Nonlinear Super Integrable Couplings of Super Broer-Kaup-Kupershmidt Hierarchy and Its Super Hamiltonian Structures

Abstract

Nonlinear integrable couplings of super Broer-Kaup-Kupershmidt hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identity, and the conserved functionals were proved to be in involution in pairs under the defined Poisson bracket. As its reduction, nonlinear integrable couplings of the classical integrable hierarchy were obtained.

1. Introduction

With the development of soliton theory, super integrable systems associated with Lie super algebra have aroused growing attentions by many mathematicians and physicists. It was known that super integrable systems contained the odd variables, which would provide more prolific fields for mathematical researchers and physical ones. Several super integrable systems, including super AKNS hierarchy, super KdV hierarchy, super NLS-MKdV hierarchy, super Tu hierarchy, super Broer-Kaup-Kupershmidt hierarchy, have been studied in [1–8]. There are some interesting results on the super integrable systems, such as Darboux transformation in [9], super Hamiltonian structures in [10–12] binary nonlinearization in [13], and reciprocal transformation in [14].

The research of integrable couplings of the well-known integrable hierarchy has received considerable attention in [15–23] A few approaches to construct linear integrable couplings of the classical soliton equation are presented by permutation, enlarging spectral problem, using matrix Lie algebra [24] constructing new loop Lie algebra and creating semidirect sums of Lie algebra. Recently, Ma [25] and Ma and Zhu [26] presented a scheme for constructing nonlinear continuous and discrete integrable couplings using the block type matrix algebra. However, there is one interesting question for us is how to generate nonlinear super integrable couplings for the super integrable hierarchy.

In this paper, we would like to construct nonlinear super integrable couplings of the super soliton equations through enlarging matrix Lie super algebra. We take the Lie algebra 𝐵(0,1) as an example to illustrate the approach for extending Lie super algebras. Based on the enlarged Lie super algebra 𝑔𝑙(6,2), we work out nonlinear super integrable Hamiltonian couplings of the super Broer-Kaup-Kupershmidt hierarchy. Finally, we will reduce the nonlinear super Broer-Kaup-Kupershmidt integrable Hamiltonian couplings to some special cases.

2. Enlargement of Lie Super Algebra 𝐵(0,1)

Consider the Lie super algebra 𝐵(0,1). Its basis is 𝐸₁=1000−10000,𝐸₂=,𝐸010000000₃=000100000,𝐸₄=,𝐸0010000−10₅=,000001100 (1) where 𝐸₁, 𝐸₂, and 𝐸₃ are even elements and 𝐸₄, and 𝐸₅ are odd elements. Their nonzero (anti)commutation relations are 𝐸₁,𝐸₂=2𝐸₂,𝐸₁,𝐸₃=−2𝐸₃,𝐸₁,𝐸₄=𝐸₄,𝐸₁,𝐸₅=−𝐸₅,𝐸₂,𝐸₃=𝐸₁,𝐸₂,𝐸₅=𝐸₄,𝐸₃,𝐸₄=𝐸₅,𝐸₄,𝐸₄=−2𝐸₂,𝐸₄,𝐸₅=𝐸₁,𝐸₅,𝐸₅=2𝐸₃. (2) Let us enlarged the Lie super algebra 𝐵(0,1) to the Lie super algebra 𝑔𝑙(6,2) with a basis 𝑒₁=100000−100000100000−1000000,𝑒₂=,𝑒0100000000000100000000000₃=0000010000000000010000000,𝑒₄=,𝑒00100000−1000100000−1000000₅=0001000000000100000000000,𝑒₆=,𝑒0000000100000000010000000₇=000010000000000000000−1010,𝑒₈=,0000000001000000000010−100 (3) where 𝑒₁, 𝑒₂, 𝑒₃, 𝑒₄, 𝑒₅, and 𝑒₆ are even, and 𝑒₇, and 𝑒₈ are odd.

