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

Abstract:

      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.