Two expanding integrable systems and quasi-Hamiltonian function associated with an equation hierarchy

摘要

A Lie algebra sl(2) which is isomorphic to the known Lie algebra A, is introduced for which an isospectral Lax pair is presented, whose compatibility condition leads to a soliton-equa-tion hierarchy. By using the trace identity, its Hamiltonian structure is obtained. Especially, as its reduction cases, a Sine equation and a complex modified KdV(cmKdV) equation are obtained.respectively. Then we enlarge the sl(2) into a bigger Lie algebra sl(4) so that a type of expanding integrable model of the hierarchy is worked out. However, the soliton-equa-tion hierarchy is not integrable couplings. In order to generate the integrable couplings, an isospectral Lax pair is introduced. Under the frame of the zero curvature equation, we generate an integrable coupling whose quasi-Hamiltonian function is derived by employ-ing the variational identity. Finally, two types of computing formulas of the constant y are obtained, respectively.