3″d″,bx‴=u3c‴−u2g‴−u1f‴+u4d‴+u3‴c−u2‴g−u1‴f+u4‴d+βu3‴c′−u2‴g′−u1‴f′+u4‴d′+βu3′c‴−u2′g‴−u1′f‴+u4′d‴+u3′c′−u2′g′−u1′f′+u4′d′+vu3″c″−u2″g″−u1″f″+u4″d″,cx‴=λf‴−u4a‴−u3b‴−u4‴a−u3‴b−βu4‴a′+u3‴b′−βu4′a‴+u3′b‴−u4′a′−u3′b′−vu4″a″+u3″b″,dx‴=λg‴−u3a‴−u4b‴−u3‴a−u4‴b−βu3‴a′+u4‴b′−βu3′a‴+u4′b‴−u3′a′−u4′b′−vu3″a″+u4″b″,fx‴=−λc‴+u1b‴+u2a‴+u1‴b+u2‴a+βu1‴b′+u2‴a′+βu1′b‴+u2′a‴+u1′b′+u2′a′+vu1″b″+u2″a″,gx‴=−λd‴+u1a‴+u2b‴+u1‴a+u2‴b+βu1‴a′+u2‴b′+βu1′a‴+u2′b‴+u1′a′+u2′b′+vu1″a″+u2″b″.$(27)

Now, we define the enlarged Lax matrices ${\overline{V}}_{2}^{\left[m\right]}=\left({\lambda }^{m}{\overline{V}}_{2}{\right)}_{+}={M}_{3}\left({V}^{\left[m\right]},{V}_{1}^{\left[m\right]},{V}_{2}^{\left[m\right]},{V}_{3}^{\left[m\right]}\right),m\ge 0,$ where V[m] is defined as ${V}^{\left[m\right]}=\left({\lambda }^{m}V\right)+,\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{V}_{i}^{\left[m\right]}=\left({\lambda }^{m}{V}_{i}\right)+,i=1,2,3.$

Solving the enlarged zero curvature equations ${\overline{U}}_{2{t}_{m}}-{\overline{V}}_{2x}^{\left[m\right]}+\left[{\overline{U}}_{2},{\overline{V}}_{2}^{\left[m\right]}\right]=0,m\ge 0,$ we get tri-integrable couplings of the soliton hierarchy in [16]: $u¯tm=utmu1tmu2tmu3tm=J10000J10000J10000J1P1mP2mP3mP4m=J10000J10000J10000J1P2,m+1,$(28)

where $utm=u1u2u3u4tm=fm+1gm+1−cm+1−dm+1,u1tm=u1′u2′u3′u4′=fm+1′gm+1′−cm+1′−dm+1′,u2tm=u1″u2″u3″u4″tm=fm+1″gm+1″−cm+1″−dm+1″,u3tm=u1‴u2‴u3‴u4‴=fm+1‴gm+1‴−cm+1‴−dm+1‴,$ $J1=00−120000−121200001200,$

and $P1m=−2cm+1−2dm+1−2fm+1−2gm+1,P2m=−2cm+1′−2dm+1′−2fm+1′−2gm+1′,P3m=−2cm+1″−2dm+1″−2fm+1″−2gm+1″,P4m=−2cm+1‴−2dm+1‴−2fm+1‴−2gm+1‴.$

## 3.2 Hamiltonian structures

In this section, in order to generate the Hamiltonian structure of the hierarchy (28), we also use the corresponding variational identity [19] $δδu¯∫{V¯,U¯λ}dx=λ−γ∂∂λλγ{V¯,U¯u},$

where {·,·} is a required bilinear form, which is symmetric, non-degenerate, and invariant under the Lie bracket.