The generator of Lie super algebra 𝑔𝑙(6,2), 𝑒_𝑖, 0≤𝑖≤8, satisfy the following (anti)commutation relations: 𝑒₁,𝑒₂=2𝑒₂,𝑒₁,𝑒₃=−2𝑒₃,𝑒₁,𝑒₅=2𝑒₅,𝑒₁,𝑒₆=−2𝑒₆,𝑒₁,𝑒₇=𝑒₇,𝑒₁,𝑒₈=−𝑒₈,𝑒₂,𝑒₃=𝑒₁,𝑒₂,𝑒₄=−2𝑒₅,𝑒₂,𝑒₆=𝑒₄,𝑒₂,𝑒₈=𝑒₇,𝑒₃,𝑒₄=2𝑒₆,𝑒₃,𝑒₅=−𝑒₄,𝑒₃,𝑒₇=𝑒₈,𝑒₄,𝑒₅=2𝑒₅,𝑒₄,𝑒₆=−2𝑒₆,𝑒₅,𝑒₆=𝑒₄,𝑒₇,𝑒₈=𝑒₁−𝑒₄,𝑒₇,𝑒₇=2𝑒₅−2𝑒₂,𝑒₈,𝑒₈=2𝑒₃−2𝑒₆,𝑒₁,𝑒₄=𝑒₂,𝑒₅=𝑒₂,𝑒₇=𝑒₃,𝑒₆=𝑒₃,𝑒₈=𝑒₄,𝑒₇=𝑒₄,𝑒₈=𝑒₅,𝑒₇=𝑒₅,𝑒₈=𝑒₆,𝑒₇=𝑒₆,𝑒₈=0. (4) Define a loop super algebra corresponding to the Lie superalgebra 𝑔𝑙(6,2), denote by 𝑔𝑙(6,2)=𝑔𝑙(6,2)⊗ℂ𝜆,𝜆⁻¹=𝑒_𝑖𝜆^𝑚,𝑒_𝑖()}.∈𝑔𝑙6,2,𝑖=1,2,…,8,𝑚=0,±1,±2,… (5) The corresponding (anti)commutative relations are given as 𝑒_𝑖𝜆^𝑚,𝑒_𝑗𝜆^𝑛=𝑒_𝑖,𝑒_𝑗𝜆^𝑚+𝑛,∀𝑒_𝑖,𝑒_𝑗∈𝑔𝑙(6,2). (6)

3. Nonlinear Super Integrable Couplings of Super Broer-Kaup-Kupershmidt Hierarchy

Let us start from an enlarged spectral problem associated with 𝑔𝑙(6,2): 𝜙_𝑥=𝑈(𝑢,𝜆)𝜙,𝑈=𝑒₁(1)+𝑟𝑒₁(0)+𝑠𝑒₂(0)+𝑒₃(0)+𝑢₁𝑒₄(0)+𝑢₂𝑒₅(0)+𝛼𝑒₇(0)+𝛽𝑒₈(0)=𝜆+𝑟𝑠𝑢₁𝑢₂𝛼1−𝜆−𝑟0−𝑢₁𝛽00𝜆+𝑟+𝑢₁𝑠+𝑢₂0001−𝜆−𝑟−𝑢₁0,𝛽−𝛼−𝛽𝛼0 (7) where 𝑟, 𝑠, 𝑢₁, and 𝑢₂ are even potentials but 𝛼 and 𝛽 are odd ones.

