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

  • Jian Zhang EMAIL logo , Chiping Zhang and Yunan Cui
Published/Copyright: March 11, 2017
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Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

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.

Keywords: Bi-integrable couplings; Tri-integrable couplings; Zero curvature equations; Hamiltonian structures
MSC 2010: 37K05; 37K10; 35Q53

1 Introduction

An integrable coupling of a given system

(1) u t = K ( u ) = K ( x , t , u , u x , u t , u x x , u x t , u t t , ⋯ )

is a triangular integrable system of the following form [1]:

u t = K ( u ) , v t = S ( u , v ) ,

where u is a function of variables t and x, u x = ∂ u ∂ x and u t = ∂ u ∂ 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]:

u t = K ( u ) , u 1 , t = S 1 ( u , u 1 ) , u 2 , t = S 2 ( u , u 1 , u 2 ) .

Similarly, by a tri-integrable coupling, we mean an enlarged triangular integrable system of the following form [2]:

u t = K ( u ) , u 1 , t = S 1 ( u , u 1 ) , u 2 , t = S 2 ( u , u 1 , u 2 ) , u 3 , t = S 3 ( u , u 1 , u 2 , u 3 ) .

Integrable couplings correspond to non-semisimple Lie algebras ḡ, and such Lie algebras can be written as semi-direct sums [3]:

g ¯ = g ⊎ g c , g − s e m i s i m p l e , g c − s o l v a b l e .

The notion of semi-direct sums g ¯ = g ⊎ g c means that g and gc satisfy [g, gc] ⊆ gc, where [g, gc] = {[A, B] | A ∈ g, B ∈ gc}, with [·,·] denoting the Lie bracket of ḡ. Obviously, gc is an ideal of ḡ. 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, gcg ⊆ gc, where ggc = {AB | A ∈ g, B ∈ gc}.

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 [4–6]. 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 [9–11].

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 [15–17].

2 Bi-integrable couplings and Hamiltonian structures

2.1 Bi-integrable couplings associated with SO(4)

We take the class of 3 × 3 block matrices in the form of

(2) M 2 ( A , A 1 , A 2 ) = A A 1 A 2 0 A + α A 1 A 1 + α A 2 0 0 A + α A 1 ,

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 ḡ by a semi-direct sum

g ¯ ( λ ) = g ⊎ g C ,

with

g = { M 2 ( A , 0 , 0 ) | A ∈ S O ( 4 ) ~ } , g c = { M 2 ( 0 , A 1 , A 2 ) | A 1 , A 2 ∈ S O ( 4 ) ~ } ,

where the loop algebra S O ( 4 ) ~ is defined by

S O ( 4 ~ ) = { A ( λ ) ∈ S O ( 4 ) | entries o f A ( λ ) − Laurent series in λ } .

The corresponding matrix product reads

[ M 2 ( A , A 1 , A 2 ) , M 2 ( B , B 1 , B 2 ) ] = M 2 ( C , C 1 , C 2 ) ,

with C, C1 and C2 being defined by

C = [ A , B ] , C 1 = [ A , B 1 ] + [ A 1 , B ] + α [ A 1 , B 1 ] , C 2 = [ A , B 2 ] + [ A 2 , B ] + [ A 1 , B 1 ] + α [ A 1 , B 2 ] + α [ A 2 , B 1 ] .

We consider the Lie algebra G = {e1, e2, e3, e4, e5, e6} in SO(4) [18], where

e 1 = 0 0 0 − 1 0 0 0 0 0 0 0 0 1 0 0 0 , e 2 = 0 0 0 0 0 0 − 1 0 0 1 0 0 0 0 0 0 , e 3 = 0 − 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 , e 4 = 0 0 0 0 0 0 0 0 0 0 0 − 1 0 0 1 0 , e 5 = 0 0 1 0 0 0 0 0 − 1 0 0 0 0 0 0 0 , e 6 = 0 0 0 0 0 0 0 − 1 0 0 0 0 0 1 0 0 .

The soliton hierarchy introduced in [16] has a spectral problem

ϕ x = U ϕ = U ( u , λ ) ϕ , u = ( u 1 , u 2 , u 3 , u 4 ) T ,

with the spectral matrix U given as

(3) U = U ( u , λ ) = λ e 2 + u 1 e 3 + u 2 e 4 + u 3 e 5 + u 4 e 6 = 0 − u 1 u 3 0 u 1 0 − λ − u 4 − u 3 λ 0 − u 2 0 u 4 u 2 0 .

Based on this special non-semisimple Lie algebra ḡ(λ)we choose the corresponding enlarged spectral matrix Ū1 = M2(U, U1, U2) and the following supplementary spectral matrices:

(4) U 1 = U 1 ( u 1 , λ ) = 0 − u 1 ′ u 3 ′ 0 u 1 ′ 0 0 − u 4 ′ − u 3 ′ 0 0 − u 2 ′ 0 u 4 ′ u 2 ′ 0 , u 1 = u 1 ′ u 2 ′ u 3 ′ u 4 ′ ,

(5) U 2 = U 2 ( u 2 , λ ) = 0 − u 1 ″ u 3 ″ 0 u 1 ″ 0 0 − u 4 ″ − u 3 ″ 0 0 − u 2 ″ 0 u 4 ″ u 2 ″ 0 , u 2 = u 1 ″ u 2 ″ u 3 ″ u 4 ″ .

In order to solve the enlarged stationary zero curvature equation V ¯ 1 x = [ U ¯ 1 , V ¯ 1 ] , we take V ¯ 1 = M 2 ( V , V 1 , V 2 ) , where V is defined as in [16]

(6) V = V ( u , λ ) = 0 − c f − a c 0 − b − g − f b 0 − d a g d 0 = ∑ i ≥ 0 0 − c i f i − a i c i 0 − b i − g i − f i b i 0 − d i a i g i d i 0 λ − i ,

V1 and V2 are defined similarly

(7) V 1 = V 1 ( u , u 1 , λ ) = 0 − c ′ f ′ − a ′ c ′ 0 − b ′ − g ′ − f ′ b ′ 0 − d ′ a ′ g ′ d ′ 0 = ∑ i ≥ 0 0 − c i ′ f i ′ − a i ′ c i ′ 0 − b i ′ − g i ′ − f i ′ b i ′ 0 − d i ′ a i ′ g i ′ d i ′ 0 λ − i ,

(8) V 2 = V 2 ( u , u 1 , u 2 , λ ) = 0 − c ″ f ″ − a ″ c ″ 0 − b ″ − g ″ − f ″ b ″ 0 − d ″ a ″ g ″ d ″ 0 = ∑ i ≥ 0 0 − c i ″ f i ″ − a i ″ c i ″ 0 − b i ″ − g i ″ − f i ″ b i ″ 0 − d i ″ a i ″ g i ″ d i ″ 0 λ − i .

It now follows from the enlarged stationary zero curvature equation V ¯ 1 x = [ U ¯ 1 , V ¯ 1 ] that

(9) V x = [ U , V ] , V 1 x = [ U , V 1 ] + [ U 1 , V ] + α [ U 1 , V 1 ] , V 2 x = [ U , V 2 ] + [ U 2 , V ] + [ U 1 , V 1 ] + α [ U 1 , V 2 ] + α [ U 2 , V 1 ] .

