扩展的可积模型及其约化与达布变换

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.

Expanding Integrable Models and Their Some Reductions as Well as Darboux Transformations

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 $\tilde H$ and $\tilde T$, respectively. By using the loop algebra $\tilde H$ 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 $\tilde T$, 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 algebra $\bar H$ of the Lie algebra $H$ from which a new Liouville-integrable hierarchy with 5-potential functions is produced and reduced to a complex mKdV equation, whose 3-Hamiltonian structure can be obtained by using the trace identity. A new approach is then given for deriving multi-Hamiltonian structures of integrable hierarchies. Finally, we extend the loop algebra $\tilde H$ to obtain an integrable hierarchy with variable coefficients.