In ordr to obtain super integrable couplings of super integrable hierarchy, we first solve the adjoint representation of (7), 𝑉_𝑥=[𝑈,𝑉], (8) with 𝑉=𝐴𝑒₁(0)+𝐵𝑒₂(0)+𝐶𝑒₃(0)+𝐸𝑒₄(0)+𝐹𝑒₅(0)+𝐺𝑒₆(0)+𝜌𝑒₇(0)+𝛿𝑒₈(0)=,𝐴𝐵𝐸𝐹𝜌𝐶−𝐴𝐺−𝐸𝛿00𝐴+𝐸𝐵+𝐹000𝐶+𝐺−𝐴−𝐸0𝛿−𝜌−𝛿𝜌0 (9) where 𝐴, 𝐵, 𝐶, 𝐸, 𝐹, and 𝐺 are commuting fields and 𝜌, and 𝛿 are anticommuting fields. Then we obtain 𝐴_𝑥𝐵=−𝐵+𝑠𝐶+𝛽𝜌+𝛼𝛿,_𝑥𝐶=−2𝑠𝐴+2𝜆𝐵+2𝑟𝐵−2𝛼𝜌,_𝑥𝐸=2𝐴−2𝜆𝐶−2𝑟𝐶+2𝛽𝛿,_𝑥=𝑢₂𝐶−𝐹+𝑠𝐺+𝑢₂𝐹𝐺−𝛼𝛿−𝛽𝜌,_𝑥=−2𝑢₂𝐴+2𝑢₁𝐵−2𝑠𝐸−2𝑢₂𝐸+2𝜆𝐹+2𝑟𝐹+2𝑢₁𝐺𝐹+2𝛼𝜌,_𝑥=−2𝑢₁𝐶+2𝐸−2𝜆𝐺−2𝑟𝐺−2𝑢₁𝜌𝐺−2𝛽𝛿,_𝑥𝛿=−𝛼𝐴−𝛽𝐵+𝜆𝜌+𝑟𝜌+𝑠𝛿,_𝑥=𝛽𝐴−𝛼𝐶+𝜌−𝜆𝛿−𝑟𝛿. (10) Substituting 𝐴=_𝑚≥0𝐴_𝑚𝜆^−𝑚,𝐵=_𝑚≥0𝐵_𝑚𝜆^−𝑚,𝐶=_𝑚≥0𝐶_𝑚𝜆^−𝑚,𝐸=_𝑚≥0𝐸_𝑚𝜆^−𝑚,𝐹=_𝑚≥0𝐹_𝑚𝜆^−𝑚,𝐺=_𝑚≥0𝐺_𝑚𝜆^−𝑚,𝜌=_𝑚≥0𝜌_𝑚𝜆^−𝑚,𝛿=_𝑚≥0𝛿_𝑚𝜆^−𝑚, (11) into the previous equation gives the following recursive formulas: 𝐴_𝑚,𝑥=−𝐵_𝑚+𝑠𝐶_𝑚+𝛽𝜌_𝑚+𝛼𝛿_𝑚,𝐵_𝑚,𝑥=−2𝑠𝐴_𝑚+2𝐵_𝑚+1+2𝑟𝐵_𝑚−2𝛼𝜌_𝑚,𝐶_𝑚,𝑥=2𝐴_𝑚−2𝐶_𝑚+1−2𝑟𝐶_𝑚+2𝛽𝛿_𝑚,𝐸_𝑚,𝑥=𝑢₂𝐶_𝑚−𝐹_𝑚+𝑠𝐺_𝑚+𝑢₂𝐺_𝑚−𝛼𝛿_𝑚−𝛽𝜌_𝑚,𝐹_𝑚,𝑥=−2𝑢₂𝐴_𝑚+2𝑢₁𝐵_𝑚−2𝑠𝐸_𝑚−2𝑢₂𝐸_𝑚+2𝐹_𝑚+1+2𝑟𝐹_𝑚+2𝑢₁𝐹_𝑚+2𝛼𝜌_𝑚,𝐺_𝑚,𝑥=−2𝑢₁𝐶_𝑚+2𝐸_𝑚−2𝐺_𝑚+1−2𝑟𝐺_𝑚−2𝑢₁𝐺_𝑚−2𝛽𝛿_𝑚,𝜌_𝑚,𝑥=−𝛼𝐴_𝑚−𝛽𝐵_𝑚+𝜌_𝑚+1+𝑟𝜌_𝑚+𝑠𝛿_𝑚,𝛿_𝑚,𝑥=𝛽𝐴_𝑚−𝛼𝐶_𝑚+𝜌_𝑚−𝛿_𝑚+1−𝑟𝛿_𝑚. (12) From these equations, we can successively deduce 𝐴₀=1,𝐵₀=𝐶₀=𝐹₀=𝐺₀=𝜌₀=𝛿₀𝐸=0,₀=𝜀=constant,𝐴₁=0,𝐵₁=𝑠,𝐶₁𝐸=1,₁=0,𝐹₁=𝑢₂+𝜀𝑢₂+𝜀𝑠,𝐺₁𝜌=𝜀,₁=𝛼,𝛿₁=𝛽,𝐴₂1=−2𝐵𝑠−𝛼𝛽,₂=12𝑠_𝑥−𝑟𝑠,𝐶₂𝐸=−𝑟,₂1=−2𝑢₂−12𝜀𝑢₂−12𝐹𝜀𝑠+𝛼𝛽,₂=12𝑢_2𝑥+12𝜀𝑢_2𝑥+12𝜀𝑠_𝑥−𝑠𝑢₁−𝑟𝑢₂−𝜀𝑟𝑢₂−𝜀𝑟𝑠−𝑢₁𝑢₂−𝜀𝑢₁𝑢₂−𝜀𝑠𝑢₁,𝐺₂=−𝑢₁−𝜀𝑢₁−𝜀𝑟,𝜌₂=𝛼_𝑥𝛿−𝑟𝛼,₂=−𝛽_𝑥−𝑟𝛽,𝐴₃1=−4𝑠_𝑥+𝑟𝑠+𝛼𝛽_𝑥−𝛼_𝑥𝐵𝛽+2𝑟𝛼𝛽,₃=14𝑠_𝑥𝑥−12𝑟_𝑥𝑠−𝑟𝑠_𝑥−12𝑠²−𝑠𝛼𝛽+𝑟²𝑠+𝛼𝛼_𝑥,𝐶₃=12𝑟_𝑥−12𝑠−𝛼𝛽+𝑟²−𝛽𝛽_𝑥,𝐸₃=(𝜀+1)𝑟𝑢₂+𝑠𝑢₁+𝑢₁𝑢₂−14𝑢_2𝑥1+𝜀𝑟𝑠−4𝜀𝑠_𝑥−𝛼𝛽_𝑥+𝛼_𝑥𝐹𝛽−2𝑟𝛼𝛽,₃=()×−1𝜀+12𝑠𝑢_1𝑥−𝑠_𝑥𝑢₁−12𝑟_𝑥𝑢₂−𝑟𝑢_2𝑥−12𝑢_1𝑥𝑢₂−𝑢₁𝑢_2𝑥+14𝑢_2𝑥𝑥−𝑠𝑢₂−12𝑢²₂+𝑟²𝑢₂+𝑠𝑢²₁+𝑢²₁𝑢₂+2𝑟𝑢₁𝑢₂+2𝑟𝑠𝑢₁1+𝑠𝛼𝛽+4𝜀𝑠_𝑥𝑥−12𝜀𝑠²−𝜀𝑟𝑠_𝑥−12𝜀𝑟_𝑥𝑠+𝜀𝑟²𝑠−𝛼𝛼_𝑥,𝐺₃=12𝑢_1𝑥+12𝜀𝑢_1𝑥+12𝜀𝑟_𝑥+2𝑟𝑢₁−12𝑢₂−12𝜀𝑢₂−12𝜀𝑠+𝛼𝛽+2𝜀𝑟𝑢₁+𝜀𝑟²+𝑢²₁+𝜀𝑢²₁+𝛽𝛽_𝑥,𝜌₃=𝛼_𝑥𝑥−2𝑟𝛼_𝑥−𝑟_𝑥1𝛼−21𝑠𝛼+2𝑠_𝑥𝛽+𝑠𝛽_𝑥+𝑟²𝛿𝛼,₃=𝛽_𝑥𝑥+𝑟_𝑥𝛽+2𝑟𝛽_𝑥−12𝑠𝛽+𝛼_𝑥+𝑟²𝛽. (13) Equations (12) can be written as 4𝐴_𝑚+1+2𝐸_𝑚+12𝐶_𝑚+1+𝐺_𝑚+12𝐴_𝑚+1+2𝐸_𝑚+1𝐶_𝑚+1+𝐺_𝑚+12𝛿_𝑚+1−2𝜌_𝑚+1=𝐿4𝐴_𝑚+2𝐸_𝑚2𝐶_𝑚+𝐺_𝑚2𝐴_𝑚+2𝐸_𝑚𝐶_𝑚+𝐺_𝑚2𝛿_𝑚−2𝜌_𝑚, (14) where1𝐿=2𝜕−𝜕⁻¹𝑟𝜕−𝑠−𝜕⁻¹𝑠𝜕−𝜕⁻¹𝑢₁𝜕−𝑢₂−𝜕⁻¹𝑢₂1𝜕−2𝛼−𝜕⁻¹1𝛼𝜕2𝛽−𝜕⁻¹1𝛽𝜕2−12𝜕−𝑟0−2𝑢₁12𝛽000𝐿₃₃𝐿₃₄100002−12𝜕−𝑟−𝑢₁00𝛽−2𝛼−𝛽𝛼−𝜕−𝑟−1−𝛼+𝛽𝜕−2𝑠𝛽𝛼−𝛽𝜕2𝑠𝛽𝑠+𝛼𝛽𝜕−𝑟. (15)

