A simple method for generating integrable hierarchies with multi-potential functions

A simple efficient method for obtaining integrable hierarchies of evolution equations with multi-potential functions is proposed. By making use of the concept of cycled numbers, a new loop algebra A∼1∗ is constructed. Taking account of applicable convenience, a few subalgebras of the loop algebra A∼1∗ are presented again, from one of them, two new integrable Hamiltonian hierarchies with multi-potential functions are obtained. Finally, a 3 × 3 loop algebra G∼ with five dimensions is established, from which a coupled integrable coupling system of one of the above integrable hierarchies is derived. The approach presented in this paper can be used generally.