A New Non-Isospectral Integrable Hierarchy and Some Associated Symmetries

Abstract:

      By applying a subgroup of the Lie group $M_nC$ we introduce a linear nonisospectral problem whose compatibility condition gives rise to a nonisospectral integrable hierarchy of evolution equations, which reduces to a generalized nonisospectral integrable hierarchy (GNIH). The GNIH further reduces to the standard nonlinear Schr\"{o}dinger equation and the KdV equation which have important applications in physics science. Based on this, we discuss the $K$ symmetries and the $\tau$ symmetries of the generalized AKNS hierarchy $u_t=K_m(u)$ with isospectral condition coming from the GNIH. Furthermore, we also consider the $K$ symmetries and the $\tau$ symmetries of the nonisospectral AKNS hierarchy $u_t=\tau_{N+1}^l$. Finally, we obtain the symmetry Lie algebras for the both integrable hierarchies, and present some applications for the symmetries and the Lie algebras, which means that some Lie groups of transformations and the infinitesimal operators of reduced equations are generated.