In this paper we first present a 3-dimensional Lie algebra H and enlarge it into a 6-dimensional Lie algebra T with corresponding loop algebras？H and？T, respectively. By using the loop algebra？H and the Tu scheme, ...In this paper we first present a 3-dimensional Lie algebra H and enlarge it into a 6-dimensional Lie algebra T with corresponding loop algebras？H and？T, respectively. By using the loop algebra？H and the Tu scheme, we obtain an integrable hierarchy from which we derive a new Darboux transformation to produce a set of exact periodic solutions. With the loop algebra？T, a new integrable-coupling hierarchy is obtained and reduced to some variable-coefficient nonlinear equations, whose Hamiltonian structure is derived by using the variational identity. Furthermore, we construct a higher-dimensional loop algebraˉH of the Lie algebra H from which a new Liouville-integrable hierarchy with 5-potential functions is produced and reduced to a complex m Kd V equation, whose 3-Hamiltonian structure can be obtained by using the trace identity. A new approach is then given for deriving multiHamiltonian structures of integrable hierarchies. Finally, we extend the loop algebra？H to obtain an integrable hierarchy with variable coefficients.展开更多
基金Supported by the Research Grant Council of the Hong Kong Special Administrative Region(Grant No.City U101211)the National Natural Science Foundation of China(Grant No.11371361)the Natural Science Foundation of Shandong Province(Grant No.ZR2013AL016)
文摘In this paper we first present a 3-dimensional Lie algebra H and enlarge it into a 6-dimensional Lie algebra T with corresponding loop algebras？H and？T, respectively. By using the loop algebra？H and the Tu scheme, we obtain an integrable hierarchy from which we derive a new Darboux transformation to produce a set of exact periodic solutions. With the loop algebra？T, a new integrable-coupling hierarchy is obtained and reduced to some variable-coefficient nonlinear equations, whose Hamiltonian structure is derived by using the variational identity. Furthermore, we construct a higher-dimensional loop algebraˉH of the Lie algebra H from which a new Liouville-integrable hierarchy with 5-potential functions is produced and reduced to a complex m Kd V equation, whose 3-Hamiltonian structure can be obtained by using the trace identity. A new approach is then given for deriving multiHamiltonian structures of integrable hierarchies. Finally, we extend the loop algebra？H to obtain an integrable hierarchy with variable coefficients.
