In this paper, we establish a family of symplectic integrators for a class
of high order Schrödinger equations with trapped terms. First, we find its symplectic structure and reduce it to a finite dimensional Hamilton system via spatial discretization.
Then we apply the symplectic Euler method to the Hamiltonian system.
It is demonstrated that the scheme not only preserves symplectic geometry structure
of the original system, but also does not require to resolve coupled nonlinear
algebraic equations which is different with the general implicit symplectic schemes.
The linear stability of the symplectic Euler scheme and the errors of the numerical
solutions are investigated. It shows that the semi-explicit scheme is conditionally
stable, first order accurate in time and 2lth order accuracy in space. Numerical tests
suggest that the symplectic integrators are more effective than non-symplectic ones,
such as backward Euler integrators.