with 𝐿₃₃=12𝜕−𝜕⁻¹𝑟𝜕−𝜕⁻¹𝑢₁𝐿𝜕,₃₄=−𝑠−𝑢₂−𝜕⁻¹𝑠𝜕−𝜕⁻¹𝑢₂𝜕. (16) Then, let us consider the spectral problem (7) with the following auxiliary spectral problem: 𝜙_𝑡𝑛=𝑉^(𝑛)𝜙, (17) with 𝑉^(𝑛)=^𝑛_𝑗=0𝐴_𝑗𝐵_𝑗𝐸_𝑗𝐹_𝑗𝜌_𝑗𝐶_𝑗−𝐴_𝑗𝐺_𝑗−𝐸_𝑗𝛿_𝑗00𝐴_𝑗+𝐸_𝑗𝐵_𝑗+𝐹_𝑗000𝐶_𝑗+𝐺_𝑗−𝐴_𝑗−𝐸_𝑗0𝛿_𝑗−𝜌_𝑗−𝛿_𝑗𝜌_𝑗0𝜆^𝑛−𝑗+−𝐶_𝑛+10−𝐺_𝑛+1000𝐶_𝑛+10𝐺_𝑛+1000−𝐶_𝑛+1−𝐺_𝑛+100000𝐶_𝑛+1+𝐺_𝑛+10.00000 (18)