$\mathrm{\forall }a=\left({a}_{1},\cdots ,{a}_{6},{a}_{1}^{\prime },\cdots ,{a}_{6}^{\prime },{a}_{1}^{″},\cdots ,{a}_{6}^{″},{a}_{1}^{‴},\cdots ,{a}_{6}^{‴}\right)\in {\mathrm{R}}^{24},b=\left({b}_{1},\cdots ,{b}_{6},{b}_{1}^{\prime },\cdots {b}_{6}^{\prime },{b}_{1}^{″},\cdots ,{b}_{6}^{″},{b}_{1}^{‴},\cdots ,{b}_{6}^{‴}\right)\in {\mathrm{R}}^{24},$ we define the Lie bracket [·,·] on R24 as follows: $\left[a,b\right]={a}^{T}{\overline{R}}_{2}\left(b\right),$ where $R¯2(b)=R(b)R1(b)R2(b)R3(b)0R(b)+βR1(b)βR2(b)R1(b)+βR3(b)00R(b)+βR1(b)+μR2(b)vR2(b)000R(b)+βR1(b),$

with R(b), R1(b)and R2.(b)being defined by (15), (16) and (17), and $R3(b)=00b6‴b5‴−b4‴−b3‴00b5‴b6‴−b3‴−b4‴−b6‴−b5‴00b2‴b1‴−b5‴−b6‴00b1‴b2‴b4‴b3‴−b2‴−b1‴00b3‴b4‴−b1‴−b2‴00.$

Following the properties of the matrix ${F}_{2}:{F}_{2}\left({\overline{R}}_{2}\left(b\right){\right)}^{T}={\overline{R}}_{2}\left(b\right){F}_{2}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{F}_{2}={F}_{2}^{T},$ we have $F2=η1η2η3η4η2βη2+η4βη3βη4η3βη3μη3+vη40η4βη400⊗r1r20000r2r1000000r1r20000r2r1000000r1r20000r2r1,$

where ⊗ is the Kronecker product and η1, η2, η3, η4, r1, r2 are arbitrary constants. We choose r1 ±r2, then we easily have $det(F2)=η412(β2η1−βη2+η4)6(μη3+vη4)6(r12−r22)12≠0.$

In order to get the Hamiltonian structure of the Lax integrable system, we define a bilinear form {a, b} on R24 as follows: ${a,b}=aTF2b.$(29)

Now, it is easy to compute that ${V¯2,U¯2,λ}=(r2a+r1b)η1+(r2a′+r1b′)η2+(r2a″+r1b″)η3+(r2a‴+r1b‴)η4,{V¯2,U¯2,λ}=(r2a+r1b)η1+(r2a′+r1b′)η2+(r2a″+r1b″)η3+(r2a‴+r1b‴)η4,{V¯2,U¯2,u1}=(r2d+r1c)η1+(r2d′+r1c′)η2+(r1c″+r2d″)η3+(r2d‴+r1c‴)η4,{V¯2,U¯2,u2}=(r2c+r1d)η1+(r2c′+r1d′)η2+(r2c″+r1d″)η3+(r1d‴+r2c‴)η4,{V¯2,U¯2,u3}=(r2g+r1f)η1+(r2g′+r1f′)η2+(r2g″+r1f″)η3+(r1f‴+r2g‴)η4,{V¯2,U¯2,u4}=(r2f+r1g)η1+(r2f′+r1g′)η2+(r2f″+r1g″)η3+(r1g‴+r2f‴)η4,{V¯2,U¯2,u1′}=[(r2d+r1c)+β(r1c′+r2d′)]η2+β(r2d″+r1c″)η3+[(r2d′+r1c′)+β(r1c‴+r2d‴)]η4,{V¯2,U¯2,u2′}=[(r2c+r1d)+β(r2c′+r1d′)]η2+β(r2c″+r1d″)η3+[(r2c′+r1d′)+β(r2c‴+r1d‴)]η4,{V¯2,U¯2,u3′}=[(r2g+r1f)+β(r2g′+r1f′)]η2+β(r2g″+r1f″)η3+[(r2g′+r1f′)+β(r2g‴+r1f‴)]η4,{V¯2,U¯2,u4′}=[(r2f+r1g)+β(r2f′+r1g′)]η2+β(r2f″+r1g″)η3+[(r2f′+r1g′)+β(r2f‴+r1g‴)]η4,{V¯2,U¯2,u1″}=[(r2d+r1c)+β(r2d′+r1c′)+μ(r2d″+r1c″)]η3+v(r2d″+r1c″)η4,{V¯2,U¯2,u2″}=[(r2c+r1d)+β(r2c′+r1d′)+μ(r2c″+r1d″)]η3+v(r2c″+r1d″)η4,{V¯2,U¯2,u3″}=[(r2g+r1f)+β(r2g′+r1f′)+μ(r2g″+r1f″)]η3+v(r2g″+r1f″)η4,{V¯2,U¯2,u4″}=[(r2f+r1g)+β(r2f′+r1g′)+μ(r2f″+r1g″)]η3+v(r2f″+r1g″)η4,{V¯2,U¯2,u1‴}=[(r2d+r1c)+β(r2d′+r1c′)]η4,{V¯2,U¯2,u2‴}=[(r2c+r1d)+β(r2c′+r1d′)]η4,{V¯2,U¯2,u3‴}=[(r2g+r1f)+β(r2g″+r1f″)]η4,{V¯2,U¯2,u4‴}=[(r2f+r1g)+β(r2f″+r1g″)]η4.$