The above equation system equivalently leads to

(10) a x = u 4 c − u 2 f − u 1 g + u 3 d , b x = u 3 c − u 2 g − u 1 f + u 4 d , c x = λ f − u 4 a − u 3 b , d x = λ g − u 3 a − u 4 b , f x = − λ c + u 1 b + u 2 a , g x = − λ d + u 1 a + u 2 b ,

(11) a x ′ = u 4 c ′ − u 2 f ′ − u 1 g ′ + u 3 d ′ + u 4 ′ c − u 2 ′ f − u 1 ′ g + u 3 ′ d + α ( u 4 ′ c ′ − u 2 ′ f ′ − u 1 ′ g ′ + u 3 ′ d ′ ) , b x ′ = u 3 c ′ − u 2 g ′ − u 1 f ′ + u 4 d ′ + u 3 ′ c − u 2 ′ g − u 1 ′ f + u 4 ′ d + α ( u 3 ′ c ′ − u 2 ′ g ′ − u 1 ′ f ′ + u 4 ′ d ′ ) , c x ′ = λ f ′ − u 4 a ′ − u 3 b ′ − u 4 ′ a − u 3 ′ b − α ( u 4 ′ a ′ + u 3 ′ b ′ ) , d x ′ = λ g ′ − u 3 a ′ − u 4 b ′ − u 3 ′ a − u 4 ′ b − α ( u 3 ′ a ′ + u 4 ′ b ′ ) , f x ′ = − λ c ′ + u 1 b ′ + u 2 a ′ + u 1 ′ b + u 2 ′ a + α ( u 1 ′ b ′ + u 2 ′ a ′ ) , g x ′ = − λ d ′ + u 1 a ′ + u 2 b ′ + u 1 ′ a + u 2 ′ b + α ( u 2 ′ b ′ + u 1 ′ a ′ ) ,

and

(12) a x ″ = u 4 c ″ − u 2 f ″ − u 1 g ″ + u 3 d ″ + u 4 ″ c − u 2 ″ f − u 1 ″ g + u 3 ″ d + α ( u 4 ′ c ″ − u 2 ′ f ″ − u 1 ′ g ″ + u 3 ′ d ″ ) + u 4 ′ c ′ − u 2 ′ f ′ − u 1 ′ g ′ + u 3 ′ d ′ + α ( u 4 ″ c ′ − u 2 ″ f ′ − u 1 ″ g ′ + u 3 ″ d ′ ) , b x ″ = u 3 c ″ − u 2 g ″ − u 1 f ″ + u 4 d ″ + u 3 ″ c − u 2 ″ g − u 1 ″ f + u 4 ″ d + α ( u 3 ′ c ″ − u 2 ′ g ″ − u 1 ′ f ″ + u 4 ′ d ″ ) + u 3 ′ c ′ − u 2 ′ g ′ − u 1 ′ f ′ + u 4 ′ d ′ + α ( u 3 ″ c ′ − u 2 ″ g ′ − u 1 ″ f ′ + u 4 ″ d ′ ) , c x ″ = λ f ″ − u 4 a ″ − u 3 b ″ − u 4 ″ a − u 3 ″ b − α ( u 4 ′ a ″ + u 3 ′ b ″ ) − u 4 ′ a ′ − u 3 ′ b ′ − α ( u 4 ″ a ′ + u 3 ″ b ′ ) , d x ″ = λ g ″ − u 3 a ″ − u 4 b ″ − u 3 ″ a − u 4 ″ b − α ( u 3 ′ a ″ + u 4 ′ b ″ ) − u 3 ′ a ′ − u 4 ′ b ′ − α ( u 3 ″ a ′ + u 4 ″ b ′ ) , f x ″ = − λ c ″ + u 1 b ″ + u 2 a ″ + u 1 ″ b + u 2 ″ a + α ( u 1 ′ b ″ + u 2 ′ a ″ ) + u 1 ′ b ′ − u 2 ′ a ′ + α ( u 1 ″ b ′ + u 2 ″ a ′ ) , g x ″ = − λ d ″ + u 1 a ″ + u 2 b ″ + u 1 ″ a + u 2 ″ b + α ( u 1 ′ a ″ + u 2 ′ b ″ ) + u 1 ′ a ′ + u 2 ′ b ′ + α ( u 1 ″ a ′ + u 2 ″ b ′ ) .

Now, we define the enlarged Lax matrices V ¯ 1 [ m ] = ( λ m V ¯ 1 ) + = M 2 ( V [ m ] , V 1 [ m ] , V 2 [ m ] ) , m ≥ 0 , where V[m]is defined as V [ m ] = ( λ m V ) + ,  and V i [ m ] = ( λ m V i ) + , i = 1 , 2.

Solving the enlarged zero curvature equations U ¯ 1 t m − V ¯ 1 x [ m ] + [ U ¯ 1 , V ¯ 1 [ m ] ] = 0 , m ≥ 0 , we get bi-integrable couplings of the soliton hierarchy in [16]:

(13) u ¯ t m = u t m u 1 t m u 2 t m = J 1 0 0 0 J 1 0 0 0 J 1 P 1 m P 2 m P 3 m = J 1 0 0 0 J 1 0 0 0 J 1 P 1 , m + 1 ,

where

u t m = u 1 u 2 u 3 u 4 t m = f m + 1 g m + 1 − c m + 1 − d m + 1 , u 1 t m = u 1 ′ u 2 ′ u 3 ′ u 4 ′ = f m + 1 ′ g m + 1 ′ − c m + 1 ′ − d m + 1 ′ , u 2 t m = u 1 ″ u 2 ″ u 3 ″ u 4 ″ t m = f m + 1 ″ g m + 1 ″ − c m + 1 ″ − d m + 1 ″ , J 1 = 0 0 − 1 2 0 0 0 0 − 1 2 1 2 0 0 0 0 1 2 0 0 ,

and

P 1 m = − 2 c m + 1 − 2 d m + 1 − 2 f m + 1 − 2 g m + 1 , P 2 m = − 2 c m + 1 ′ − 2 d m + 1 ′ − 2 f m + 1 ′ − 2 g m + 1 ′ , P 3 m = − 2 c m + 1 ″ − 2 d m + 1 ″ − 2 f m + 1 ″ − 2 g m + 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]

(14) δ δ u ¯ ∫ { V ¯ , U ¯ λ } d x = λ − γ ∂ ∂ λ λ γ { V ¯ , U ¯ u } ,

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

∀ a = ( a 1 , ⋯ , a 6 , a 1 , ′ ⋯ , a 6 ′ , a 1 ″ , ⋯ , a 6 ″ ) ∈ R 18 , b = ( b 1 , ⋯ , b 6 , b 1 ′ , ⋯ , b 6 ′ , b 1 ″ , ⋯ , b 6 ″ ) ∈ R 18 , we define the Lie bracket {·,·} on R18 as follows:

[ a , b ] = a T R ¯ 1 ( b ) , R ¯ 1 ( b ) = R ( b ) R 1 ( b ) R 2 ( b ) 0 R ( b ) + α R 1 ( b ) R 1 ( b ) + α R 2 ( b ) 0 0 R ( b ) + α R 1 ( b ) ,

where

(15) R b = 0 0 b 6 b 5 − b 4 − b 3 0 0 b 5 b 6 − b 3 − b 4 − b 6 − b 5 0 0 b 2 b 1 − b 5 − b 6 0 0 b 1 b 2 b 4 b 3 − b 2 − b 1 0 0 b 3 b 4 − b 1 − b 2 0 0 ,

(16) R 1 b = 0 0 b 6 ′ b 5 ′ − b 4 ′ − b 3 ′ 0 0 b 5 ′ b 6 ′ − b 3 ′ − b 4 ′ − b 6 ′ − b 5 ′ 0 0 b 2 ′ b 1 ′ − b 5 ′ − b 6 ′ 0 0 b 1 ′ b 2 ′ b 4 ′ b 3 ′ − b 2 ′ − b 1 ′ 0 0 b 3 ′ b 4 ′ − b 1 ′ − b 2 ′ 0 0 ,

and

(17) R 2 b = 0 0 b 6 ″ b 5 ″ − b 4 ″ − b 3 ″ 0 0 b 5 ″ b 6 ″ − b 3 ″ − b 4 ″ − b 6 ″ − b 5 ″ 0 0 b 2 ″ b 1 ″ − b 5 ″ − b 6 ″ 0 0 b 1 ″ b 2 ″ b 4 ″ b 3 ″ − b 2 ″ − b 1 ″ 0 0 b 3 ″ b 4 ″ − b 1 ″ − b 2 ″ 0 0 .