From the compatible condition, 𝜙_{𝑥,𝑡𝑛}=𝜙_{𝑡𝑛,𝑥}, according to (7) and (17), we get the zero curvature equation: 𝑈_𝑡𝑛−𝑉_𝑥^(𝑛)+𝑈,𝑉^(𝑛)=0, (19) which gives a nonlinear Lax super integrable hierarchy:𝑢_𝑡𝑛=𝑟𝑠𝑢₁𝑢₂𝛼𝛽_𝑡𝑛=−𝐶_𝑛+1,𝑥2𝐵_𝑛+1−2𝑠𝐶_𝑛+1−𝐺_𝑛+1,𝑥−2𝑢₂𝐶_𝑛+1+2𝐹_𝑛+1−2𝑠𝐺_𝑛+1−2𝑢₂𝐺_𝑛+1𝜌_𝑛+1−𝛼𝐶_𝑛+1−𝛿_𝑛+1+𝛽𝐶_𝑛+1. (20) The super integrable hierarchy (20) is a nonlinear super integrable couplings for the super BKK hierarchy: 𝑢_𝑡𝑛=𝑟𝑠𝛼𝛽_𝑡𝑛=−𝐶_𝑛+1,𝑥2𝐵_𝑛+1−2𝑠𝐶_𝑛+1𝜌_𝑛+1−𝛼𝐶_𝑛+1−𝛿_𝑛+1+𝛽𝐶_𝑛+1. (21)

4. Super Hamiltonian Structures

A direct calculation reads Str𝑈_𝜆,𝑉=4𝐴+2𝐸,Str𝑈_𝑟𝑈,𝑉=4𝐴+2𝐸,Str_𝑠,𝑉=2𝐶+𝐺,Str𝑈_𝑢1𝑈,𝑉=𝐶+𝐺,Str_𝑢2,𝑉=𝐵+𝐹,Str𝑈_𝛼𝑈,𝑉=2𝛿,Str_𝛽,𝑉=−2𝜌. (22)

Substituting the above results into the super trace identity in [11, 12], 𝛿𝛿𝑢Str𝛿𝑈𝑉𝛿𝜆d𝑥=𝜆^−𝛾𝜕𝜆𝜕𝜆^𝛾Str𝛿𝑈𝑉𝛿𝑢 (23) yields that 𝛿()𝛿𝑢4𝐴+2𝐸d𝑥=𝜆^−𝛾𝜕𝜆𝜕𝜆^𝛾.4𝐴+2𝐸2𝐶+𝐺2𝐴+2𝐸𝐶+𝐺2𝛿−2𝜌 (24)

Comparing the coefficients of 𝜆^−𝑛−1 on both side of (24), 𝛿𝛿𝑢4𝐴_𝑛+1+2𝐸_𝑛+1d𝑥=𝜆^−𝛾𝜕𝜆𝜕𝜆^𝛾4𝐴_𝑛+2𝐸_𝑛2𝐶_𝑛+𝐺_𝑛2𝐴_𝑛+2𝐸_𝑛𝐶_𝑛+𝐺_𝑛2𝛿_𝑛−2𝜌_𝑛,𝑛≥0. (25) From the initial values in (12), we obtain 𝛾=0. Thus we have ℋ_𝑛=𝛿𝑢4𝐴_𝑛+2𝐸_𝑛2𝐶_𝑛+𝐺_𝑛2𝐴_𝑛+2𝐸_𝑛𝐶_𝑛+𝐺_𝑛2𝛿_𝑛−2𝜌_𝑛,ℋ_𝑛1=−𝑛+14𝐴_𝑛+1+2𝐸_𝑛+1d𝑥,𝑛≥0. (26)

