TY - JOUR
T1 - Integrable hamiltonian hierarchies associated with the equation of heat conduction
AU - Zhang, Yufeng
AU - Tam, Honwah
AU - Mei, Jianqin
N1 - Copyright:
Copyright 2010 Elsevier B.V., All rights reserved.
PY - 2010/6/10
Y1 - 2010/6/10
N2 - Using a 4-dimensional Lie algebra g, an isospectral Lax pair is introduced, whose compatibility condition is equivalent to a soliton hierarchy of evolution equations with three components of potential functions. Its Hamiltonian structure is obtained by employing the quadratic-form identity proposed by Guo and Zhang. In order to obtain explicit Hamiltonian functions, a detailed computing formula for the constant appearing in the quadratic-form identity is obtained. One type of reduction equations of the hierarchy is also produced, which is further reduced to the standard equation of heat conduction. By introducing a loop algebra of the Lie algebra g, we obtain a soliton hierarchy with an arbitrary parameter which can be reduced to the previous equation hierarchy obtained, whose quasi-Hamiltonian structure is also worked out by the quadratic-form identity. Finally, we extend the Lie algebra g into a higher-dimensional Lie algebra so that a new integrable Hamiltonian hierarchy, which comprise integrable couplings, is produced; its reduced equations in particular contain two arbitrary parameters.
AB - Using a 4-dimensional Lie algebra g, an isospectral Lax pair is introduced, whose compatibility condition is equivalent to a soliton hierarchy of evolution equations with three components of potential functions. Its Hamiltonian structure is obtained by employing the quadratic-form identity proposed by Guo and Zhang. In order to obtain explicit Hamiltonian functions, a detailed computing formula for the constant appearing in the quadratic-form identity is obtained. One type of reduction equations of the hierarchy is also produced, which is further reduced to the standard equation of heat conduction. By introducing a loop algebra of the Lie algebra g, we obtain a soliton hierarchy with an arbitrary parameter which can be reduced to the previous equation hierarchy obtained, whose quasi-Hamiltonian structure is also worked out by the quadratic-form identity. Finally, we extend the Lie algebra g into a higher-dimensional Lie algebra so that a new integrable Hamiltonian hierarchy, which comprise integrable couplings, is produced; its reduced equations in particular contain two arbitrary parameters.
KW - Hamiltonian structure
KW - Lie algebra
KW - Quadratic-form identity
UR - http://www.scopus.com/inward/record.url?scp=77953559700&partnerID=8YFLogxK
U2 - 10.1142/S0217984910023360
DO - 10.1142/S0217984910023360
M3 - Journal article
AN - SCOPUS:77953559700
SN - 0217-9849
VL - 24
SP - 1573
EP - 1594
JO - Modern Physics Letters B
JF - Modern Physics Letters B
IS - 14
ER -