We use the formula (19), and find that γ = 0. Applying the corresponding variational identity, we obtain the following Hamiltonian structure for the hierarchy of tri-integrable couplings (28): $u¯tm=K¯2,m(u¯)=J¯2δH¯2,mδu¯,m≥0,$

where the Hamiltonian operator is $J¯2=η1η2η3η4η2βη2+η4βη3βη4η3βη3μη3+vη40η4βη400−1⊗00−120000−121200001200,$

and the Hamiltonian functionals read $H¯2,m=−∫(r2am+2+r1bm+2)η1+(r2am+2′+r1bm+2′)η2m+1dx−∫(r2am+2″+r1bm+2″)η3+(r2am+2‴+r1bm+2‴)η4m+1dx,m≥0.$

Based on (24), (25), (26), (27), a direct computation yields to the recursion relation $P2,m+1=L¯2P2,m,$

where the recursion operator ${\overline{L}}_{2}$ is given by $L¯2=M3T(L1,L11,L31,L41)=L1000L11L1+βL1100L31βL31L1+βL11+μL310L41L1+βL41vL31L1+βL11,$

with L1 and ${L}_{1}^{1}$ being given as in (20), and $L31=x11x12x13x14x21x22x23x24x31x32x33x34x41x42x43x44,L41=y11y12y13y14y21y22y23y24y31y32y33y34y41y42y43y44,$

