Two (2 + 1)-dimensional hierarchies of evolution equations and their Hamiltonian structures
Introduction
Employing the Lax pair method is a current approach for generating (2 + 1)-dimensional hierarchies of evolution equations. For example, Ablowitz et al. [1] made use of a self-dual Yang–Mills hierarchy and its reductions to have obtained some various isospectral problems. By choosing different potential functions, some (1 + 1)-dimensional and (2 + 1)-dimensional integrable equations were obtained, such as the KP equation and the DS integrable system, and so on. Chakravarty et al. [2] further reduced the self-dual Yang–Mills equation by the symmetry and gauge transformations to obtain some matrix equations. By reducing such equations, the (2 + 1)-dimensional Burgers equation, the KP equation and the DS equation were all produced. Tu Guizhang et al. [3] presented a scheme for generating (2 + 1)-dimensional hierarchies of evolution equations by introducing a residue operator on an associative algebra which consists of all pseudodifferential operators , where the operator is defined byhere is an associative algebra over the field or C. In what follows, we briefly recall the scheme which is called the TAH scheme.
- (1)
Introduce a residue operator
- (2)
Fix a matrix operator , where is a parameter, is a vector function. Solving the matrix operator equation for Vwhere , yields the recurrence relations among , where comes from the expansionhere .
- (3)
We try to find an operator J and form the hierarchy
- (4)
In terms of the trace identity
We could obtain the Hamiltonian structure of Eq. (4). As comparison, we recall the Tu scheme which is for generating (1 + 1)-dimensional integrable hierarchies of evolution equations and the resulting Hamiltonian structures [4]:
- (1)
Choose a matrix loop Lie algebra L with basis .
- (2)
Construct a couple of matrices U and , where are potentials, i.e., functions of x and t.
- (3)
Solve the stationary matrix equation .
- (4)
Construct a sequence of matrices such that .
- (5)
Then will generate a sequence of integrable systems. The trace identity proposed by Tu which presents that , where , can be used to further find the corresponding Hamiltonian structure.
Section snippets
A new (2 + 1)-dimensional hierarchy and its Hamiltonian structure
We have known the Lie algebraalong with the following commutative operations
.
A loop algebra of G can be defined aswhereConsider the isospectral problemsSolving the stationary matrix Eq. (3) gives rise to
Another (2 + 1)-dimensional hierarchy of evolution equations
In the section we shall introduce a pair of Lax matrices and try to deduce a (2 + 1)-dimensional hierarchy which is different from Eq. (16), from which we want to know how to introduce the appropriate Lax matrices so that we could derive interesting (2 + 1)-dimensional hierarchies of evolution equations. Setwhich are different from those in Eqs. (6), (7). By making use of Eq. (3) we haveChoose
Acknowledgement
Yufeng Zhang is grateful to professor Tu Guizhang for his help and guidance.
This work was supported by the National Natural Science Foundation of China (Grant No. 11371361), the Fundamental Research Funds for the Central Universities (2013XK03) and the Natural Science Foundation of Shandong Province (Grant No. ZR2013AL016).
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