Following the properties of the matrix F 1 : F 1 ( R ¯ 1 ( b ) ) T = − R ¯ 1 ( b ) F 1 and F 1 = F 1 T , we have

F 1 = η 1 η 2 2 η 3 η 2 α η 2 + 2 η 3 2 α η 3 2 η 3 2 α η 3 0 ⊗ r 1 r 2 0 0 0 0 r 2 r 1 0 0 0 0 0 0 r 1 r 2 0 0 0 0 r 2 r 1 0 0 0 0 0 0 r 1 r 2 0 0 0 0 r 2 r 1 ,

where η1, η2, η3, r1, r2 are arbitrary constants. We choose r1 ≠ ±r2, then we easily have

det ( F 1 ) = 4096 ( α 2 η 1 − α η 2 + 2 η 3 ) 6 η 3 12 ( r 1 2 − r 2 2 ) 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:

(18) { a , b } = a T F 1 b .

Now we can compute that

{ V ¯ 1 , U ¯ 1 , λ } = ( r 2 a + r 1 b ) η 1 + ( r 2 a ′ + r 1 b ′ ) η 2 + 2 ( r 2 a ″ + r 1 b ″ ) η 3 , { V ¯ 1 , U ¯ 1 , u 1 } = ( r 1 c + r 2 d ) η 1 + ( r 1 c ′ + r 2 d ′ ) η 2 + 2 ( r 1 c ″ + r 2 d ″ ) η 3 , { V ¯ 1 , U ¯ 1 , u 2 } = ( r 2 c + r 1 d ) η 1 + ( r 2 c ′ + r 1 d ′ ) η 2 + 2 ( r 2 c ″ + r 1 d ″ ) η 3 , { V ¯ 1 , U ¯ 1 , u 3 } = ( r 1 f + r 2 g ) η 1 + ( r 1 f ′ + r 2 g ′ ) η 2 + 2 ( r 1 f ″ + r 2 g ″ ) η 3 , { V ¯ 1 , U ¯ 1 , u 4 } = ( r 2 f + r 1 g ) η 1 + ( r 2 f ′ + r 1 g ′ ) η 2 + ( 2 r 2 f ″ + 2 r 1 g ″ ) η 3 , { V ¯ 1 , U ¯ 1 , u 1 ′ } = [ r 1 c + r 2 d + α ( r 1 c ′ + r 2 d ′ ) ] η 2 + 2 [ r 1 c ′ + r 2 d ′ + α ( r 1 c ″ + r 2 d ″ ) ] η 3 , { V ¯ 1 , U ¯ 1 , u 2 ′ } = [ r 2 c + r 1 d + α ( r 2 c ′ + r 1 d ′ ) ] η 2 + 2 [ r 2 c ′ + r 1 d ′ + α ( r 2 c ″ + r 1 d ″ ) ] η 3 , { V ¯ 1 , U ¯ 1 , u 3 ′ } = [ r 1 f + r 2 g + α ( r 1 f ′ + r 2 g ′ ) ] η 2 + 2 [ r 1 f ′ + r 2 g ′ + α ( r 1 f ″ + r 2 g ″ ) ] η 3 , { V ¯ 1 , U ¯ 1 , u 4 ′ } = [ r 2 f + r 1 g + α ( r 2 f ′ + r 1 g ′ ) ] η 2 + 2 [ r 2 f ′ + r 1 g ′ + α ( r 2 f ″ + r 1 g ″ ) ] η 3 , { V ¯ 1 , U ¯ 1 , u 1 ″ } = 2 [ r 1 c + r 2 d + α ( r 1 c ′ + r 2 d ′ ) ] η 3 , { V ¯ 1 , U ¯ 1 , u 2 ″ } = 2 [ r 2 c + r 1 d + α ( r 2 c ′ + r 1 d ′ ) ] ) η 3 , { V ¯ 1 , U ¯ 1 , u 3 ″ } = 2 [ r 1 f + r 2 g + α ( r 1 f ′ + r 2 g ′ ) ] η 3 , { V ¯ 1 , U ¯ 1 , u 4 ″ } = 2 [ r 2 f + r 1 g + α ( r 2 f ′ + r 1 g ′ ) ] η 3 ,

and furthermore, we use the formula [19]:

(19) γ = − λ 2 d d λ ln ⁡ | { V ¯ , V ¯ } | ,

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 ¯ t m = K ¯ 1 , m ( u ¯ ) = J ¯ 1 δ H ¯ 1 , m δ u ¯ ; m ≥ 0 ,

with the Hamiltonian operator

J ¯ 1 = η 1 η 2 2 η 3 η 2 α η 2 + 2 η 3 2 α η 3 2 η 3 2 α η 3 0 − 1 ⊗ 0 0 − 1 2 0 0 0 0 − 1 2 1 2 0 0 0 0 1 2 0 0 ,

and the Hamiltonian functionals

H ¯ 1 , m = − ∫ ( r 2 a m + 2 + r 1 b m + 2 ) η 1 + ( r 2 a m + 2 ′ + r 1 b m + 2 ′ ) η 2 + ( 2 r 2 a m + 2 ″ + 2 r 1 b m + 2 ″ ) η 3 m + 1 d x , m ≥ 0.

Based on (10), (11), (12), a direct computation yields a recursion relation

P 1 , m + 1 = L ¯ 1 P 1 , m ,

where

L ¯ 1 = M 2 T ( L 1 , L 1 1 , L 2 1 ) = L 1 0 0 L 1 1 L 1 + α L 1 1 0 L 2 1 L 1 1 + α L 2 1 L 1 + α L 1 1 ,

with L 1 , L 1 1 and L 2 1 being defined by

(20) L 1 = l 11 l 12 l 13 l 14 l 21 l 22 l 23 l 24 l 31 l 32 l 33 l 34 l 41 l 42 l 43 l 44 , L 1 1 = l 11 ′ l 12 ′ l 13 ′ l 14 ′ l 21 ′ l 22 ′ l 23 ′ l 24 ′ l 31 ′ l 32 ′ l 33 ′ l 34 ′ l 41 ′ l 42 ′ l 43 ′ l 44 ′ ,

(21) L 2 1 = l 11 ″ l 12 ″ l 13 ″ l 14 ″ l 21 ″ l 22 ″ l 23 ″ l 24 ″ l 31 ″ l 32 ″ l 33 ″ l 34 ″ l 41 ″ l 42 ″ l 43 ″ l 44 ″ ,