and $x11=x22=(u2+μu2″+βu2′)∂−1u4″+(u1+μu1″+βu1′)∂−1u3″+βu1″∂−1u3′+βu2″∂−1u4′+u1″∂−1u3+u2″∂−1u4,x12=x21=(u2+μu2″+βu2′)∂−1u3″+(u1+μu1″+βu1′)∂−1u4″+βu1″∂−1u4′+βu2″∂−1u3′+u1″∂−1u4+u2″∂−1u3,x13=x24=−(u2+μu2″+βu2′)∂−1u2″−(u1+μu1″+βu1′)∂−1u1″−βu1″∂−1u1′−βu2″∂−1u2′−u1″∂−1u1−u2″∂−1u2,x14=x23=−(u2+μu2″+βu2′)∂−1u1″−(u1+μu1″+βu1′)∂−1u2″−βu1″∂−1u2′−βu2″∂−1u1′−u1″∂−1u2−u2″∂−1u1,x31=x42=(u4+μu4″+βu4′)∂−1u4″+(u3+μu3″+βu3′)∂−1u3″+βu4″∂−1u4′+βu3″∂−1u3′+u4″∂−1u4+u3″∂−1u3,x32=x41=(u4+μu4″+βu4′)∂−1u3″+(u3+μu3″+βu3′)∂−1u4″+βu4″∂−1u3′−βu3″∂−1u4′+u4″∂−1u3+u3″∂−1u4,x33=x44=−(u4+μu4″+βu4′)∂−1u2″−(u3+μu3″+βu3′)∂−1u1″−βu4″∂−1u2′−βu3″∂−1u1′−u4″∂−1u2−u3″∂−1u1,x34=x43=−(u4+μu4″+βu4′)∂−1u1″−(u3+μu3″+βu3′)∂−1u2″−βu4″∂−1u1′−βu3″∂−1u2′−u4″∂−1u1−u3″∂−1u2,y11=y22=(u2+βu2′)∂−1u4‴+(u1+βu1′)∂−1u3‴+vu2″∂−1u4″+vu1″∂−1u3″+(u2′+βu2‴)∂−1u4′+(u1′+βu1‴)∂−1u3′+u2‴∂−1u4+u1‴∂−1u3,$ $y12=y21=(u2+βu2′)∂−1u3‴+(u1+βu1′)∂−1u4‴+vu2″∂−1u3″+vu1″∂−1u4″+(u2′+βu2‴)∂−1u3′+(u1′+βu1‴)∂−1u4′+u2‴∂−1u3+u1‴∂−1u4,y13=y24=−(u2+βu2′)∂−1u2‴−(u1+βu1′)∂−1u1‴−vu2″∂−1u2″−vu1″∂−1u1″−(u2′+βu2‴)∂−1u2′−(u1′+βu1‴)∂−1u1′−u2‴∂−1u2−u1‴∂−1u1,y14=y23=−(u2+βu2′)∂−1u1‴−(u1+βu1′)∂−1u2‴−vu2″∂−1u1″−vu1″∂−1u2″−(u2′+βu2‴)∂−1u1′−(u1′+βu1‴)∂−1u2′−u2‴∂−1u1−u1‴∂−1u2,y31=y42=(u4+βu4′)∂−1u4‴+(u3+βu3′)∂−1u3‴+vu4″∂−1u4″+vu3″∂−1u3″+(βu4‴+u4′)∂−1u4′+(βu3‴+u3′)∂−1u3′+u4‴∂−1u4+u3‴∂−1u3,y32=y41=(u4+βu4′)∂−1u3‴+(u3+βu3′)∂−1u4‴+vu4″∂−1u3″+vu3″∂−1u4″+(βu4‴+u4′)∂−1u3′+(βu3‴+u3′)∂−1u4′+u4‴∂−1u3+u3‴∂−1u4,y33=y44=−(u4+βu4′)∂−1u2‴−(u3+βu3′)∂−1u1‴−vu4″∂−1u2″−vu3″∂−1u1″−(βu4‴+u4′)∂−1u2′−(βu3‴+u3′)∂−1u1′−u4‴∂−1u2−u3‴∂−1u1,y34=y43=−(u4+βu4′)∂−1u1‴−(u3+βu3′)∂−1u2‴−vu4″∂−1u1″−vu3″∂−1u2″−(βu4‴+u4′)∂−1u1′−(βu3‴+u3′)∂−1u2′−u4‴∂−1u1−u3‴∂−1u2.$

## 4 Conclusion

In this paper, we use the non-semisimple Lie algebras consisting of 3 × 3, 4 × 4 block matrices, and apply them to the construction of bi-integrable couplings and tri-integrable couplings assiciated with SO(4), based on the enlarged zero curvature equations. According to the associated variational identities, their Hamiltonian structures can be generated.

There are many interesting aspects of integrable couplings we have not solved, for example, how we can generate integrable couplings and their Hamiltomian structures when irreducible representations of SO(3) and SO(4) are used to form matrix loop algebras. In addition, the relations between the hierarchy of tri-integrable couplings associated with SO(4) and the hierarchy of tri-integrable couplings associated with SO(3) are also very interesting problems.

