Two expanding integrable systems and quasi-Hamiltonian function associated with an equation hierarchy |
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| Institution: | 1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan 450001, China;2. School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, TX 78539, USA;1. Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, PR China;2. School of Sciences, Jiangnan University, Wuxi 214122, PR China;3. Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA;1. College of Sciences, China University of Mining and Technology, Xuzhou 221116, Jiangsu, PR China;2. College of Teacher Education, Tianjin Normal University, Tianjin 300387, PR China |
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| Abstract: | A Lie algebra sl(2) which is isomorphic to the known Lie algebra A1 is introduced for which an isospectral Lax pair is presented, whose compatibility condition leads to a soliton-equation 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-equation 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 employing the variational identity. Finally, two types of computing formulas of the constant γ are obtained, respectively. |
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