A New Nonlinear Integrable Couplings of Yang Equations Hierarchy and Its Hamiltonian Structure
2014-07-31WEIHanyuXIATiecheng
WEI Han-yu,XIA Tie-cheng
(1.College of Mathematics and Statistics,Zhoukou Normal University,Zhoukou 466001,China;2.Department of Mathematics,Shanghai University,Shanghai 200444,China)
A New Nonlinear Integrable Couplings of Yang Equations Hierarchy and Its Hamiltonian Structure
WEI Han-yu1,2,XIA Tie-cheng2
(1.College of Mathematics and Statistics,Zhoukou Normal University,Zhoukou 466001,China;2.Department of Mathematics,Shanghai University,Shanghai 200444,China)
Based on a kind of non-semisimple Lie algebras,we establish a way to construct nonlinear continuous integrable couplings.Variational identities over the associated loop algebras are used to furnish Hamiltonian structures of the resulting continuous couplings. As an illustrative example of the scheme is given nonlinear continuous integrable couplings of the Yang hierarchy.
zero curvature equations;integrable couplings;variational identities
§1.Introduction
Integrable couplings[12]are coupled systems of integrable equations,which has been introduced when we study of Virasoro symmetric algebras.It is an important problems to look for integrable couplings because integrable couplings have much richer mathematical structures and physical meanings.In recent years,many methods of searching for integrable couplingshave been developed[39],for example,the spectral matrices as follows
where the sub-spectral matrix U is associated with the given integrable equation ut=K(u). However,researcher f i nd out that obtained integrable couplings is relative simple.So in order to get better integrable couplings of the known integrable system,we need to introduce an enlarged relatively complex spectral matrix[10-11]
From a zero curvature representation
where
and¯u consist of u and v,we can obtain
This is an integrable couplings of Eq(1),and it is a nonlinear integrable coupling because the commutator[Ua,Va]can generate nonlinear terms.
Let us take a solution¯W to the enlarged stationary zero curvature equation
Then,we use the variational identity[6-7]
to search for the Hamiltonian structures of the integrable couplings[7].In the variational identity(6),〈.,.〉is non-degenerate,symmetric and ad-invariant bilinear form over the Lie algebra
In what follows,we will make an application of this general scheme to the Yang hierarchy.
§2.The Integrable Couplings of the Yang Equations Hierarchy
2.1Yang Hierarchy
From the spectral problem[12]
Setting
The stationary zero curvature equation Vx=[U,V]yields that
Choose the initial data
then we have
From the compatibility conditions of the following problems
The Hamiltonian operator J,the hereditary recursion operator L and the Hamiltonian functions Hnas follows
(1.8)
2.2Integrable Couplings
Let us begin with an enlarged spectral matrix
with the help of the corresponding enlarged stationary zero curvature equation¯Vx=[¯U,¯V],we have
which generates
Setting
Then,the Eq(1.11)can be transformed into
We choose the initial data
then we have
By using the zero curvature equation,
Then,we have the following results
Obviously,when p1=p2=p3=0 in Eq(1.18),the above results become Eq(1.7).So we can say Eq(1.18)is the integrable couplings of Yang hierarchy.When n=2,the Eq(1.18)can reduced to b
where
So,we can say that the system in(1.18)with n≥2 provide nonlinear integrable couplings of the Yang hierarchy.
§3.Hamiltonian Structures of the Integrable Couplings of the Yang Equations Hierarchy
To construct Hamiltonian structures of the integrable couplings obtained,we need to compute non-degenerate,symmetric and invariant bilinear forms on the following Lie algebra
For computations convenience,we transform this Lie algebra¯g into a vector from through the mapping
The mapping δ induces a Lie algebraic structure on R6,isomorphic to the matrix Lie algebra ¯g above.It is easy to see that the corresponding commutator[.,.]on R6is given by
where
Def i ne a bilinear form on R6as follows
where F is a constant matrix,which is main idea by Zhang and Guo presented in 2005[5].
Then the symmetric property〈a,b〉=〈b,a〉and the ad-invariance property under the Lie product
requires that FT=F and
So we can obtain
where η1and η2are arbitrary constants.Therefore,a bilinear form on the underlying Lie algebra¯g is def i ned by
where
It is non-degenerate if and only if
Based on(2.12)and Eqs(1.9)∼(1.10),we can easily compute that
By applying the operator Γn+2to both sides of variational identity(6)
So we obtain that equation hierarchy(1.18)possess the Hamiltonian structures
where the Hamiltonian operator¯J and the Hamiltonian functional Hnare given by
and
With the help of(1.16),we can see a recursion relation
with
where L is given by(1.8)and
Up to now,we have already obtained Hamiltonian structure for integrable couplings of the Yang hierarchy.We must point out that by changing the nonlinear coupling terms of equations, more nonlinear integrable couplings with physical meaning can be obtained.So,with the help of this method,more meaningful results of other integrable hierarchies can be generated.
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tion:35Q51
CLC number:O175.9Document code:A
1002–0462(2014)02–0180–09
date:2012-04-16
Supported by the Natural Science Foundation of China(11271008,61072147,11071159); Supported by the First-class Discipline of Universities in Shanghai;Supported by the Shanghai University Leading Academic Discipline Project(A13-0101-12-004)
Biography:WEI Han-yu(1982-),male,native of Zhoukou,Henan,a lecturer of Zhoukou Normal University, Ph.D.,engages in solitons and integrable systems.
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