Three nonlinear integrable couplings of the nonlinear Schrödinger equations

Based on two types of expanding Lie algebras of a Lie algebra G, three isospectral problems are designed. Under the framework of zero curvature equation, three nonlinear integrable couplings of the nonlinear Schröding equations are generated. With the help of variational identity, we get the Hamiltonian structure of one of them. Furthermore, we get the result that the hierarchy is also integrable in sense of Liouville.

► We designed three different isospectral problems based on a well known Lie algebra G.
► Three integrable couplings are generated, which are different from the traditional ones.
► Then we can get the Hamiltonian structure of one hierarchy and verify that it is also Liouville integrable with the help of variational identity.