It then follows that the nonlinear super integrable couplings (20) possess the following super Hamiltonian form 𝑢_𝑡𝑛=𝐾_𝑛(𝑢)=𝐽𝛿ℋ_𝑛,𝛿𝑢 (27) where 1𝐽=0−𝜕0𝜕00−𝜕0𝜕0𝛼−𝛽0𝜕0−2𝜕00𝜕0−2𝜕0−𝛼𝛽0−𝛼0𝛼0−210𝛽0−𝛽−20. (28) is a super Hamiltonian operator and ℋ_𝑛 (𝑛≥0) are Hamiltonian functions.

It can be verified that 𝐽^∗=10𝜕0−𝜕00𝜕0−𝜕0−𝛼𝛽0−𝜕02𝜕00−𝜕02𝜕0𝛼−𝛽0𝛼0−𝛼0−210−𝛽0𝛽−20,𝐽^∗𝐿=𝐿^∗𝐽^∗, (29) where 𝐽^∗ and 𝐿^∗ are conjugate operators of 𝐽 and 𝐿, respectively.

If we define the following Poisson bracket ℋ_𝑛,ℋ_𝑚=𝛿ℋ_𝑛𝛿𝑢,𝐽𝛿ℋ_𝑚=𝛿𝑢𝛿ℋ_𝑛,𝛿𝑟𝛿ℋ_𝑛,𝛿𝑠𝛿ℋ_𝑛𝛿𝑢₁,𝛿ℋ_𝑛𝛿𝑢₂,𝛿ℋ_𝑛,𝛿𝛼𝛿ℋ_𝑛𝛿𝛽×𝐽𝛿ℋ_𝑚𝛿𝑟𝛿ℋ_𝑚𝛿𝑠𝛿ℋ_𝑚𝛿𝑢₁𝛿ℋ_𝑚𝛿𝑢₂𝛿ℋ_𝑚𝛿𝛼𝛿ℋ_𝑚𝛿𝛽d𝑥, (30) where (⋅,⋅) denotes the inner product, as (𝛿ℋ_𝑛/𝛿𝛼)(𝛿ℋ_𝑚/𝛿𝛽)=−(𝛿ℋ_𝑚/𝛿𝛽)(𝛿ℋ_𝑛/𝛿𝛼), we can obtain ℋ_𝑛,ℋ_𝑚=𝛿ℋ_𝑚,𝛿𝑟𝛿ℋ_𝑚,𝛿𝑠𝛿ℋ_𝑚𝛿𝑢₁,𝛿ℋ_𝑚𝛿𝑢₂,𝛿ℋ_𝑚,𝛿𝛼𝛿ℋ_𝑚𝛿𝛽×𝐽^∗𝛿ℋ_𝑛𝛿𝑟𝛿ℋ_𝑛𝛿𝑠𝛿ℋ_𝑛𝛿𝑢₁𝛿ℋ_𝑛𝛿𝑢₂𝛿ℋ_𝑛𝛿𝛼𝛿ℋ_𝑛ℋ𝛿𝛽d𝑥=−_𝑚,ℋ_𝑛. (31) From (14) and (31), we have 𝛿ℋ_𝑚+1𝛿𝑢=𝐿𝛿ℋ_𝑚,𝛿𝑢𝛿ℋ_𝑚𝛿𝑢=𝐿^𝑚𝛿ℋ₀.𝛿𝑢 (32)