and

l 11 = l 22 = u 1 ∂ − 1 u 3 + u 2 ∂ − 1 u 4 , l 12 = l 21 = u 1 ∂ − 1 u 4 + u 2 ∂ − 1 u 3 , l 13 = l 24 = − ∂ − u 1 ∂ − 1 u 1 − u 2 ∂ − 1 u 2 , l 14 = l 23 = − u 1 ∂ − 1 u 2 − u 2 ∂ − 1 u 1 , l 31 = l 42 = ∂ + u 4 ∂ − 1 u 4 + u 3 ∂ − 1 u 3 , l 32 = l 41 = u 4 ∂ − 1 u 3 + u 3 ∂ − 1 u 4 , l 33 = l 44 = − u 4 ∂ − 1 u 2 − u 3 ∂ − 1 u 1 , l 34 = l 43 = − u 4 ∂ − 1 u 1 − u 3 ∂ − 1 u 2 , l 11 ′ = l 22 ′ = u 1 ∂ − 1 u 3 ′ + u 2 ∂ − 1 u 4 ′ + u 1 ′ ∂ − 1 u 3 + u 2 ′ ∂ − 1 u 4 + α u 1 ′ ∂ − 1 u 3 ′ + α u 2 ′ ∂ − 1 u 4 ′ , l 12 ′ = l 21 ′ = u 1 ∂ − 1 u 4 ′ + u 2 ∂ − 1 u 3 ′ + u 1 ′ ∂ − 1 u 4 + u 2 ′ ∂ − 1 u 3 + α u 1 ′ ∂ − 1 u 4 ′ + α u 2 ′ ∂ − 1 u 3 ′ , l 13 ′ = l 24 ′ = − u 1 ∂ − 1 u 1 ′ − u 2 ∂ − 1 u 2 ′ − u 1 ′ ∂ − 1 u 1 − u 2 ′ ∂ − 1 u 2 − α u 1 ′ ∂ − 1 u 1 ′ − α u 2 ′ ∂ − 1 u 2 ′ , l 14 ′ = l 23 ′ = − u 1 ∂ − 1 u 2 ′ − u 2 ∂ − 1 u 1 ′ − u 1 ′ ∂ − 1 u 2 − u 2 ′ ∂ − 1 u 1 − α u 1 ′ ∂ − 1 u 2 ′ − α u 2 ′ ∂ − 1 u 1 ′ , l 31 ′ = l 42 ′ = u 4 ∂ − 1 u 4 ′ + u 3 ∂ − 1 u 3 ′ + u 4 ′ ∂ − 1 u 4 + u 3 ′ ∂ − 1 u 3 + α u 3 ′ ∂ − 1 u 3 ′ + α u 4 ′ ∂ − 1 u 4 ′ , l 32 ′ = l 41 ′ = u 4 ∂ − 1 u 3 ′ + u 3 ∂ − 1 u 4 ′ + u 4 ′ ∂ − 1 u 3 + u 3 ′ ∂ − 1 u 4 + α u 3 ′ ∂ − 1 u 4 ′ + α u 4 ′ ∂ − 1 u 3 ′ , l 33 ′ = l 44 ′ = − u 4 ∂ − 1 u 2 ′ − u 3 ∂ − 1 u 1 ′ − u 4 ′ ∂ − 1 u 2 − u 3 ′ ∂ − 1 u 1 − α u 3 ′ ∂ − 1 u 1 ′ − α u 4 ′ ∂ − 1 u 2 ′ , l 34 ′ = l 43 ′ = − u 4 ∂ − 1 u 1 ′ − u 3 ∂ − 1 u 2 ′ − u 4 ′ ∂ − 1 u 1 − u 3 ′ ∂ − 1 u 2 − α u 3 ′ ∂ − 1 u 2 ′ − α u 4 ′ ∂ − 1 u 1 ′ , l 11 ″ = l 22 ″ = u 1 ∂ − 1 u 3 ″ + u 2 ∂ − 1 u 4 ″ + u 1 ″ ∂ − 1 u 3 + u 2 ″ ∂ − 1 u 4 + u 1 ′ ∂ − 1 u 3 ′ + u 2 ′ ∂ − 1 u 4 ′ + α ( u 1 ′ ∂ − 1 u 3 ″ + u 2 ′ ∂ − 1 u 4 ″ + u 1 ″ ∂ − 1 u 3 ′ + u 2 ″ ∂ − 1 u 4 ′ ) , l 12 ″ = l 21 ″ = u 1 ∂ − 1 u 4 ″ + u 2 ∂ − 1 u 3 ″ + u 1 ″ ∂ − 1 u 4 + u 2 ″ ∂ − 1 u 3 + u 1 ′ ∂ − 1 u 4 ′ + u 2 ′ ∂ − 1 u 3 ′ + α ( u 1 ′ ∂ − 1 u 4 ″ + u 2 ′ ∂ − 1 u 3 ″ + u 1 ″ ∂ − 1 u 4 ′ + u 2 ″ ∂ − 1 u 3 ′ ) , l 13 ″ = l 24 ″ = − u 1 ∂ − 1 u 1 ″ − u 2 ∂ − 1 u 2 ″ − u 1 ″ ∂ − 1 u 1 − u 2 ″ ∂ − 1 u 2 − u 1 ′ ∂ − 1 u 1 ′ − u 2 ′ ∂ − 1 u 2 ′ + α ( − u 1 ′ ∂ − 1 u 1 ″ − u 2 ′ ∂ − 1 u 2 ″ − u 1 ″ ∂ − 1 u 1 ′ − u 2 ″ ∂ − 1 u 2 ′ ) , l 14 ″ = l 23 ″ = − u 1 ∂ − 1 u 2 ″ − u 2 ∂ − 1 u 1 ″ − u 1 ″ ∂ − 1 u 2 − u 2 ″ ∂ − 1 u 1 − u 1 ′ ∂ − 1 u 2 ′ − u 2 ′ ∂ − 1 u 1 ′ + α ( − u 1 ′ ∂ − 1 u 2 ″ − u 2 ′ ∂ − 1 u 1 ″ − u 1 ″ ∂ − 1 u 2 ′ − u 2 ″ ∂ − 1 u 1 ′ ) , l 31 ″ = l 42 ″ = u 4 ∂ − 1 u 4 ″ + u ∂ − 1 u 3 ″ + u 4 ″ ∂ − 1 u 4 + u 3 ″ ∂ − 1 u + u 4 ′ ∂ − 1 u 4 ′ + u 3 ′ ∂ − 1 u 3 ′ + α ( u 4 ′ ∂ − 1 u 4 ″ + u 3 ′ ∂ − 1 u 3 ″ + u 4 ″ ∂ − 1 u 4 ′ + u 3 ″ ∂ − 1 u 3 ′ ) , l 32 ″ = l 41 ″ = u 4 ∂ − 1 u ″ 3 + u 3 ∂ − 1 u ″ 4 + u ′ 4 ′ − 1 ∂ u 3 + u 3 ″ ∂ − 1 u 4 + u 4 ′ ∂ − 1 u 3 ′ + u 3 ′ ∂ − 1 u 4 ′ + α ( u 4 ′ ∂ − 1 u 3 ″ + u 3 ′ ∂ − 1 u 4 ″ + u 4 ″ ∂ − 1 u 3 ′ + u 3 ″ ∂ − 1 u 4 ′ ) , l 33 ″ = l 44 ″ = − u 4 ∂ − 1 u 2 ″ − u 3 ∂ − 1 u 1 ″ − u 4 ″ ∂ − 1 u 2 − u 3 ″ ∂ − 1 u 1 − u 4 ′ ∂ − 1 u 2 ′ − u 3 ′ ∂ − 1 u 1 ′ + α ( − u 4 ′ ∂ − 1 u 2 ″ − u 3 ′ ∂ − 1 u 1 ″ − u 4 ″ ∂ − 1 u 2 ′ − u 3 ″ ∂ − 1 u 1 ′ ) , l 34 ″ = l 43 ″ = − u 4 ∂ − 1 u 1 ″ − u 3 ∂ − 1 u 2 ″ − u 4 ″ ∂ − 1 u 1 − u 3 ″ ∂ − 1 u 2 − u 4 ′ ∂ − 1 u 1 ′ − u 3 ′ ∂ − 1 u 2 ′ + α ( − u 4 ′ ∂ − 1 u 1 ″ − u 3 ′ ∂ − 1 u 2 ″ − u 4 ″ ∂ − 1 u 1 ′ − u 3 ″ ∂ − 1 u 2 ′ ) ,

where ∂ = d d x  and ∂ − 1 = ∫ d d x d x .

3 Tri-integrable couplings and Hamiltonian structures

3.1 Tri-integrable couplings associated with SO(4)

We take the class of 4 × 4 block matrices in the form of

(22) M 3 ( A , A 1 , A 2 , A 3 ) = A A 1 A 2 A 3 0 A + β A 1 β A 2 A 1 + β A 3 0 0 A + β A 1 + μ A 2 v A 2 0 0 0 A + β A 1 ,

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 ⊎ g C ,

with

g = { M 3 ( A , 0 , 0 , 0 ) | A ∈ S O ( 4 ~ ) } , g c = { M 3 ( 0 , A 1 , A 2 , A 3 ) | A 1 , A 2 , A 3 ∈ S O ( 4 ~ ) } ,

where the loop algebra S O ( 4 ) ~ is defined by

S O ( 4 ) ~ = { A ( λ ) ∈ S O ( 4 ) | entries of A ( λ ) − Laurent series in λ } .

The corresponding matrix product reads

[ M 3 ( A , A 1 , A 2 , A 3 ) , M 3 ( B , B 1 , B 2 , B 3 ) ] = M 3 ( C , C 1 , C 2 , C 3 ) ,

with C, C1,C2 and C3 being defined by

C = [ A , B ] , C 1 = [ A , B 1 ] + [ A 1 , B ] + β [ A 1 , B 1 ] , C 2 = [ A , B 2 ] + [ A 2 , B ] + μ [ A 2 , B 2 ] + β [ A 1 , B 2 ] + β [ A 2 , B 1 ] , C 3 = [ A , B 3 ] + [ A 3 , B ] + β [ A 3 , B 1 ] + β [ A 1 , B 3 ] + [ A 1 , B 1 ] + v [ A 2 , B 2 ] .