## Acknowledgement

This work was supported by NNSF of China (Nos.11171055 and 11471090) and Scientific Research Fund of Heilongjiang provincial Education Department (No. 12541184).

## References

• [1]

Ma W. X., Fuchssteiner B., Integrable theory of the perturbation equations, Chaos, Solitons Fractals, 1996, 7, 1227-1250.

• [2]

Ma W. X., Meng J. H., Zhang H. Q., Integrable couplings, variational identities and Hamiltonian formulations, Global J. Math. Sci., 2012, 1, 1-17. Google Scholar

• [3]

Frappat L., Sciarrino A., Sorba P., Dictionary on Lie Algebras and Superalgebras, Academic, San Diego, CA, 2000. Google Scholar

• [4]

Ma W. X., Xu X. X., Zhang Y. F., Semi-direct sums of Lie algebras and continuous integrable couplings, J. Math. Phys., 2006, 351, 125-130. Google Scholar

• [5]

Ma W. X., Loop algebras and bi-integrable couplings, Chin. Ann. Math. Ser. B, 2012, 33, 207-224.

• [6]

Ma W. X., Enlarging spectral problems to construct integrable couplings of soliton equations, Phys. Lett. A, 2003, 316, 72-76.

• [7]

Tu G. Z., The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems, J. Math. Phys., 1989, 30, 330-338.

• [8]

Tu G. Z., On Liouville integrability of zero-curvature equations and the Yang hierarchy, J. Phys. A: Math. Gen., 1999, 22, 2375-2392. Google Scholar

• [9]

Zhang Y. F., Guo F. K., Two pairs of Lie algebras and the integrable couplings as well as the Hamiltonian structure of the Yang hierarchy, Chaos, Solitons Fractals, 2007, 34, 490-495.

• [10]

Zhang Y. F., A generalized multi-component Glachette-Johnson(GJ) hierarchy and its integrable coupling system, Chaos, Solitons Fractals, 2004, 21, 305-310.

• [11]

Zhang Y. F., Fan E. G., Tam H., A few expanding Lie algebras of the Lie algebra A1 and applications, Phys. Lett. A, 2006, 359, 471-480.

• [12]

Ma W. X., Meng J. H., Shekhtman B., Tri-integrable couplings of the KdV hierarchy associated with a non-semisimple Lie algebra, Global J. Math. Sci., 2013, 2, 9-19. Google Scholar

• [13]

Meng J. H., Ma W. X., Hamiltonian tri-integrable couplings of the AKNS hierarchy, Commun. Theor. Phys., 2013, 59, 385-392.

• [14]

Manukure S., Ma W. X., Bi-integrable couplings of a new soliton hierarchy associated with a non-semisimple Lie algebra, Appl. Math. Comput., 2014, 245, 44-52.

• [15]

Cao Y., Chen L. Y., Bi-integrable couplings of a new soliton hierarchy associated with SO(4). Adv. Math. Phys., 2015, 2015, 857684.

• [16]

He B. Y., Chen L. Y., Ma L. L., Two soliton hierarchies associated with SO(4) and the applications of SU(2) ⊗ SU(2) ≅ SO(4), J. Math. Phys., 2014, 55, 093510.

• [17]

He B. Y., Chen L. Y., Cao Y., Bi-integrable couplings and tri-integrable couplings of the modified Ablowitz-Kaup-Newell-Segur hierarchy with self-consistent sources, J. Math. Phys., 2015, 56, 013502.

• [18]

Beveren E. V., The orthogonal group in four dimensions, Some notes on group theory, Chap. 12 (see http://cft.fis.uc.pt/eef/evbgroups.pdf).

• [19]

Ma W. X., Variational identities and applications to Hamiltonian structures of soliton equations, Nonlinear Anal., 2009, 71, e1716-e1726. Google Scholar

Accepted: 2017-01-26

Published Online: 2017-03-11

Conflict of interestConflict of interests: The authors declare that there is no conflict of interests regarding the publication of this paper.

Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 203–217, ISSN (Online) 2391-5455,

Export Citation