A Few Super-Integrable Hierarchies and Some Reductions, Super-Hamiltonian Structures

Abstract

Super-integrable equations provide some important physical models. In the paper, we have introduced two bases of the super Lie algebra B(0, 1). One of them is used to link three super hierarchies of evolution equations under the 0-curvature equations. We have constructed its two kinds of super loop algebras. By employing the first loop algebra, an isospectral problem was introduced for which a super (1 + 1)-dimensional integrable hierarchy is obtained, whose super Hamiltonian structure was derived from the super-trace identity proposed by Tu et al. As the reduction cases, we have obtained two different explicit super Schrödinger and mKdV systems. Further reduced equations are the well-known nonlinear Schrödinger equation, the super KdV equations and the mKdV equation. By employing the second loop algebra, we have obtained a new super integrable hierarchy containing 5 even superfunctions and 4 odd ones, which possesses super bi-Hamiltonian structure. As the reduction case, we have obtained an explicit super integrable system which is further reduced to 3 super integrable coupled equations, especially we have got a super coupled constrained equation with sources with respect to the odd superfunctions. Finally, we have used the TAH scheme to obtain a super (2+1)-dimensional hierarchy of evolution equations, whose Hamiltonian structure was generated by a new trace identity developed by us in the paper, which are new results, to the best of our knowledge.