We will adopt the following enlarged spectral matrix to construct tri-integrable couplings for SO(4) hierarchy

U ¯ 2 = U ¯ 2 ( u ¯ , λ ) = M 3 ( U , U 1 , U 2 , U 3 ) ∈ 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

U 3 = U 3 ( u 3 , λ ) = 0 − u 1 ‴ u 3 ‴ 0 u 1 ″ 0 0 − u 4 ‴ − u 3 ‴ 0 0 − u 2 ‴ 0 u 4 ‴ u 2 ‴ 0 , u 3 = u 1 ‴ u 2 ‴ u 3 ‴ u 4 ‴ .

We take

V ¯ 2 = V ¯ 2 ( u ¯ , λ ) = M 3 ( V , V 1 , V 2 , V 3 ) = V V 1 V 2 V 3 0 V + β V 1 β V 2 V 1 + β V 3 0 0 V + β V 1 + μ V 2 v V 2 0 0 0 V + β V 1 ,

where V, V1, V2 are defined by (6), (7) and (8), also V3 reads

(23) V 3 = V 3 ( u , u 1 , u 2 , u 3 , λ ) = 0 − c ‴ f ‴ − a ‴ c ‴ 0 − b ‴ − g ‴ − f ‴ b ‴ 0 − d ‴ a ‴ g ‴ d ‴ 0 ,

and

a ‴ = ∑ i ≥ 0 a i ‴ λ − i , b ″ = ∑ i ≥ 0 b i ‴ λ − i , c ‴ = ∑ i ≥ 0 c i ‴ λ − i , f ‴ = ∑ i ≥ 0 f i ‴ λ − i , g ‴ = ∑ i ≥ 0 g i ‴ λ − i .

It now follows from the enlarged stationary zero curvature equation V ¯ 2 x = [ U ¯ 2 , V ¯ 2 ] that

V x = [ U , V ] , V 1 x = [ U , V 1 ] + [ U 1 , V ] + β [ U 1 , V 1 ] , V 2 x = [ U , V 2 ] + [ U 2 , V ] + μ [ U 2 , V 2 ] + β [ U 1 , V 2 ] + β [ U 2 , V 1 ] , V 3 x = [ U , V 3 ] + [ U 3 , V ] + β [ U 3 , V 1 ] + [ U 1 , V 1 ] + β [ U 1 , V 3 ] + v [ U 2 , V 2 ] .

The above equation system equivalently leads to

(24) a x = u 4 c − u 2 f − u 1 g + u 3 d , b x = u 3 c − u 2 g − u 1 f + u 4 d , c x = λ f − u 4 a − u 3 b , d x = λ g − u 3 a − u 4 b , f x = − λ c + u 1 b + u 2 a , g x = − λ d + u 1 a + u 2 b ,

(25) a x ′ = u 4 c ′ − u 2 f ′ − u 1 g ′ + u 3 d ′ + u 4 ′ c − u 2 ′ f − u 1 ′ g + u 3 ′ d + β ( u 4 ′ c ′ − u 2 ′ f ′ − u 1 ′ g ′ + u 3 ′ d ′ ) , b x ′ = u 3 c ′ − u 2 g ′ − u 1 f ′ + u 4 d ′ + u 3 ′ c − u 2 ′ g − u 1 ′ f + u 4 ′ d + β ( u 3 ′ c ′ − u 2 ′ g ′ − u 1 ′ f ′ + u 4 ′ d ′ ) , c x ′ = λ f ′ − u 4 a ′ − u 3 b ′ − u 4 ′ a − u 3 ′ b − β ( u 4 ′ a ′ + u 3 ′ b ′ ) , d x ′ = λ g ′ − u 3 a ′ − u 4 b ′ − u 3 ′ a − u 4 ′ b − β ( u 3 ′ a ′ + u 4 ′ b ′ ) , f x ′ = − λ c ′ + u 1 b ′ + u 2 a ′ + u 1 ′ b + u 2 ′ a + β ( u 1 ′ b ′ + u 2 ′ a ′ ) , g x ′ = − λ d ′ + u 1 a ′ + u 2 b ′ + u 1 ′ a + u 2 ′ b + β ( u 2 ′ b ′ + u 1 ′ a ′ ) ,

(26) a x ″ = u 4 c ″ − u 2 f ″ − u 1 g ″ + u 3 d ″ + u 4 ″ c − u 2 ″ f − u 1 ″ g + u 3 ″ d + μ u 4 ″ c ″ − u 2 ″ f ″ − u 1 ″ g ″ + u 3 ″ d ″ + β u 4 ″ c ′ − u 2 ″ f ′ − u 1 ″ g ′ + u 3 ″ d ′ , + β u 4 ′ c ″ − u 2 ′ f ″ − u 1 ′ g ″ + u 3 ′ d ″ b x ″ = u 3 c ″ − u 2 g ″ − u 1 f ″ + u 4 d ″ + u 3 ″ c − u 2 ″ g − u 1 ″ f + u 4 ″ d + μ u 3 ′ c ′ − u 2 ′ g ′ − u 1 ′ f ′ + u 4 ′ d ′ + β u 3 ″ c ′ − u 2 ′ g ′ − u 1 ″ f ′ + u 4 ″ d ′ , + β u 3 ′ c ″ − u 2 ′ g ″ − u 1 ′ f ″ + u 4 ′ d ″ c x ″ = λ f ″ − u 4 a ″ − u 3 b ″ − u 4 ″ a − u 3 ″ b − β u 4 ′ a ″ + u 3 ′ b ″ − μ u 4 ″ a ″ + u 3 ″ b ″ − β u 4 ″ a ′ + u 3 ″ b ′ , d x ″ = λ g ″ − u 3 a ″ − u 4 b ″ − u 3 ″ a − u 4 ″ b − β u 3 ′ a ″ + u 4 ′ b ″ − μ u 3 ″ a ″ + u 4 ″ b ″ − β u 3 ″ a ′ + u 4 ″ b ′ , f x ″ = − λ c ″ + u 1 b ″ + u 2 a ″ + u 1 ″ b + u 2 ″ a + β u 1 ′ b ″ + u 2 ′ a ″ + μ u 1 ″ b ″ + u 2 ″ a ″ + β u 1 ″ b ′ + u 2 ″ a ′ , g x ″ = − λ d ″ + u 1 a ″ + u 2 b ″ + u 1 ″ a + u 2 ″ b + β u 1 ′ a ″ + u 2 ′ b ″ + μ u 1 ″ a ″ + u 2 ″ b ″ + β u 1 ″ a ′ + u 2 ″ b ′ ,

