On generating linear and nonlinear integrable systems with variable coefficients

Under an isospectral Lax pair, a new integrable hierarchy of evolution equations is obtained by starting from a given Lie algebra T  , which can be reduce to a new coupled integrable equation similar to the long wave equation, but it is not the standard long wave equation. By making use of an enlarged Lie algebra T1T1 of the Lie algebra T  , we obtain a type of equation hierarchy (called a linear hierarchy). The corresponding Hamiltonian structure of the equation hierarchy is derived from the variational identity. As we all know that nonlinear equations with variable coefficients can be used to describe some real phenomena in physical and engineering fields. It is an interesting and important topic to consider how to generate variable-coefficient nonlinear integrable equations from the mathematical viewpoint. In the paper, we construct another enlarged Lie algebra T2T2 of the Lie algebra T for which an integrable hierarchy (called the nonlinear hierarchy) of nonlinear integrable equations with variable coefficients is obtained. Furthermore, the Hamiltonian structure of the integrable hierarchy is produced by using the variational identity again. As long as the linear hierarchy and the nonlinear hierarchy are derived, following their reductions, some linear and nonlinear evolution equations with variable coefficients are obtained, respectively. The corresponding Hamiltonian structures of such reduced equations with variable coefficients are followed to present.