Suppose 𝑘,𝑙 are arbitrary nonnegative integers consider ℋ_𝑘,ℋ_𝑘+𝑙=𝐿^𝑘𝛿ℋ₀𝛿𝑢,𝐽𝐿^𝑘+𝑙𝛿ℋ₀.𝛿𝑢 (33) By using the properties of 𝐽 and 𝐿 in (29), we can obtain ℋ_𝑘,ℋ_𝑘+𝑙=𝐿^𝑘𝛿ℋ₀𝛿𝑢,𝐽𝐿^𝑘+𝑙𝛿ℋ₀=𝐿𝛿𝑢^𝑘𝛿ℋ₀,()𝐿𝛿𝑢𝐽𝐿^{𝑘+𝑙−1}𝛿ℋ₀=()𝛿𝑢𝐽𝐿^∗𝐿^𝑘𝛿ℋ₀𝛿𝑢,𝐿^{𝑘+𝑙−1}𝛿ℋ₀=𝐿𝛿𝑢_∗𝐽^∗𝐿^𝑘𝛿ℋ₀𝛿𝑢,𝐿^{𝑘+𝑙−1}𝛿ℋ₀=𝐽𝛿𝑢_∗𝐿^𝑘+1𝛿ℋ₀𝛿𝑢,𝐿^{𝑘+𝑙−1}𝛿ℋ₀=𝐿𝛿𝑢_𝑘+1𝛿ℋ₀𝛿𝑢,𝐽𝐿^{𝑘+𝑙−1}𝛿ℋ₀=𝛿𝑢𝛿ℋ_𝑘+1𝛿𝑢,𝐽𝛿ℋ_{𝑘+𝑙−1}=ℋ𝛿𝑢^𝑘+1,ℋ_{𝑘+𝑙−1}. (34) Ascending the subscript of left factor ℋ_𝑘, then we have ℋ_𝑘,ℋ_𝑘+𝑙=ℋ_𝑘+𝑙,ℋ_𝑘. (35) By using the antisymmetric law of Poisson bracket (31), we have ℋ_𝑘,ℋ_𝑘+𝑙=0. (36) So the conserved functionals {ℋ_𝑛}^∞₁ are in involution in pairs under the Poisson bracket (30).

5. Reductions

Taking 𝛼=𝛽=0, the hierarchy (27) reduces to a nonlinear integrable couplings of the Broer-Kaup-Kupershmidt hierarchy.

