TY - JOUR
T1 - Spectral radius analysis of matrices and the associated with integrable systems
AU - TAM, Hon Wah
AU - Zhang, Yufeng
N1 - Funding Information:
∗This work was supported by Hong Kong Research Grant Council grant 2016/05p and The National Science Foundation of China (10471139).
PY - 2009/9/30
Y1 - 2009/9/30
N2 - An isospectral problem is introduced, a spectral radius of the corresponding spectral matrix is obtained, which enlightens us to set up an isospectral problem whose compatibility condition gives rise to a zero curvature equation in formalism, from which a Lax integrable soliton equation hierarchy with constraints of potential functions is generated along with 5 parameters, whose reduced cases present three integrable systems, i.e., AKNS hierarchy, Levi hierarchy and D-AKNS hierarchy. Enlarging the above Lie algebra into two bigger ones, the two integrable couplings of the hierarchy are derived, one of them has Hamiltonian structure by employing the quadratic-form identity or variational identity. The corresponding integrable couplings of the reduced systems are obtained, respectively. Finally, as comparing study for generating expanding integrable systems, a Lie algebra of antisymmetric matrices and its corresponding loop algebra are constructed, from which a great number of enlarging integrable systems could be generated, especially their Hamiltonian structure could be computed by the trace identity.
AB - An isospectral problem is introduced, a spectral radius of the corresponding spectral matrix is obtained, which enlightens us to set up an isospectral problem whose compatibility condition gives rise to a zero curvature equation in formalism, from which a Lax integrable soliton equation hierarchy with constraints of potential functions is generated along with 5 parameters, whose reduced cases present three integrable systems, i.e., AKNS hierarchy, Levi hierarchy and D-AKNS hierarchy. Enlarging the above Lie algebra into two bigger ones, the two integrable couplings of the hierarchy are derived, one of them has Hamiltonian structure by employing the quadratic-form identity or variational identity. The corresponding integrable couplings of the reduced systems are obtained, respectively. Finally, as comparing study for generating expanding integrable systems, a Lie algebra of antisymmetric matrices and its corresponding loop algebra are constructed, from which a great number of enlarging integrable systems could be generated, especially their Hamiltonian structure could be computed by the trace identity.
KW - Hamiltonian structure
KW - Integrable couplings
KW - Lie algebra
KW - Spectral radius analysis
UR - http://www.scopus.com/inward/record.url?scp=70350331785&partnerID=8YFLogxK
U2 - 10.1142/S0217979209053138
DO - 10.1142/S0217979209053138
M3 - Journal article
AN - SCOPUS:70350331785
SN - 0217-9792
VL - 23
SP - 4855
EP - 4879
JO - International Journal of Modern Physics B
JF - International Journal of Modern Physics B
IS - 24
ER -