(27) a x ‴ = u 4 c ‴ − u 2 f ‴ − u 1 g ‴ + u 3 d ‴ + u 4 ‴ c − u 2 ‴ f − u 1 ‴ g + u 3 ‴ d + β u 4 ‴ c ′ − u 2 ‴ f ′ − u 1 ‴ g ′ + u 3 ‴ d ′ + β u 4 ′ c ‴ − u 2 ′ f ‴ − u 1 ′ g ‴ + u 3 ′ d ‴ + u 4 ′ c ′ − u 2 ′ f ′ − u 1 ′ g ′ + u 3 ′ d ′ + v u 4 ″ c ″ − u 2 ″ f ″ − u 1 ″ g ″ + u 3 ″ d ″ , b x ‴ = u 3 c ‴ − u 2 g ‴ − u 1 f ‴ + u 4 d ‴ + u 3 ‴ c − u 2 ‴ g − u 1 ‴ f + u 4 ‴ d + β u 3 ‴ c ′ − u 2 ‴ g ′ − u 1 ‴ f ′ + u 4 ‴ d ′ + β u 3 ′ c ‴ − u 2 ′ g ‴ − u 1 ′ f ‴ + u 4 ′ d ‴ + u 3 ′ c ′ − u 2 ′ g ′ − u 1 ′ f ′ + u 4 ′ d ′ + v u 3 ″ c ″ − u 2 ″ g ″ − u 1 ″ f ″ + u 4 ″ d ″ , c x ‴ = λ f ‴ − u 4 a ‴ − u 3 b ‴ − u 4 ‴ a − u 3 ‴ b − β u 4 ‴ a ′ + u 3 ‴ b ′ − β u 4 ′ a ‴ + u 3 ′ b ‴ − u 4 ′ a ′ − u 3 ′ b ′ − v u 4 ″ a ″ + u 3 ″ b ″ , d x ‴ = λ g ‴ − u 3 a ‴ − u 4 b ‴ − u 3 ‴ a − u 4 ‴ b − β u 3 ‴ a ′ + u 4 ‴ b ′ − β u 3 ′ a ‴ + u 4 ′ b ‴ − u 3 ′ a ′ − u 4 ′ b ′ − v u 3 ″ a ″ + u 4 ″ b ″ , f x ‴ = − λ c ‴ + u 1 b ‴ + u 2 a ‴ + u 1 ‴ b + u 2 ‴ a + β u 1 ‴ b ′ + u 2 ‴ a ′ + β u 1 ′ b ‴ + u 2 ′ a ‴ + u 1 ′ b ′ + u 2 ′ a ′ + v u 1 ″ b ″ + u 2 ″ a ″ , g x ‴ = − λ d ‴ + u 1 a ‴ + u 2 b ‴ + u 1 ‴ a + u 2 ‴ b + β u 1 ‴ a ′ + u 2 ‴ b ′ + β u 1 ′ a ‴ + u 2 ′ b ‴ + u 1 ′ a ′ + u 2 ′ b ′ + v u 1 ″ a ″ + u 2 ″ b ″ .

Now, we define the enlarged Lax matrices V ¯ 2 [ m ] = ( λ m V ¯ 2 ) + = M 3 ( V [ m ] , V 1 [ m ] , V 2 [ m ] , V 3 [ m ] ) , m ≥ 0 , where V[m] is defined as V [ m ] = ( λ m V ) + , and V i [ m ] = ( λ m V i ) + , i = 1 , 2 , 3.

Solving the enlarged zero curvature equations U ¯ 2 t m − V ¯ 2 x [ m ] + [ U ¯ 2 , V ¯ 2 [ m ] ] = 0 , m ≥ 0 , we get tri-integrable couplings of the soliton hierarchy in [16]:

(28) u ¯ t m = u t m u 1 t m u 2 t m u 3 t m = J 1 0 0 0 0 J 1 0 0 0 0 J 1 0 0 0 0 J 1 P 1 m P 2 m P 3 m P 4 m = J 1 0 0 0 0 J 1 0 0 0 0 J 1 0 0 0 0 J 1 P 2 , m + 1 ,

where

u t m = u 1 u 2 u 3 u 4 t m = f m + 1 g m + 1 − c m + 1 − d m + 1 , u 1 t m = u 1 ′ u 2 ′ u 3 ′ u 4 ′ = f m + 1 ′ g m + 1 ′ − c m + 1 ′ − d m + 1 ′ , u 2 t m = u 1 ″ u 2 ″ u 3 ″ u 4 ″ t m = f m + 1 ″ g m + 1 ″ − c m + 1 ″ − d m + 1 ″ , u 3 t m = u 1 ‴ u 2 ‴ u 3 ‴ u 4 ‴ = f m + 1 ‴ g m + 1 ‴ − c m + 1 ‴ − d m + 1 ‴ ,

J 1 = 0 0 − 1 2 0 0 0 0 − 1 2 1 2 0 0 0 0 1 2 0 0 ,

and

P 1 m = − 2 c m + 1 − 2 d m + 1 − 2 f m + 1 − 2 g m + 1 , P 2 m = − 2 c m + 1 ′ − 2 d m + 1 ′ − 2 f m + 1 ′ − 2 g m + 1 ′ , P 3 m = − 2 c m + 1 ″ − 2 d m + 1 ″ − 2 f m + 1 ″ − 2 g m + 1 ″ , P 4 m = − 2 c m + 1 ‴ − 2 d m + 1 ‴ − 2 f m + 1 ‴ − 2 g m + 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]

δ δ u ¯ ∫ { V ¯ , U ¯ λ } d x = λ − γ ∂ ∂ λ λ γ { V ¯ , U ¯ u } ,

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

∀ a = ( a 1 , ⋯ , a 6 , a 1 ′ , ⋯ , a 6 ′ , a 1 ″ , ⋯ , a 6 ″ , a 1 ‴ , ⋯ , a 6 ‴ ) ∈ R 24 , b = ( b 1 , ⋯ , b 6 , b 1 ′ , ⋯ b 6 ′ , b 1 ″ , ⋯ , b 6 ″ , b 1 ‴ , ⋯ , b 6 ‴ ) ∈ R 24 , we define the Lie bracket [·,·] on R24 as follows: [ a , b ] = a T R ¯ 2 ( b ) , where

R ¯ 2 ( b ) = R ( b ) R 1 ( b ) R 2 ( b ) R 3 ( b ) 0 R ( b ) + β R 1 ( b ) β R 2 ( b ) R 1 ( b ) + β R 3 ( b ) 0 0 R ( b ) + β R 1 ( b ) + μ R 2 ( b ) v R 2 ( b ) 0 0 0 R ( b ) + β R 1 ( b ) ,

with R(b), R1(b)and R2.(b)being defined by (15), (16) and (17), and

R 3 ( b ) = 0 0 b 6 ‴ b 5 ‴ − b 4 ‴ − b 3 ‴ 0 0 b 5 ‴ b 6 ‴ − b 3 ‴ − b 4 ‴ − b 6 ‴ − b 5 ‴ 0 0 b 2 ‴ b 1 ‴ − b 5 ‴ − b 6 ‴ 0 0 b 1 ‴ b 2 ‴ b 4 ‴ b 3 ‴ − b 2 ‴ − b 1 ‴ 0 0 b 3 ‴ b 4 ‴ − b 1 ‴ − b 2 ‴ 0 0 .

Following the properties of the matrix F 2 : F 2 ( R ¯ 2 ( b ) ) T = R ¯ 2 ( b ) F 2 and F 2 = F 2 T , we have

F 2 = η 1 η 2 η 3 η 4 η 2 β η 2 + η 4 β η 3 β η 4 η 3 β η 3 μ η 3 + v η 4 0 η 4 β η 4 0 0 ⊗ r 1 r 2 0 0 0 0 r 2 r 1 0 0 0 0 0 0 r 1 r 2 0 0 0 0 r 2 r 1 0 0 0 0 0 0 r 1 r 2 0 0 0 0 r 2 r 1 ,

where ⊗ is the Kronecker product and η1, η2, η3, η4, r1, r2 are arbitrary constants. We choose r1 ≠ ±r2, then we easily have

det ( F 2 ) = η 4 12 ( β 2 η 1 − β η 2 + η 4 ) 6 ( μ η 3 + v η 4 ) 6 ( r 1 2 − r 2 2 ) 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:

(29) { a , b } = a T F 2 b .