When 𝑛=2 in (27), we obtain the nonlinear super couplings of the second super Broer-Kaup-Kupershmidt equations: 𝑟_𝑡21=−2𝑟_𝑥𝑥−2𝑟𝑟_𝑥+12𝑠_𝑥+𝛼_𝑥𝛽+𝛼𝛽_𝑥+𝛽𝛽_𝑥𝑥,𝑠_𝑡2=12𝑠_𝑥𝑥−2𝑟_𝑥𝑠−2𝑟𝑠_𝑥+2𝛼𝛼_𝑥+2𝑠𝛽𝛽_𝑥,𝑢_1,𝑡2=()−1𝜀+12𝑢_1𝑥𝑥−2𝑟_𝑥𝑢₁−2𝑟𝑢_1𝑥+12𝑢_2𝑥−2𝑢₁𝑢_1𝑥−12𝑟_𝑥𝑥+12𝑠_𝑥−2𝑟𝑟_𝑥+12𝑟_𝑥𝑥−12𝑠_𝑥−𝛼_𝑥𝛽−𝛼𝛽_𝑥+2𝑟𝑟_𝑥−𝛽𝛽_𝑥𝑥,𝑢_2,𝑡2=()1𝜀+12𝑢_2𝑥𝑥−2𝑠_𝑥𝑢₁−2𝑠𝑢_1𝑥−2𝑟_𝑥𝑢₂−2𝑟𝑢_2𝑥−2𝑢_1𝑥𝑢₂−2𝑢₁𝑢_2𝑥+12𝑠_𝑥𝑥−2𝑟_𝑥𝑠−2𝑟𝑠_𝑥−12𝑠_𝑥𝑥+2𝑟_𝑥𝑠+2𝑟𝑠_𝑥−2𝛼𝛼_𝑥−2𝑠𝛽𝛽_𝑥,𝛼_𝑡2=𝛼_𝑥𝑥−2𝑟𝛼_𝑥−32𝑟_𝑥1𝛼+2𝑠_𝑥𝛽+𝑠𝛽_𝑥+𝛼𝛽𝛽_𝑥,𝛽_𝑡2=−𝛽_𝑥𝑥−12𝑟_𝑥𝛽−2𝑟𝛽_𝑥−𝛼_𝑥. (37) Particularly, taking 𝛼=𝛽=0 in (37), we can get the nonlinear integrable couplings of the second order Broer-Kaup-Kupershmidt equations: 𝑟_𝑡21=−2𝑟_𝑥𝑥−2𝑟𝑟_𝑥+12𝑠_𝑥,𝑠_𝑡2=12𝑠_𝑥𝑥−2𝑟_𝑥𝑠−2𝑟𝑠_𝑥,𝑢_1,𝑡2=()−1𝜀+12𝑢_1𝑥𝑥−2𝑟_𝑥𝑢₁−2𝑟𝑢_1𝑥+12𝑢_2𝑥−2𝑢₁𝑢_1𝑥−12𝑟_𝑥𝑥+12𝑠_𝑥−2𝑟𝑟_𝑥+12𝑟_𝑥𝑥−12𝑠_𝑥+2𝑟𝑟_𝑥,𝑢_2,𝑡2=()×1𝜀+12𝑢_2𝑥𝑥−2𝑠_𝑥𝑢₁−2𝑠𝑢_1𝑥−2𝑟_𝑥𝑢₂−2𝑟𝑢_2𝑥−2𝑢_1𝑥𝑢₂−2𝑢₁𝑢_2𝑥+12𝑠_𝑥𝑥−2𝑟_𝑥𝑠−2𝑟𝑠_𝑥−12𝑠_𝑥𝑥+2𝑟_𝑥𝑠+2𝑟𝑠_𝑥. (38) Let 𝜀=0 in (37); we have 𝑟_𝑡21=−2𝑟_𝑥𝑥−2𝑟𝑟_𝑥+12𝑠_𝑥+𝛼_𝑥𝛽+𝛼𝛽_𝑥+𝛽𝛽_𝑥𝑥,𝑠_𝑡2=12𝑠_𝑥𝑥−2𝑟_𝑥𝑠−2𝑟𝑠_𝑥+2𝛼𝛼_𝑥+2𝑠𝛽𝛽_𝑥,𝑢_1,𝑡21=−2𝑢_1𝑥𝑥−2𝑟_𝑥𝑢₁−2𝑟𝑢_1𝑥+12𝑢_2𝑥−2𝑢₁𝑢_1𝑥−𝛼_𝑥𝛽−𝛼𝛽_𝑥−𝛽𝛽_𝑥𝑥,𝑢_2,𝑡2=12𝑢_2𝑥𝑥−2𝑠_𝑥𝑢₁−2𝑠𝑢_1𝑥−2𝑟_𝑥𝑢₂−2𝑟𝑢_2𝑥−2𝑢_1𝑥𝑢₂−2𝑢₁𝑢_2𝑥−2𝛼𝛼_𝑥−2𝑠𝛽𝛽_𝑥,𝛼_𝑡2=𝛼_𝑥𝑥−2𝑟𝛼_𝑥−32𝑟_𝑥1𝛼+2𝑠_𝑥𝛽+𝑠𝛽_𝑥+𝛼𝛽𝛽_𝑥,𝛽_𝑡2=−𝛽_𝑥𝑥−12𝑟_𝑥𝛽−2𝑟𝛽_𝑥−𝛼_𝑥. (39) If setting 𝜀=−1, 𝑢₁=−𝑟, and 𝑢₂=−𝑠 in (37), we obtain the second order super Broer-Kaup-Kupershmidt equations: 𝑟_𝑡21=−2𝑟_𝑥𝑥−2𝑟𝑟_𝑥+12𝑠_𝑥+𝛼_𝑥𝛽+𝛼𝛽_𝑥+𝛽𝛽_𝑥𝑥,𝑠_𝑡2=12𝑠_𝑥𝑥−2𝑟_𝑥𝑠−2𝑟𝑠_𝑥+2𝛼𝛼_𝑥+2𝑠𝛽𝛽_𝑥,𝛼_𝑡2=𝛼_𝑥𝑥−2𝑟𝛼_𝑥−32𝑟_𝑥1𝛼+2𝑠_𝑥𝛽+𝑠𝛽_𝑥+𝛼𝛽𝛽_𝑥,𝛽_𝑡2=−𝛽_𝑥𝑥−12𝑟_𝑥𝛽−2𝑟𝛽_𝑥−𝛼_𝑥. (40)

6. Remarks

In this paper, we introduced an approach for constructing nonlinear integrable couplings of super integrable hierarchy. Zhang [27] once employed two kinds of explicit Lie algebra 𝐹 and 𝐺 to obtain the nonlinear integrable couplings of the GJ hierarchy and Yang hierarchy, respectively. It is easy to see that Lie algebra 𝐹 given in [27] is isomorphic to the Lie algebra span {𝑒₁,𝑒₂,𝑒₃,𝑒₄,𝑒₅,𝑒₆} in 𝑔𝑙(6,2). So we can obtain nonlinear integrable couplings of super GJ and Yang hierarchy easily. The method in this paper can be applied to other super integrable systems for constructing their integrable couplings.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 61072147), the Natural Science Foundation of Henan Province (no. 132300410202), the Science and Technology Key Research Foundation of the Education Department of Henan Province (no. 12A110017) and the Youth Research Foundation of Shangqiu Normal University (no. 2011QN12).