Elsevier

Applied Mathematics and Computation

Volume 243, 15 September 2014, Pages 601-606
Applied Mathematics and Computation

Two (2 + 1)-dimensional hierarchies of evolution equations and their Hamiltonian structures

https://doi.org/10.1016/j.amc.2014.06.012Get rights and content

Abstract

Two kinds of appropriate isospectral problems are introduced by using a Lie algebra. With the help of the TAH scheme, we generate two new (2 + 1)-dimensional hierarchies of evolution equations, whose Hamiltonian structures are derived from the trace identity proposed by Tu Guizhang, Andrushkiw R.I. and Huang X.C. Finally, we propose some problems worth thinking about.

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 A[ξ] which consists of all pseudodifferential operators i=-Naiξi, where the operator ξ is defined byξf=fξ+(yf),fA,here A is an associative algebra over the field K=R or C. In what follows, we briefly recall the scheme which is called the TAH scheme.

  • (1)

    Introduce a residue operatorR:A[ξ]A,Raiξi=a-1.

  • (2)

    Fix a matrix operator U=U(λ+ξ,u)[ξ], where λ is a parameter, u=(u1,,up)T is a vector function. Solving the matrix operator equation for VVx=[U,V]where V=Vnλ-n, yields the recurrence relations among Vn:g(n)(g1(n),,gp(n))T, where gi(n) comes from the expansionV,Uui=ngi(n)λ-n,here a,b=trR(ab),a,bA[ξ].

  • (3)

    We try to find an operator J and form the hierarchyut=Jg(n).

  • (4)

    In terms of the trace identityδδuiV,Uλ=λ-γλλγV,Uui,i=1,2,,p,

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 e1(λ),e2(λ),,en(λ).

  • (2)

    Construct a couple of matrices U and V:U=u1e1(λ)+u2e2(λ)++upep(λ)+B(λ), where u1,,up are potentials, i.e., functions of x and t.

  • (3)

    Solve the stationary matrix equation Vx=[U,V].

  • (4)

    Construct a sequence of matrices V(n) such that Vx(n)-[U,V(n)]Ce1++Cep.

  • (5)

    Then Ut-Vx(n)+[U,V(n)]=0 will generate a sequence of integrable systems. The trace identity proposed by Tu which presents that δδuiV,Uλ=λ-γλλγV,Uui,i=1,2,,p, where a,b=tr(ab),a,bL, can be used to further find the corresponding Hamiltonian structure.

By employing the Tu scheme, some interesting integrable hierarchies and their properties were obtained, such as the beautiful results in [5], [6], [7], [8], [9].

Section snippets

A new (2 + 1)-dimensional hierarchy and its Hamiltonian structure

We have known the Lie algebraG=span{h1,h2,e,f},h1=1000,h2=0001,e=0100,f=0010,along with the following commutative operations

[h1,h2]=0,[h1,e]=e,[h2,e]=-e,[h1,f]=-f,[h2,f]=f,[e,f]=h1-h2.

A loop algebra of G can be defined asG̃=span{h1(n),h2(n),e(n),f(n)},wherehi(n)=hiλn,i=1,2;e(n)=eλn,f(n)=fλn,nZ.Consider the isospectral problemsφx=Uφ,U=(λ+ξ)h1(0)-(λ+ξ)h2(0)+qe(1)+rf(0),φt=Vφ,V=Ah1(1)+Dh2(1)+Be(1)+Cf(0).Solving the stationary matrix Eq. (3) gives rise toAx=(λ+ξ)A+qC-A(λ+ξ)-Br,Bx=(λ+ξ)B+qDλ-Aqλ+B(λ

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. SetU=01uλ+ξ+v,V=ABCD,which are different from those in Eqs. (6), (7). By making use of Eq. (3) we haveAm,x=Cm-Bmu,Bm,x=Dm-Am-Bm+1-Bmξ-Bmv,Cm,x=uAm-Dmu+Cm+1+ξCm+vCm,Dm,x=uBm+vDm-Dmv+Dm,y-Cm.Choose B0

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).

References (11)

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