Now, it is easy to compute that

{ V ¯ 2 , U ¯ 2 , λ } = ( r 2 a + r 1 b ) η 1 + ( r 2 a ′ + r 1 b ′ ) η 2 + ( r 2 a ″ + r 1 b ″ ) η 3 + ( r 2 a ‴ + r 1 b ‴ ) η 4 , { V ¯ 2 , U ¯ 2 , λ } = ( r 2 a + r 1 b ) η 1 + ( r 2 a ′ + r 1 b ′ ) η 2 + ( r 2 a ″ + r 1 b ″ ) η 3 + ( r 2 a ‴ + r 1 b ‴ ) η 4 , { V ¯ 2 , U ¯ 2 , u 1 } = ( r 2 d + r 1 c ) η 1 + ( r 2 d ′ + r 1 c ′ ) η 2 + ( r 1 c ″ + r 2 d ″ ) η 3 + ( r 2 d ‴ + r 1 c ‴ ) η 4 , { V ¯ 2 , U ¯ 2 , u 2 } = ( r 2 c + r 1 d ) η 1 + ( r 2 c ′ + r 1 d ′ ) η 2 + ( r 2 c ″ + r 1 d ″ ) η 3 + ( r 1 d ‴ + r 2 c ‴ ) η 4 , { V ¯ 2 , U ¯ 2 , u 3 } = ( r 2 g + r 1 f ) η 1 + ( r 2 g ′ + r 1 f ′ ) η 2 + ( r 2 g ″ + r 1 f ″ ) η 3 + ( r 1 f ‴ + r 2 g ‴ ) η 4 , { V ¯ 2 , U ¯ 2 , u 4 } = ( r 2 f + r 1 g ) η 1 + ( r 2 f ′ + r 1 g ′ ) η 2 + ( r 2 f ″ + r 1 g ″ ) η 3 + ( r 1 g ‴ + r 2 f ‴ ) η 4 , { V ¯ 2 , U ¯ 2 , u 1 ′ } = [ ( r 2 d + r 1 c ) + β ( r 1 c ′ + r 2 d ′ ) ] η 2 + β ( r 2 d ″ + r 1 c ″ ) η 3 + [ ( r 2 d ′ + r 1 c ′ ) + β ( r 1 c ‴ + r 2 d ‴ ) ] η 4 , { V ¯ 2 , U ¯ 2 , u 2 ′ } = [ ( r 2 c + r 1 d ) + β ( r 2 c ′ + r 1 d ′ ) ] η 2 + β ( r 2 c ″ + r 1 d ″ ) η 3 + [ ( r 2 c ′ + r 1 d ′ ) + β ( r 2 c ‴ + r 1 d ‴ ) ] η 4 , { V ¯ 2 , U ¯ 2 , u 3 ′ } = [ ( r 2 g + r 1 f ) + β ( r 2 g ′ + r 1 f ′ ) ] η 2 + β ( r 2 g ″ + r 1 f ″ ) η 3 + [ ( r 2 g ′ + r 1 f ′ ) + β ( r 2 g ‴ + r 1 f ‴ ) ] η 4 , { V ¯ 2 , U ¯ 2 , u 4 ′ } = [ ( r 2 f + r 1 g ) + β ( r 2 f ′ + r 1 g ′ ) ] η 2 + β ( r 2 f ″ + r 1 g ″ ) η 3 + [ ( r 2 f ′ + r 1 g ′ ) + β ( r 2 f ‴ + r 1 g ‴ ) ] η 4 , { V ¯ 2 , U ¯ 2 , u 1 ″ } = [ ( r 2 d + r 1 c ) + β ( r 2 d ′ + r 1 c ′ ) + μ ( r 2 d ″ + r 1 c ″ ) ] η 3 + v ( r 2 d ″ + r 1 c ″ ) η 4 , { V ¯ 2 , U ¯ 2 , u 2 ″ } = [ ( r 2 c + r 1 d ) + β ( r 2 c ′ + r 1 d ′ ) + μ ( r 2 c ″ + r 1 d ″ ) ] η 3 + v ( r 2 c ″ + r 1 d ″ ) η 4 , { V ¯ 2 , U ¯ 2 , u 3 ″ } = [ ( r 2 g + r 1 f ) + β ( r 2 g ′ + r 1 f ′ ) + μ ( r 2 g ″ + r 1 f ″ ) ] η 3 + v ( r 2 g ″ + r 1 f ″ ) η 4 , { V ¯ 2 , U ¯ 2 , u 4 ″ } = [ ( r 2 f + r 1 g ) + β ( r 2 f ′ + r 1 g ′ ) + μ ( r 2 f ″ + r 1 g ″ ) ] η 3 + v ( r 2 f ″ + r 1 g ″ ) η 4 , { V ¯ 2 , U ¯ 2 , u 1 ‴ } = [ ( r 2 d + r 1 c ) + β ( r 2 d ′ + r 1 c ′ ) ] η 4 , { V ¯ 2 , U ¯ 2 , u 2 ‴ } = [ ( r 2 c + r 1 d ) + β ( r 2 c ′ + r 1 d ′ ) ] η 4 , { V ¯ 2 , U ¯ 2 , u 3 ‴ } = [ ( r 2 g + r 1 f ) + β ( r 2 g ″ + r 1 f ″ ) ] η 4 , { V ¯ 2 , U ¯ 2 , u 4 ‴ } = [ ( r 2 f + r 1 g ) + β ( r 2 f ″ + r 1 g ″ ) ] η 4 .

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 ¯ t m = K ¯ 2 , m ( u ¯ ) = J ¯ 2 δ H ¯ 2 , m δ u ¯ , m ≥ 0 ,

where the Hamiltonian operator is

J ¯ 2 = η 1 η 2 η 3 η 4 η 2 β η 2 + η 4 β η 3 β η 4 η 3 β η 3 μ η 3 + v η 4 0 η 4 β η 4 0 0 − 1 ⊗ 0 0 − 1 2 0 0 0 0 − 1 2 1 2 0 0 0 0 1 2 0 0 ,

and the Hamiltonian functionals read

H ¯ 2 , m = − ∫ ( r 2 a m + 2 + r 1 b m + 2 ) η 1 + ( r 2 a m + 2 ′ + r 1 b m + 2 ′ ) η 2 m + 1 d x − ∫ ( r 2 a m + 2 ″ + r 1 b m + 2 ″ ) η 3 + ( r 2 a m + 2 ‴ + r 1 b m + 2 ‴ ) η 4 m + 1 d x , m ≥ 0.

Based on (24), (25), (26), (27), a direct computation yields to the recursion relation

P 2 , m + 1 = L ¯ 2 P 2 , m ,

where the recursion operator L ¯ 2 is given by

L ¯ 2 = M 3 T ( L 1 , L 1 1 , L 3 1 , L 4 1 ) = L 1 0 0 0 L 1 1 L 1 + β L 1 1 0 0 L 3 1 β L 3 1 L 1 + β L 1 1 + μ L 3 1 0 L 4 1 L 1 + β L 4 1 v L 3 1 L 1 + β L 1 1 ,

with L1 and L 1 1 being given as in (20), and

L 3 1 = x 11 x 12 x 13 x 14 x 21 x 22 x 23 x 24 x 31 x 32 x 33 x 34 x 41 x 42 x 43 x 44 , L 4 1 = y 11 y 12 y 13 y 14 y 21 y 22 y 23 y 24 y 31 y 32 y 33 y 34 y 41 y 42 y 43 y 44 ,

