Generations of integrable hierarchies and exact solutions of related evolution equations with variable coefficients

We first propose a way for generating Lie algebras from which we get a few kinds of reduced Lie algebras, denoted by R ⁶, R ⁸ and R ₁ ⁶ ,R ₂ ⁶ , respectively. As for applications of some of them, a Lax pair is introduced by using the Lie algebra R ⁶ whose compatibility gives rise to an integrable hierarchy with 4-potential functions and two arbitrary parameters whose corresponding Hamiltonian structure is obtained by the variational identity. Then we make use of the Lie algebra R ₁ ⁶ to deduce a nonlinear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is also obtained. Again,via using the Lie algebra R ₂ ⁶ , we introduce a Lax pair and work out a linear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is obtained. Finally, we get some reduced linear and nonlinear equations with variable coefficients and work out the elliptic coordinate solutions, exact traveling wave solutions, respectively.