and

x 11 = x 22 = ( u 2 + μ u 2 ″ + β u 2 ′ ) ∂ − 1 u 4 ″ + ( u 1 + μ u 1 ″ + β u 1 ′ ) ∂ − 1 u 3 ″ + β u 1 ″ ∂ − 1 u 3 ′ + β u 2 ″ ∂ − 1 u 4 ′ + u 1 ″ ∂ − 1 u 3 + u 2 ″ ∂ − 1 u 4 , x 12 = x 21 = ( u 2 + μ u 2 ″ + β u 2 ′ ) ∂ − 1 u 3 ″ + ( u 1 + μ u 1 ″ + β u 1 ′ ) ∂ − 1 u 4 ″ + β u 1 ″ ∂ − 1 u 4 ′ + β u 2 ″ ∂ − 1 u 3 ′ + u 1 ″ ∂ − 1 u 4 + u 2 ″ ∂ − 1 u 3 , x 13 = x 24 = − ( u 2 + μ u 2 ″ + β u 2 ′ ) ∂ − 1 u 2 ″ − ( u 1 + μ u 1 ″ + β u 1 ′ ) ∂ − 1 u 1 ″ − β u 1 ″ ∂ − 1 u 1 ′ − β u 2 ″ ∂ − 1 u 2 ′ − u 1 ″ ∂ − 1 u 1 − u 2 ″ ∂ − 1 u 2 , x 14 = x 23 = − ( u 2 + μ u 2 ″ + β u 2 ′ ) ∂ − 1 u 1 ″ − ( u 1 + μ u 1 ″ + β u 1 ′ ) ∂ − 1 u 2 ″ − β u 1 ″ ∂ − 1 u 2 ′ − β u 2 ″ ∂ − 1 u 1 ′ − u 1 ″ ∂ − 1 u 2 − u 2 ″ ∂ − 1 u 1 , x 31 = x 42 = ( u 4 + μ u 4 ″ + β u 4 ′ ) ∂ − 1 u 4 ″ + ( u 3 + μ u 3 ″ + β u 3 ′ ) ∂ − 1 u 3 ″ + β u 4 ″ ∂ − 1 u 4 ′ + β u 3 ″ ∂ − 1 u 3 ′ + u 4 ″ ∂ − 1 u 4 + u 3 ″ ∂ − 1 u 3 , x 32 = x 41 = ( u 4 + μ u 4 ″ + β u 4 ′ ) ∂ − 1 u 3 ″ + ( u 3 + μ u 3 ″ + β u 3 ′ ) ∂ − 1 u 4 ″ + β u 4 ″ ∂ − 1 u 3 ′ − β u 3 ″ ∂ − 1 u 4 ′ + u 4 ″ ∂ − 1 u 3 + u 3 ″ ∂ − 1 u 4 , x 33 = x 44 = − ( u 4 + μ u 4 ″ + β u 4 ′ ) ∂ − 1 u 2 ″ − ( u 3 + μ u 3 ″ + β u 3 ′ ) ∂ − 1 u 1 ″ − β u 4 ″ ∂ − 1 u 2 ′ − β u 3 ″ ∂ − 1 u 1 ′ − u 4 ″ ∂ − 1 u 2 − u 3 ″ ∂ − 1 u 1 , x 34 = x 43 = − ( u 4 + μ u 4 ″ + β u 4 ′ ) ∂ − 1 u 1 ″ − ( u 3 + μ u 3 ″ + β u 3 ′ ) ∂ − 1 u 2 ″ − β u 4 ″ ∂ − 1 u 1 ′ − β u 3 ″ ∂ − 1 u 2 ′ − u 4 ″ ∂ − 1 u 1 − u 3 ″ ∂ − 1 u 2 , y 11 = y 22 = ( u 2 + β u 2 ′ ) ∂ − 1 u 4 ‴ + ( u 1 + β u 1 ′ ) ∂ − 1 u 3 ‴ + v u 2 ″ ∂ − 1 u 4 ″ + v u 1 ″ ∂ − 1 u 3 ″ + ( u 2 ′ + β u 2 ‴ ) ∂ − 1 u 4 ′ + ( u 1 ′ + β u 1 ‴ ) ∂ − 1 u 3 ′ + u 2 ‴ ∂ − 1 u 4 + u 1 ‴ ∂ − 1 u 3 ,

y 12 = y 21 = ( u 2 + β u 2 ′ ) ∂ − 1 u 3 ‴ + ( u 1 + β u 1 ′ ) ∂ − 1 u 4 ‴ + v u 2 ″ ∂ − 1 u 3 ″ + v u 1 ″ ∂ − 1 u 4 ″ + ( u 2 ′ + β u 2 ‴ ) ∂ − 1 u 3 ′ + ( u 1 ′ + β u 1 ‴ ) ∂ − 1 u 4 ′ + u 2 ‴ ∂ − 1 u 3 + u 1 ‴ ∂ − 1 u 4 , y 13 = y 24 = − ( u 2 + β u 2 ′ ) ∂ − 1 u 2 ‴ − ( u 1 + β u 1 ′ ) ∂ − 1 u 1 ‴ − v u 2 ″ ∂ − 1 u 2 ″ − v u 1 ″ ∂ − 1 u 1 ″ − ( u 2 ′ + β u 2 ‴ ) ∂ − 1 u 2 ′ − ( u 1 ′ + β u 1 ‴ ) ∂ − 1 u 1 ′ − u 2 ‴ ∂ − 1 u 2 − u 1 ‴ ∂ − 1 u 1 , y 14 = y 23 = − ( u 2 + β u 2 ′ ) ∂ − 1 u 1 ‴ − ( u 1 + β u 1 ′ ) ∂ − 1 u 2 ‴ − v u 2 ″ ∂ − 1 u 1 ″ − v u 1 ″ ∂ − 1 u 2 ″ − ( u 2 ′ + β u 2 ‴ ) ∂ − 1 u 1 ′ − ( u 1 ′ + β u 1 ‴ ) ∂ − 1 u 2 ′ − u 2 ‴ ∂ − 1 u 1 − u 1 ‴ ∂ − 1 u 2 , y 31 = y 42 = ( u 4 + β u 4 ′ ) ∂ − 1 u 4 ‴ + ( u 3 + β u 3 ′ ) ∂ − 1 u 3 ‴ + v u 4 ″ ∂ − 1 u 4 ″ + v u 3 ″ ∂ − 1 u 3 ″ + ( β u 4 ‴ + u 4 ′ ) ∂ − 1 u 4 ′ + ( β u 3 ‴ + u 3 ′ ) ∂ − 1 u 3 ′ + u 4 ‴ ∂ − 1 u 4 + u 3 ‴ ∂ − 1 u 3 , y 32 = y 41 = ( u 4 + β u 4 ′ ) ∂ − 1 u 3 ‴ + ( u 3 + β u 3 ′ ) ∂ − 1 u 4 ‴ + v u 4 ″ ∂ − 1 u 3 ″ + v u 3 ″ ∂ − 1 u 4 ″ + ( β u 4 ‴ + u 4 ′ ) ∂ − 1 u 3 ′ + ( β u 3 ‴ + u 3 ′ ) ∂ − 1 u 4 ′ + u 4 ‴ ∂ − 1 u 3 + u 3 ‴ ∂ − 1 u 4 , y 33 = y 44 = − ( u 4 + β u 4 ′ ) ∂ − 1 u 2 ‴ − ( u 3 + β u 3 ′ ) ∂ − 1 u 1 ‴ − v u 4 ″ ∂ − 1 u 2 ″ − v u 3 ″ ∂ − 1 u 1 ″ − ( β u 4 ‴ + u 4 ′ ) ∂ − 1 u 2 ′ − ( β u 3 ‴ + u 3 ′ ) ∂ − 1 u 1 ′ − u 4 ‴ ∂ − 1 u 2 − u 3 ‴ ∂ − 1 u 1 , y 34 = y 43 = − ( u 4 + β u 4 ′ ) ∂ − 1 u 1 ‴ − ( u 3 + β u 3 ′ ) ∂ − 1 u 2 ‴ − v u 4 ″ ∂ − 1 u 1 ″ − v u 3 ″ ∂ − 1 u 2 ″ − ( β u 4 ‴ + u 4 ′ ) ∂ − 1 u 1 ′ − ( β u 3 ‴ + u 3 ′ ) ∂ − 1 u 2 ′ − u 4 ‴ ∂ − 1 u 1 − u 3 ‴ ∂ − 1 u 2 .

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.

  1. Conflict of interest

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

Acknowledgement

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

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Received: 2016-8-30
Accepted: 2017-1-26
Published Online: 2017-3-11

© 2017 Zhang et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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https://doi.org/10.1515/math-2017-0017
Keywords for this article
Bi-integrable couplings; Tri-integrable couplings; Zero curvature equations; Hamiltonian structures
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  247. Special Issue on Recent Developments in Differential Equations
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  257. Special Issue on Recent Developments in Differential Equations
  258. Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities
  259. Special Issue on Recent Developments in Differential Equations
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  264. Integro-differential systems with variable exponents of nonlinearity
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  268. Multiplicity solutions of a class fractional Schrödinger equations
  269. Special Issue on Recent Developments in Differential Equations
  270. Determining of right-hand side of higher order ultraparabolic equation
  271. Special Issue on Recent Developments in Differential Equations
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  277. Topical Issue on Cyber-security Mathematics
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  279. Topical Issue on Cyber-security Mathematics
  280. Expert knowledge and data analysis for detecting advanced persistent threats
  281. Topical Issue on Cyber-security Mathematics
  282. Feedback equivalence of convolutional codes over finite rings
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